Advances in Discrete Dynamical Systems, Difference Equations and Applications: 26th ICDEA, Sarajevo, Bosnia and Herzegovina, July 26-30, 2021 3031252241, 9783031252242

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Springer Proceedings in Mathematics & Statistics

Saber Elaydi Mustafa R. S. Kulenović Senada Kalabušić   Editors

Advances in Discrete Dynamical Systems, Difference Equations and Applications 26th ICDEA, Sarajevo, Bosnia and Herzegovina, July 26–30, 2021

Springer Proceedings in Mathematics & Statistics Volume 416

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Saber Elaydi · Mustafa R. S. Kulenovi´c · Senada Kalabuši´c Editors

Advances in Discrete Dynamical Systems, Difference Equations and Applications 26th ICDEA, Sarajevo, Bosnia and Herzegovina, July 26–30, 2021

Editors Saber Elaydi Department of Mathematics Trinity University San Antonio, TX, USA

Mustafa R. S. Kulenovi´c University of Rhode Island Kingston, RI, USA

Senada Kalabuši´c University of Sarajevo Sarajevo, Bosnia and Herzegovina

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-031-25224-2 ISBN 978-3-031-25225-9 (eBook) https://doi.org/10.1007/978-3-031-25225-9 Mathematics Subject Classification: 39-XX, 37-XX, 37N25, 92Bxx, 92D25, 93C55 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

A Discrete-Time Predator-Prey Model with Selection and Mutation . . . . Azmy S. Ackleh, Sankar Sikder, and Aijun Zhang On the Dynamics and Asymptotic Behaviour of the Mean Square of Scalar Linear Stochastic Difference Equations . . . . . . . . . . . . . . . . . . . . . John A. D. Appleby and Emmet Lawless Border Collision and Heteroclinic Bifurcations in a 2D Piecewise Smooth Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viktor Avrutin, Laura Gardini, Iryna Sushko, Zhanybai T. Zhusubaliyev, and Ulanbek A. Sopuev

1

25

61

Using Homotopy Link Function with Lipschitz Threshold in Studying Synchronized Fluctuations in Hierarchical Models . . . . . . . . Tahmineh Azizi

75

Solving Third-Order Linear Recurrence Relations with Applications to Number Theory and Combinatorics . . . . . . . . . . . . . . Armen G. Bagdasaryan

97

A Survey on Max-Type Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . 123 Antonio Linero-Bas and Daniel Nieves-Roldán Catalan Numbers Recurrence as a Stationary State Equation of the Probabilistic Cellular Automaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Mariusz Białecki Oscillation of Second Order Impulsive Neutral Difference Equations of Non-canonical Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 G. N. Chhatria and A. K. Tripathy On the Robustness Property of Nonuniform Exponential Dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Davor Dragiˇcevi´c

v

vi

Contents

Implicit Linear First Order Difference Equations Over Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Sergey Gefter, Anna Goncharuk, and Aleksey Piven’ Global Attraction and Repulsion of a Heteroclinic Limit Cycle in Three Dimensional Kolmogorov Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Zhanyuan Hou Bifurcation and Stability of a Ricker Host-Parasitoid Model with a Host Constant Refuge and General Escape Function . . . . . . . . . . . . 233 Senada Kalabuši´c, Džana Drino, and Esmir Pilav SageMath Tools for Stability and Bifurcation for Discrete Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Sinan Kapçak Pullback Attractors of Nonautonomous Lattice Difference Equations . . . 299 Peter E. Kloeden Global Dynamics of Modified Discrete Lotka-Volterra Model . . . . . . . . . . 309 M. R. S. Kulenovi´c and Sarah Van Beaver Nonwandering Sets and Special α-limit Sets of Monotone Maps on Regular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Habib Marzougui and Aymen Daghar Asymptotic Stability, Bifurcation Analysis and Chaos Control in a Discrete Evolutionary Ricker Population Model with Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Karima Mokni and Mohamed Ch-Chaoui Weighted Norms In Advanced Volterra Difference Equations . . . . . . . . . . 405 Youssef N. Raffoul Comparison of Tests for Oscillations in Delay/Advanced Difference Equations with Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Andrea Rožnjik, Hajnalka Péics, and George E. Chatzarakis Krause Mean Processes Generated by Cubic Stochastic Matrices IV: Off-Diagonally Uniformly Positive Nonautonomous Cubic Stochastic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Mansoor Saburov and Khikmat Saburov Linearization for Difference Equations with Infinite Delay . . . . . . . . . . . . 461 Lokesh Singh A Method to Derive Discrete Population Models . . . . . . . . . . . . . . . . . . . . . . 473 Sabrina H. Streipert and Gail S. K. Wolkowicz

Contents

vii

Reproduction Number Versus Turnover Number in Structured Discrete-Time Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Horst R. Thieme

A Discrete-Time Predator-Prey Model with Selection and Mutation Azmy S. Ackleh, Sankar Sikder, and Aijun Zhang

Abstract We study a discrete-time predator-prey system with selection and mutation in the prey population where individuals are distributed over a finite number of phenotypic traits. For the pure selection case, we establish conditions for competitive exclusion between individuals with different traits in the prey population and we show that the system converges to a boundary equilibrium representing the predator and the fittest prey trait. For the full selection mutation model, we explore coexistence through establishing persistence of more than one trait. Finally, we perform numerical simulations that support and complement the theoretical results and provide additional insights into possible model dynamics. Keywords Discrete-time predator-prey model · Selection and mutation · Competitive exclusion · Coexistence · Persistence

1 Introduction Selection-mutation models have been the focus of many studies in recent years [1, 2, 5–9, 11–17, 19]. These are models where individuals in the population are distributed over discrete [1, 5, 6] or continuous [2, 7, 9] trait spaces and where any individual may produce individuals of the same (selection) or different (mutation) traits. Some of these models consider time to be continuous [2, 5, 12, 19] and others consider time to be discrete [1, 8]. Here the focus is on a selection-mutation predator-prey model with discrete time. Thus, in what follows, we discuss the most relevant models and results in the literature. In 2005, Ackleh et al. [1] considered the following discrete-time Beverton–Holt selection model:

A. S. Ackleh (B) · S. Sikder · A. Zhang University of Louisiana at Lafayette, Lafayette, LA 70504, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_1

1

2

A. S. Ackleh et al.

n i (t + 1) =

(1 + ai )n i (t) 1 + mi

k 

, i = 1, . . . , k.

(1)

n j (t)

j=1

Here, n i represents the density of individuals with trait i, where it is assumed that individual of trait i produce individuals of trait i only, and the constants ai , m i > 0. Local stability analysis was performed for this k dimensional model in [1], and the global stability of the competitive exclusion equilibrium where n 1 → ma11 and n i → 0 as t → ∞ was only established for the case k = 2 under the condition ma11 > ma22 . In [8, 14], the global stability of the competitive exclusion equilibrium was established for a general number k under the condition ma11 > maii , i = 2, . . . , k using two completely different approaches. In [8], the selection model in [1] was extended to the following discrete selectionmutation model:  (1 + γii ai )n i (t) + γi j a j n j (t) i= j

n i (t + 1) = 1 + mi

k 

,

(2)

n j (t)

j=1

where ai , m i > 0, and  = (γi j ) satisfies

k 

γi j = 1, j = 1, . . . , k. Here,  repre-

i=1

sents the selection-mutation matrix where a fraction γi j of the offspring of individuals of type j belongs to type i. Observe that for k = 1, the models (1) and (2) reduce to the classical Beverton–Holt model given by n 1 (t + 1) =

(1 + a1 )n 1 (t) . 1 + m 1 n 1 (t)

In [8], the authors provided conditions for competitive exclusion and coexistence amongst the traits for model (2). In particular, they showed that if ma11 = · · · = al > mal+1 ≥ · · · . ≥ makk , then n i (t) → 0 as t → ∞ for i = l + 1, . . . , k and the soluml l+1 tion (n 1 (t), . . . , n k (t)) → n¯ = (n¯ 1 , . . . , n¯ l , 0, . . . , 0), where n¯ is an initial condition dependent equilibrium satisfying |n| ¯ = ma11 . Here | · | is the norm given by |n| = k  |n i | for n ∈ Rk . They also show that if the selection-mutation matrix  = (γi j ) i=1

is irreducible then there exists an () > 0 such that lim inf min (n i (t)) >  and, t→∞ 1≤i≤k

therefore, all traits coexist. Then they establish a global stability result for the case where mutation is small (i.e., the off diagonal elements of  are small) by using perturbation results from [21].

A Discrete-Time Predator-Prey Model with Selection and Mutation

3

In [3, 4] the following predator-prey model was considered: n(t + 1) = φ(n(t))n(t)(1 − f ( p(t)) p(t)) p(t + 1) = sp(t) + b(n(t))n(t) f ( p(t)) p(t), where n and p are the densities of the prey and predator population, respectively. The constant 0 < s < 1 represents the predator survival while the three nonlinearities φ, f and b represent density dependent prey growth, predator consumption of prey, and the conversion of consumed prey into new predators. In particular, f is defined to be the probability that an individual prey is consumed by an individual predator. The local stability analysis was then established including the local stability of the unique interior equilibrium which represents coexistence between the prey and the predator population. In [10], the authors provided sufficient conditions for the global stability r0 . of the interior equilibrium for the case φ(n) = 1+mn In this paper, we extend the models studied in [3, 4, 8] to a predator-prey model where the prey population is distributed over a discrete trait space i = 1, . . . , k. The main goal is to understand the impact of predator-prey interaction on the dynamics of a selection-mutation prey model. The paper is organized as follows. In Sect. 2, we introduce the selection-mutation predator-prey model. We show the system is point dissipative. In Sect. 3, we consider the pure selection model. In Sect. 4, we study the coexistence and persistence of the full selection-mutation predator-prey model. In Sect. 5, we perform some numerical experiments to support and complement the theoretical results. Finally, in Sect. 6, we give some concluding remarks.

2 The Model Assume that the prey population is distributed over a discrete trait space i = 1, . . . , k, and let n i (t) denote the number of individuals having trait i at time t. Let p(t) denote the number of individuals in the predator population at time t. We consider the following discrete-time predator-prey model: (1 + γii ai )n i (t) +



γi j a j n j (t)

i= j

n i (t + 1) = 1 + mi

k 

n j (t)

j=1

p(t + 1) = sp +

k  j=1

b j (|n(t)|)n j (t) f j ( p) p,

 1 − f i ( p) p), i = 1, . . . , k (3)

4

where ai > 0, m i > 0,

A. S. Ackleh et al. k 

γi j = 1, j = 1, . . . , k, and |n(t)| =

i=1

k 

|n j (t)|. The func-

j=1

tions n i (t) and p(t) represent the population densities of the ith prey and predator, respectively, at time t. The function f i ( p) represents the probability that an individual predator consumes an individual prey with trait i, where 0 ≤ f i ( p) ≤ 1. Thus, f i ( p) p is the total fraction of the prey subpopulation with trait i consumed by p predators where 0 ≤ f i ( p) p ≤ 1, and 1 − f i ( p) p gives the fraction of the ith prey that managed to escape from being consumed. Moreover, s is the survival probability of the predator, where 0 < s < 1, and the function bi (|n|) represents a conversion factor that gives the number of new offspring of the predators resulting from consuming a prey of type i. Note that the model (3) extends the models studied in [3, 4, 8]. We assume, bi (·) and f i (·) are functions that belong to the following set X :   X := v ∈ C 1 [0, ∞)|v  < 0, (v(x)x) > 0, lim v(x) = 0, and lim v(x)x < ∞ . x→∞

x→∞

a Note that a nonlinear function of the Beverton–Holt form v(x) = 1+bx is in X for positive a and b. We begin by establishing that the system is point dissipative. For the definition of a point dissipative system, we refer the reader to Definition 2.3 in [18].

Theorem 1 The system (3) is point dissipative. i} Proof It is easy to follow that |n| ≤ 1+max{a . By the arguments of Lemma 2.3 in [3], min{m i } max bi (D)D i} with D = 1+max{a which establishes the result. we have lim sup p ≤ min{m i } 1 − s t→∞ 

We recall Theorem 2.6 in [8] regarding the persistence in the prey population. For i 1 , . . . , i j ∈ {1, . . . , k} with i 1 < i 2 < · · · < i j , let i1 ,...,i j be the matrix ˆ = (γˆ lm ) defined by γˆ lm = γil im with 1 ≤ l, m ≤ j. Then this result follows. Theorem 2 Assume that there exist j ∈ {1, . . . , k} and i 1 , . . . , i j ∈ {1, . . . , k} with i 1 < i 2 < · · · < i j , such that i1 ,...,i j is irreducible. Then the system (3) is persistent in the prey population. That is, there exists  = () > 0 such that lim inf |n(t)| >  t→∞

for every non-zero solution of (3).

3 The Pure-Selection Model Without Mutation In this section, we consider the pure selection model without mutation. Specifically, we assume that γii = 1 and γi j = 0 for i = j. Thus, we obtain

A Discrete-Time Predator-Prey Model with Selection and Mutation

n i (t + 1) =

5

(1 + ai )n i (t)  1 − f i ( p) p), i = 1, . . . , k k  1 + mi n j (t) (4)

j=1

p(t + 1) = sp +

k 

b j (|n(t)|)n j (t) f j ( p) p,

j=1

where the variables and the parameters are the same as in (3).

3.1 Equilibria and Their Stabilities An equilibrium (n ∗1 , . . . , n ∗k , p ∗ ) of system (4) must satisfy the following: ni =

(1 + ai )n i  1 − f i ( p) p), i = 1, . . . , k k  1 + mi nj (5)

j=1

p = sp +

k 

b j (|n|)n j f j ( p) p.

j=1

We first establish the existence and local stability for an equilibrium describing competitive exclusion amongst the prey where every prey population equals zero except for one that is positive, i.e., an equilibrium of the form (n ∗ , p ∗ ) = (0, . . . , n ∗j , . . . , 0, p ∗ ) for some j ∈ {1, . . . , k}, where n ∗j > 0 and p ∗ > 0. a

a

Theorem 3 Let j ∈ {1, . . . , k} be fixed. Assume that s + b j ( mjj ) mjj f j (0) > 1. Then there exists a unique equilibrium of the form (n ∗ , p ∗ ) = (0, . . . , n ∗j , . . . , 0, p ∗ ), 1+ai where n ∗j > 0 and p ∗ > 0. Furthermore, assume that f j ≤ − f j2 and 1+m ∗ (1 − in j

f i ( p ∗ ) p ∗ ) < 1 for all i = j and asymptotically stable. a

a

1+a j (1 1+m j n ∗j

− f j ( p ∗ ) p ∗ ) = 1. Then (n ∗ , p ∗ ) is locally

Proof Since s + b j ( mjj ) mjj f j (0) > 1, the existence of a unique equilibrium point of the form (n ∗ , p ∗ ) = (0, . . . , n ∗j , . . . , 0, p ∗ ), where n ∗j > 0 and p ∗ > 0, easily follows from Theorem 2.2 in [3]. Next we assume, without loss of generality, that the equilibrium is of the form (n ∗1 , 0, . . . , 0, p ∗ ), i.e., only trait 1 component of the prey is positive. By Eq. (5), (1+a )n ∗ 1+a1 ∗ ∗ n ∗1 = 1+m11 n ∗1 (1 − f 1 ( p ∗ ) p ∗ ). Note for n ∗1 > 0, 1+m ∗ (1 − f 1 ( p ) p ) = 1. 1 n1 1 For convenience, we exchange the order equations of the predator and the prey in variables. In other words, instead of the order (n 1 , n 2 , . . . , n k , p) we consider the order ( p, n 1 , n 2 , . . . , n k ), and obtain that the Jacobian matrix J = (Ji j ),

6

A. S. Ackleh et al.

 b j (|n|)n j ( f j ( p) p) , J1i = ∂n∂i−1 kj=1  (1+a i )n i − f i ( p) p) for 2 ≤ i ≤ k + b j (|n|)n j ( f j ( p) p) for 2 ≤ i ≤ k + 1, Ji1 = 1+m i j nj  i+1 )n i+1 1 − f i+1 ( p) p) for 2 ≤ i, j ≤ k + 1. Thus, 1, Ji j = ∂n∂j−1 (1+a 1+m i+1 |n| 1 ≤ i, j ≤ k + 1, where J11 = s +

k

j=1

J ( p ∗ , n ∗1 , 0, . . . , 0) =



 AB , 0 C

where



s + (1 − s) ( f fp) A=  1 )n 1 − (1+a 1+m 1 n 1 ( f p)

p(1−s) (bn 1 ) n1 b 1 1+m 1 n 1



 , p= p∗ ,n 1 =n ∗1

B=

J13 , . . . , J1(k+1) J23 , . . . , J2(k+1)

 p= p∗ ,n 1 =n ∗1

and 1 + ak 1 + a2 ∗ ∗ ∗ ∗ (1 − f 2 ( p ) p ), . . . , (1 − f k ( p ) p ) . C = diag 1 + m 2 n ∗1 1 + m k n ∗1 

Note that the eigenvalues of J are the same as the eigenvalues of A and C. By the arguments of Theorem 2.2 as in [3], checking the two dimensional Jury’s conditions, the absolute values of the eigenvalues of A are less than 1. The eigenvalues of C are 1+ai (1 − f i ( p ∗ ) p ∗ ) < 1 for 2 ≤ i ≤ k. Thus, this establishes the local asymptotic 1+m i n ∗1 stability of (n ∗ , p ∗ ).  Corollary 1 Assume that all ma11 = · · · = makk , f i ( p) = f ( p) for all i and f  ≤ − f 2 . If a1 > ai for all 2 ≤ i ≤ k and s + b1 ( ma11 ) ma11 f (0) > 1, then the unique equilibrium (n ∗1 , 0, . . . , 0, p ∗ ), with n ∗1 > 0 and p ∗ > 0, is locally asymptotically stable. Proof The existence of an equilibrium (n ∗1 , 0, . . . , 0, p ∗ ) follows from the arguments of Theorem 2.2 in [3]. To prove its local asymptotic stability, it suffices to 1+ai ai ∗ ∗ = c and thus ai = prove that 1+m ∗ (1 − f ( p ) p ) < 1 for all 2 ≤ i ≤ k. Let mi in cm i . Let g(x) =

1

1+cx (1 1+n ∗1 x

− f ( p ∗ ) p ∗ ). Note that g(m i ) =

1+ai (1 1+m i n ∗1 ∗

− f ( p ∗ ) p ∗ ) and

1+a1 1+a1 ∗ ∗ > 1. g(m 1 ) = 1. Since g(m 1 ) = 1+m ∗ (1 − f ( p ) p ) = 1 and p > 0, 1+m 1 n ∗1 1 n1 a1 ∗ ∗  Thus, m 1 > n 1 , which implies c > n 1 . The derivative of g(x) is given by g (x) =

c−n ∗1 (1 − f ( p ∗ ) p ∗ ) > 0, where its positivity follows from the fact that c > (1+n ∗1 x)2 ∗ n 1 . Since a1 > ai , m 1 > m i for all 2 ≤ i ≤ k, then g(m i ) < g(m 1 ) = 1. That is, 1+ai (1 − f ( p ∗ ) p ∗ ) < 1 for all 2 ≤ i ≤ k. The result now follows from Theorem 3. 1+m i n ∗1



A Discrete-Time Predator-Prey Model with Selection and Mutation

7

3.2 Persistence In [8], the authors proved the following result for model (1): a

a

Theorem 4 Assume that ma11 = ma22 = · · · = mkk0 > mkk0 +1 ≥ · · · ≥ makk for some k0 ∈ 0 0 +1 1, . . . , k. Then n i (t) → 0 as t → ∞ for all i = k0 + 1, . . . , k and n(t) → n ∗ as t → ∞, where n ∗ is an equilibrium with |n ∗ | = a1 /m 1 . Let n ∗ be as in Theorem 4. Then we have the following theorem regarding the persistence in predator. Theorem 5 (Persistence in Predator) Let the assumption in Theorem 4 be satisfied. k  If, in addition, s + b j (|n ∗ |)n ∗j f j (0) > 1, then (4) is persistent in the predator. j=1

That is, there exists  > 0 such that lim inf p(t) >  for every non-zero solution with t→∞

n(0), p(0) > 0.

Proof Let U = {n ∈ Rk+ , p ∈ R+ | p = 0} on which the subsystem is given by (1). i} Let V = {n ∈ Rk+ , p ∈ R+ | ≤ |n| ≤ N , 0 ≤ p ≤ P} where N = 1+max{a and P = min{m i }

i} with D = 1+max{a . By Theorems 1 and 2, for any solution (n(t), p(t)) min{m i } with initial n(0), p(0) > 0, there exists a T > 0 such that for t > T , (n(t), p(t)) ∈ V. Let Q = U ∩ V. By Theorem 4, the omega limit set of Q is (Q) = {(n ∗ , 0)| |n ∗ | = k  b j (|n|)n j f j ( p). Then for y = (n ∗ , 0) ∈ (Q), we a1 /m 1 }. Let A(n, p) = s +

max bi (D)D 1−s

j=1

have A(y) = s +

k 

b j (|n ∗ |)n ∗j f j (0) > 1. Then, clearly A(y) is primitive and its

j=1

spectral radius r (A(y)) > 1. Furthermore, A(z) p > 0 for all z ∈ Q and p > 0. By Corollary 1 in [20], Q is a uniformly weak repeller. Then by Theorem 2.3 [20] (or Theorem 3.7 in [18]), there exists  > 0 such that lim inf p(t) ≥ , ∀n(0), p(0) > 0. t→∞  Next, we explore some sufficient conditions for the competitive exclusion in model (4). Theorem 6 The following hold: (a) If m i ≥ m 1 , ai < a1 and ci ≥ c1 for all i > 1, then n i (t) → 0 as t → ∞ for i > 1. k  (b) Assume that all f i ( p) = f ( p), s + b j (|n ∗ |)n ∗j f j (0) > 1, m 1 > m i for i = 1, j=1 a

a

= ma22 = · · · = mkk0 > mkk0 +1 ≥ ··· ≥ 0 0 +1 0 as t → ∞ for all i > 1. a1 m1

ak mk

for some k0 ∈ 1, . . . , k. Then n i (t) →

8

A. S. Ackleh et al.

Proof We begin by proving the first assertion. (a) Since m i ≥ m 1 , and ci ≥ c1 for all i > 1, we have k n i (t + 1) 1 + ai 1 + m 1 j=1 n j (t) 1 − f i ( p) p n i (t) = , i = 2, . . . , k  n 1 (t + 1) 1 + a1 1 + m i kj=1 n j (t) 1 − f 1 ( p) p n 1 (t) ≤

1 + ai n i (t) . 1 + a1 n 1 (t)

n i (t) = 0 for i = 2, . . . , k. n 1 (t) Since the system is point dissipative and persistent in the prey according to Theorems 1 and 2, we must have n i (t) → 0 as t → ∞ for i > 1. Next, we prove the second assertion. (b) Let 

a1 k . S = n ∈ R+ , p ∈ R+ ||n| ≤ m1 Since ai < a1 , we have

1+ai 1+a1

< 1, and therefore lim

t→∞

We claim that S is also positively invariant. Rewrite Eq. (1) as n˜ i (t + 1) = (1+ai )n˜ i (t) , i = 1, . . . , k. For Eq. (1), Ackleh et al. in [8] proved that S˜ = ˜  1+m i |n(t)| n˜ ∈ Rk+ ||n| ˜ ≤ ma11 is positively invariant. Now for any (n(0), p(0)) ∈ S, let ˜ i (0) i )n i (0) i )n n(0) ˜ = n(0). Since 1 − f i ( p) p ≤ 1, we have, n i (1) ≤ (1+a = (1+a = 1+m i |n(0)| 1+m i |n(0)| ˜ a ˜ |n(1)| n˜ i (1), i = 1, . . . , k. By the invariance of S, ˜ ≤ 1 and thus |n(1)| ≤ a1 . It m1

then follows from an induction argument that |n(t)| ≤ S is also positively invariant.

a1 m1

m1

for t > 0. Therefore,

Now for any (n(0), p(0)) ∈ S with n 1 (0) > 0 and p(0) > 0, it follows that n 1 (t) > 0 and p(t) > 0 for all t > 0. We shall prove two cases: ma11 = maii for i ≤ k0 and ma11 > maii for i > k0 . Case I : ma11 = maii for i ≤ k0 . Let β =

a1 m1

and g(x) =

that |n| ≤ β. Therefore, (1+ai ) (1 (1+m i |n|)

By the invariance of S, we have f i ( p) p) ≥ 0. Note that g(m i ) =

− f i ( p) p). We have, n i (t + 1) =

and

(1+βx) (1 − f i ( p) p). (1+|n|x) (β−|n|)  g (x) = (1+|n|x) 2 (1 −

(1 + ai )(1 − f i ( p) p) n i (t) = g(m i )n i (t), 1 + m i |n|

A Discrete-Time Predator-Prey Model with Selection and Mutation

n i (t + 1) = n 1 (t + 1) =

(1+ai )(1− f i ( p) p) 1+m i |n| (1+a1 )(1− f 1 ( p) p) 1+m 1 |n|

9

n i (t) n 1 (t)

g(m i ) n i (t) , i ≤ k0 . g(m 1 ) n 1 (t)

Now g  (x) ≥ 0 implies that g is nondecreasing and thus g(m i ) ≤ g(m 1 ) because n i (t) g(m i ) exists for i ≤ k0 . ≤ 1. So we have lim of m 1 > m i . This gives us g(m 1) t→∞ n 1 (t) n i (t) = l for some l ≥ 0. Obviously, if l = 0, we must have n i (t) → 0 Let lim t→∞ n 1 (t) as t → ∞. If l > 0, then we have that implies |n| →

a1 . m1

g(m i ) g(m 1 )

→ 1 as t → ∞. Hence

(1+ai ) 1+m i |n| (1+a1 ) 1+m 1 |n|

→ 1,

By Theorem 5, lim inf | p(t)| >  for some positive . t→∞

i )(1− f i ( p) p) n i (t) ≤ (1 − f i ())n i (t), and thus Then as t → ∞, n i (t + 1) = (1+a1+m i |n| n i (t) → 0. Furthermore we claim that the solutions eventually enter S and remain there for any (n(0), p(0)) ∈ / S, i.e., for any (n(0), p(0)) satisfying |n(0)| > ma11 = (1+ai ) β. Now we show for (n(0), p(0)) ∈ / S, g(m i ) = (1+m (1 − f i ( p) p) < 1. i |n(0)|) Note that, ai < m i |n(0)|. Adding 1 to the both sides, we get, 1 + ai < 1 + i < 1 and m i |n(0)|. Dividing both sides by 1 + m i |n(0)|, we have 1+m1+a i |n(0)| 1+ai thus, 1+m i |n(0)| (1 − f i ( p) p) < 1. So n i (1) < n i (0) and by induction, n i (t + 1) < n i (t) for all t ≥ 0. Therefore, lim n i (t) ≥ β. Thus, lim n i (t + 1) = t→∞ t→∞ 1 + ai 1 + ai (1 − f i ( p) p)n i gives us lim (1 − f i ( p) p) = 1. Howlim t→∞ 1 + m i |n| t→∞ 1 + m i |n| i ever, 1+m 1+a lim |n| (1 − f i ( p) p) ≤ (1 − f i ( p) p) < 1, which is a contradiction. i

t→∞

This completes the proof of the claim. Case II : ma11 > maii for i > k0 . Now again, we have n i (t + 1) = n 1 (t + 1)

(1+ai )(1− f i ( p) p) 1+m i |n| (1+a1 )(1− f 1 ( p) p) 1+m 1 |n|

n i (t) , i > k0 . n 1 (t)

Let aˆ i be such that maii = maˆ i1 . Since m 1 > m i , aˆ i > ai . From ma11 > maii = maˆ i1 , we have a1 > aˆ i . Combining these two, we have a1 > aˆ i > ai . By using arguments similar to those above used to proof Case I, for maii = maˆ i1 , we get (1+aˆ i )(1− f i ( p) p) 1+m 1 |n|

≥

(1+ai )(1− f i ( p) p) . 1+m i |n|

On the other hand since a1 > aˆ i , we have, (1+ai )(1− f i ( p) p) n i (t) 1+m i |n| (1+a1 )(1− f 1 ( p) p) (1+aˆ i )(1− f i ( p) p) =l > . Thus ≤ 1 and so lim (1+a 1 )(1− f 1 ( p) p) 1+m 1 |n| 1+m 1 |n| t→∞ n 1 (t) 1+m 1 |n| for i > k0 for some l ≥ 0. By the same arguments as before, we have n i (t) → 0  as t → ∞ for i > k0 .

10

A. S. Ackleh et al.

We now have the following corollary, the proof of which easily follows from the results in Theorems 5 and 6. Corollary 2 Assume that all f i ( p) = f ( p), s +

k 

b j (|n ∗ |)n ∗j f j (0) > 1, m 1 >

j=1

m i for all i > 1, and ma11 > ma22 ≥ · · · ≥ makk . Then there exists  > 0 such that lim inf min{|n 1 (t)|, | p(t)|} >  for every non-zero solution of (4), and n i (t) → 0 t→∞

as t → ∞ for all i = 2, . . . , k. By Theorem 6 (b) together with Theorem 2.1 in [10], we have the following corollary regarding the global stability of (4): Corollary 3 Let f i ( p) = f 1 ( p) p, and

c , 1+cp

 G(n 1 , p) = 1 −

F1 =

(1+a1 )n 1 (1 1+m 1 n 1

∂ F1 ∂ F2 + ∂n 1 ∂p

+

− f 1 ( p) p), F2 = sp + b1 (n 1 )n 1

∂ F1 ∂ F2 ∂ F1 ∂ F2 + . ∂n 1 ∂ p ∂ p ∂n 1

Assume the following hold: (a) s +

k 

b j (|n ∗ |)n ∗j f j (0) > 1;

j=1

(b) m 1 > m i for all i > 1; (c) ma11 > ma22 ≥ · · · ≥ makk ; (d) G(n ∗1 , p ∗ ) > 0. Then for every non-zero solution of (4), p(t) → p ∗ , n 1 (t) → n ∗1 , and n i (t) → 0 as t → ∞ for all i = 2, . . . , k, where (n ∗1 , 0, . . . , 0, p ∗ ) is the equilibrium of (4).

4 The Full Model with Mutation In this section, we consider the full model with mutation.

4.1 Coexistence and Persistence In this subsection, we shall prove uniform persistence [18, 20] for the full model under certain conditions. Theorem 7 If  is irreducible, then the system (3) is uniformly persistent in the prey. That is, there exists an  > 0 such that lim inf min{n i (t)} > , t→∞

i

A Discrete-Time Predator-Prey Model with Selection and Mutation

11

for every nonnegative nontrivial solution (n(t), p(t)) of system (3). Proof Rewrite the system (3) as n(t + 1) = A n(t) p(t + 1) = sp(t) +

k 

(6)

b j (n(t))n j (t) f j ( p(t)) p(t),

j=1

  where A = ai j (1 − f i ( p) p)

1≤i, j≤k

for i = j. Consider   where A˜  = ai j

with ai j =

(1+γii ai ) 1+m i |n|

for i = j and ai j =

γi j a j 1+m i |n|

n(t + 1) = A˜  n(t),

1≤i, j≤k

(7)

. By Lemma 2.7 in [8], there exists an ˆ > 0 such that lim inf min{n˜ i (t)} > ˆ, t→∞

i

for every nonnegative nontrivial solution n(t) ˜ of system (7). By comparison A ≥ ¯ p), ¯ we have that n(t) ≥ n(t), ˜ and thus A˜  (1 − f ( p) ¯ p)ˆ ¯ , lim inf min{n i (t)} > (1 − f ( p) t→∞

i

bi (D)D where (n(t), p(t)) is the solution of (6) provided n(0) = n(0), ˜ and p¯ = max1−s is the upper bound of p in Theorem 1. Letting  = (1 − f ( p) ¯ p)ˆ ¯ , that finishes the proof. 

We have the following theorem regarding the persistence in predator. Theorem 8 (Persistence in the Predator) Assume that n(t) ˜ → n ∗ as t → ∞, where ∗ n(t) ˜ is a solution with n(0) ˜ > 0, and n is an interior equilibrium of (7). If s + k  b j (|n ∗ |)n ∗j f j (0) > 1, then (3) is persistent in the predator. That is, there exists j=1

 > 0 such that lim inf p(t) >  for every non-zero solution with n(0), p(0) > 0. t→∞

Proof The proof follows by a similar argument as in Theorem 5.



Theorem 8 requires the global stability of system (7). However, system (7) is still a complicated system and its global stability remains an open question. In [8], the authors established global stability of system (7) for some particular cases. Based on the results in [8], we have the following corollaries. Corollary 4 Assume that m 1 = · · · = m k . If s +

k  j=1

is persistent in the predator.

b j (|n ∗ |)n ∗j f j (0) > 1, then (3)

12

A. S. Ackleh et al.

Proof By Lemma 2.7 in [8], we have the global stability of system (7) if m 1 = · · · =  m k . Then, using Theorem 8, the result follows. Corollary 5 Assume that a1 /m 1 > ai /m i for all i = 2, . . . , k. Then there exist δ,  > 0 such that for every  irreducible and  − I < δ, there exists an interior k  equilibrium n ∗ satisfying ||n ∗ | − a1 /m 1 | < . Moreover, if s + b j (|n ∗ |)n ∗j f j (0) > 1, then (3) is persistent in the predator.

j=1

Proof By Theorem 2.9 in [8], we have the global stability of system (7). Then the result follows from Theorem 8.  Corollary 6 Assume that k = 2 in (3) and (i) γ11 a1 + 1 ≥ γ12 a2 and γ22 a2 + 1 ≥ γ21 a1 ; (ii) γ11 a1 = γ12 a2 and γ22 a2 = γ21 a1 . In any of the following cases: (a) (b) (c) (d) (e) (f) (g)

γ11 a1 > γ12 a2 > 0 and γ22 a2 > γ21 a1 > 0; γ11 a1 > γ12 a2 > 0 and γ212a1 < γ22 a2 < γ21 a1 ; γ22 a2 > γ21 a1 > 0 and γ122a2 < γ11 a1 < γ12 a2 ; γ21 = 0, γ11 a1 > γ12 a2 > 0 and γ11b1a1 < γ22b2a2 ; γ12 = 0, γ22 a2 > γ21 a1 > 0 and γ11b1a1 > γ22b2a2 ; γ21 = 0, γ122a2 < γ11 a1 < γ12 a2 and γ11b1a1 < γ22b2a2 ; γ12 = 0, γ212a1 < γ22 a2 < γ21 a1 and γ11b1a1 > γ22b2a2 ,

there exists an interior equilibrium n ∗ of system (7) that is asymptotically stable 2  and attracts all solutions with n j (0) > 0 for j = 1, 2. Moreover, if s + b j (|n ∗ |) n ∗j f j (0) > 1, then (3) is persistent in the predator.

j=1

Proof By Corollary 2.12 in [8], we have the global stability of system (7). Then the result follows from Theorem 8. 

5 Numerical Simulations In this section, we provide numerical simulations that support and complement the theoretical results established in the previous sections.

A Discrete-Time Predator-Prey Model with Selection and Mutation

13

5.1 Numerical Simulations for the Pure Selection Case We perform numerical experiments for the model under the pure selection case. In the first set of numerical examples, we demonstrate the effect of the predator on the dynamics of the prey population. ci with c1 = Example 1 (Effect of predator) We let k = 3, s = 0.5, f i ( p) = 1+c ip b¯i c2 = c3 = 0.99, and bi (|n|) = 1+bˆ |n| , with b¯1 = 4, b¯2 = 1, b¯3 = 5, bˆ1 = 0.8, bˆ2 = i 0.9, bˆ3 = 1.6. We take the initial condition to be n 1 (0) = 10, n 2 (0) = 20, n 3 (0) = 25, p(0) = 22, and consider three scenarios as follows: (a) In the first scenario, we set the parameters

a1 = 100, a2 = 150, a3 = 200, m 1 = 2, m 2 = 5, m 3 = 2. = 50, ma22 = 150 = 30, ma33 = 200 = 100. Observe that In this case, we have, ma11 = 100 2 5 2 ai the third prey type which has the largest ratio m i wins the competition (Fig. 1). (b) In the second scenario, we let f i ( p), bi (|n|) and s be the same as in part (a). We choose the other parameter values as follows: a1 = 100, a2 = 150, a3 = 50, m 1 = 2, m 2 = 3, m 3 = 1. = 50, ma22 = 150 = 50, ma33 = 50 = 50. In this case, the system (4) Here, ma11 = 100 2 3 1 has seven non-negative equilibrium points. In Table 1, we list all the non-negative equilibria of system (4) and their corresponding spectral radius to determine local stability. Moreover, numerical simulations suggest that the equilibrium (0, 23.6216, 0, 1.1123) is globally asymptotically stable. Thus, the second prey type which has the maximum ai value wins the competition and all other prey types go extinct (Fig. 2).

Table 1 Local stability for the equilibria of system (4) Equilibria Spectral radius (0, 0, 0, 0) (50, 0, 0, 0) (0, 50, 0, 0) (0, 0, 50, 0) (0, 0, 7.8949, 4.7814) (0, 23.6216, 0, 1.1123) (5.7164, 0, 0, 7.1956)

151.0 5.3293 1.5761 3.5556 1.0669 0.9964 1.0242

Local stability Unstable Unstable Unstable Unstable Unstable Stable Unstable

14

A. S. Ackleh et al. 30

First prey Second prey Third prey Predator

25

Population

20

15

10

5

0 5

10

15

20

25

30

35

40

45

50

Time

Fig. 1 Competitive exclusion outcome where the prey with the largest ratio maii wins the competition First prey Second prey Third prey Predator

30

25

Population

20

15

10

5

0 200

400

600

800

1000

1200

1400

1600

Time

Fig. 2 Competitive exclusion outcome for populations with equal ratios

ai mi

1800

2000

A Discrete-Time Predator-Prey Model with Selection and Mutation

15

30

First prey Second prey Third prey Predator

25

Population

20

15

10

5

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time

Fig. 3 Competitive exclusion outcome where the prey with the smallest ratio tition

ai mi

wins the compe-

(c) For the third scenario, we let f i ( p), bi (|n|) and s be the same as in part (a). We choose the other parameter values as follows: a1 = 100.001, a2 = 150, a3 = 50.001, m 1 = 2, m 2 = 3, m 3 = 1. In this case, ma33 > ma11 > ma22 . Even though ma33 has the largest ratio, the second prey type which has the smallest ratio wins the competition (Figs. 3). Note that the results in (a) coincide with those in [8] in the sense that the species which has the largest ratio maii wins the competition. However, the results in (b) and (c) differ from those in [8]. In particular, in part (b), we do not observe coexistence between the different prey types even though the ratios maii are all equal, and in part (c) the species which has the smallest ratio maii wins the competition. These numerical simulations demonstrate that predator-prey interaction may change the dynamics of the system and presents an advantage to a prey phenotypic trait that would have gone extinct without the presence of the predator. Next, we offer another example on the effect of predator on the overall system dynamics. Example 2 (Effect of predator on consumption of prey) Let k = 3, s = 0.5, f i ( p) = ¯ ci with c1 = 0.94 and c3 = 0.99 and bi (|n|) = 1+bbˆi |n| , with b¯1 = 4, b¯2 = 1, b¯3 = 1+ci p i 5, bˆ1 = 0.8, bˆ2 = 0.9, bˆ3 = 1.6. Here we choose the rest of the parameter values as

16

A. S. Ackleh et al.

Fig. 4 Bifurcation diagram for c2

follows: a1 = 100, a2 = 150, a3 = 50, m 1 = 2, m 2 = 3, m 3 = 1. = 50, ma22 = 150 = 50, ma33 = 50 = 50. We obtain a Notice we have ma11 = 100 2 3 1 bifurcation diagram with respect to c2 with step size 0.01 from 0 to 1. Choose the initials with n 1 (0) = 10, n 2 (0) = 20, n 3 (0) = 30, and p(0) = 50. For each c2 value, we simulate the model over the time interval t ∈ [0, 10000] and plot the last 500 points of each simulation. We present the resulting bifurcation diagram in Fig. 4. By observation, the first bifurcation occurs around 0.45, where the predator starts to persist. The second bifurcation occurs at around 0.95 where the second prey type starts to die out while the first prey type persists and wins the competition. In the following numerical example, we present an interesting phenomenon where competitive exclusion occurs and the long term behavior is periodic. Example 3 (Competitive exclusion with cycles) Let k = 3, s = 0.5, and we choose the same function f ( p) as in Example 3.1 in [4]. In particular, let ⎧ c1 ⎪ p ≤1−δ ⎨ 1+c2 p , c0 , 1−δ < p 0, we can construct such an f ∈ X . Now for this piece-wise function, we consider the system (4) with the parameter values given by c0 = 1.48, c = 0.5, c1 = 0.99, c2 = 0.000034125, c3 = 145.098, δ = 0.01. ¯ Let bi (|n|) = 1+bbˆi |n| , with b¯1 = 1.013513513, b¯2 = 1.01, b¯3 = 0.9, bˆ1 = 1, bˆ2 = i 1.1, bˆ3 = 1.5. We set the initial population to the following values:

n 1 (0) = 10, n 2 (0) = 20, n 3 (0) = 25, p(0) = 22, and consider the following three cases: Table 2 Vertices for the six-cycle n1 0.3199954543 0.1998772248 3.5204119502 49.8631124031 39.0520432064 0.9100938184

p 1.2469359076 0.8678150334 0.5789534468 0.7418690958 1.1006566067 1.5323876197

18

A. S. Ackleh et al. 1.1

1.05

1

Predator

0.95

0.9

0.85

0.8

0.75

0.7 0

5

10

15

20

25

30

35

40

45

50

55

Third prey

Fig. 6 An eight-cycle limit of the solution solution to (4)

(a) For the first case, we choose the following parameter values: a1 = 149, a2 = 175, a3 = 100, m 1 = 1, m 2 = 2, m 3 = 1. In this case, the solution converges to a 6-cycle, as shown in Fig. 5 with vertices as in Table 2. (b) For this case, we choose a1 = 149, a2 = 75, a3 = 200, m 1 = 1, m 2 = 2, m 3 = 1, and keep the rest of the parameter values as before. Here, the solution converges to an 8-cycle as shown in Fig. 6 with vertices as in Table 3. (c) In the third case, we choose a1 = 149, a2 = 600, a3 = 100, m 1 = 1, m 2 = 2, m 3 = 1, and keep the rest of the parameter values as before. The solution converges to the equilibrium (n 1 , n 2 , n 3 , p) = (0, 1.3739755355, 0, 1.0982527754) as shown in Fig. 7.

5.2 Numerical Simulations for a Selection-Mutation Case We consider the long-term dynamics for the selection mutation model with an irreducible matrix. Example 4 (Long-term dynamics of the selection-mutation model with an irreci with c1 = 0.99, c2 = 0.99, and ducible matrix) Let k = 3, s = 0.5, f i ( p) = 1+c ip b¯i c3 = 0.99 and bi (|n|) = , with b¯1 = 4, b¯2 = 1, b¯3 = 5, bˆ1 = 0.8, bˆ2 = 0.9, 1+bˆi |n|

bˆ3 = 1.6. We set the initial population to the following values:

A Discrete-Time Predator-Prey Model with Selection and Mutation Table 3 Vertices for eight-cycle n3

19

p

0.6869603875 8.2400674085 50.0520124062 47.9926075545 35.1170378895 21.0717100907 6.4949123922 1.1359449841

0.9084404697 0.7280635376 0.7641213252 0.8299709634 0.9012199741 0.9759451538 1.0498868509 1.0655411136

2.5

2

Predator

1.5

1

0.5

0 0

0.5

1

1.5

2

Second prey

Fig. 7 An equilibrium limit of the solution of (4)

n 1 (0) = 10, n 2 (0) = 20, n 3 (0) = 25, p(0) = 22, and choose the rest of parameter values as follows: a1 = 300, a2 = 100, a3 = 400, m 1 = 10, m 2 = 4, m 3 = 2, and

⎛

⎞ 0.35 0.50 0.15  = ⎝0.25 0.45 0.3 ⎠ . 0.4 0.05 0.55

2.5

20

A. S. Ackleh et al. 25

First prey Second prey Third prey Predator

Population

20

15

10

5

0 10

20

30

40

50

60

70

80

90

100

Time

Fig. 8 Coexistence of the predator and all prey types

Clearly, the results in Fig. 8 demonstrate that in this case all prey-types and the predator coexist. Next, we shall study the long-term dynamics of the selection mutation model with a reducible matrix. Example 5 (Effect of predator on the selection-mutation model with a reducible matrix) We let k = 5, i.e., we consider a six dimensional system. We assume c b¯ that the functions b j (|n|) = 1+bˆ j |n| and f j ( p) = 1+cj j p , j = 1, . . . , 5 for the system j (4). Let c1 = c2 = c3 = c4 = c5 = 0.99 and b¯1 = 4, b¯2 = 1, b¯3 = 5, b¯4 = 4, b¯5 = 2, bˆ1 = 0.8, bˆ2 = 0.9, bˆ3 = 1.6, bˆ4 = 1.5, bˆ5 = 0.85. Let s = 0.5 and ⎛ ⎞ 0.6 0.75 0 0 0 ⎜0.4 0.25 0 0 0 ⎟ ⎜ ⎟ ⎟ =⎜ ⎜ 0 0 0.35 0.2 0.3⎟ , ⎝ 0 0 0.15 0.6 0.5⎠ 0 0 0.5 0.2 0.2 which is a reducible matrix. We set the initial population to the following values: n 1 (0) = 10, n 2 (0) = 12, n 3 (0) = 14, n 4 (0) = 15, n 5 (0) = 16, p(0) = 10, and consider the following two cases:

A Discrete-Time Predator-Prey Model with Selection and Mutation

21

18

First prey Second prey Third prey Fourth prey Fifth prey Predator

16

14

Population

12

10

8

6

4

2

0 5

10

15

20

25

30

35

40

Time

Fig. 9 Competitive exclusion between blocks with a reducible  where the second block wins the competition

(a) First, we choose the following parameter values: a1 = 100, a2 = 150, a3 = 200, a4 = 300, a5 = 490, m 1 = 2, m 2 = 5, m 3 = 2, m 4 = 10, m 5 = 7. In this case, = 50, ma22 = 150 = 30, ma33 = 200 = 100, ma44 = 300 = 30, ma55 = we have, ma11 = 100 2 5 2 10 490 = 70. The results in Fig. 9 show that the prey types in the second block of the 7 matrix γ win the competition. That is the third, fourth and fifth prey types persist while the first and second prey type go to extinction. (b) Next, we choose the parameters ai and m i as follows: a1 = 700, a2 = 150, a3 = 50, a4 = 75, a5 = 110, m 1 = 1, m 2 = 3, m 3 = 1, m 4 = 1.5, m 5 = 2. We 75 = 700, ma22 = 150 = 50, ma33 = 50 = 50, ma44 = 1.5 = 50, ma55 = have that ma11 = 700 1 3 1 110 = 55. In this case, the prey types in the first block (i.e., the first two prey types) 2 win the competition (Fig. 10).

22

A. S. Ackleh et al. First prey Second prey Third prey Fourth prey Fifth prey Predator

60

50

Population

40

30

20

10

0 5

10

15

20

25

30

35

40

45

50

Time

Fig. 10 Competitive exclusion between blocks with a reducible  where the first block wins the competition

6 Concluding Remarks In the paper, we study the long term-dynamics of a discrete-time predator-prey model with selection and mutation in the prey population where individuals in the prey population are distributed over a finite number of phenotypes. First we prove that the system is dissipative and persistent under certain conditions. Then for the pure selection model, we provide conditions for competitive exclusion and global stability of a boundary equilibrium representing coexistence between the predator and the fittest prey. For the full selection-mutation model with an irreducible matrix , we show that the system is persistent with lim inf min{n i (t)} >  representing coexistence of t→∞

i

all prey types. Finally, we offer some numerical examples and demonstrate that the system with a reducible (block diagonal) matrix , attains competitive exclusion dynamics with one block winning the competition.

A Discrete-Time Predator-Prey Model with Selection and Mutation

23

References 1. Ackleh, A.S., Dib, Y.M., Jang, S.R.J.: A discrete-time beverton-holt competition model. In: Difference Equations and Discrete Dynamical Systems, pp. 1–10 (2005). https://doi.org/10. 1142/9789812701572_0001 2. Ackleh, A.S., Fitzpatrick, B.G., Thieme, H.R.: Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete Cont. Dyn. Syst.-B 5(4), 917–928 (2005) 3. Ackleh, A.S., Hossain, M.I., Veprauskas, A., Zhang, A.: Persistence and stability analysis of discrete-time predator- prey models: a study of population and evolutionary dynamics. J. Differ. Equ. Appl. 25(11), 1568–1603 (2019). https://doi.org/10.1080/10236198.2019.1669579 4. Ackleh, A.S., Hossain, M.I., Veprauskas, A., Zhang, A.: Long-term dynamics of discrete-time predator-prey models: stability of equilibria, cycles and chaos. J. Differ. Equ. Appl. 26(5), 693–726 (2020). https://doi.org/10.1080/10236198.2020.1786818 5. Ackleh, A.S., Hu, S.: Comparison between stochastic and deterministic selection-mutation models. Math. Biosci. Eng. 4(2), 133–157 (2007) 6. Ackleh, A.S., Marshall, D.F., Heatherly, H.E.: Extinction in a generalized lotka-volterra predator-prey model. J. Appl. Math. Stoch. Anal. 13, 287–297 (2000) 7. Ackleh, A.S., Marshall, D.F., Heatherly, H.E., Fitzpatrick, B.G.: Survival of the fittest in a generalized logistic model. Math. Models Methods Appl. Sci. 9(09), 1379–1391 (1999) 8. Ackleh, A.S., Sacker, R.J., Salceanu, P.: On a discrete selection-mutation model. J. Differ. Equ. Appl. 20(10), 1383–1403 (2014). https://doi.org/10.1080/10236198.2014.933819 9. Ackleh, A.S., Saintier, N.: Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete Con. Dyn. Syst.-B 26(3), 1469–1497 (2021) 10. Ackleh, A.S., Salceanu, P., Veprauskas, A.: A nullcline approach to global stability in discretetime predator-prey models. J. Differ. Equ. Appl. 27(8), 1120–1133 (2021) 11. Calsina, À., Cuadrado, S.: Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics. J. Math. Biol. 48(2), 135–159 (2004) 12. Calsina, À., Cuadrado, S.: Asymptotic stability of equilibria of selection-mutation equations. J. Math. Biol. 54(4), 489–511 (2007) 13. Calsina, A., Cuadrado, S., Desvillettes, L., Raoul, G.: Asymptotics of steady states of a selection-mutation equation for small mutation rate. Proc. R. Soc. Edinb.: Sect. A Math. 143(6), 1123–1146 (2013). https://doi.org/10.1017/S0308210510001629 14. Chow, Y., Hsieh, J.: On multidimensional discrete-time beverton-holt competition models. J. Differ. Equ. Appl. 19(3), 491–506 (2013) 15. Chow, Y., Palmer, K.: On a discrete three-dimensional leslie-gower competition model. Discrete Cont. Dyn. Syst.-B 24(8), 4367–4377 (2019) 16. Iglesias, S.F., Mirrahimi, S.: Selection and mutation in a shifting and fluctuating environment. Commun. Math. Sci. 19(7), 1761–1798 (2021) 17. Lorenzi, T., Chisholm, R.H., Desvillettes, L., Hughes, B.D.: Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments. J. Theor. Biol. 386, 166–176 (2015) 18. Magal, P., Zhao, X.Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37(1), 251–275 (2005) https://doi.org/10.1137/ S0036141003439173 19. Raoul, G.: Long time evolution of populations under selection and vanishing mutations. Acta Appl. Math. 114(1), 1–14 (2011) 20. Salceanu, P., Smith, H.: Lyapunov exponents and persistence in discrete dynamical systems. Discrete Cont. Dyn. Syst. - Ser. B 12(1), 187–203 (2009). https://doi.org/10.3934/dcdsb.2009. 12.187 21. Smith, H., Waltman, P.: Perturbation of a globally stable steady state. Proc. Amer. Math. Soc. 127(2), 447–453 (1999)

On the Dynamics and Asymptotic Behaviour of the Mean Square of Scalar Linear Stochastic Difference Equations John A. D. Appleby and Emmet Lawless

Abstract In this paper, we show that the mean square of the solution of a scalar autonomous linear stochastic difference equation of finite order can be written in terms of the solution of an auxiliary deterministic Volterra summation equation. The dynamics of the covariance of the process can also be written in terms of this deterministic equation. As an application, we determine necessary and sufficient conditions for the mean square stability of the equilibrium solution in the case that the underlying deterministic equation is of first order, and determine the exact real exponential rate of growth or decay in the mean–square. Keywords Stochastic difference equations · Linear difference equations · Mean square · Mean square stability · Asymptotic stability · Characteristic equation · Volterra equation · Discrete renewal theorem

1 Introduction This paper is concerned with giving a deterministic description of the dynamics of the mean square E[X 2 (n)] for the solution X (n) of the linear stochastic difference equation 

p j=1



q j=1

 β j X (n − j) ξ(n),

n≥1 ψ(n), n = − p  + 1, . . . , 0, (1) where p  = max( p, q), and then applying these results to give a very precise characterisation of the asymptotic behaviour of the mean square in the case p = 1. Standard (and natural) independence, measurability, and integrability conditions, X (n) =

α j X (n − j) +

J. A. D. Appleby (B) · E. Lawless School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland e-mail: [email protected] E. Lawless e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_2

25

26

J. A. D. Appleby and E. Lawless

detailed below, are imposed on the initial sequence ψ and the (zero mean and unit variance) noise sequence ξ to enable this to happen. The α’s and β’s are deterministic constants, and p and q are fixed deterministic integers. It is to be noted that if ψ(n) = 0 for all n ≤ 0, then the solution of (1) is X (n) = 0 for all n ≥ 0, so it is of interest to study the asymptotic behaviour of E[X 2 (n)] as n → ∞ in order to study the mean–square asymptotic stability of the zero solution. The asymptotic behaviour of such linear equations is a topic of great interest (see e.g. [14, 21]). We refer the reader to the monograph by Shaikhet [22] for much of the discrete stability theory. A closely related topic is viewing the stochastic difference equation as a discretisation of stochastic differential equations, and considering the connection between their dynamics (see e.g. [4, 5, 7, 10, 11]), and the results we present here can clearly be applied to study the discretisation of stochastic delay differential equations of Itô type. The approach we use to study stability here can be considered as a discrete–time analogue of the approach applied in [2] which uses the renewal equation to determine exact asymptotic behaviour. Indeed, although there are many results in the literature concerning sufficient conditions for mean square asymptotic stability, to the best of our knowledge there are comparatively few which detail necessary and sufficient conditions for equations with delay. For continuous equations, some examples are [1, 6, 18], and results on the junction of continuous and discrete time, which also seek to characterise stability include [24, 25]. To the best of our knowledge, the earliest work on the characterisation of mean square stability for delayed equations in continuous time appears to be [20]. An alternative approach to obtaining necessary and sufficient conditions for mean square stability in one–step linear difference equations goes back to Bellman [3]. As mentioned above, an interesting and comprehensive monograph on Liapunov techniques for studying the stability of stochastic difference equations is Shaikhet [22]; a corresponding volume for continuous analysis is [23]. Many results in the theory of stochastic dynamics in discrete time are analogues and developments of corresponding results in the deterministic theory. A comprehensive exposition of this theory, which underpins stochastic results both in [23], and in the current work, can be found in the book of Kulenvoic and Ladas [15]. The literature on mean square dynamics in continuous–time is vast, and although not directly connected to this work, provides some context for the questions we address. Definitive monographs which cover this topic, and are often devoted to studying exponential moment estimates on solutions include those by Mao [16, 17] and Kolmanovskii and Myshkis [13]. A number of elementary techniques and notions from deterministic difference equations are employed in this paper, to which we refer the reader to Elaydi [8]. Much of this paper revolves around the study of Volterra summation equations; for more details on this topic the reader may consult [19]. We summarise briefly the results of the paper. First, we show that E[X 2 (n)] can be written entirely in terms of the solution of a certain linear deterministic convolution Volterra summation equation, along with some other sequences related to the solution of the stochastically–unforced variant of (1), namely

On the Dynamics and Asymptotic Behaviour of the Mean Square …

x(n) =

p 

α j x(n − j), n ≥ 1; x(n) = ψ(n), n ≤ 0.

27

(2)

j=1

Thus, to a certain degree, the principal result of the paper is to reduce a problem in stochastic dynamics to a quite classical problem in deterministic dynamics, albeit one with some particular features. We sketch how this result can be used to write down an expression for the covariance of the process X , in the form Cov(X (n), X (n + k)) for k ≥ 0. We hope that this type of expression can be useful in later work concerning equations which do not have equilibrium solutions, and in which “stability” is characterised not by for example, the mean square tending to zero, but instead by the solution (or the solution, when suitably re–scaled) being an asymptotically stationary process. A full asymptotic analysis of the Volterra equations in the case when p = 2 and for general p > 2 will be given in later works, but to show the merit of expressing the mean square in terms of Volterra equations, in this paper, we give a complete asymptotic analysis in the case that p = 1. The situation is simpler for p = 1, since certain calculations can be performed directly and explicitly. On the other hand, it can be guessed on the basis of the equations satisfied by the mean square for general p, that at the very least a more indirect approach is needed in the general case. In this work, in the case where p = 1, our representative result has the following character: the mean square is asymptotic to a real positive geometric sequence, in the sense that E[X 2 (n)] ∼ Cμ2n , n → ∞ where C is a positive constant (depending on ψ) and μ > 0 is ψ–independent, and is the unique real zero of an explicit nonlinear equation. In fact, it can be shown that this zero is also a real zero of a polynomial which can be written down exactly. According as to whether μ > 1, μ = 1 or μ < 1, we have exponential growth, convergence to a finite and non–trivial limit, or exponential convergence to zero, of the mean square, and it turns out that we can deduce easily–checked necessary and sufficient conditions for the mean square asymptotic stability. These conditions, which can be formulated directly in terms of the parameters in (1), have some value, since they avoid the need to compute μ explicitly, or approximate its value numerically. Lastly, we note that the asymptotic behaviour of E[X 2 (n)] is real exponential, irrespective of the form of the noise perturbation. This gives a tantalising hint that the leading order asymptotic behaviour of the mean square of linear stochastic equations might be real exponential more often than might be naively suspected. We hope to explore this further in later works. To whet the reader’s appetite, we finish the introduction with the equation obeyed by the mean–square. Introduce r (n) =

p  j=1

α j r (n − j), n ≥ 1; r (0) = 1; r (n) = 0; n ≤ −1,

(3)

28

J. A. D. Appleby and E. Lawless

and let x be the sequence given by (2), noting that although x is stochastic, its dynamics is deterministic, since the randomness in x is confined to its initial sequence. Now introduce the following sequences depending on x and r G(rn−1 ) :=

q 

β j r (n − j), n ≥ 1,

j=1

G(xn−1 ) :=

q 

β j x(n − j), n ≥ 1.

j=1

Then the sequence Y defined by Y (n − 1) :=

q 

β j X (n − j), n ≥ 1,

j=1

has mean square E[Y 2 (n)] satisfying the deterministic Volterra equation E[Y 2 (n)] = E[G 2 (xn )] +

n 

G 2 (rn− j )E[Y 2 ( j − 1)], n ≥ 1,

j=1

(with E[Y 2 (0)] = E[G 2 (x0 )]) and the mean square of X can be related explicitly to that of Y via the relation E[X 2 (n)] = E[x 2 (n)] +

n 

r 2 (n − j)E[Y 2 ( j − 1)], n ≥ 1.

j=1

It is the authors’ belief that the approach in this paper can be used successfully to deduce Volterra type–equations for the mean square of a variety of linear stochastic difference equations, and even equations with unbounded delay, including those in higher dimensions, non–autonomous equations, and equations which possess forcing terms. We note also that the method of proof in this work is a discrete analogue of results deduced in continuous time, which should enhance the prospects for this work to throw light on the performance of numerical methods.

2 Standing Assumptions and Precise Formulation of the Problem Let p, q ∈ N and define p  := max( p, q) ∈ N. At first, we do not restrict p ≥ 1. Later, to derive asymptotic results, we will specialise to the case when p = 1. We now give a precise formulation of the problem, which is quite standard; the reader can refer to [22] for details. Let (Ω, F, P) be a complete probability space

On the Dynamics and Asymptotic Behaviour of the Mean Square …

29

(Ω, F, P) equipped with a filtration, and consider the equation  p X (n) =

j=1 α j X (n − j) +

 q

 β X (n − j) ξ(n), n ≥ 1 j j=1

n = − p  + 1, . . . , 0. (4) In order to impose hypotheses on the data in this equation, we give some standard notation and terminology. The expectation of an F–measurable random variable Z (when this is well–defined) will be denoted by E[Z ]. For m > 0, we say that an F–measurable random variable obeys Z ∈ L m (Ω) if E[|X |m ] < +∞. The random variables U and V are independent in this setting with respect to the probability measure P: this is equivalent to P[U ≤ u, V ≤ v] = P[U  ≤ u][V ≤ v] for all u, v ∈ R. We say that the sequence (an )n∈K is in m (K) if n∈K |an |m < +∞. Typically here m = 1 or 2, and K is N or Z+ . We will define carefully in a moment the filtration (F(n))n≥0 with respect to which a process X = {X (n) : n ≥ − p  + 1} obeying (4) will be (F(n))–adapted. Such a process will be called a solution to (4). p q Let {α j } j=1 , {β j } j=1 be real sequences with βq = 0 and α p = 0, so that the number of lags in both the deterministic and noisy part of (4) are indeed p and q respectively. We assume ψ(n),

{ξ(n)}n≥1 is a sequence of independent and identically distributed random variables with zero mean and variance one. (5) Let F ξ (n) = σ ({ξ(n) : 1 ≤ j ≤ n}), n ≥ 1, so that (F ξ (n))n≥1 is the natural filtration of ξ. Since we wish to study the mean–square of (4), it is natural to prescribe some integrability conditions on the initial values of (4). Thus, we assume {ψ(n)}0n=− p +1 is a sequence of square integrable random variables, independent of F ξ (∞).

(6)

Let G = σ({ψ(n) : n ≤ 0}). Let (F(n))n0 be the filtration generated by the random variables {ξ(n) : 1 ≤ j ≤ n} and {ψ(n) : n ≤ 0} with F(0) := G to ensure measurability of (4) for all n ∈ Z. To emphasise the square integrability of the initial sequence, we will sometimes use the notation ψ ∈ L 2 (Ω), while understanding that ψ is a process, rather than a single random variable. In this framework, there is a unique (F(n))–adapted process satisfying (4). Note that each of the X (n) for n ≤ 0 is F(0)–measurable, while X (n) is F(n)–measurable for n ≥ 1. Since we wish to study the behaviour of E[X 2 (n)], the mean square of X (n), as n → ∞, we lastly note that the assumption (6), which guarantees that the initial sequence ψ has finite second moments, also ensures that E[X 2 (n)] < +∞ for all n ≥ − p  + 1.

30

J. A. D. Appleby and E. Lawless

In the paper, we shall often be concerned about the possibility that solutions will converge to steady state solutions, in an appropriate manner. In the light of this, we note that if the initial sequence ψ is zero i.e. if ψ(n) = 0 for all n ∈ {− p  + 1, . . . , 0}, then X (n) = 0 for all n ≥ 0. This means that zero is an equilibrium solution of the equation. Since the behaviour of the solution is well–understood in this case, we assume hereinafter that ⎤ ⎡ 0  ψ 2 ( j)⎦ > 0. (7) E⎣ j=− p +1

This is also equivalent to P[ψ ≡ 0] < 1. Thus, we can sometimes allow the initial sequence to be zero (and therefore generate a corresponding zero solution), but not always. It is frequently apt to emphasise that the solution depends on the initial sequence and write X (n; ψ) for the solution of (4) with initial condition ψ. For linear equations, it is to be hoped that the asymptotic behaviour (especially asymptotic stability) is not strongly dependent on the initial conditions, expect for some relatively small set of initial sequences. This motivates the following (standard) definition: we say that the zero solution is globally mean square asymptotically stable if lim E[X 2 (n; ψ)] = 0

for all ψ ∈ L 2 (Ω).

n→∞

There is also a notion of mean square stability: this means that for every  > 0 there exists δ = δ() > 0 such that E[max− p +1≤n≤0 ψ 2 (n)] < δ implies E[X 2 (n; ψ)] <  for all n ≥ 0. We shall investigate the second type of stability much less: however, it is interesting to identify conditions under which one has the latter type of stability, but not the former. The analysis in this paper essentially shows that if the former asymptotic stability prevails, then the latter stability condition is always fulfilled. Lastly, it is very natural, especially for linear equations, to determine whether the convergence to the zero solution is exponentially fast, and this leads to a notion of exponential mean square asymptotic stability. Focusing on the global exponential mean square asymptotic stability, a standard definition is that there exists a K > 0 and a η ∈ (0, 1), both independent of ψ, such that  E[X 2 (n; ψ)] ≤ K E

max 

− p +1≤ j≤0

ψ 2 ( j) η 2n , n ≥ 0,

for all ψ ∈ L 2 (Ω). We discussed above the existence and possible stability of the zero solution, without dwelling on other possible equilibria. In fact, ψ ≡ 0 is the only constant solution, except in the case that q  j=1

β j = 0,

p  j=1

α j = 1,

(8)

On the Dynamics and Asymptotic Behaviour of the Mean Square …

31

in which case every constant c ∈ R is a solution of (4) i.e. if ψ(n) = c for n ∈ {− p  + 1, . . . , 0}, then X (n) = c for all n ≥ 0. We note that the second condition in (8) means that the underlying deterministic equation x(n) =

p 

α j x(n − j), n ≥ 1

j=1

does not have an asymptotically stable equilibrium at zero since λ = 1 is a solution of the characteristic equation ϕ(λ) = 0, where ϕ(λ) := λ p −

p 

α j λ p− j .

j=1

As it transpires, a necessary condition for the mean square stability of (4) is the asymptotic stability of the underlying deterministic equation, so for the study of such stability, the question of multiple equilibria is precluded. We also note that the condition (8) is connected with a certain type of degeneracy in the noise term, to which we devote some effort. However, we do not analyse the stability of non–zero constant solutions, which are present under (8), in this work. Finally, we employ the standard Landau notation (for sequences) at various points in the paper. Recall that if a and b are sequences defined on Z+ say, and b is positive, we write a(n) = O(b(n)) as n → ∞ if there exists a constant K > 0 such that |a(n)| ≤ K b(n) for all n ≥ 0.

3 Volterra Summation Equations for the Mean Square Next we introduce the so called resolvent, r (n), from which all solutions of the stochastically unperturbed equation can be found. r is defined as follows: r (n) =

p 

α j r (n − j), n ≥ 1; r (0) = 1; r (n) = 0; n ≤ −1.

(9)

j=1

It is useful to have notation for the solution to (4) in the absence of stochastic terms: to this end, we introduce the sequence x given by x(n) =

p 

α j x(n − j), n ≥ 1; x(n) = ψ(n), n ≤ 0.

(10)

j=1

For p ≥ 2, by using z–transforms, it is possible to show that x can be written in terms of r and ψ according to

32

J. A. D. Appleby and E. Lawless

  r (n)ψ(0) + nj=1 r (n − j)b( j), n ≥ 1, x(n) = ψ(n), n ≤ 0,  p

where b(n) =

j=n+1

0,

(11)

α j ψ(n − j), 1 ≤ n ≤ p − 1, n ≥ p.

In the case when p = 1, we have a much simpler representation than (11) for x, namely  r (n)ψ(0) = α1n ψ(0), n ≥ 1, x(n) = (12) ψ(n), n ≤ 0. We now wish to rewrite (4) in terms of (9) and (10). Making use of z–transforms we can derive a variation of constants formula for X (n) by defining Y (n − 1) :=

q 

β j X (n − j), n ≥ 1.

(13)

j=1

Then X (n) =

p 

α j X (n − j) + Y (n − 1)ξ(n), n ≥ 1.

j=1

By variation of constants, X obeys 

x(n) + X (n) = ψ(n),

n

j=1 r (n

− j)Y ( j − 1)ξ( j), n ≥ 1, n ≤ 0.

(14)

The Eq. (14) hints at the following strategy: first, try to formulate the dynamics of Y in terms of Y directly, without mention of X . This will be done by substituting (14) into the righthand side of (13). Then, use the resulting summation equation for Y to deduce another summation equation for the mean square of Y . Finally, once this is done, use (14) one more time to get a relation between the mean square of Y and the mean square of X . The first step of the programme is to show that E[Y 2 ] obeys a linear convolution Volterra summation equation. This follows from the fact that Y itself obeys a stochastic Volterra summation equation. In the proof, and subsequently, we find it useful to define q  β j x(n − j), n ≥ 1 (15) G(xn−1 ) := j=1

and

On the Dynamics and Asymptotic Behaviour of the Mean Square …

G(rn−1 ) :=

q 

β j r (n − j), n ≥ 1.

33

(16)

j=1

Proposition 1 Suppose that X is the solution of (4) and that ψ and ξ obey (6) and (5) respectively. Suppose that Y is given by (13), that x and r are given by (10) and (9), and therefore G(r· ) and G(x· ) are defined by (16) and (15), respectively. Then Y obeys Y (n) = G(xn ) +

n 

G(rn−k )Y (k − 1)ξ(k), n ≥ 1; Y (0) = G(x0 ).

(17)

k=1

Moreover, E[Y 2 (n)] is well–defined for n ≥ 0 and given by E[Y 2 (n)] = E[G 2 (xn )] +

n 

G 2 (rn− j )E[Y 2 ( j − 1)], n ≥ 1.

(18)

j=1

with E[Y 2 (0)] = E[G 2 (x0 )]. E[Y 2 ] obeys a linear Volterra convolution summation equation with a non–negative kernel and non–negative forcing term. The asymptotic behaviour of such equations is well–understood (see e.g. [8]). In a subsequent result, we show that the mean square of X can be found in terms r and the mean square of Y and x, by exploiting the relation (14). The proof of Proposition 1 below is given in a lot of detail. One reason for this is that the arguments used therein can be reused almost verbatim to give representations for E[X 2 (n)] and Cov(X (n), X (n + k)) for k ≥ 0. Accordingly, by investing some care in the proof of Proposition 1, we are able to give very brief details of the proofs of the representations involving the mean square and covariance of X . Proof (of Proposition 1) First we prove (17): it is slightly easier to do this by reindexing to establish instead Y (n − 1) = G(xn−1 ) +

n−1 

G(rn−k−1 )Y (k − 1)ξ(k), n ≥ 2.

k=1

This consolidated equation is obtained by considering separately the cases when n ≥ q + 1 and n ∈ {2, . . . , q}. We obtain an expression for Y (n − 1) when n = 1 also. Take n ≥ q + 1. In (13), the sum ranges from j = 1, . . . , q and so n ≥ q + 1 ≥ j + 1. Therefore, using (14) in (13) in the form X (n − j) = x(n − j) +

n− j  k=1

r (n − j − k)Y (k − 1)ξ(k)

34

J. A. D. Appleby and E. Lawless

the definition (15), and exchanging the order of the finite sums, we get Y (n − 1) =

q 

β j x(n − j) +

q n− j  

j=1

β j r (n − j − k)Y (k − 1)ξ(k)

j=1 k=1

= G(xn−1 ) +

q n− j  

β j r (n − j − k)Y (k − 1)ξ(k)

j=1 k=1

= G(xn−1 ) +

⎧ n−1 ⎨min(q,n−k)   k=1

⎩

j=1

⎫ ⎬

β j r (n − k − j) Y (k − 1)ξ(k). ⎭

If k ≤ n − q in the sum, min(q, n − k) = q, and the term in brackets is G(rn−k−1 ), by (16). If k > n − q, then min(q, n − k) = n − k and the term in brackets is min(q,n−k) 

β j r (n − k − j) =

n−k 

j=1

β j r (n − k − j) =

j=1

q 

β j r (n − k − j) = G(rn−k−1 ),

j=1

since r (n − k − j) = 0 for j > n − k. Hence the term in brackets is always G(rn−k−1 ), and so we get Y (n − 1) = G(xn−1 ) +

n−1 

G(rn−k−1 )Y (k − 1)ξ(k), n ≥ q + 1.

(19)

k=1

Next we consider the case where n ∈ {2, . . . , q}. By (13), since X (n) = x(n) for n ≤ 0, we have Y (n − 1) =

n−1 

β j X (n − j) +

j=1

=

n−1 

β j X (n − j)

j=n

 βj

q 

x(n − j) +

n− j 

j=1

 β j r (n − j − k)Y (k − 1)ξ(k)

k=1

+

q 

β j X (n − j)

j=n

= G(xn−1 ) +

n− j n−1  

β j r (n − j − k)Y (k − 1)ξ(k)

j=1 k=1

= G(xn−1 ) +

⎧ n−1 ⎨ n−k  k=1

⎩

j=1

⎫ ⎬

β j r (n − k − j) Y (k − 1)ξ(k), ⎭

On the Dynamics and Asymptotic Behaviour of the Mean Square …

35

where we exchanged the order of summation at the last step. Now, since n ≤ q, and k ≥ 1 in the sum, n − k < q. Therefore as r is zero for negative indices, we get n−k 

β j r (n − k − j) =

j=1

q 

β j r (n − k − j) = G(rn−k−1 ).

j=1

Therefore Y (n − 1) = G(xn−1 ) +

n−1 

G(rn−k−1 )Y (k − 1)ξ(k), n ∈ {2, . . . , q}.

(20)

k=1

Finally, the initial value Y (0) is given by Y (0) =

q 

β j X (1 − j) =

j=1

q 

β j x(1 − j) = G(x0 ).

j=1

Thus, taking (20) and (19) together and re–indexing, we get (17). Now we deduce the Eq. (18) for the mean–square of Y . Consider for n ≥ 1, Y 2 (n) = G 2 (xn ) + 2G(xn )  +

n 

G(rn−k )Y (k − 1)ξ(k)

k=1 n 

2

G(rn−k )Y (k − 1)ξ(k)

(21) .

k=1

We take expectations of each term on the righthand side of (21). For the first term we get E[G 2 (xn )]. To deal with the second, fix an n 0 ∈ N and define for n ≥ n 0 , M3 (n) := 2G(xn 0 )

n 

G(rn 0 −k )Y (k − 1)ξ(k) =:

k=1

n 

Z 1 (k − 1)ξ(k),

k=1

where Z 1 (k − 1) := 2G(xn 0 )G(rn 0 −k )Y (k − 1) where 1 ≤ k ≤ n. Note Z 1 ∈ L 1 (Ω) because G(rn 0 −k ) is deterministic and G(xn 0 ) and Y (k − 1) are both in L 2 (Ω). Since G(xn 0 ) is F(0)–measurable, we have that Z 1 is F(k − 1)–measurable and independent of ξ(k). This allows us to conclude  E[M3 (n)] = E

n  k=1

 Z 1 (k − 1)ξ(k) =

n 

E[Z 1 (k − 1)]E[ξ(k)] = 0.

k=1

Letting n = n 0 we see that E[M3 (n 0 )] = 0, but since n 0 was chosen arbitrarily this gives

36

J. A. D. Appleby and E. Lawless

 E 2G(xn )

n 

 G(rn−k )Y (k − 1)ξ(k) = 0, for all n ≥ 1.

(22)

k=1

Now we turn our attention to the third term on the righthand side of (21). Writing the square as an iterated sum, taking expectations, and using the fact that G(r· ) is deterministic, we get ⎡ 2 ⎤ n n   E⎣ G(rn−k )Y (k − 1)ξ(k) ⎦ = G 2 (rn−k )E[Y 2 (k − 1)ξ 2 (k)] k=1

k=1

+2

n−1 

n 

G(rn− j )G(rn−k )E[Y ( j − 1)ξ( j)Y (k − 1)ξ(k)]. (23)

j=1 k= j+1

Now, note for k ≥ j + 1 that ξ(k) and Y ( j − 1)ξ( j)Y (k − 1) are independent. Thus E[Y ( j − 1)ξ( j)Y (k − 1)ξ(k)] = 0 provided that E[|Y ( j − 1)ξ( j)Y (k − 1)|] < +∞ for k ≥ j + 1. Note that E[Y 2 (k − 1)] < +∞, so by the Cauchy–Schwarz inequality, we have that E[|Y ( j − 1)ξ( j)Y (k − 1)|] < +∞ provided E[Y 2 ( j − 1)ξ 2 ( j)] < +∞. But this is true by the independence and integrability of Y 2 ( j − 1) and ξ 2 ( j). Therefore, the mixed summation term is zero. Since Y 2 (k − 1) and ξ 2 (k) are independent and in L 1 (Ω), and E[ξ 2 (k)] = 1, we have that E[Y 2 (k − 1)ξ 2 (k)] = E[Y 2 (k − 1)]. We are then left with ⎡ E⎣

n 

2 ⎤ n    ⎦ G(rn−k )Y (k − 1)ξ(k) G 2 (rn−k )E Y 2 (k − 1) . =

k=1

k=1

Finally, from this equality and (22), taking expectations across (21) gives (18) as required. We can now use (14) to can deduce a representation for the mean square of X (n) in terms of Y (n), (9) and (10). Proposition 2 Suppose that X is the solution of (4) and that ψ and ξ obey (6) and (5) respectively. Let r be defined by (9), x be defined by (10) and Y be defined by (13). Then E[X 2 (n)] = E[x 2 (n)] +

n 

r 2 (n − j)E[Y 2 ( j − 1)], n ≥ 1.

(24)

j=1

Before turning to the proof of (24), we summarise our progress. In Proposition 1, we showed that E[Y 2 ] satisfies a deterministic linear convolution Volterra equation with kernel G 2 (r· ) and forcing term E[G 2 (x· )]. Thus, if good information is available about these sequences, the asymptotic behaviour of E[Y 2 ] can be determined. If this can be done, it ought to be quite straightforward to determine the asymptotic

On the Dynamics and Asymptotic Behaviour of the Mean Square …

37

behaviour of E[X 2 ] using (24), since the asymptotic behaviour of the sequences x and r , which are the only other ones present in (24), can be found using classical results from the theory of deterministic linear difference equations (see e.g. [8]). Although this looks like good progress, there is a potential stumbling block, and it is present in (18) rather than in (24). The price that we have paid for the elegant representation (18) is that the kernel G 2 (r· ) and forcing term E[G 2 (x· )] are no longer part of the problem data, but sequences about which appropriate properties must be established in terms of the problem data. A little reflection shows that the forcing term E[G 2 (x· )] is probably not especially problematic, provided reasonably good asymptotic information is available about r . This is because the asymptotic behaviour of x depends almost entirely on r , and thus the asymptotic behaviour of G(x· ) should be governed by r too. Of course, it is not unreasonable to assume that very good information about the asymptotic behaviour of r should be quite readily available: for instance, one can use the Schur–Cohn conditions (see e.g. [8]) to determine necessary and sufficient conditions under which r tends to zero, and if this is the case, the convergence of r to zero is exponentially fast. Considering the convergence of E[X 2 ] to zero, we see that convergence of E[x 2 ] to zero is necessary. In turn this forces the zeros of the characteristic equation of (10) to all have modulus less than 1, which is the situation that is captured by the Schur–Cohn conditions. Looking at (24), it then seems desirable to characterise when E[Y 2 ] tends to zero, since this should now be sufficient to force E[X 2 ] to zero. Turning back to (18), we see that conditions which force r to tend to zero, should also force E[G 2 (x· )] to tend to zero exponentially fast. Finally, summing on both sides of (18), it becomes clear that a condition of the type G ∗ :=

∞ 

G 2 (rn−1 ) < 1

(25)

n=1

will be sufficient, and possibly quite close to necessary, in order that E[Y 2 ] should tend to zero. But for general equations when p > 2, it is perhaps impossible to get a closed–form expression for r because it is very difficult to obtain expressions for the zeros of the characteristic polynomial of x directly in terms of the coefficients of the characteristic polynomial. Thus, closed–form expressions for G(r· ) (and hence the summand in G ∗ ) seem elusive, and so finding an expression for G ∗ that can be computed in terms of the problem data looks difficult. However, the computational complications which are present for general p essentially disappear in the case when p = 1, since simple closed–form expressions for r , and hence G(r· ), can be found. Accordingly, we will study the asymptotic behaviour of the mean square of X when p = 1 in the next section. Proof (of Proposition 2) We need only consider the case when n ≥ 1, then by squaring (14) for n ≥ 1, we get

38

J. A. D. Appleby and E. Lawless

X 2 (n) = x 2 (n) + 2x(n)

n 

r (n − j)Y ( j − 1)ξ( j)

j=1

⎛ ⎞2 n  +⎝ r (n − j)Y ( j − 1)ξ( j)⎠ . (26) j=1

It remains to take expectations on both sides of (26). However, the expectation of each of the terms on the right hand side can be tackled using an identical argument to that used to deal with the corresponding terms on the righthand side of (21) in the proof of Proposition 1. Doing this, we find that the expectation of the first term is E[x 2 (n)], the expectation of the second term is zero, and the expectation of the last term is ⎡⎛ E ⎣⎝

n 

⎞2 ⎤ r (n − j)Y ( j − 1)ξ( j)⎠ ⎦ =

n 

j=1

r 2 (n − j)E[Y 2 ( j − 1)].

j=1

Taking expectations across (26) and using these results, the result is proven. Finally, in this section, we get an expression for Cov(X (n), X (n + k)) where k ≥ 0, again in terms of E[Y 2 ]. Recall for U, V ∈ L 2 (Ω) that the covariance of U and V , denoted by Cov(U, V ), is given by Cov(U, V ) := E[(U − E[U ])(V − E[V ])].

Proposition 3 Suppose that X is the solution of (4) and that ψ and ξ obey (6) and (5) respectively. Let r by defined by (9), x be defined by (10) and Y be defined by (13). Let k ≥ 0. Then E[X (n)X (n + k)] = E[x(n)x(n + k)] +

n 

r (n − j)r (n − j + k)E[Y 2 ( j − 1)], n ≥ 1,

j=1

(27) and Cov(X (n), X (n + k)) = Cov(x(n), x(n + k)) +

n 

r (n − j)r (n − j + k)E[Y 2 ( j − 1)], n ≥ 1. (28)

j=1

Notice that when k = 0 in (27), we recover (24). Thus, this result generalises Proposition 2. It also suggests that it is a good strategy, in order to understand the dynamics of the variance, mean square or covariance of X , is to first determine the properties of the sequence E[Y 2 ] using the Volterra equation (18), and then to work

On the Dynamics and Asymptotic Behaviour of the Mean Square …

39

out explicitly the behaviour of the variance, mean square or covariance using (24) or (27). This should be achievable, since the main terms in these expressions are convolutions of E[Y 2 ] with deterministic sequences involving r , whose properties can be well–understood by classic deterministic techniques for finite–order linear difference equations. Proof (of Proposition 3) Since n ≥ 1, we have X (n) = x(n) + M1 (n) where M1 (n) =

n 

r (n − j)Y ( j − 1)ξ( j).

j=1

Using the ideas of Proposition 1, we can show that E[M1 (n)] = 0, and so E[X (n)] = E[x(n)]. Therefore, by the definition of the covariance, we see that (28) is a direct consequence of (27). To prove (27), note that n ≥ 1 gives n + k ≥ 1, so by means of the variation of constants formula for X , we have E[X (n)X (n + k)] = E[x(n)x(n + k)] + E[x(n)M1 (n + k)] + E[x(n + k)M1 (n)] + E[M1 (n)M1 (n + k)]. (29) The first term on the righthand side is in the correct form. The second and third terms on the right hand side are both zero: the argument that secures this is the same as that showing the expectation of the second term on the righthand side of (21) is zero. For the third term we write M1 (n)M1 (n + k) =

n  n+k 

r (n − j)r (n + k − l)Y ( j − 1)Y (l − 1)ξ( j)ξ(l),

j=1 l=1

so E[M1 (n)M1 (n + k)] =

n  n+k 

r (n − j)r (n + k − l)E[Y ( j − 1)Y (l − 1)ξ( j)ξ(l)]

j=1 l=1

=

n 

r (n − j)r (n + k − j)E[Y 2 ( j − 1)ξ 2 ( j)]

j=1

+

n+k n  

r (n − j)r (n + k − l)E[Y ( j − 1)Y (l − 1)ξ( j)ξ(l)].

j=1 l=1,l = j

Now, we may use the argument used to simplify (23) to show that each of the expectations in the second sum is zero, and that E[Y 2 ( j − 1)ξ 2 ( j)] = E[Y 2 ( j − 1)]. Hence

40

J. A. D. Appleby and E. Lawless

E[M1 (n)M1 (n + k)] =

n 

r (n − j)r (n + k − j)E[Y 2 ( j − 1)].

j=1

Inserting this into (29), we arrive at the desired formula (27).

4 Asymptotic Behaviour, p = 1 4.1 The Simple Case q = 1 In the case when p = 1, the stochastic difference equation reads X (n) = αX (n − 1) +

q 

β j X (n − j)ξ(n), n ≥ 1;

X (n) = ψ(n), n ≤ 0,

j=1

(30) where we have relabelled α := α1 to simplify notation. In this section, we take α = 0. Notice that x(n) = αn ψ(0) and r (n) = αn for all n ≥ 0. We take q ≥ 2, because the case when q = 1 is well–understood. Indeed, if q = 1 we have X (n) = (α + β1 ξ(n))X (n − 1), n ≥ 1, and the sequence ψ collapses to the random variable ψ(0), which has a finite second moment. Thus n  (α + β1 ξ( j))ψ(0), n ≥ 1. X (n) = j=1

Using the independence of the ξ, and the independence of ξ from ψ(0), we have E[X 2 (n)] =

n 

E[(α + β1 ξ( j))2 ]E[ψ 2 (0)] = (α2 + β12 )n E[ψ 2 (0)],

j=1

since the ξ’s have zero mean and unit variance. Thus, in this simple case, we see that the asymptotic behaviour of the mean square is real exponential, and the rate of growth or decay can be found in terms of the parameters of the original difference equation. The rest of this section is devoted to demonstrating that these properties remain unaltered in the case when q is an arbitrary integer. Thus, as far as mean square behaviour goes at least, arbitrary delay structure in the noise is unable to introduce oscillatory behaviour to the solution.

On the Dynamics and Asymptotic Behaviour of the Mean Square …

41

4.2 Preparatory Results We start by proving some preparatory results. We find it useful to introduce the polynomial q  β j λq− j . (31) φ(λ) = j=1

Notice that φ is nontrivial, since βq = 0 by hypothesis. It is a polynomial of degree at most q − 1. First, if we disallow the trivial initial condition, we disallow x being trivial. Lemma 1 x(n) = 0 for all n ≥ −q + 1 if and only if ψ(n) = 0 for all n ≥ −q + 1. Proof By the definition of (10), if ψ ≡ 0, then x ≡ 0. Conversely if x(n) = 0 for all n ≥ 0, then αn ψ(0) = 0 for all n ≥ 0, and thus ψ(0) = 0. Since ψ(n) = x(n) for all n ≤ 0, this ensures ψ(n) ≡ 0. Next, we show that G(r· ) cannot be the zero sequence. Lemma 2 There exists n ≥ 1 such that G(rn ) = 0. Proof Firstly, let n ≥ q and suppose to the contrary that G(rn−1 ) = 0 for all n ≥ q. Then as j ∈ {1, . . . , q} obeys n ≥ q ≥ j, we have for n ≥ q, G(rn−1 ) =

q 

β j r (n − j) =

j=1

q 

β j αq− j αn−q = φ(α)αn−q .

j=1

Hence G(rn−1 ) = 0 for all n ≥ q implies φ(α) = 0. Next observe when n ∈ {1, . . . , q − 1}, we have n < q and so, G(rn−1 ) =

n 

β j r (n − j) +

j=1

q 

β j r (n − j) =

j=n+1

Thus G(rn−1 ) = φ(α)αn−q −

n 

β j αq− j αn−q .

j=1

q 

β j αq− j αn−q .

j=n+1

Since φ(α) = 0 we get q 

β j αq− j αn−q = 0, n ∈ {1, . . . , q − 1}.

j=n+1

Putting n = q − 1 yields βq αq−1−q = 0, so as α = 0, we must have βq = 0. But this contradicts the hypothesis that βq = 0. Hence we cannot have G(rn−1 ) ≡ 0.

42

J. A. D. Appleby and E. Lawless

By Lemma 2 we see that G(rn−1 ) ≡ 0, and consequently from the proof of Lemma 2 we obtain a formula for G(rn−1 ) in terms of the problem data  φ(α)αn−q , n ≥ q + 1, G(rn−1 ) = n n− j β α , n = 1, . . . , q. j=1 j

(32)

Since r (n) = αn for n ≥ 0, there is a K > 0 such that G 2 (rn ) ≤ K 2 α2n for n ≥ 0. Hence the function ∞  Γ (λ) := λ−2(n+1) G 2 (rn ) (33) n=0

is well–defined for λ > |α|. We determine the maximal domain of Γ in the positive reals presently. It should be noted that we can compute Γ (λ) in a reasonably explicit form. Indeed for λ > |α| we have, Γ (λ) =

q−1 

λ−2( j+1) G 2 (r j ) +

j=0





q

=

λ−2 j

j=1

∞ 

λ−2( j+1) φ2 (α)α2( j+1−q)

j=q



2

j

βk α j−k

+

k=1

φ2 (α) α2 . λ2 − α2 λ2q

Clearly, since G(r· ) is non–trivial by Lemma 2, Γ is decreasing and continuous on its domain, and Γ (λ) → 0 as λ → ∞. If φ(α) = 0, by Lemma 2 we have that Γ (λ) → ∞ as λ → |α|+ , so there is a unique μ ∈ (|α|, ∞) such that Γ (μ) = 1. If φ(α) = 0, then  j 2 q   −2 j j−k λ βk α , Γ (λ) = j=1

k=1

and Γ is well–defined on (0, ∞). Note that Γ (λ) → ∞ as λ → 0+ unless m 

β j αm− j = 0, m = 1, 2, . . . , q.

j=1

For m = 1, this forces β1 = 0. For m = 2, we have 0 = β1 α + β2 = β2 , and continuing in this manner, we arrive at β1 = β2 = . . . = βq = 0, a contradiction. Therefore, if φ(α) = 0, we have Γ (λ) → ∞ as λ → 0+ , Γ is decreasing, and Γ (λ) → 0 as λ → ∞. Hence there is a unique μ ∈ (0, ∞) such that Γ (μ) = 1. Since Γ is decreasing, if Γ (|α|) > 1, then Γ (μ) = 1 < Γ (|α|), so μ > |α|. Finally, we compute

On the Dynamics and Asymptotic Behaviour of the Mean Square …

Γ (|α|) =

 j q   j=1

2 βk α

−k

 j q−1  

=

k=1

j=1

43

2 βk α

−k

,

(34)

k=1

using the fact that φ(α) = 0 at the second step. We summarise this discussion in a Lemma. Lemma 3 Let Γ be the function defined by (33), and φ be given by (31). (i) If φ(α) = 0, then Γ : (|α|, ∞) → (0, ∞) and lim Γ (λ) = +∞.

λ↓|α|

Moreover, there exists a unique μ ∈ (|α|, ∞) such that Γ (μ) = 1, which implies 1=

q 

−2 j

μ

j=1

 j 

2 βk α

+

j−k

k=1

φ2 (α) α2 . μ2 − α2 μ2q

(35)

(ii) If φ(α) = 0, then Γ : (0, ∞) → (0, ∞) and lim Γ (λ) = +∞. λ↓0

Moreover, there exists a unique μ ∈ (0, ∞) such that Γ (μ) = 1. This implies 1=

q−1  j=1

−2 j

μ

 j 

2 βk α

j−k

.

(36)

k=1

Finally, we have μ > |α| if Γ (|α|) > 1, μ = |α| if Γ (|α|) = 1 and μ < |α| if Γ (|α|) < 1. The significance of the number μ, is that, in most cases it gives the exponential rate of growth or decay: E[X 2 (n)] ∼ Cμ2n , as n → ∞ We notice in each case that μ can be viewed as the solution of a certain polynomial equation. In the first case, the polynomial is degree 2q + 2, and in the second, the degree is 2q − 2. Thus we may view (35) and (36) as characteristic equations which identify the leading order exponent. One final preliminary result is required. Perusal of the Eq. (18) reveals that if E[G 2 (xn−1 )] = 0 for all n ≥ 1,

(37)

then E[Y 2 (n)] = 0 for all n ≥ 0. Indeed, this implies that Y (n) = 0 for all n ≥ 1, a.s. Therefore, X obeys X (n) = αn ψ(0) for all n ≥ 0 a.s., and hence we

44

J. A. D. Appleby and E. Lawless

have E[X 2 (n)] = α2n E[ψ 2 (0)] for n ≥ 1. The condition (37) switches off the noise term completely, making the solution of the equation a.s. deterministic. It also means, that although the asymptotic behaviour of the mean square is exponential, it is not determined by the characteristic Eqs. (35) or (36). Clearly, it would be useful to know how typical such degenerate behaviour is, and to characterise when it occurs. The next result achieves such a characterisation, and shows that it is very exceptional. Lemma 4 Suppose that P[ψ ≡ 0] < 1. Then the following are equivalent: (i) E[G 2 (xn−1 )] = 0 for all n ≥ 1; (ii) G(xn−1 ) = 0 for all n ≥ 1, a.s.; (iii) φ(α) = 0 and n 

β j αn− j ψ(0) +

j=1

q 

β j ψ(n − j) = 0, n = 1, . . . , q − 1, a.s.;

j=n+1

(iv) φ(α) = 0, ψ( j) = α j ψ(0),

j = −q + 1, . . . , 0, a.s.

(38)

Proof It is evident that (i) implies (ii). Suppose (ii) holds; we attempt to prove (iii). Let n ≥ q. Then G(xn−1 ) =

q 

β j x(n − j) =

j=1

q 

β j αn− j ψ(0) = φ(α)αn−q ψ(0).

j=1

Thus G(xn−1 ) = 0 for all n ≥ q if and only if φ(α) = 0 or ψ(0) = 0 a.s., since α = 0 by assumption. Suppose that ψ(0) = 0 a.s. along with the stipulation that G(xn−1 ) ≡ 0. Now consider the case when n ∈ {1, . . . , q − 1}, so that G(xn−1 ) =

n 

β j αn− j ψ(0) +

j=1

q 

β j ψ(n − j), n ∈ {1, . . . , q − 1}.

(39)

j=n+1

Hence by hypothesis 0=

q 

β j ψ(n − j),

for n = 1, . . . , q − 1, a.s.

j=n+1

Taking n = q − 1 gives 0 = βq ψ(−1). Since βq = 0 by assumption, we must have ψ(−1) = 0 a.s. Now take n = q − 2: we get 0 = βq−1 ψ(−1) + βq ψ(−2), which forces ψ(−2) = 0 a.s. Continuing in this fashion we get ψ(n) = 0 for all n ∈ {−q + 1, . . . , 0}, a.s.

On the Dynamics and Asymptotic Behaviour of the Mean Square …

45

This contradicts the standing assumption that ψ ≡ 0. Hence we must have ψ(0) = 0 with positive probability and so φ(α) = 0. Therefore, from (39) we have 0=

n 

q 

β j αn− j ψ(0) +

j=1

β j ψ(n − j), n ∈ {1, . . . , q − 1}, a.s.,

(40)

j=n+1

which completes the proof of statement (iii). Next, assume (iii) in order to show (iv). Take n = q − 1 in (40). Then a.s., we have q−1 

β j αq− j ψ(0) + αβq ψ(−1) = 0.

j=1

q Since 0 = φ(α) = j=1 β j αq− j , this implies βq (−ψ(0) + αψ(−1)) = 0 a.s. Since βq = 0, we have ψ(−1) = α−1 ψ(0) a.s. Now take n = q − 2 in (40). This gives a.s. α−2

q−2 

β j αq− j ψ(0) + βq−1 ψ(−1) + βq ψ(−2) = 0,

j=1

and using φ(α) = 0 again we get a.s. α−2 (−βq − βq−1 α)ψ(0) + βq−1 ψ(−1) + βq ψ(−2) = 0. Since ψ(−1) = α−1 ψ(0) a.s., this yields βq (−α−2 ψ(0) + ψ(−2)) = 0 a.s. so ψ(−2) = α−2 ψ(0) a.s. Continuing in this way, we can show that ψ( j) = α j ψ(0) for j = −1, −2, . . . , −q + 1 a.s., which proves (iv). Finally, if (iv) holds, then ψ( j) = α j ψ(0) for j ≤ 0 a.s. and φ(α) = 0. From the latter, we get G(xn−1 ) = 0 for all n ≥ q automatically. For n ∈ {1, . . . , q − 1} we have a.s. G(xn−1 ) =

n 

βjα

n− j

ψ(0) +

j=1

=

n  j=1

=

q 

q 

β j ψ(n − j)

j=n+1

βjα

n− j

ψ(0) +

q 

β j αn− j ψ(0)

j=n+1

β j αq− j αn−q ψ(0) = φ(α)αn−q ψ(0) = 0.

j=1

Hence G(xn−1 ) = 0 for all n ≥ 1 a.s., which implies that E[G 2 (xn−1 )] = 0 for all n ≥ 1, proving (i). Hence (i)–(iv) are equivalent.

46

J. A. D. Appleby and E. Lawless

One implication of this result is that degenerate noise is impossible provided φ(α) = 0. This is a very weak constraint on the data. We notice that it also rules out the behaviour in part (ii) of Lemma 3. It transpires that we can give a complete description of the dynamics by considering cases: first, the typical case when φ(α) = 0. Second, the case where φ(α) = 0, but the initial data does not give rise to degeneracy: this case is certainly rarer, but does not place many constraints on the initial data. The third case is when φ(α) = 0, and the initial conditions produce degeneracy. We have shown above that this occurs only in the case where the initial function is the geometric sequence ψ( j) = α j ψ(0) for all j ≤ 0 a.s., and in this case we have X (n) = αn ψ(0) for all n ≥ −q + 1 a.s. This in turn leads to E[X 2 (n)] = α2n E[ψ 2 (0)]. Since this last atypical case is now dealt with, our asymptotic analysis will be devoted to the first two cases.

4.3 Asymptotic Behaviour The result that enables us to determine the exponential asymptotic behaviour is the following discrete version of the renewal theorem. For the result see e.g. Feller [9, Chaps. 12 and 13] or Karlin and Taylor [12, Chap. 3]. To make our presentation self–contained and explicit, we state the result next. Theorem Theorem) Let (γ(n)) ¯ n≥1 be a non-negative sequence  1 (Discrete Renewal ¯ = 1 such that ∞ ¯ < ∞ and let (u(n))n≥0 be defined by with ∞ n=1 γ(n) n=1 n γ(n) u(n) =

n 

γ(k)u(n ¯ − k), n ≥ 1; u(0) = 1,

(41)

k=1

and let (y(n))n≥0 be the solution to y(n) = g(n) +

n 

γ(k)y(n ¯ − k), n ≥ 1; y(0) = g(0).

k=1

Then

1 , ¯ n=1 n γ(n)

lim u(n) = ∞

n→∞

y(n) =

n k=0

g(k)u(n − k) and if g ∈ 1 (Z+ ) then ∞ g(n) lim y(n) = ∞n=0 . n→∞ n ¯ n=1 γ(n)

We are now in a position to prove two theorems.

(42)

On the Dynamics and Asymptotic Behaviour of the Mean Square …

47

Theorem 2 Let X be the solution to (30). Let φ be given by (31) and suppose that φ(α) = 0. Then there exists μ > |α| such that μ is the unique solution of Γ (μ) = 1, and satisfies (35) i.e. 1=

q 

−2 j

μ

 j 

j=1

2 βk α

k=1

j−k

+

φ2 (α) α2 . μ2 − α2 μ2q

Moreover, there exists C = C(ψ) ∈ (0, ∞) such that E[X 2 (n; ψ)] = C(ψ) n→∞ μ2n lim

for all ψ for which P[ψ ≡ 0] < 1. Proof We employ the notation γ(n − 1) := G 2 (rn−1 ), n ≥ 1, γ(n) ¯ := γ(n − 1), n ≥ 1. By re–indexing and reversing the order of summation, (18) can be re–written as E[Y (n)] = E[G (xn )] + 2

2

n 

G 2 (rk−1 )E[Y 2 (n − k)], n ≥ 1,

k=1

recalling also that E[Y 2 (0)] = E[G 2 (x0 )]. Let μ be given by (35). Write yμ (n) :=

E[Y 2 (n)] E[G 2 (xn )] γ(n) ¯ , g (n) := , n ≥ 0, γμ (n) := 2n , n ≥ 1. μ 2n 2n μ μ μ

Hence by rescaling by μ−2n we get yμ (n) = gμ (n) +

n 

γμ (k)yμ (n − k), n ≥ 1.

(43)

k=1

Now, we have that γμ (n) ≥ 0 for all n ≥ 1. Also by the definition of μ > |α| we have ∞  n=1

γμ (n) =

∞ 

μ−2n G 2 (rn−1 ) = Γ (μ) = 1.

n=1

Next since r (n) = αn , it follows that E[x 2 (n)] = α2n E[ψ 2 (0)], E[G 2 (xn )] = O(α2n ) and G 2 (rn ) = O(α2n ) as n → ∞. Since μ > |α|, we see that gμ ∈ 1 (Z+ ). Finally, since γ(n) ¯ γ(n − 1) G 2 (rn−1 ) = , γμ (n) = 2n = 2n μ μ μ2n

48

J. A. D. Appleby and E. Lawless

2 2 n and ∞ setting η := α /μ ∈ (0, 1), we see that γμ (n) = O(η ) as n → ∞. Hence n=1 nγμ (n) < +∞. Therefore, Theorem 1 applies to yμ and we get

∞ gμ (n) lim yμ (n) = ∞n=0 . n→∞ n=1 nγμ (n) This can be written ∞ −2n μ E[G 2 (xn )] E[Y 2 (n)] n=0 =: C  . = ∞ 2n −2n G 2 (r n→∞ μ nμ ) n−1 n=1 lim

Since G(rn−1 ) is not identically zero, the denominator is strictly positive and finite. Since φ(α) = 0, (38) fails to hold, and so Lemma 4 allows us to conclude that E[G 2 (x)] is not identically zero. Therefore the numerator is strictly positive and finite. Thus C  ∈ (0, ∞). Next, scaling (24) by μ−2n yields n E[x 2 (n)] 1  r 2 (n − j) E[Y 2 ( j − 1)] E[X 2 (n)] = + · · . μ2n μ2n μ2 j=1 μ2(n− j) μ2( j−1)

The first term on the righthand side tends to zero, because μ > |α|. The sequence rμ2 (n) := r 2 (n)/μ2n = O(η n ) as n → ∞ and therefore rμ2 is in 1 (Z+ ). Therefore the limit of the righthand side is given by ∞ ∞ E[X 2 (n)] 1  r 2 (l)  1  l  1 = · C = ηC = 2 C  =: C, n→∞ μ2n μ2 l=0 μ2l μ2 l=0 μ − α2

lim

and the constant on the righthand side is finite and non–zero, since C  ∈ (0, ∞). We deal with the last outstanding case, in which α is a zero of φ, and in which the initial condition prevents the noise switching off entirely. Now it is possible for E[X 2 ] to inherit the asymptotic behaviour of r 2 . Theorem 3 Let X be the solution to (30). Suppose that P[ψ ≡ 0] < 1. Let φ be given by (31), suppose that φ(α) = 0 and that (38) fails to hold. Then Γ (|α|) is given by (34) i.e.,  j 2 q−1   −k Γ (|α|) = βk α , j=1

k=1

and the following are true: (i) If Γ (|α|) > 1, then there exists a unique μ ∈ (|α|, ∞) such that Γ (μ) = 1. Moreover, μ obeys (36) i.e.

On the Dynamics and Asymptotic Behaviour of the Mean Square …

1=

q−1 

−2 j

μ

j=1

 j 

49

2 βk α

j−k

,

k=1

and there is a positive constant C = C(ψ) such that E[X 2 (n; ψ)] = C(ψ). n→∞ μ2n lim

(ii) If Γ (|α|) < 1, then there exists a positive constant C = C(ψ) such that E[X 2 (n; ψ)] = C(ψ). n→∞ α2n lim

(iii) If Γ (|α|) = 1, then there exists a positive constant C = C(ψ) such that E[X 2 (n; ψ)] = C(ψ). n→∞ nα2n lim

Proof If we are in case (i), the proof of Theorem 2 can be followed verbatim. We start the arguments that secure the proofs in cases (ii) and (iii) in tandem; only at a very late stage is it necessary to argue the cases separately. By hypothesis, we have that there is a unique μ ∈ (0, |α|] such that Γ (μ) = 1. Write ¯ := γ(n − 1), n ≥ 1, γ(n − 1) := G 2 (rn−1 ), n ≥ 1, γ(n) and define γμ (n) :=

γ(n) ¯ , n ≥ 1. μ2n

Now, we have that γμ (n) ≥ 0 for all n ≥ 1. Also by the definition of μ > 0 we have ∞ 

γμ (n) =

n=1

∞ 

μ−2n G 2 (rn−1 ) = Γ (μ) = 1.

n=1

Next, G(rn−1 ) = φ(α)αn−q for n ≥ q. Hence G(rn−1 ) = 0 for n ≥ q. Thus for n ≥ q, we have γ(n) ¯ G 2 (rn−1 ) γμ (n) = 2n = = 0. μ μ2n This implies that defined by

∞ n=1

u μ (n) =

nγμ (n) = n  k=1

q−1 n=1

nγμ (n) < +∞. Hence the sequence u μ

γμ (k)u μ (n − k), n ≥ 1; u μ (0) = 1

(44)

50

J. A. D. Appleby and E. Lawless

obeys lim u μ (n) = q−1

n→∞

1

n=1

nγμ (n)

=: u ∗ ∈ (0, ∞),

by Theorem 1. Note that the non–triviality of G(r· ), which is a consequence of Lemma 2, ensures that the denominator is non–zero. Now write yμ (n) :=

E[Y 2 (n)] E[G 2 (xn )] , g (n) := , n ≥ 0. μ μ2n μ2n

Then by Theorem 1, we can write yμ (n) =

n 

gμ (k)u μ (n − k), n ≥ 0,

k=0

or equivalently E[Y 2 (n)]/μ2n =

n 

μ−2k E[G 2 (xk )]u μ (n − k).

k=0

Since E[G 2 (xn )] = 0 for all n ≥ q − 1, this becomes E[Y 2 (n)]/μ2n =

q−2 

μ−2k E[G 2 (xk )]u μ (n − k).

k=0

Since u μ tends to a positive limit u ∗ , we have E[Y 2 (n)]  −2k = μ E[G 2 (xk )]u ∗ =: C  , n→∞ μ2n k=0 q−2

lim

and C  ∈ (0, ∞), because, by the assumption that (38) fails to hold, Lemma 4 allows us to conclude that E[G 2 (x)] is not identically zero. Next, scaling (24) by α−2n yields for n ≥ 1 n 1  E[Y 2 ( j − 1)] μ2( j−1) E[X 2 (n)] 2 = E[ψ (0)] + 2 . α2n α j=1 μ2( j−1) α2( j−1)

(45)

Now, we specialise the argument to deal with cases (ii) and (iii). In case (ii), we have μ < |α|, so the sum in (45) is the convolution of an absolutely summable sequence and a convergent sequence. Hence

On the Dynamics and Asymptotic Behaviour of the Mean Square …

51

∞

E[X 2 (n)] 1   μ2 j 2 = E[ψ (0)] + C =: C ∈ (0, ∞), n→∞ α2n α2 α2 j j=0 lim

as required. In case (iii), we have μ = |α|. Then for n ≥ 1, from (45) we have n E[X 2 (n)] 1 1 1  E[Y 2 ( j − 1)] 2 E[ψ = (0)] + . nα2n n α2 n j=1 μ2( j−1)

Since the summand tends to C  , we have that E[X 2 (n)] 1 = 2 C  =: C ∈ (0, ∞), n→∞ nα2n α lim

as required. Theorems 2 and 3 give precise rates of growth of decay of the mean square. However, sometimes we are satisfied simply knowing whether the zero solution of (4) is globally asymptotically stable, which is to say that lim E[X 2 (n; ψ)] = 0, for all ψ ∈ L 2 (Ω).

n→∞

We now give an explicit characterisation of this phenomenon in terms of the problem data. We start by noticing that when |α| < 1 that Γ (1) is always finite and given by Γ (1) =

 j q   j=1

k=1

2 βk α

j−k

+

φ2 (α)α2 1 − α2

Notice when q = 1, the expression on the right hand side becomes Γ (1) = β12 +

β12 α2 β12 = . 1 − α2 1 − α2

Thus Γ (1) < 1 is equivalent to β12 + α2 < 1. Earlier in this section, we were able to show by elementary arguments that this is condition is necessary and sufficient to the solution of (4) to tend to zero in mean square in the case when p = q = 1, because E[X 2 (n)] = (α2 + β12 )n E[ψ 2 (0)]. We now show that this is still the case for p = 1 when q is any integer. Theorem 4 Let X be the solution to (30). Let φ be given by (31). Then the following are equivalent for solutions of (4) with p = 1, q ≥ 2 and α1 := α:

52

J. A. D. Appleby and E. Lawless

(i) |α| < 1,

 j q   j=1

2 βk α

+

j−k

k=1

φ2 (α)α2 < 1; 1 − α2

(46)

(ii) lim E[X 2 (n; ψ)] = 0, for all ψ ∈ L 2 (Ω).

n→∞

Proof Suppose (i) holds. Since, E[x 2 (n)] = α2n E[ψ 2 (0)], we have E[x 2 (n)] → 0 for all ψ. Moreover, because r 2 (n) = α2n is in 1 (Z+ ), by (24), we have that E[X 2 (n; ψ)] → 0 as n → ∞ provided E[Y 2 (n)] → 0 as n → ∞. Notice that this is implied by E[Y 2 ] being summable. Take (18) and sum on both sides from n = 1, . . . , N : N 

E[Y 2 (n)] =

n=1

N 

E[G 2 (xn )] +

n=1

N  n 

G 2 (rn− j )E[Y 2 ( j − 1)].

n=1 j=1

Change the order of summation in the second sum to get N 

E[Y (n)] = 2

n=1

N  n=1

≤

N  n=1

E[G (xn )] + 2

N N  

G 2 (rn− j )E[Y 2 ( j − 1)]

j=1 n= j

E[G 2 (xn )] + Γ (1)

N 

E[Y 2 ( j − 1)],

j=1

 2 n−q ψ(0) for where we note that Γ (1) = ∞ n=0 G (r n ). Notice that G(x n−1 ) = φ(α)α 2 n ≥ q, so as |α| < 1, we have that E[G (x)] is a summable sequence, and thus the first sum on the righthand side is uniformly bounded, by G 1 , say. Adding E[Y 2 (0)] N E[Y 2 (n)] we get for N ≥ 1 that to both sides and using the notation S N := n=0 S N ≤ E[Y 2 (0)] + G 1 + Γ (1)S N −1 ≤ G 1 + Γ (1)S N , since (S N ) is non–decreasing. Since Γ (1) < 1, (S N ) is uniformly bounded above, and so E[Y 2 ] is absolutely summable. As indicated above, this is sufficient to give (ii), as desired. We now show that (ii) implies (i). Assume (ii). We show the first part of (46) is necessary. Pick the initial sequence ψ(n) = 1 for all n ≤ 0. Then E[x 2 (n)] = α2n for n ≥ 0. By (24), we have E[X 2 (n)] ≥ E[x 2 (n)] = α2n . But since E[X 2 (n)] → 0 as n → ∞ for every choice of initial sequence, this implies α2n → 0 as n → ∞. Hence |α| < 1, as required. Let the initial sequence ψ be such that E[G 2 (x)] is not the zero sequence. This can always be achieved, by virtue of Lemma 4. Suppose that Γ (1) > 1. Then, there is a μ > 1 such that Γ (μ) = 1, and by Theorems 2 and 3 we have E[X 2 (n)]/μ2n →

On the Dynamics and Asymptotic Behaviour of the Mean Square …

53

C > 0 as n → ∞. But this is inconsistent with E[X 2 (n)] → 0 as n → ∞ for every choice of initial sequence. Thus, we must have Γ (1) ≤ 1. Now suppose Γ (1) = 1. Once again, following the argument in Theorems 2 and 3, we see that we must have lim E[X 2 (n)] = C ∈ (0, ∞),

n→∞

with the positivity assured by the fact that E[G 2 (x)] is not the zero sequence. But we are assumed to have E[X 2 (n; ψ)] → 0 as n → ∞ for every initial sequence, there is again a contradiction. Hence (ii) implies |α| < 1 and Γ (1) < 1. Finally, note the explicit expression for Γ (1) to conclude (i).

4.4 Covariance Dynamics We recall that a process Z is weakly stationary if it has constant mean, and there is a function γ such that Cov(Z (n), Z (n + k)) = γ(k) for all k ≥ 0 and t ≥ 0. It is asymptotically weakly stationary if E[Z (n)] tends to a finite limit as n → ∞ and moreover lim Cov(Z (n), Z (n + k)) = γ(k). n→∞

In the case that φ(α) = 0, we automatically have μ > |α|, and can show that the process X˜ (n) = X (n)/μn is asymptotically weakly stationary by appealing to Proposition 3. If φ(α) = 0, it can still be the case that μ > |α| (which we know holds if and only if Γ (|α|) > 1) and once again X˜ (n) = X (n)/μn is asymptotically weakly stationary. These results are of greatest interest when μ > 1, in which the solution grows in mean square, but however, by suitable scaling, an asymptotically stationary process results. As an example of this kind of result, we give the argument when μ > |α|. Then, automatically, E[ X˜ (n)] = (α/μ)n E[ψ(0)] for all n ≥ 1, and therefore we have E[ X˜ (n)] → 0 as n → ∞. Next, rescale (28), and use the fact that r (n) = αn when p = 1, to get for n ≥ 1 the relation Cov( X˜ (n), X˜ (n + k)) =

1 μ2n+k

Cov(x(n), x(n + k))

+

1 μ2

 k  n  2n−2 j α α 1 E[Y 2 ( j − 1)]. 2( j−1) μ μ μ j=1

Next, we have Cov(x(n), x(n + k)) = Var[ψ(0)]α2n+k ,

54

J. A. D. Appleby and E. Lawless

so the first term on the righthand side tends to zero. As to the second term, we note that E[Y 2 (n)]/μ2n has a finite non–zero limit C  which is positive unless φ(α) = 0 and the initial function obeys ψ( j) = α j ψ(0) for all j ≥ 0 a.s. Therefore, excluding these cases we have that 1 lim Cov( X˜ (n), X˜ (n + k)) = 2 n→∞ μ

 k  ∞  2 j α α C  =: γ(k), μ μ j=0

as required.

5 The Case When α = 0 In the last section, we considered the dynamics of the mean square of (4) when p = 1, so that X obeys X (n) = αX (n − 1) +

q 

β j X (n − j)ξ(n), n ≥ 1,

j=1

when α = 0. One way of interpreting this equation is as a stochastic perturbation of the deterministic equation x(n) = αx(n − 1). This is especially pertinent if we are interested in studying a stable deterministic equation (where |α| < 1) and asking how much noise can be added to the system before it loses stability (in the sense of the mean square). As |α| becomes smaller, the underlying deterministic equation becomes more stable, in the sense that solutions of the unperturbed equation tend to zero. Now, we suppose that the zero solution of the underlying deterministic system is very strongly attracting, by setting α = 0. Then, the deterministic equation reads x(n) = 0, n ≥ 1; x(n) = ψ(n), n = −q + 1, . . . , 0. In this limiting case we see that all solutions collapse to zero at time n = 1, no matter how far away from zero the initial sequence ψ is, and stays there forever. Thus, the corresponding stochastic difference equation now reads X (n) =

q 

β j X (n − j)ξ(n), n ≥ 1;

X (n) = ψ(n), n ∈ {−q + 1, . . . , 0}.

j=1

(47) We will now determine the asymptotic behaviour of the mean square of X . The details of the proofs will be considerably shorter than in Sect. 4, partly because this case is easier to analyse, and partly because the results we have already established can be reused to tackle Eq. (47).

On the Dynamics and Asymptotic Behaviour of the Mean Square …

55

Assume that ψ and ξ have the same properties as in Sect. 2, and also βq = 0, so that the maximal delay q ∈ N is correctly identified. We shall take q ≥ 2, since the argument in the case q = 1 is covered in Sect. 4.1 already: we note that when q = 1 that E[X 2 (n; ψ)] = β12n E[ψ 2 (0)], n ≥ 1, so that once again real exponential dynamics ensue, and there is global mean square asymptotic stability if and only if β12 < 1. In fact, we will show that the dynamics are to leading order real exponential, and that there is global mean square asymptotic stability if and only if q  β 2j < 1. (48) j=1

To do this, define as before Y by (13), so that X (n) = Y (n − 1)ξ(n), n ≥ 1.

(49)

Since Y (n − 1) and ξ(n) are independent for each n ≥ 1, we have immediately that E[X 2 (n)] = E[Y 2 (n − 1)], n ≥ 1.

(50)

Hence there is no need for an analogue of Proposition 2. Thus determining the asymptotic behaviour of the mean square of Y immediately secures that of X . We follow the programme from Sect. 3 by deducing a Volterra summation equation for Y . Once this is done, we will be able to read off a Volterra summation equation for the mean square automatically, by appealing to the proof of Proposition 1: specifically, once an equation of the form (17) is available for Y , we can immediately conclude an equation of the form (18) for E[Y 2 ]. To do this, we need analogues of the sequences G(r· ) and G(x· ) which played a major role in Proposition 1. These analogues are β˜ and H , respectively, given by ˜ ˜ := 0, n ≥ q + 1, β(n) := βn , n ∈ {1, . . . , q}, β(n)

(51)

and H (n − 1) :=

q 

β j ψ(n − j), n ∈ {1, . . . , q},

H (n − 1) := 0, n ≥ q + 1.

j=n

(52) Proposition 4 Suppose that X is the solution of (4) and that ψ and ξ obey (6) and (5) respectively. Suppose that Y is given by (13). Then Y obeys

56

J. A. D. Appleby and E. Lawless

Y (n) = H (n) +

n 

˜ + 1 − k)Y (k − 1)ξ(k), n ≥ 1; Y (0) = H (0). (53) β(n

k=1

Moreover, E[Y 2 (n)] is well–defined for n ≥ 0 and given by E[Y 2 (n)] = E[H 2 (n)] +

n 

β˜ 2 (n + 1 − j)E[Y 2 ( j − 1)], n ≥ 1.

(54)

j=1

with E[Y 2 (0)] = E[H 2 (0)]. Proof (of Proposition 4) Taking into account the F(0)–measurability of H and the fact that β˜ is deterministic, we see that proving (53) suffices to secure (54) also. This is because deducing (54) from (53) follows the proof of Proposition 1 from (21) to the end, with essentially only notational changes being required. Let n ≥ q + 2. For j ∈ {1, . . . , q}, we have n ≥ j + 1, so X (n − j) = Y (n − j − 1)ξ(n − j). Hence for n ≥ q + 1, Y (n − 1) =

q 

β j Y (n − j − 1)ξ(n − j) =

j=1

n−1 

βn−k Y (k − 1)ξ(k).

k=n−q

˜ − k) = 0 for 1 ≤ k ≤ n − q − 1, we have Then for n ≥ q + 2, since β(n 

n−q−1

Y (n − 1) =

˜ − k)Y (k − 1)ξ(k) + β(n

k=1

=

n−1 

n−1 

βn−k Y (k − 1)ξ(k)

k=n−q

˜ − k)Y (k − 1)ξ(k) β(n

k=1

= H (n − 1) +

n−1 

˜ − k)Y (k − 1)ξ(k), β(n

k=1

since H (n − 1) = 0 for all n ≥ q + 1. Thus Y (n) = H (n) +

n 

˜ + 1 − k)Y (k − 1)ξ(k), n ≥ q + 1. β(n

k=1

For n = q + 1, and j ≤ q, n − j ≥ 1, so Y (n − 1) =

q  j=1

β j X (n − j) =

q 

˜ j)Y (n − 1 − j)ξ(n − j) = β(

j=1

Hence by replacing n − 1 by n, we have for n = q

n−1  k=n−q

˜ − k)Y (k − 1)ξ(k). β(n

On the Dynamics and Asymptotic Behaviour of the Mean Square …

Y (n) =

n 

˜ + 1 − k)Y (k − 1)ξ(k) = H (n) + β(n

k=n+1−q

n 

57

˜ + 1 − k)Y (k − 1)ξ(k), β(n

k=1

noting that H (q) = 0. If n ∈ {2, . . . , q}, if j ≤ n − 1, then n − j ≥ 1 and so Y (n − 1) =

n−1 

β j X (n − j) +

q 

j=1

=

n−1 

β j X (n − j)

j=n

β j Y (n − j − 1)ξ(n − j) +

j=1

=

q 

q 

β j ψ(n − j)

j=n

β j ψ(n − j) +

j=n

n−1 

βn−k Y (k − 1)ξ(k)

k=1

= H (n − 1) +

n−1 

˜ − k)Y (k − 1)ξ(k). β(n

k=1

Hence Y (n) = H (n) +

n 

˜ + 1 − k)Y (k − 1)ξ(k), n = 1, . . . , q − 1, β(n

k=1

so we have established (53). Finally, Y (0) =

q j=1

β j ψ(1 − j) = H (0), as required.

The asymptotic behaviour of (54) can be analysed using Theorem 1. However, we can alternatively exploit the special structure of the equation to give a more direct proof using the theory of finite order linear difference equations, and we do this now. ˜ Consider next (54) for n ≥ q + 1. Then E[H 2 (n)] = 0. Since β(n) = 0 for n ≥ q + 1, we have E[Y 2 (n)] =

n 

β˜ 2 (n + 1 − j)E[Y 2 ( j − 1)] +

j=n−q+1

=

n 

β˜ 2 (n + 1 − j)E[Y 2 ( j − 1)] =

j=n−q+1

n−q 

β˜ 2 (n + 1 − j)E[Y 2 ( j − 1)]

j=1 q 

βk2 E[Y 2 (n − k)].

k=1

Thus E[Y 2 ] obeys a qth order linear difference equation with non–negative coefficients (it is authentically qth order, since βq = 0). Consider the polynomial φ0 (λ) = λq −

q  j=1

β 2j λ(q− j)

(55)

58

J. A. D. Appleby and E. Lawless

which is degree q in λ. Since the β 2 ’s are non–negative, and βq2 > 0, the zero of φ0 with largest modulus is real and positive. Call it μ2 . One can see that μ2 is the unique positive zero of the strictly increasing function λ → φ0 (λ)/λq . Furthermore, we 2 is less than one, equal to one, or greater than one according have that the value of μ q as to whether S := j=1 β 2j is less than 1, equal to one, or greater than one. Indeed, we have that E[Y 2 (n)] =: C  ∈ [0, ∞). lim n→∞ μ2n Next, we observe that C  is zero if and only if H (n − 1) = 0 for all n ∈ {1, . . . , q} a.s. Suppose H (q − 1) = 0. Putting n = q in (52) yields 0 = βq ψ(0), so ψ(0) = 0 a.s. Consider H (n − 1) = 0 for n = q − 1: this simplifies to 0 = βq−1 ψ(0) + βq ψ(−1) = βq ψ(−1). Hence ψ(−1) = 0 a.s. Continuing in this manner, we arrive at ψ( j) = 0 for all j ≤ 0 a.s., which contradicts the hypothesis that ψ is non–zero. Therefore, we have that C  > 0 and hence that E[X 2 (n)] E[Y 2 (n − 1)] 1 C = lim = 2 =: C ∈ (0, ∞). 2n 2(n−1) 2 n→∞ n→∞ μ μ μ μ lim

Therefore, we have proven the following result on exponential asymptotic behaviour. Theorem 5 Suppose X is the solution of (47). ψ and ξ obey (6) and (5) with P[ψ ≡ 0] < 1. Suppose βq = 0. Let φ0 be given by (55). Then there is a unique positive zero μ2 of φ0 and there exists C = C(ψ) > 0 such that E[X 2 (n)] = C(ψ) ∈ (0, ∞). n→∞ μ2n lim

We have also given a characterisation of the global mean square asymptotic stability: Theorem 6 Suppose X is the solution of (47). Let ψ and ξ obey (6) and (5) and βq = 0. Suppose that ψ is non–zero. Then the following are equivalent: q 2 (i) j=1 β j < 1; (ii) lim E[X 2 (n; ψ)] = 0 for all ψ ∈ L 2 (Ω). n→∞

Acknowledgements The authors wish to thank the conference organisers for the opportunity to present their work. We are also grateful to the anonymous referee for their careful reading of our manuscript, and for suggesting the inclusion of further relevant literature. EL is partially supported by Science Foundation Ireland (16/IA/4443).

On the Dynamics and Asymptotic Behaviour of the Mean Square …

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References 1. Appleby, J.A.D.: Mean square characterisation of a stochastic Volterra integrodifferential equation with delay. Int. J. Dyn. Syst. Differ. Equ. 11(3/4), 194–226 (2021) 2. Appleby, J.A.D., Mao, X., Riedle, M.: Geometric Brownian motion with delay: mean square characterisation. Proc. Amer. Math. Soc. 137, 339–348 (2009) 3. Bellman, R.: Stochastic transformations and functional equations. Proc. Sympos. Appl. Math. 16, 171–177 (1964) 4. Buckwar, E., Kelly, C.: Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations. SIAM J. Numer. Anal. 48(1), 298–321 (2010) 5. Buckwar, E., Kelly, C.: Non-normal drift structures and linear stability analysis of numerical methods for systems of stochastic differential equations. Comput. Math. Appl. 64(7), 2282– 2293 (2012) 6. Buckwar, E., Notarangelo, G.: A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete Cont. Dyn. Syst. Series B 18(6), 1521–1531 (2013) 7. Buckwar, E., Sickenberger, T.: A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. Math. Comput. Simul. 81(6), 1110–1127 (2011) 8. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005) 9. Feller, W.: An Introduction to Probability Theory and its Applications, vol. 1. Wiley, New York (1950) 10. Higham, D.J.: A-stability and stochastic mean-square stability. BIT Numer. Math. 40(2), 404– 409 (2000) 11. Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000) 12. Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Academic, San Diego (1975) 13. Kolmanovskii, V.B., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht (1999) 14. Kolmanovskii, V.B., Shaikhet, L.E.: Some peculiarities of the general method of Lyapunov functionals construction. Appl. Math. Lett. 15(3), 355–360 (2002). https://doi.org/10.1016/ S0893-9659(01)00143-4 15. Kulenovic, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures. Chapman and Hall/CRC, Boca Raton, FL (2002) 16. Mao, X.: Exponential Stability of Stochastic Differential Equations. Pure Applied Mathematics. Marcel Dekker, New York (1994) 17. Mao, X.: Stochastic Differential Equations and their Applications, 2nd edn. Horwood Publishing Limited, Chichester (2008) 18. Mackey, M.C., Nechaeva, I.G.: Solution moment stability in stochastic differential delay equations. Phys. Rev. E 3(52), 3366–3376 (1995) 19. Raffoul, Y.: Qualitative Theory of Volterra Difference Equations. Springer, Cham (2018) 20. Shaikhet, L.E.: Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect. Theory of Stochastic Processes (Russian), vol. 6, no. 136, pp. 120–123. Naukova Dumka, Kiev (1978) 21. Shaikhet, L.E.: Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations. Appl. Math. Lett. 10(3), 111–115 (1997) 22. Shaikhet, L.E.: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer, Heidelberg (2011) 23. Shaikhet, L.E.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Heidelberg (2013)

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Border Collision and Heteroclinic Bifurcations in a 2D Piecewise Smooth Map Viktor Avrutin, Laura Gardini, Iryna Sushko, Zhanybai T. Zhusubaliyev, and Ulanbek A. Sopuev

Abstract We consider a 2D piecewise smooth map originating from an application (acting as a discrete-time model of a DC/DC converter with pulse-width modulated multilevel control). We focus on several non-trivial transformations occurring in the phase space of this map under parameter variation. In particular, we describe the effect of a fold border collision bifurcation leading to the appearance of a pair of cycles, an attracting and a saddle one, a sequence of transformations of the basins of coexisting attractors, as well as heteroclinic bifurcations which result first in the destruction of an attracting closed resonant curve and then in the creation of another one such curve. Keywords Piecewise smooth maps · Border collision bifurcations · Homoclinic bifurcations · Heteroclinic bifurcations · Basin boundaries · DC/DC converters · Multilevel converters

V. Avrutin Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany e-mail: [email protected] L. Gardini (B) Department of Economics, Society, Politics, University of Urbino, A. Saffi st. 42, 61029 Urbino, Italy e-mail: [email protected] I. Sushko Institute of Mathematics, NAS of Ukraine, Tereshchenkivska st. 3, Kyiv 01024, Ukraine e-mail: [email protected] Kyiv School of Economics, Mykoly Shpaka st. 3, Kyiv 03113, Ukraine Z. T. Zhusubaliyev · U. A. Sopuev Department of Mathematics, Osh State University, Lenin st. 331, 723500 Osh, Kyrgyzstan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_3

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1 Introduction For a long time, piecewise smooth dynamical systems attract attention of researches both from applied and theoretical points of view. Nowadays, it is well accepted that such systems are useful for description of many real world processes, in particular, those characterized by sharp switching between states, various kinds of impacts, frictions, sliding, etc. (see, e.g. [10]). On the other hand, piecewise smooth systems possess many properties and exhibit many effects which cannot be studied by the standard tools of nonlinear dynamics theory. The primary reason for that is the existence of one or several borders in the phase space (often called switching manifolds) along which the system changes its definition. Under parameter variation, an invariant set can collide with such a border, and this collision can lead to a drastic change of the dynamics. Possible outcomes of such bifurcations include direct transitions from an attracting fixed point to a stable cycle of any period or to a chaotic attractor (see [19, 20]), the appearance of one or several closed invariant curves [7], multiple or even infinitely many attractors [6, 13, 16, 23]. This kind of transformations are collected under the term border collision bifurcation, and at present there are many studies related to investigation of border collision bifurcations in both theoretical and applied context (see, e.g., [5, 8, 9, 11, 18, 22, 24, 27, 29]). The aim of our present study is twofold. Considering a 2D map which comes from an application, we first discuss the effect of a fold border collision bifurcation leading to the appearance of a pair of cycles, an attracting one and a saddle. Recall that by contrast to a standard fold bifurcation, a fold border collision bifurcation is not related to an eigenvalue equal to +1 but to the collision with a border: at the bifurcation, these cycles coincide and have one point at the border. Thereafter, we present examples of heteroclinic tangles leading to destruction and creation of attracting resonant closed invariant curves. In fact, there are several publications describing heteroclinic tangles in smooth systems (see, e.g., [4, 12, 31]. Specifically for noninvertible maps we refer also to [1–3, 14, 15]). Our examples are interesting because of peculiar properties of the invariant sets involved in the tangles. These sets are non-smooth due to intersections with the switching manifolds, moreover, they can be smooth at the tangency points at a heteroclinic bifurcation as it occurs in smooth systems, but can also have cusp contact points. The paper is organized as follows. In Sect. 2, the investigated map is introduced. Then, in Sect. 3, we first describe a fold border collision bifurcation and then two heteroclinic tangles: the first one leading to the destruction of a closed invariant curve, and the second one causing a new closed invariant curve to appear. Section 4 concludes.

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2 Investigated Map We consider a 2D piecewise-smooth map F : R2 → R2 , (xn+1 , yn+1 ) = F(xn , yn )

(1)

defined on five partitions as follows:

F(x, y) =

⎧ FR (x, y) ⎪ ⎪ ⎪ ⎪ ⎪ (1) ⎪ ⎪ ⎪ FM (x, y) ⎨

FM(2) (x, y) ⎪ ⎪ ⎪ (3) ⎪ ⎪ ⎪ FM (x, y) ⎪ ⎪ ⎩ FL (x, y)

if (x, y) ∈ I R (1) if (x, y) ∈ I M (2) if (x, y) ∈ I M (3) if (x, y) ∈ I M

if (x, y) ∈ I L

where    λ x e 1 FR : → λ2 y e  λ   e 1 x (i) FM : → y e λ2

 · (x + Γ ) − Γ , · (y + Γ ) − Γ 

· x − 3i + Γ + (i−1) + 13 · eλ1 (i−3·α·ϕ/P) − Γ 3 , i = 1, 2, 3,

· y − 3i + Γ + (i−1) + 13 · eλ2 (i−3·α·ϕ/P) − Γ 3     λ x e 1 · (x − 1 + Γ ) + 1 − Γ . FL : → λ2 y e · (y − 1 + Γ ) + 1 − Γ

The partitions I R = {(x, y) | ϕ(x, y) < 0} (i) P = (x, y) | 3α (i − 1)  ϕ(x, y) < IM

I L = (x, y) | ϕ(x, y)  αP

P i 3α

, i = 1, 2, 3

are separated by the borders s (i) = (x, y) | ϕ(x, y) =

P i 3α

, i = 0, 1, 2, 3

and the function ϕ(x, y) is given by ϕ(x, y) = Q + (1 − χ) · y + (χ − ϑ) · x. Here λ1 , λ2 , Γ , P, α, Q, χ are real parameters, 0 < Γ < 1, α > 0, and ϑ = λ2 /λ1 .

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As mentioned in the Introduction, map F represents a discrete-time model of a DC/DC converter with pulse-width modulated multilevel control (for details see [28, 30]). In the present work, as in [28], the following parameter values are fixed: λ1 = −0.15660377358, λ2 = −0.2, P = 0.977, Q = 0.4885,

(2)

Γ = 0.467946108, χ = 0.0575, and the parameter α is varied.

3 Phase Portrait Transformations As a starting point of our discussion, let us describe the phase portrait of map F at α = 29.1377 shown in Fig. 1a. In this case, map F has  a repelling fixed point (focus) denoted by O; q

 an attracting closed invariant curve denoted by C1 . Here, the upper index q refers to the quasiperiodic dynamics on the curve which for short is called a quasiperiodic q curve; the basin of C1 is shown in pink; q

 a repelling closed invariant quasiperiodic curve C2 constituting the basin boundary q of C1 . This curve is non-smooth, since it intersects the borders s (1) and s (2) ;

Fig. 1 a Phase portrait of map F at α = 29.1377; b a magnification of the rectangle indicated in (a). Other parameter values are fixed as in (2)

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 an attracting closed invariant curve C3r formed by the unstable invariant set of a saddle 6-cycle Q 6 = {q0 , . . . , q5 } approaching points of an attracting 6-cycle (focus) P6 = { p0 , . . . , p5 }. Here, the upper index r in C3r refers to periodic dynamics on the curve which for short is called a resonant curve. The unstable invariant set of Q 6 is denoted by W u (Q 6 ), and its branches associated with the point qi , i = 0, . . . , 5, are denoted by W+u (qi ) and W−u (qi ) (see Fig. 1b where the branches W+u (qi ) and W−u (qi ) are indicated for i = 0, 1). In this way, the saddle-focus connection C3r is given by the branches W+u (qi ) and W−u (qi ) approaching the related points of the attracting cycle Q 6 (with the basin shown in light green). The stable invariant set of Q 6 is denoted by W s (Q 6 ) and its branches associated with the point qi are denoted by W+s (qi ) and W−s (qi ) (see the branches W+s (qi ) and W−s (qi ), i = 0, 1, in Fig. 1b); notice that the ‘inner’ branches of W s (Q 6 ) denoted q by W+s (qi ) are winding (in backward iterations) on the curve C2 , or, more preq s cisely, the curve C2 is the α-limit set for the points of W+ (qi ). For increasing values of the parameter α, the phase portrait described above undergoes several non-trivial transformations as illustrated in Fig. 2. In particular, a new attracting 6-cycle appears (in Fig. 2, its basin is shown in yellow), then the ‘external tails’ of this basin appear which are very thin (almost invisible) in Fig. 2a and much thicker (well visible) in Fig. 2b. With further increasing α, these tails gradually increase in size, while the size of the basin shown in green decreases (see Fig. 2c– e), up to disappearance of the ‘external tails’ of that basin (see Fig. 2f). Comparing Fig. 2a and e, we see that under variation of α the two basins shown in yellow and green interchange their dominant role. Our aim is to explain the mechanisms of these transformations.

3.1 Fold Border Collision Bifurcation The first transformation is related to a fold border collision bifurcation leading to the appearance of two new 6-cycles. In fact, as one can see in Fig. 3, at α = 29.1378 there are still all the invariant sets described above (see Fig. 1a), in particular, the curve C3r containing the saddle cycle Q 6 and the attracting cycle P6 . In Fig. 2, the basin of C3r is shown in green. However, besides these sets, there is a new pair of 6-cycles, namely the saddle cycle Q 6 = {q0 , . . . , q5 } and an attracting cycle P6 = { p0 , . . . , p5 }. In Fig. 2, the basin of P6 is shown in yellow. Note that at the beginning the yellow basin is much smaller than the green one. At the border collision bifurcation, the cycles Q 6 and P6 coincide, i.e., qi = pi , i = 0, . . . , 5, and one periodic point, say q0 = p0 , belongs to the border s (2) . After this bifurcation, the basin boundary of P6 is formed by the branches W+s (qi ) and W−s (qi ), i = 0, . . . , 5, of the stable invariant set W s (Q 6 ) of the saddle cycle Q 6 (see Fig. 3c, where the branches W+s (q0 ) and W−s (q0 ) are indicated, as well as the branches W+u (q0 ) and W−u (q0 )).

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Fig. 2 Basin of an attracting closed invariant curve (in pink) and of the two coexisting attracting 6-cycles (in green and yellow) at a α = 29.141, b α = 29.143, c α = 29.145, d α = 29.147, e α = 29.148, f α = 29.149

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Fig. 3 a Phase portrait of map F at α = 29.1378. b, c Magnified rectangles indicated in (a) and (b), respectively

At the first sight, in Fig. 3a the basin of P6 consists of six (yellow) islands so that the pair of branches W+s (qi ) and W−s (qi ), i = 0, . . . , 5, constituting the boundary of each island, forms a closed loop. However, such a basin boundary is possible only in noninvertible maps (see, e.g. [17, 21]), while map F is invertible. Indeed, the q branches W+s (qi ) and W−s (qi ) never intersect: they are winding on the curve C2 , that q is, their points tend (in backward iterations) to C2 , similar to the mentioned above branches W+s (qi ) of the stable invariant set W s (Q 6 ) of the cycle Q 6 . This means, in q particular, that the repelling closed invariant curve C2 still forms the basin boundary q of the attracting closed invariant curve C1 , but it does not belong any longer to the q basin boundary of the attracting cycle P6 , and in any external neighborhood of C2 there are points converging to the cycle P6 and points converging to P6 .

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3.2 Heteroclinic Tangles: Destruction and Creation of a Closed Invariant Curve To explain the mechanism of appearance and disappearance of the ‘external tails’ of the basins illustrated in Fig. 2, let us first compare the phase portraits of map F at α = 29.1378 (see Fig. 3) and at α = 29.145 (see Fig. 4):  Considering, for example, the point q1 of the saddle cycle Q 6 , one can clearly see that in Fig. 3b the branch W+u (q1 ) connects q1 with p0 , while in Fig. 4b this branch connects q1 with p0 . Thus, the attracting saddle-focus connection C3r existing at α = 29.1378 does not exist any longer at α = 29.145.

Fig. 4 a Phase portrait of map F at α = 29.145. b, c Magnified rectangles indicated in (a) and (b), respectively

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 At α = 29.1378, the basin of the cycle P6 is bounded by the stable invariant set W s (Q 6 ) of the saddle cycle Q 6 (see Fig. 3), while at α = 29.145 this basin is bounded, besides W s (Q 6 ), also by the stable invariant set W s (Q 6 ) of the saddle cycle Q 6 (see Fig. 4).  Comparing, for example, the branches W+u (q1 ) and W+u (q0 ) in Figs. 3b and 4b, one can notice that these branches (as well as other similar pairs) change their location with respect to each other. Similar transformations may be associated with a heteroclinic tangle. In our case, it is illustrated in Fig. 5 where the phase portrait near the point q1 of the saddle cycle Q 6 is shown. At α ≈ 29.1399483 the first heteroclinic bifurcation occurs at which unstable invariant set W u (Q 6 ) and stable invariant set W s (Q 6 ) become tangent to each other: in Fig. 5a, the tangency points of W+u (q1 ) and W+s (q0 ) are clearly visible. Then, for increasing α, the invariant sets W u (Q 6 ) and W s (Q 6 ) become transversely intersecting at infinitely many points: see Fig. 5b where α = 29.1399525. Finally, at α ≈ 29.1399575 the last heteroclinic bifurcation occurs (see Fig. 5c), after which the basin boundary of the cycle P6 is constituted not only by the stable invariant set W s (Q 6 ) but also by the stable invariant set W s (Q 6 ). In particular, the boundaries of the ‘external tails’ of the basin of P6 , appearing due to the heteroclinic tangle, are bounded by the branches W−s (qi ) and W+s (qi ), i = 0, . . . , 5. In this way, the closed invariant curve C3r is destroyed. It is worth noticing a peculiar property of the last heteroclinic bifurcation described above, related to the non-smoothness of the map F. Besides the fact that the involved stable and unstable invariant sets are nonsmooth due to their intersections with the borders s (1) and s (2) , one more peculiarity can be seen in Fig. 5c: at the last heteroclinic bifurcation the stable invariant set W+s (q5 ) has cusps at the contact points with W+u (q0 ), differently from the case of the first heteroclinic bifurcation associated with a standard smooth tangency. In fact, similar cusp tangency points at homo- or heteroclinic bifurcations are characteristic for piecewise linear maps (see, e.g. [25]). One more qualitative transformation of the phase portrait which is also associated with a heteroclinic tangle is illustrated in Fig. 6. In fact, as illustrated in Fig. 2, for further increasing α the basin of P6 (shown in yellow) continues to increase in size, while the size of the basin of P6 (shown in green) decreases. The ‘external tails’ of the basin of the cycle Q 6 which are still visible in Fig. 2e, do not exists any longer in Fig. 2f. This transformation occurs through a heteroclinic tangle involving the same branches of the stable invariant set W s (Q 6 ), as in the previous case, namely, the branches W+s (qi ), i = 0, . . . , 5, but different branches of the unstable invariant set W u (Q 6 ), namely, the branches W−u (qi ). Figure 6a shows the phase portrait before the first heteroclinic bifurcation, in Fig. 6b the branches W+s (qi ) and W+u (qi ) are tangent to each other at the first heteroclinic bifurcation, in Fig. 6c they are transversely intersecting, in Fig. 6d the branches W+s (qi ) and W+u (qi ) are again tangent to each other at the last heteroclinic bifurcation, and Fig. 6e shows the phase portrait after the last heteroclinic bifurcation. As a result, the branches W+s (qi ) and W+u (qi ) changed their location with respect to each other, and the stable invariant set W u (Q 6 ) do not belong any longer to the basin boundary of P6 .

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Fig. 5 Phase portrait of map F near the point q1 of the saddle cycle Q 6 a at the first heteroclinic bifurcation, α = 29.1399483; b the unstable invariant set W u (Q 6 ) and the stable invariant set W s (Q 6 ) are transversely intersecting, α = 29.1399525; c at the last heteroclinic bifurcation, α = 29.1399575

As a result, a new attracting closed invariant resonant curve C4r is created formed by the unstable invariant set W u (Q 6 ) of the saddle cycle Q 6 approaching points of the attracting cycle P6 .

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Fig. 6 Phase portrait of map F near the point q0 of the saddle cycle Q 6 a before the heteroclinic bifurcation, α = 29.15; b at the first heteroclinic bifurcation, α = 29.15314; c at the heteroclinic intersection, α = 29.1532; d at the last heteroclinic bifurcation, α = 29.153401; e after the heteroclinic tangle, α = 29.16

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4 Conclusion The considered map is important from the applied point of view and has very complicated dynamics. In fact, there are several papers where various results are presented related to the attractors, their bifurcations as well as bifurcation structures in the parameter space of this map [26, 28, 30]. However, many problems remain open, and it is a challenging task to describe the dynamics of the considered map completely. Our present study contributes to the understanding of the role of nonsmoothness of the map, in particular, for the creation of new cycles via border collision bifurcation. In the present work, we discussed the role of heteroclinic tangles for transformations of basins of attraction as well as for destruction and creation of attracting resonant closed invariant curves. We emphasize also specific properties of these curves, namely, their nonsmoothness caused by the intersection with switching manifolds, as well as cusp tangency points at the heteroclinic bifurcations. Acknowledgements The work of V. Avrutin was supported by the German Research Foundation within the scope of the project “Generic bifurcation structures in piecewise-smooth maps with extremely high number of borders in theory and applications for power converter systems – 2”. Zh. T. Zhusubaliyev and U. A. Sopuev acknowledge the support by the grant 14-22 of the Osh State University.

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Using Homotopy Link Function with Lipschitz Threshold in Studying Synchronized Fluctuations in Hierarchical Models Tahmineh Azizi

Abstract In this chapter, we use homotopy and Lipschitz concepts to study synchronization effects in a triangular discrete-time dynamical systems. We have generalized the previous results in this area by defining a new link function with Lipschitz threshold. Using this new link function we continuously deform the orbits of the original model into the coupled model such that we preserve qualitative properties such as stability and periodicity. We analytically obtain this Lipschitz synchronization threshold and we prove that for less than this threshold, two systems are completely synchronized, i.e. they eventually evolve identically in time. We apply this method to a one-dimensional Ricker type population model, whose trajectories as is well known, can be chaotic. We use some qualitative dynamical systems tools such as Poincare section, spectrum and time series to detect the chaotic dynamics and chaotic signals for different synchronization Lipschitz thresholds s and growth rates r . Finally, we numerically find the Lipschitz synchronization threshold for different growth rates using mean phase and amplitude differences. Keywords Discrete-time dynamical system · Chaotic dynamics · Synchronization · Lipschitz threshold · Poincare section · Spectrum

1 Introduction The triangular discrete-time dynamical systems have been studied widely by different researchers during recent years [1–4]. In such models, it is assumed that one of the species, x1 is the dominant species and have the following form: x1 (n + 1) = f (x1 (n)) x2 (n + 1) = g(x1 (n), x2 (n)) T. Azizi (B) Florida State University, Tallahassee, FL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_4

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where x1 and x2 represent the population sizes for species 1 and 2. Here, the Jacobian matrix is lower triangular and in the biology literature, this is called hierarchical models. The local and global dynamics of a two-species hierarchical competition with a strong Allee effect have been discussed in [1]. Moreover, Elaydi et al. in [2] provided examples of their application to competition models and they proved that if the triangular model does not have periodic orbits of prime period 2 and it has a unique fixed point, then the fixed point is globally asymptotically stable. In this study, we are interested to look at the complex dynamics of these types models, such as chaos and chaos synchronization. Before 1961, when Edward Lorenz discovered sensitive dependence on initial conditions or the butterfly effect by working on a simplified version of atmospheric transfer model, the study of the long term behavior of a dynamical system was restricted to models of ordinary differential equations with bounded solutions. Lorenz’s finding showed the main property of chaotic dynamical systems is sensitivity to initial conditions, i.e. that small perturbations in initial conditions make a large difference in the long time behavior of solutions [5]. Since then, chaos as a complex nonlinear phenomenon has been studied by many scientists and engineers [6–10]. One of the most important achievement in chaos theory is chaos synchronization. According to its definition, chaos synchronization occurs when there exists a strong correlation between the synchronized system and the original system that makes two systems have the same dynamical behavior. In general, synchronization takes place when the error between solutions of two systems vanishes and approaches to zero when time approaches infinity. Chaos synchronization has many applications in physics, biology and engineering [11–16]. In 1990, Pecora and Carroll [14, 16] showed two Lorenz systems with the property of sensitive dependence on the initial conditions could be synchronized. In 2020, Azizi and Kerr introduced a new mathematical framework in the study of chaos synchronization a novel discrete-time dynamical system obtained using a convex link function with a contraction constant [17, 18]. They developed a new concept in synchronization theory called the stable synchronization threshold and they analytically and numerically proved that after that threshold is reached the drive-response system loses complete synchronization properties and solutions become divergent from each other. The advantage of this new coupling method is that qualitative dynamics are preserved and two systems show the identical behavior. This is true for chaotic regimes with small values of the threshold. They applied this coupling method to synchronized cycles of one dimensional and two dimensional population dynamics with different ranges of dynamics, ranging from stable steady states to periodic orbits and chaos. In this chapter, we extend these results to a general case in discrete-time dynamical systems theory in which we do not need to have a contraction constant as synchronization threshold to couple two discrete-time dynamical systems. We define a new link function called the Homotopy link function with Lipschitz threshold to couple two discrete-time systems. Using this new approach we remove the need to have a contraction mapping and we can construct a drive-response system for general cases when we have Lipschitz function. We proved that large synchronization Lipschitz

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thresholds complete synchronization fails in a chaotic regime, while for small threshold is holds. We apply our results to a simple Ricker type population model by using various qualitative dynamical tools, such as time-series analysis and we obtain the Lipschitz synchronization threshold for different growth rates by using mean phase and amplitude differences. Finally, by means of Poincare sections and power spectra, we demonstrate the occurrence of chaotic dynamics and different chaotic signals.

2 Analytical Derivation of Drive-Response System In this section, we study the complete synchronization in a general discrete-time drive-response system. Here, we use a convex function to build the proposed driveresponse system. To begin with, consider the following discrete-time dynamical system: X n+1 = f (X n )

(1)

where X ∈ Rn is the state vector of drive system at time n, f is a mapping from Rn to itself and is continuously differentiable. Next step is to find a perfect linking such that the system (1) and new coupled one remain in step with each other in time. To model the response system or coupled system, we use a convex link function as the form H (X, Y ) := (1 − s) X + s Y where H : R2n → Rn and X, Y ∈ Rn are the state vectors of response system at time n, and 0 < s ≤ 1 is synchronization threshold. Therefore, for Hn := (1 − s) X n + s Yn , the response system has the form: Yn+1 := f (Hn ) = f ((1 − s) X n + s Yn )

(2)

and we demonstrate the error between the solutions of the drive system (1) and the response system (2) by e(n) = ||Yn − X n ||.

2.1 Complete Synchronization Using Contraction Mapping Theorem To explain the complete synchronization between two systems (1) and (2), we need to recall some known concepts which are crucial part of the proposed coupling method: Definition 2.1 We say that the drive system (1) and response system (2) are in complete synchronization if lim e(n) = lim ||Yn − X n || = 0

n→∞

n→∞

(3)

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means that two systems eventually evolve identically in time. Definition 2.2 Let E be a Banach space. Then, the map F : E → E is called a contraction mapping if there exists a constant 0 ≤ α < 1 such that for every pair of points X, Y ∈ E, we have ||F(X ) − F(Y )|| ≤ α||X − Y ||, where α is called a contraction constant of F on E. The error between the drive and response system (1) and (2) has the following form: e(n + 1) = Yn+1 − X n+1 = f ((1 − s) X n + s Yn ) − f (X n )

(4)

We can easily see that for 0 < s ≤ 1: ||((1 − s) X n + s Yn ) − X n || ≤ s ||Yn − X n || Here, we assume that f is a contraction mapping. Then, for the Eq. (4) we can write: ||e(n + 1)|| = || f ((1 − s) X n + s Yn ) − f (X n )|| ≤ β||Yn − X n || = β||e(n)|| where, β is a contraction constant. As we defined before, to get complete synchronization, we need to have limn→∞ ||e(n)|| = 0. Therefore, for contraction constant 0 ≤ β < 1, lim ||e(n + 1)|| = lim ||Yn+1 − X n+1 || = 0

n→∞

n→∞

which means that the drive-response system (1)–(2) satisfies the complete synchronization properties. We will find β in Theorem (2.3). Theorem 2.3 Given the non-linear coupled dynamical system (1) and (2), where 1 the map f : R2n → Rn , and for the values s < s˜ = , we get ρA + α lim ||Yn+1 − X n+1 || = 0

n→∞

means that passing the synchronization threshold s˜ makes the drive-response system (1) and (2) lose the complete synchronization properties. Proof Suppose the following C r maps which have a fixed point at the origin: X n+1 = A X n + F(X n ), Yn+1 = A((1 − s) X n + s Yn ) + F((1 − s) X n + s Yn ),

(5) (6)

where the contraction mapping F(X n ) = F2 (X n ) + · · · + Fr −1 (X n ) + O(|(X n )|r ), includes the vector-valued homogeneous polynomials of degree 2, . . . , r . Consider the following equation for the error:

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e(n + 1) = Yn+1 − X n+1 = (1 − s) A X n + s A Yn − A X n + F((1 − s) X n + s Yn ) − F(X n ) = s A (Yn − X n ) + F((1 − s) X n + s Yn ) − F(X n )

Since, we assumed that F is a contraction mapping, it satisfies the following inequality: ||F(Y ) − F(X )|| ≤ α ||Y − X || where, α is a contraction constant. By triangular inequality we can write: ||e(n + 1)|| = ||s A (Yn − X n ) + F((1 − s) X n + s Yn ) − F(X n )|| ≤ s ρ A ||Yn − X n || + s α ||Yn − X n || = s ρ A ||e(n)|| + s α ||e(n)|| where, 0 < s ≤ 1 and ρ A is the spectral radius of A which is equal to ρ A = max |λi | where λ is the root of characteristic polynomial or eigenvalue for A. Since, 0 ≤ α < 1, therefore ||e(n + 1)|| ≤ s ρ A ||e(n)|| + s α ||e(n)|| = s (ρ A + α) ||e(n)|| We know that for complete synchronization, the error between the solutions should converge toward zero. Thus, lim n→∞ ||e(n)|| = limn→∞ ||Yn − X n || = 0. As a result, for s (ρ A + α) < 1 we have lim ||e(n + 1)|| = lim ||Yn+1 − X n+1 || = 0

n→∞

n→∞

1 = s˜ . Here, s˜ = β, which we discussed in the beginning of ρA + α this section. After passing s˜ , we lose the complete synchronization between (1) and (2). 

for which, s
0 such that ||Z ∗ − Z || < δ implies that ||h  (Z ) − Z ∗ || <  for  > 0.

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Proof The proof is straightforward.

Remark 4 In Theorem (2.6), hyperbolicity is a robust property and it is one of the most important assumptions. It has been experimentally observed that there may be some situations in which the response system is stable but the response system has complex dynamics and the reason is using non differentiable link function or any non-differentiable transport system. In this study, we take the advantages of using a continuous convex link function which can completely control over the behavior of response system and we will numerically show that the response system inherits the same qualitative dynamics as its drive system and even for smaller synchronization threshold, the response system and drive system are almost completely equivalent.

3 Numerical Results In studying complex population oscillations in natural systems, synchronization has been used widely by different researchers researchers [19]. In 2000, when Bernd Blasius and Lewi Stone were studying the chaotic UPCA foodweb model, they discovered the importance of spatio-temporal structures associated with phase synchronization in conservation ecology. They claimed that it is true that perturbation of a local patch population can make them to get extincted, to buffer the endangered population by colonizers, the periodicity of spatial phase synchronization would be effective. They proposed that even though population synchronization can make a global population extinction [20], phase synchronization would help to maintain species persistence. According to this finding, synchronization would be effective in shaping the distribution and abundance of species in continental scale. The Ricker model is one of the most widely-used ecological models which displays regular and irregular complex nonlinear dynamics [21] and its coupled system as the following form: x(n + 1) = x(n) er (1−x(n)) y(n + 1) = H (x(n), y(n)) e

(10) r (1−H (x(n),y(n)))

(11)

where H (x(n), y(n)) := (1 − s) x(n) + s y(n)

(12)

Here, x demonstrates the population size of drive system, y represents the population size of response system, r is the intrinsic growth rate s is Lipschitz synchronization threshold and H (x, y) : R2 → R is Homotopy link function with Lipschitz constant which has been used to couple (10) and (11).

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To analyze the complete synchronization for drive-response system (10)–(11) we will use some qualitative methods which have been used frequently to detect chaos. In order to understand some dynamical behaviors of system we picked a single parameter r , which demonstrates how dependence is the dynamics of the systems on this certain parameter. To find Lipschitz synchronization threshold of system, we apply mean phase difference and mean amplitude difference. We use these two types of synchronizations for drive-response population model (10)–(11) which have been coupled using the proposed Homotopy link function with Lipschitz constant. We assume the oscillations of the drive-response system (10)–(11) would be synchronized if their mean phase differences coincide repeatedly and they have the same mean amplitude difference. These two types of synchronization have been used widely in science and engineering [23–28]. Definition 3.1 We define two systems are in phase synchronization if they have equivalent mean phase. In other word, if they have a constant difference in phase. We define mean phase for two oscillators as n ≡ ||xn+1 − xn ||

⇒ < τ >=

Nτ

n=1

Nτ

n

(13)

where, Nτ is the number of cycles within a time τ . To put it another way, for two non-identical oscillators, phase synchronization occurs when their phases evolve in synchrony but their amplitude remain unsynchronized. Definition 3.2 We define two systems are in amplitude synchronization if they have identical mean amplitude. We define mean amplitude for two oscillators as the following < A >=

Nτ

n=1

Nτ

An

(14)

Remark 5 Note that the mean amplitude synchronization and mean phase synchronization are qualitatively similar with each other. Figures 1, 6, 11, and 16 display the mean phase difference, i.e. | R2 −  R1 | and the mean amplitude difference, i.e. |A R2 − A R1 | for different growth rates for coupled Ricker model (10) and (11). As we can see, using these two tools, we can numerically find the Lipschitz synchronization threshold that after that value the systems (10) and (11) lose the phase synchronization and amplitude synchronization properties.

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Fig. 1 The mean phase difference and the mean amplitude difference for drive-response system (10)–(11) when r = 2.9

Fig. 2 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.05 and r = 2.9, drive system (red color) and response system (black color)

The Poincare section and power spectrum of drive-response system (10)–(11) have been displayed in Figs. 2, 4, 7, 9, 12, 14, 17, and 19 for different growth rate r and Lipschitz synchronization threshold s. Basically, the Poincare section can be obtained by sampling the phase portrait and it helps to simplify the complicated dynamics. To interpret these results, periodic behaviors may be displayed with a fixed point in Poincare section and also chaotic dynamics can be recognized by set of distinct points in Poincare map. In addition, we can use the frequency spectra to detect the wideband chaotic and periodic signals.

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Fig. 3 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.05 and r = 2.9, drive system (blue color) and response system (red color)

Fig. 4 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.95 and r = 2.9, drive system (red color) and response system (black color)

Finally, we have done time series analysis with different different growth rate r and Lipschitz synchronization threshold s to demonstrate the occurrence of unpredictable, irregular and chaotic oscillations. See Figs. 3, 5, 8, 10, 13, 15, 18, 20.

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Fig. 5 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.95 and r = 2.9, drive system (blue color) and response system (red color)

Fig. 6 The mean phase difference and the mean amplitude difference for drive-response system (10)–(11) when r = 3

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Fig. 7 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.05 and r = 3, drive system (red color) and response system (black color)

Fig. 8 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.05 and r = 3, drive system (blue color) and response system (red color)

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Fig. 9 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.95 and r = 3, drive system (red color) and response system (black color)

Fig. 10 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.95 and r = 3, drive system (blue color) and response system (red color)

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Fig. 11 The mean phase difference and the mean amplitude difference for drive-response system (10)–(11) when r = 3.5

Fig. 12 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.05 and r = 3.5, drive system (red color) and response system (black color)

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Fig. 13 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.05 and r = 3.5, drive system (blue color) and response system (red color)

Fig. 14 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.95 and r = 3.5, drive system (red color) and response system (black color)

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Fig. 15 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.95 and r = 3.5, drive system (blue color) and response system (red color)

Fig. 16 The mean phase difference and the mean amplitude difference for drive-response system (10)–(11) when r = 3.9

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Fig. 17 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.05 and r = 3.9, drive system (red color) and response system (black color)

Fig. 18 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.05 and r = 3.9, drive system (blue color) and response system (red color)

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Fig. 19 Poincare section and Spectrum for Ricker model and its coupled with corresponding error for s = 0.95 and r = 3.9, drive system (red color) and response system (black color)

Fig. 20 Evolution of host population x and its coupled y in time with two different initial conditions for drive-response system (10)–(11) when s = 0.95 and r = 3.9, drive system (blue color) and response system (red color)

4 Conclusion In this study, we have developed a new drive-response system by defining a Homotopy link function with Lipschitz threshold s to map the orbits of the drive system into the orbits of its associated synchronized system. One of the most important advantages of this type coupling is preserving the same qualitative properties such as stability and periodicity of the original system. Also, using Lipschitz threshold s improved the

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previous study in 2020 [17, 18] to be a generalized framework to couple two discretetime dynamical systems starting from different initial conditions and we applied this coupling method on a one-dimensional population model to get drive-response system (10)–(11). Analytically and numerically, we could obtain the synchronization 1 1 and we proved that for s < s˜ = , we have Lipschitz threshold s˜ = ρA + α ρA + α limn→∞ ||yn+1 − xn+1 || = 0 which means that for the values less than s˜ two systems eventually evolve identically in time. We have shown numerically that for driveresponse system (10)–(11) we have the conditions for complete synchronization if the synchronization Lipschitz threshold has smaller values. However, when we changed the values of Lipschitz threshold s to be larger, the response system (11) is not completely synchronized with its original drive system (10). We also discovered numerically that in chaotic regime with increasing the Lipschitz threshold, we can still preserve the qualitative behaviors of both systems even though we do not have complete synchronization between the solutions of drive and response system (10)– (11). We have also applied two other types of synchronization, mean phase and mean amplitude synchronization to find numerically the Lipschitz threshold for different growth rate r in chaotic regime. Finally, qualitative dynamical systems tools such as Poincare section and Spectrum helped to detect the chaotic dynamics and chaotic signals for different synchronization Lipschitz threshold s and different growth rate r. Acknowledgements The great comments provided by Dr. Saber Elaydi in ICDEA 2021 and during reviewing process are greatly appreciated.

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Solving Third-Order Linear Recurrence Relations with Applications to Number Theory and Combinatorics Armen G. Bagdasaryan

Abstract In this paper we develop a novel matrix method for solving linear recurrence relations and present explicit formulae for the general solution of the thirdorder linear homogeneous recurrence relations with variable coefficients. We obtain a summatory formula for the general solution of the recurrence relation in the special case. We review some known results and then consider some particular cases of the recurrence and examples with applications to combinatorics, especially to number sequences and polynomials. Finally, we briefly discuss further generalization of the method for higher order linear recurrence relations. Keywords Linear recurrence relations · General solution · Matrix method · Variable coefficients · Modified companion matrix · Special sequences of numbers and polynomials · Summatory formula · Fibonacci numbers · Tribonacci numbers Mathematics Subject Classification (2010) 11B37, 39A06, 11B83, 39A05, 39A10, 11B39, 11C20

1 Introduction Recurrence sequences have been a central part of number theory and combinatorics for many years. Many number sequences are defined as linear recurrences [1–5], e.g. Fibonacci, Lucas, and Tribonacci numbers and their generalizations, FibonacciNarayana numbers, Pell-Padovan numbers. The linear recurrence sequences have been extensively studied [1, 6–9] and solutions have been basically obtained using generating functions, shift operators, or matrix methods. Recurrence relations (or difference equations) play an important role in the explicit solution of huge number of problems, originating in different areas of science—physics (e.g. one-dimensional A. G. Bagdasaryan (B) Department of Mathematics, College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_5

97

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Ising models, quantum harmonic oscillator, applications to orthogonal polynomials), biology (epidemic models, population dynamics models), with perhaps the most famous example of Fibonacci number sequence, which is also linked to data structures and search algorithms, various optimality problems, and medicine (tumor growth models) [6, 8, 10–14]. In particular, linear recurrences have found important applications in economics (price fluctuation and market equilibrium), number theory and combinatorics (special numbers and polynomials, graph theory), and in other areas of discrete mathematics (e.g. computations, where recurrence relations are regarded as one of the main mathematical tool). The advantage of using recurrence relations, that justifies their wide applicability, comes from the fact that one can employ some of their known properties or techniques, such as generating functions, forward/backward shift operators, companion matrices, characteristic polynomials, etc. Let k be a natural number and let a0 (x), a1 (x), a2 (x), . . . , ak (x) be some given functions. Then linear difference equations are usually written in the form k 

a j (x) f (k − j + x) = g(x),

(1)

j=0

where f (x) is an unknown function of a discrete argument x ∈ Z (or N ∪ {0}) with values in C, and ak (x) ≡ 0. The solution of Eq. (1) is a function f (x) satisfying the Eq. (1), with given coefficients a j (x) and function g(x) for x ∈ N ∪ {0}, and satisfying some initial conditions, f (0) = f 0 , f (1) = f 1 , . . . , f (k − 1) = f k−1 , which uniquely define each solution. It is well-known that there is no general procedure for solving recurrence relations, that is why it is some sort of an art. In this paper, we consider Eq. (1) when k = 3, g(x) ≡ 0, and a0 (x) ≡ 1, that is, we are interested in third-order linear homogeneous difference equation f (x + 3) + a1 (x) f (x + 2) + a2 (x) f (x + 1) + a3 (x) f (x) = 0. Setting x + 3 = n and a1 (x) = pn , a2 (x) = qn , a3 (x) = rn , we get the form which is more commonly used in number theory and combinatorics. Thus, we will discuss the following recurrence relation f n + pn f n−1 + qn f n−2 + rn f n−3 = 0,

(2)

where pn , qn , and rn are given variable complex coefficients, rn = 0, and n ∈ N ∪ {0}, and we also set f n := f (n). The third-order linear Eq. (2) and general equation of kth-order, but with constant coefficients, have been studied, for example, in [15–19], in which some properties and solutions to equations with constant coefficients have been proposed. The linear homogeneous difference equations with variable coefficients have been considered in [20–23], where the general solutions for third-order and then kth-order equations have been presented. However, most of the papers in fact deal with equations having

Solving Third-Order Linear Recurrence Relations with Applications …

99

constant coefficients and propose solutions to this kind of equations which are mainly based on the method of generating functions, matrix methods, or shift operators. The aim of this paper is to propose a new matrix method for solving linear recurrence relations (2) and present explicit formulae for the general solution, the sequence f n , of third-order linear homogeneous recurrence relations with variable coefficients. For this purpose, we introduce a new companion matrix—the modification of the companion matrix. The solution seems to be more elegant and simple compared to other works. We also obtain a summatory formula for the general solution of the recurrence relation in a special case that interestingly involves Fibonacci numbers. We also consider some particular cases of the recurrence, and examples with applications to combinatorics, especially to number sequences and polynomials, will be discussed.

2 Previous Works In this section we recall some approaches to solving kth-order linear recurrence relations with variable coefficients that differ from the methods mentioned in the introduction. The solutions of the linear difference equations presented in [22] are based on the combinatorial approach and utilize the combinatorial properties of the indices of the coefficients of the difference equation. Consider nth-order difference equation yk+n =

n 

ak, j yk+n− j + xk+n , k ≥ 1, n ≥ 2

(3)

j=1

with variable complex coefficients ak, j , complex term xk+n and initial values y1 , . . . , yn . Proposition 1 ([22]) The solution of difference Eq. (3) with initial values y1 , . . . , yn is given by yk+n =

n 

dk, j yn+1− j +

j=1

where

dk, j xn+1− j + xk+n , k ≥ 1

j=2−k





r =1

(1 ,...,r ) 1≤1 ,...,r ≤N r ≥ j 1 +2 +···+r =k+ j−1

k+ j−1

dk, j =

0 

for j = 2 − k, . . . , n and k ≥ 1.



r 

m=1

 ak+m −mn=1 n , m

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It is shown that for ak,n = 0, k ≥ 1, the solutions yk+n = dk, j , j = 1, . . . , n; k ≥ −(n − 1), are linearly independent. The above result is obtained from the solution of a difference equation, of which the Eq. (3) is a special case, namely yk =

k−1 

bk,i yi + xk , k ≥ 1

(4)

i=1

with variable complex coefficients bk,1 , bk,2 , . . . , bk,k−1 , and complex forcing term xk . Proposition 2 ([22]) The solution of difference Eq. (4) is given by yk =

k−1 

ck,i xi + xk , k ≥ 1

i=1

where ck, j = bk,i +

k−i 



j=2

(1 ,..., j ) 1 ,..., j ≥1 1 +2 +···+ j =k−i

 bk,k−1

j 

  m−1 bk−n=1 n , k− m n=1 n

m=2

for i = 1, 2, . . . , k − 1, k ≥ 2. The solutions obtained in Propositions 1 and 2 demonstrate beautiful combinatorial constructions. However, the computations of dk, j and ck, j can be tedious as it contains a laborius part of finding integer partitions and requires summation over integer partitions. The solution of the difference Eq. (4) results in the solution of the nth-order equation, which, in turn, provides expressions for the product of companion matrices and the positive integral powers of a companion matrix. In [23], a uniform linear algebraic approach to solve linear nth-order difference equations with variable coefficients is used. The difference equation of the nth-order with variable coefficients is considered k 

p(k, i) yi = f (k),

p(k, k) = 0, k = 1, 2, . . .

(5)

i=1

which is commonly occuring difference equation, where f and p’s are known functions and whose solution is a sequence yk (k = 1, 2, . . .). Lemma 1 ([23]) The following statements hold true. (a) Let AX = B be a lower triangular system of linear equations with A = (ai j )n×n , aii = 0, X = [x1 , x2 , . . . , xn ]T and B = [b1 , b2 , . . . , bn ]T . Let Ck , k ≤ n, be the submatrix formed by the first k rows and k columns of the n × n matrix

Solving Third-Order Linear Recurrence Relations with Applications …

⎡

b1 b2 .. .

a11 a21 .. .

0 a22 .. .

··· ··· .. .

0 0 .. .

101

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ C =⎢ ⎥, ⎥ ⎢ ⎣ bn−1 an−1,1 an−1,2 · · · an−1,n−1 ⎦ bn an,1 an,2 · · · an,n−1 the last matrix being obtained by augmenting B as the first column with the first n − 1 columns of A. Then, for k = 1, 2, . . . , n (−1)k−1 xk = k det(Ck ). i=1 aii (b) Let AX = B be an upper triangular system of equations, C  be the matrix formed by augmenting the last n − 1 columns of A with B as the last column, and Ck be the submatrix formed by the last k rows and k columns of C  . Then, for k = 1, 2, . . . , n (−1)k−1 xn−k+1 = n det(Ck ). a ii i=n−k+1 Using the above auxiliary lemma, the author obtains the solution of Eq. (5), where each term of the solution sequence is given in the form of determinants of submatrices of a single solution matrix. Theorem 1 ([23]) Let R be the infinite matrix ⎡

f (1) p(1, 1) 0 ⎢ f (2) p(2, 1) p(2, 2) R=⎣ .. .. .. . . .

⎤ 0 ··· 0 · · ·⎥ ⎦ .. .

and Rn be the submatrix of R formed by its first n rows and n columns, where the p’s and f refer to Eq. (5). Then the solution of Eq. (5) is (−1)n−1 det (Rn ) . yn = n i=1 p(i, i) The obtained solution is quite nice and seems to be simple, though it requires the numerical evaluation of higher order determinants which, in general, can be cumbersome. Analogously, the similar statement is obtained for the solution of an initial value problem involving an nth-order linear difference equation with variable coefficients n  i=0

q(k, i + k) yi+k = g(k),

q(k, n + k) = 0, k = 1, 2, . . .

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along with the initial conditions y j = c j , j = 1, 2, . . . , n. As applications, the representations in the form of determinants of Toeplitz or Toeplitz type matrices for Fibonacci numbers and for the number of derangements are derived. Finally, let us consider the relationship between the Riemann zeta function and difference equations, without which this section would not be complete, even though it is related to the second-order difference equations. This reveals itself in the proof of irrationality of ζ(3) by Apéry, which is the special case of the general conjecture claiming that ζ(2k + 1), k = 1, 2, . . ., are irrational. Consider the Riemann zeta function ζ(s) =

∞ 

n −s , (s) > 1.

n=1

In the proof of irrationality of ζ(3) Apéry [24] showed that if x(n), x(0) = 0, x(1) = 6 and y(n), y(0) = 1, y(1) = 5 are two solutions of the second order difference equation n 3 u(n) − (34n 3 − 51n 2 + 27n − 5)u(n − 1) + (n − 1)3 u(n − 2) = 0 then lim

n→∞

x(n) = ζ(3), y(n)

where x(n) is a rational number, and y(n) is a positive integer, and because of the fast convergence it follows that the value of ζ(3) must be irrational. Moreover, consider the partial sums of the Riemann zeta function, the truncated zeta function n  1 ζn (s) = . js j=1 Then, using continued fraction expansion of ζ(2k + 1) [4, 6], it can be readily shown that the truncated zeta function ζn (2k + 1) satisfies the difference equation  u(n + 2) − 1 +



n+1 n+2

2k+1 

 u(n + 1) +

n+1 n+2

2k+1 u(n) = 0.

Of great interest and importance for the irrationality and other problems is finding other recurrence relations in line with the above for which ζ(s) could be a solution or a limit of ratio of two sequences. Concluding this section, we would like to mention the recurrence relation [25, 26] that arises in the study of Dedekind’s eta function

Solving Third-Order Linear Recurrence Relations with Applications …

η(τ ) = q 1/24

103

∞    1 − q n , q = e2πiτ . n=1

The polynomials Pn (x) of degree n, P1 (x) = x, Pn (1) = pn are the partition numbers [27], that appear as the coefficients in the infinite series expansion of the powers of η(τ ) ∞ ∞    −x 1 − qn =1+ Pn (x)q n , x ∈ C q x/24 (η(τ ))−x = n=1

n=1

satisfy the following recurrence relation x Pn (x) = n

 σ(n) +

n−1 

 σ(k)Pn−k (x) ,

k=1

 where σ(k) = d|k d is the sum of divisors function, and in turn the polynomials Pn (x) satisfy the third-order recurrence relation [26].

3 Preliminaries Let p(x) = x n + a1 x n−1 + a2 x n−2 + · · · + an−1 x + an be a polynomial with coefficients over an arbitrary field F. As is known, the matrix ⎡

⎤ 0 −an 0 −an−1 ⎥ ⎥ 0 −an−2 ⎥ ⎥ .. .. ⎥ . . ⎦ 0 0 · · · 1 −a1

0 ⎢1 ⎢ ⎢ C[ p(x)] = ⎢ 0 ⎢ .. ⎣.

0 0 1 .. .

··· ··· ··· .. .

has the property that det (xI − C) = p(x). The matrix C, or some of its modifications, is being called companion matrix of the polynomial p(x) since its characteristic polynomial is p(x). Consider the kth-order linear recurrence relation f n = c1 f n−1 + c2 f n−2 + · · · + ck f n−k , n ≥ k, c j ∈ R(C).

(6)

Since each term of a kth-order recurrence is determined by the k preceding terms, it is useful to think of it as a function on k-tuples of consecutive terms. Because this

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function is linear, it can be represented by a matrix. The k × k companion matrix C of the recurrence relation (6) is ⎡

· · · ck−1 ··· 0 ··· 0 . .. . .. 0 0 ··· 1

c1 ⎢ 1 ⎢ ⎢ C=⎢ 0 ⎢ .. ⎣ .

c2 0 1 .. .

⎤ ck 0⎥ ⎥ 0⎥ ⎥. .. ⎥ . ⎦ 0

It has the following well known property: ⎡ ⎤ ⎡ ⎤ ⎤n ⎡ c1 c2 · · · ck−1 ck f n+k−1 f k−1 ⎢ 1 0 · · · 0 0 ⎥ ⎢ f k−2 ⎥ ⎢ f n+k−2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ .. .. . . .. ⎥ . . .. ⎥ ⎢ .. ⎥ = ⎢ ⎣ . . . .. . ⎦ . ⎦ ⎣ . ⎦ ⎣ f0 fn 0 0 ··· 1 0 Modified Companion Matrix. Let the recursive sequence { f n }n≥0 be defined by the third-order linear recurrence f n = p f n−1 + q f n−2 + r f n−3 , n ≥ 3, r = 0 and the initial values f 0 = a, f 1 = b and f 2 = c. The regular companion matrix can usually be written either in a “row” form or “column” form as follows: ⎡ ⎤ ⎡ ⎤ p q r 0 0 p C = ⎣ 1 0 0⎦ or C = ⎣ 1 0 q⎦ . 0 1 0 0 1 r We introduce the modified “companion” matrix in the column form ⎡

⎤ 0 0 r K = ⎣ p 0 r⎦ 0 q r that is used further in the text to obtain the main results of this paper. The matrix can also be obtained using C by means of the transformation ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ p 0 r ( p−q) p 0 0 r 0 0 p ⎢ ⎥ ⎣ 1 0 q ⎦ ⎢ 0 q r ( p−r ) ⎥ = ⎣ p 0 r ⎦ , p = 0. p ⎦ ⎣ 0 q r 0 1 r r 0 0 p

Solving Third-Order Linear Recurrence Relations with Applications …

105

Notations. Let us introduce the notations that we use throughout this paper. The modified companion matrix of the recurrence relation with variable coefficients f n + pn f n−1 + qn f n−2 + rn f n−3 = 0 is denoted by

(7)

⎡

⎤ 0 0 rn Kn = ⎣ pn 0 rn ⎦ , 0 qn r n

n ≥ 1, rn = 0.

Also we denote Mn = K1 K2 · · · Kn ⎡ ⎤ 1 J0 = ⎣ 1⎦ , 1

and

⎡

⎤ αn Jn = Mn J0 = ⎣ βn ⎦ γn

(8)

are the column vectors.

4 Main Results In this section, we present the main results of this paper. To obtain our results, we make use of the following auxiliary statement. Proposition 3 ([7], Theorem 2, Chap. 5) Let f 1 (x), f 2 (x), . . . , f k (x) be solutions of the kth-order homogeneous difference equation f (x + k) + p1 (x) f (x + k − 1) + · · · + pk (x) f (x) = 0.

(9)

Suppose that determinant   f 1 (0)   f (0) D[ f 1 , f 2 , . . . , f k ] =  2  ...  f k (0)

f 1 (1) f 2 (1) ... f k (1)

 . . . f 1 (k − 1) . . . f 2 (k − 1) = 0. ... . . .  . . . f k (k − 1)

Then the general solution of linear homogeneous Eq. (9) is given by the function f (x) = c1 f 1 (x) + c2 f 2 (x) + · · · + ck f k (x), where c1 , c2 , . . . , ck are arbitrary constants.

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4.1 General Solution Theorem 2 Let c1 , c2 , c3 ∈ R be arbitrary constants, n ∈ N ∪ {0} be a nonnegative integer, and let Kn , Mn , Jn be defined as above. Moreover, assume that p1 q1 p2 = 0. Then the linear combination f n = c1 αn + c2 βn + c3 γn

(10)

represents the general solution to the third-order linear homogeneous recurrence relation with variable coefficients f n = rn f n−1 + qn rn−1 f n−2 + rn−2 qn−1 pn f n−3 ,

n ≥ 3.

(11)

Proof We begin with the chain of equalities ⎡

⎤ rn Jn = Mn J0 = Mn−1 Kn J0 = Mn−1 ⎣ pn + rn ⎦ qn + r n (12) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 = rn Mn−1 ⎣ 1⎦ + pn Mn−1 ⎣ 1⎦ + qn Mn−1 ⎣ 0⎦ , ∀ n ≥ 3. 1 0 1 Since ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 Mn−1 ⎣ 1⎦ = Jn−1 , Mn−1 ⎣ 1⎦ = qn−1 rn−2 Jn−3 , Mn−1 ⎣ 0⎦ = rn−1 Jn−2 , 1 0 1 we get the formula Jn = rn Jn−1 + rn−1 qn Jn−2 + rn−2 qn−1 pn Jn−3 . From the formula (13) it follows that the functions f n1 = αn , satisfy the Eq. (11). Consider the determinant

f n2 = βn ,

f n3 = γn

(13)

Solving Third-Order Linear Recurrence Relations with Applications …

J0 J1 J2      α(0) α(1) α(2)   D[ αn , βn , γn ] =  β(0) β(1) β(2)  γ(0) γ(1) γ(2)    1  r1 r1 (q2 + r2 )   r1 (q2 + r2 ) + p1r2  . =  1 p1 + r1  1 q1 + r1 r1 (q2 + r2 ) + q1 (r2 + p2 )

107

(14)

Since D[ αn , βn , γn ] = p1 q1 p2 = 0, from Proposition 3 it follows that the linear combination (10) is the general solution of Eq. (11).  Remark 1 With the appropriate notation, if pn = 0, qn = 0, rn = 0, n = 1, 2, 3 . . ., the relation (11) f n = rn f n−1 + qn rn−1 f n−2 + rn−2 qn−1 pn f n−3 can be reduced to the recurrence relation (7) f n + pn f n−1 + qn f n−2 + rn f n−3 = 0, where pn , qn , rn are given variable complex coefficients and n ∈ N ∪ {0}. Hence, the determinant of the companion matrix C11 of (11) ⎛⎡

⎤⎞ rn qn rn−1 rn−2 qn−1 pn ⎦⎠ = rn−2 qn−1 pn = 0, 0 0 det(C11 ) = det ⎝⎣ 1 0 1 0 −1 exists and one can find the required transformation from the matrix so the inverse C11 equation C11 X = C7 , where C7 is the companion matrix of (7):

⎡ −1 X = C11 C7 = ⎣

⎡ =⎣

0 0

1 0

1 rn−2 qn−1 pn

n − rn−2 qrn−1 pn

1 0

pn −rn rn−2 qn−1 pn

0 1

qn −rn−1 qn rn−2 qn−1 pn

⎤⎡

⎤ pn q n r n ⎦⎣ 1 0 0⎦ rn−1 qn − rn−2 0 1 0 qn−1 pn 0 1

0 0 rn rn−2 qn−1 pn

⎤ ⎦.

Alternatively, it can be found by using the matrix equation C11 C7 = B, where B is the matrix with the determinant

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⎛⎡

⎤⎞ rn−1 qn + rn pn rn−2 qn−1 pn + rn qn rn2 pn qn rn ⎦⎠ det(B) = det ⎝⎣ 1 0 0 = rn−2 qn−1 pn rn = 0, so that B −1 is the transformation of C11 into C7−1 , whence C7 is easily computed. For the variable complex coefficients of the third-order linear homogeneous recurrence relation, we have the following Theorem 3 For the ordered complex matrix product, there holds ⎡

⎤⎡ ⎤ rn+3 qn+3 rn+2 rn+1 qn+2 pn+3 rn+2 qn+2 rn+1 rn qn+1 pn+2 ⎣ 1 ⎦⎣ 1 ⎦··· 0 0 0 0 0 1 0 0 1 0 ⎤ ⎡ ⎤ ⎡ r 3 q 3 r 2 r 1 q 2 p3 αn+3 βn+3 γn+3 0 ⎦ = ⎣ αn+2 βn+2 γn+2 ⎦ , ···⎣ 1 0 αn+1 βn+1 γn+1 0 1 0 where n ≥ 0 and αn , βn , and γn are as defined above in (8). Proof The recurrence relation with initial values f 0 , f 1 , f 2 can be written as ⎡

⎤ ⎡ ⎤ ⎤⎡ f n+3 rn+3 qn+3 rn+2 rn+1 qn+2 pn+3 f n+2 ⎣ f n+2 ⎦ = ⎣ 1 ⎦ ⎣ f n+1 ⎦ , 0 0 f n+1 fn 0 1 0

n ≥ 0.

Therefore, we have ⎤ ⎡ ⎡ ⎤⎡ ⎤ rn+3 qn+3 rn+2 rn+1 qn+2 pn+3 f n+3 rn+2 qn+2 rn+1 rn qn+1 pn+2 ⎣ f n+2 ⎦ = ⎣ 1 ⎦⎣ 1 ⎦ 0 0 0 0 f n+1 0 1 0 0 1 0 ⎡ ⎤⎡ ⎤ r 3 q 3 r 2 r 1 q 2 p3 f2 0 ⎦ ⎣ f1 ⎦ , ···⎣ 1 0 n ≥ 0. f0 0 1 0 From the other side, from Eq. (10) in Theorem 2, setting c1 = f 2 , c2 = f 1 , and c3 = f 0 , we have ⎡

⎤ ⎡ ⎤⎡ ⎤ f n+3 αn+3 βn+3 γn+3 f2 ⎣ f n+2 ⎦ = ⎣ αn+2 βn+2 γn+2 ⎦ ⎣ f 1 ⎦ , f n+1 αn+1 βn+1 γn+1 f0

n ≥ 0.

Then, comparing the last two equalities, we get the required formula. Theorem 4 For pn = 0, qn = 0, rn = 0, the recurrence relation (11)



Solving Third-Order Linear Recurrence Relations with Applications …

f n = rn f n−1 + qn rn−1 f n−2 + rn−2 qn−1 pn f n−3 ,

109

n ≥ 3.

has three linearly independent solutions f n = αn , f n = βn , f n = γn , where αn , βn , and γn are given by the formula (8). Proof From the proof of Theorem 2 it follows that f n = αn , f n = βn , f n = γn are solutions of the recurrence relation (11) for n ≥ 0. The Casoratian of the three sequences {αn }, {βn }, and {γn }, n ≥ 0, is given by the determinant      αn+2 βn+2 γn+2   αn βn γn      αn+1 βn+1 γn+1  = −  αn+1 βn+1 γn+1       αn  αn+2 βn+2 γn+2  βn γn  ⎛⎡ ⎤ ⎡ ⎤⎞ rn+2 qn+2 rn+1 rn qn+1 pn+2 r 3 q 3 r 2 r 1 q 2 p3 ⎦···⎣ 1 0 0 0 0 ⎦⎠ = −det ⎝⎣ 1 0 1 0 0 1 0 =−

n 

rn−k+1 qn−k+2 pn−k+3 = 0,

k=1

which follows from Theorem 3 and the assumption that r1 q2 p3 , r2 q3 p4 , . . . , rn qn+1 pn+2 = 0. Therefore αn , βn , and γn are linearly independent sequences, which in turn implies that f n = αn , f n = βn , f n = γn are three linearly independent solutions of the recurrence relation. 

4.2 Summatory Formula We consider the third-order linear homogeneous recurrence relation f n + pn f n−1 + qn f n−2 + rn f n−3 = 0, n ∈ N ∪ {0} in the special case when qn = 1

and

rn = 1

and derive the summatory formula for Jn . We will need the following lemmas. Lemma 2 Let Pn , Q n , and Rn , n ≥ 1, be square matrices of the same order and let Rn = Pn + Q n . Then we have

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R1 R2 · · · Rn = P1 P2 · · · Pn + n 



k=1

1≤i 1 1; and R2 = {(x, y) : x, y ∈ [A, A1 ]} for 0 < A < 1. The strategy consists in analyzing the behaviour of the solutions outside the invariant regions.

A Survey on Max-Type Difference Equations

133

Also, the oscillatory behaviour of the solutions was established in [24]. For A > 0 we have that every nontrivial solution is strictly oscillatory about the equilibrium x. ¯ Moreover, the semicycles are described as follows: • If A ≤ 1, every positive semicycle (except for possibly the first) has two or three terms. Furthermore, if it has three terms, then the first and the third are equal to the equilibrium x. ¯ • If A > 1, every positive semicycle has at most two terms. • Every negative semicycle has at most two terms. Now, we focus on the behaviour of the solutions outside the first quadrant, that is, at least one of the initial conditions is non-positive. If we wish to characterize solutions with one or both of the initial conditions negative, it suffices to consider only the case where both initial conditions are negative. In this case, some solutions that begin in the third quadrant have subsequences diverging to negative infinity and other subsequences converging to zero. Theorem 1 ([24, Theorem 7]) Let A > 0 and let (xn ) be a solution of Eq. (17) with x−1 = α < 0 and x0 = β < 0. Assume that αβ < 1. For k ≥ 0 it holds x3k = αk β k+1 −→ 0; x3k+1 =

A 1 ; x3k+2 = k k+1 −→ −∞. αβ α β

The rest of solutions in the third quadrant are periodic of period three. Indeed, if A x−1 = α < 0, x0 = β < 0 and αβ ≥ 1, the solution is (xn ) = (. . . , α, β, αβ , α, β, A , . . .). Furthermore, in [12], new results are obtained by considering a sequence αβ of positive real numbers, (An ), with arbitrary positive initial conditions.

4.3 Case k = 2, l = 1: xn+1 = Next, we focus on xn+1 =

max{xn2 , A} xn xn−1

max{xn2 , A} . xn xn−1

(18)

Firstly, if A = 1, the equation is globally periodic with period nine as it was highlighted in Table 1. If A = 1, the solution is not periodic in general. Indeed, in [1], the authors show how a suitable change of variables transforms Eq. (18) into the well-known Gingerbreadman Equation, that is, yn+1 = |yn | − yn−1 + 1. In concrete, yn +1 the change of variables required is xn = A 2 . In particular, the Gingerbreadman Equation is known for being chaotic in certain regions of the plane and stable in others. Nevertheless, every solution is bounded.

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Although the solution is not periodic in general, it can be characterized the 3cycles by describing an invariant region of the first quadrant. In fact, a solution (xn ) of (18) is periodic with period three if and only if either A ≥ 1 and the√ point √ √ (x−1 , x0 ) ∈ S, where S is bounded by the curves x = A, y = A, y = xA ; or A < 1 and x−1 = x0 = 1, that is, 1 is the equilibrium point. Moreover, in [1] a characterization of the 24-cycles is also given.

4.4 A Generalization of the Lyness’ Max-Type Difference Equation Apart from studying Eq. (12) for different values of k and l, we can find in the literature some generalizations involving more delays. For instance, in [54], the authors study the periodicity of the positive solutions of the max equation xn+1

n xi , bn } max{an i=n−k+1 n = , x i=n−k i

(19)

where (an ) and (bn ) are sequences of positive numbers, x−k , x−k+1 , . . . , x0 ∈ (0, ∞) and k ∈ {2, 3, . . .}. In [34], under appropriate assumptions, that involve an auxiliary nonincreasing sequence, the authors establish a generalized invariant and analyze the periodic character of the solutions for some particular cases. For example, in the particular case where an = 1 and bn = A for all n ≥ 0, with A a positive constant, it can be guaranteed the periodic character of the solutions under certain assumptions. In concrete, let us consider xn+1 =

max{

n xi , A} i=n−k+1 , n i=n−k x i

(20)

and let (xn ) be a positive solution of Eq. (20) with initial conditions satisfying the following properties: 0  i=−k+i

xi ≤ A;

−1 

xi ≤ A; xi xi−1 ≥ 1, i = −k + 1, . . . , −1, 0;

i=−k

where k ≥ 2 and x−k , . . . , x0 are real positive numbers. Then (xn ) is periodic with prime period equal to k + 2.

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5 Reciprocal Difference Equation with Maximum In 1996, in view of some results for the equation  1 A , = max , xn xn−1 

xn+1

where the parameter A and the initial conditions are nonzero real numbers, that appeared in a preprint of [2] (see Eq. (22) below), Ladas, [44], proposed a set of open problems and conjectures related to the reciprocal difference equation with maximum of order k + 1, that is,   A0 Ak xn+1 = max , (21) ,..., xn xn−k where the parameters A0 , . . . , Ak are real numbers and the initial conditions x−k , . . . , x0 are nonzero real numbers. The main statement, related to the periodic character of the solutions, which has only been partially solved, is the following conjecture. Conjecture 1 Assume that A0 , . . . , Ak−1 ∈ [0, ∞) and Ak ∈ (0, ∞). Show that every positive solution of Eq. (21) is eventually periodic with period p ∈ {2, 3, . . . , 2(k + 1)}. It is interesting to highlight that the period in Conjecture 1 seems to be determined by the “dominant” parameter, this means that if the coefficient A p is dominating, i.e., if A p > max{A j : j = p, j ∈ {0, . . . , k}}, then every positive solution eventually A becomes a solution of the reciprocal difference equation xn+1 = xn−pp . Thus, the solution is eventually (2 p)-periodic. On the other hand, Szalkai studied (21) in the case of negative coefficients with initial conditions x−k = a0 , x−k+1 = a1 , . . . , x0 = ak , and he established necessary and sufficient conditions for every solution to be eventually periodic and determined the period in terms of the parameters A0 , . . . , Ak (see [71]). Theorem 2 For any k ≥ 0 and Ai < 0, Ak = 0, ai ∈ R (i ≤ k) the following statements are equivalent: • • • •

The sequence (xn ) is periodic. The sequence (xn ) is periodic with period k + 2. Ai = Ak−i for 0 ≤ i ≤ k. The sequence is bounded.

The argument followed in the proof shows that for n ≥ k + 2 the solution consists of positive semicycles of length k + 1, followed by negative semicycles of length 1 (or the other way around). Moreover, in the same paper we find the study of the case where all the coefficients are positive and equal, Ai = A > 0 (i ≤ k), obtaining that

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every solution is (k + 2)-periodic. It is interesting to highlight that in this case, where there is not a dominant coefficient, the prime period is the average mean of the periods of each term of the (2 j + 2)-periodic equation xn+1 = xAn−ii with i ∈ {0, . . . , k}. Another of the conjectures proposed by Ladas was based on the relation between boundedness and eventual periodicity. In fact, he claimed that every bounded solution of (21) is eventually periodic. This result was proved in the positive case in 2008 by Bidwell and Franke, [8]. Indeed, they showed that every bounded solution is eventually periodic for the reciprocal-max difference equation with periodic nonnegative parameters. Afterwards, in 2013, Cranston and Kent, [16], gave sufficient conditions on the parameters’ periods for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded. To deal with the main properties of Eq. (21) we divide the present section into three different parts. In the first two, we study the case of constant and periodic coefficients, respectively. Finally, in the third subsection, we analyse some generalizations by considering powers in the delays.

5.1 Constant Coefficients Firstly, we deal with the order two max-type difference equation  xn+1 = max

 1 A , , xn xn−1

(22)

where the parameter A and the initial conditions are nonzero real numbers. In [2], the authors show that every solution is eventually periodic with period 2, 3 or 4. We synthesize the results in Table 2. In those cases where the result is not direct, the authors make a change of variables to simplify Eq. (22). For example, when A ∈ (0, 1) and the initial conditions are not both negative numbers, the solution is eventually positive, so they set xn = A yn , obtaining the difference equation yn+1 = min{−yn , 1 − yn−1 } which is eventually 2-periodic. They also illustrate with an example that, in the gen  A , with a = 0, every solution is not eventually perieral case, xn+1 = max xan , xn−1 odic. For instance, taking a = −1 and A = −2, the initial conditions x−1 = 1, x0 = 2 1 , . . .}. generate the unbounded sequence {1, 2, − 12 , 2, 22 , − 212 , . . . , 2n , 2n+1 , − 2n+1 In general, they point out that for any negative pair a = A and initial conditions in R \ {0}, the solution generated is unbounded. Now, we generalize Eq. (22) to two arbitrary delays k and m. In [72], Voulov analyzes the periodic character of the positive solutions of  xn+1 = max

A

,

B

xn−k+1 xn−m+1

 ,

(23)

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137

A 0 depending on A, x−1 , x0

A > 0 and x−1 , x0 are not both negative numbers

2 3 4 2 3 4

A ∈ (0, 1) A=1 A ∈ (1, ∞) A ∈ (1, ∞) A=1 A ∈ (0, 1)

A > 0 and x−1 , x0 are both negative numbers

where A, B are arbitrary positive real numbers and k, m are positive integers. He proves that every positive solution is eventually periodic and, even, determines the period in terms of the parameters A, B, k and m. Theorem 3 Let A, B ∈ (0, ∞) and m, k ∈ N. Then there exists a positive integer T such that every positive solution of Eq. (23) is eventually periodic with period T (not necessarily prime). That period is determined as follows: • T = 2k if either A > B or A = B and m = 3k; • T = 2m if either A < B or A = B and k = 3m; • T = k + m if either A = B and neither m = 3k nor k = 3m.

√ The proof of the theorem is based on the change of variables  xn = yn D with 1 a ; or to , yn−m+1 D = max {A, B}. So, Eq. (23) is simplified to yn+1 = max yn−k+1   a 1 yn+1 = max yn−k+1 , with a ∈ (0, 1], depending on the value of D, where , yn−m+1 every positive solution is bounded. The author distinguishes the cases a < 1 and a = 1. Firstly, if a ∈ (0, 1), then every positive solution of Eq. (23) is eventually periodic of period T = 2k. Secondly, for a = 1, if m = 3k, then every positive solution is eventually periodic of period 2k; moreover, yn yn+k = 1 for every n ≥ 5k; finally, if m > k and m = 3k, then every positive solution is eventually periodic of period T = m + k. Notice that the main results from [2, 52] are particular cases of Theorem 3. Later on, [73], Voulov studied the third order reciprocal difference equation with maximum   A B C , (24) , , xn+1 = max xn xn−1 xn−2 where the coefficients A, B, C are nonnegative real numbers with A + B + C > 0, and the initial conditions, x−2 , x−1 , x0 are positive real numbers. He applies similar techniques to those used in the case of order two, and he proves that every solution of Eq. (24) is eventually periodic of (not necessarily prime) period T , where T is defined as follows:

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⎧ ⎪ ⎪ 2, ⎪ ⎪ ⎨ 3, T = 4, ⎪ ⎪ 5, ⎪ ⎪ ⎩ 6,

A > max{B, C}, A = B > C, B > max{A, C} or A = C ≥ B, B = C > A, C > max{A, B}.

The proof developed by Voulov is based on the fact that, in most cases, a positive solution of Eq. (24) eventually verifies an order two difference equation of the form (23). Then, Theorem 3 can be applied. Finally, in 2005, J. Bidwell proved in his Ph.D. dissertation, [7], that Eq. (21) is eventually periodic. Firstly, notice that, in the positive case (the coefficients Ai > 0 for all i = 0, . . . k, and the initial conditions are positive real numbers), solutions always exist, the recurrence is well-defined. We outline the proof followed by Bidwell in the positive case. He starts transforming Eq. (21) into a topologically conjugate system. Let D = max{A0 , . . . , Ak }, Ai = ADi and xi = √xiD , for i = 0, . . . , k. Then, the new system is

xn+1

 = max

i∈[0,k]

Ai

xn−i

 .

Observe that Ai ∈ [0, 1] and there exists j ∈ [0, k] such that A j = 1 and xi > 0 for all i ≥ −k. This yields to the boundedness character of the solutions. Indeed, every solution is bounded away from zero, because xn ∈ c, 1c for all n ≥ 0 with  

c = mini∈[0,k] x−i , x1 . To show that the system is eventually periodic he makes −i another conversion. In concrete, he takes logarithms. Notice that since the initial conditions are bounded away from zero, the new system will also be bounded: yi = ln xi ;

Bi =

 ln(Ai ), if Ai > 0; −∞, if Ai = 0.

Let b = maxi∈[0,k] |y−i |, so the equivalent system is yn+1 = max {Bi − yn−i }, i∈[0,k]

where Bi ∈ [−∞, 0], and there exists j such that B j = 0. Note that yn ∈ [−b, b] for all n ≥ −k. Now, we consider the equivalent dynamical system, log version, f : ! son+kcalled " = (y j )n+k+1 Rk+1 → Rk+1 , f = ( f 1 , . . . , f k+1 ), which verifies f (yi )i=n j=n+1 . Moreover, f is continuous, piecewise linear and ∂∂ yfij | y ∈ {−1, 0, 1} for all i, j ∈ {1, . . . , k + 1}, y ∈ Rk+1 . To prove that f is eventually periodic is equivalent to prove that the initial system is eventually periodic. The log version is considered as a

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dynamical system and is shown that is nonexpansive in L ∞ ,1 that is, || f (x) − f (y)||∞ ≤ ||x − y||∞ . Notice that f being nonexpansive implies that f n is also nonexpansive. For D a compact set, if a function f : D → D is nonexpansive respect to L ∞ , then for any x ∈ D, the ω-limit set2 of x, denoted ω(x; f ), is finite, [75]. Since the log version of the positive case is continuous, the result that the ω-limit set is finite implies that if P = |ω(x; f )|, then all y ∈ ω(x; f ) are periodic with period P. He also works with the opposite cases to the positive one. Notice that the case where all the parameters are negative is covered by [71]. Bidwell proves that, if the solutions of Eq. (21) are well-defined, then the system is equivalent to the positive case and, thus, the solutions are eventually periodic.

5.2 Periodic Coefficients Now, we deal with reciprocal difference equations admitting periodic parameters. We start by studying the difference equation  xn+1 = max

 1 An , , xn xn−1

(25)

where the coefficients (An ) are a periodic sequence of period p. This particular case of the reciprocal difference equation with maximum is also characterized by the eventual periodicity behaviour of its solutions. In Table 3 we sum up the known results related to the topic where we indicate the period in each case. The constant case, p = 1, was treated in the previous section based on [2]. Although, it is worth mentioning that Feuer presents an alternative proof in [26] which will inspire the case p = 4 and p = 5 for An ∈ (1, ∞). In the case where the sequence (An ) is 2-periodic, Briden et al. proved in [10] that every positive solution of Eq. (25) is eventually periodic with period 2, if A0 A1 < 1; 6, if A0 A1 = 1; and 4, if A0 A1 > 1. As a key role in the analysis of the long-term behaviour of the positive solutions, it can be assumed, without loss of generality, that x0 x−1 = 1. The proof distinguishes different cases and the results follow by induction. An alternative proof is given in [20]. The case where the sequence (An ) is p = 3-periodic was studied in [11, 33]. In concrete, the proof of the corresponding results (collected in Table 3) combines auxiliary lemmas, which follow by direct computation and an inductive process. Let Ω represent a measurable Lebesgue subset of Rn of non-zero measure; and let μ be a Lebesgue measure. Then L ∞ is the set of measurable functions, f : Ω → C, such that there exists Ω0 ⊂ Ω, measurable with μ(Ω \ Ω0 ) = 0, and f |Ω0 being bounded. See [57]. 2 Let X be a metric space and let f : X → X be a continuous function. The ω-limit set of x ∈ X is # ω(x, f ) = n∈N { f k (x) : k > n}, where the bar denotes the closure of a set. For further information on ω-limit sets of metric spaces, consult [9]. 1

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Table 3 Eventual periodicity of Eq. (25) p=1

p=3

p=5

An ∈ (0, 1) An = 1 An ∈ (1, ∞) An ∈ (0, 1) An ∈ (1, ∞) Ai+1 < 1 < Ai Other cases An ∈ (0, 1) An ∈ (1, ∞)

2-periodic 3-periodic 4-periodic 2-periodic 12-periodic unbounded solutions 3-periodic 2-periodic 20-periodic

p=2

p=4

p≥6

An ∈ (0, 1) A0 A1 = 1 An ∈ (1, ∞) An ∈ (0, 1) An ∈ (1, ∞)

2-periodic 6-periodic 4-periodic 2-periodic 8-periodic

An ∈ (0, 1)

2-periodic

Moreover, apart from determining the eventual period of the solution for An ∈ (0, 1), x¯ = 1 is the only fixed point. Equation (25) was studied in [26, 27] by taking (An ) as a sequence of real numbers greater than one of period four and five, respectively. In concrete, it is proved that every solution is eventually 8-periodic when (An ) is a 4-periodic sequence and eventually 20-periodic when the coefficients are periodic of period 5. In both cases, the authors follow an inductive process combined with the concepts of right and left semicycle adapted to Eq. (25). Definition 2 A right (left) semicycle  of (25) is a string of terms xl , . . . , xm with An−1 1 for n = l, . . . , m. Furthermore, if l ≥ 1, m ≤ ∞ such that xn = xn−2 xn = xn−1   Al−2 1 m l > 1, xl−1 = xl−2 (xl−1 = xl−3 ) and if m < ∞, xm+1 = x1m xm+1 = xAm−1 . When (An ) is 4-periodic and An ∈ (1, ∞), the results follow by doing a partition of the curve x y = 1 in the phase plane into three sections. If x N x N −1 = 1, then the zones are: (i) x NA−1 ≤ x N ≤ Ax N −1 ; (ii) x N > Ax N −1 ; and (iii) x N < x NA−1 . By considering this partition, it is easy to see that eventually every solution will have a term in (i). Thus, taking into account some auxiliary lemmas related to the length of right and left semicycles, it can be guaranteed that every solution is eventually periodic with period 8. In a similar way, when (An ) is 5-periodic and An ∈ (1, ∞), in [27], the authors prove that every solution is eventually periodic with period 20. In 2005, Chen, [14], proves that every positive solution of Eq. (25) is eventually periodic with period 2 when (An ) is a sequence of period p and An ∈ (0, 1). The clue is that every solution eventually ends verifying xn+1 = x1n for all n ≥ n 0 . To prove this claim, let a0 < a1 < . . . < al be distinct values of the periodic sequence (An ) of positive real numbers with period p ( p ≥ 2), l ≤ p − 1, a0 = min0≤n≤ p−1 An and al = max0≤n≤ p−1 An . For m = 0, 1, . . . , l − 1, he introduces the interval Im = √ √ [ am , am+1 ]. Then, the author studies the evolution of the sequence along those intervals Im to guarantee the eventual periodicity. For more information related to other particular reciprocal difference equations with maximum and periodic coefficients, see [55, 70, 74] or [41].

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In the literature we can also find results related with the general case  xn+1 = max

i∈{0,...,k}

 (Ai )n , xn−i

(26)

with positive initial conditions x−k , . . . , x0 and (Ai )n nonnegative and periodic with the i-th component having period pi . Moreover, it may be assumed that for each n there exists i ∈ {0, . . . , k} such that (Ai )n > 0. For instance, Bidwell and Franke, [8], show that for any size k every solution that is bounded must be eventually periodic. To prove that fact, they use the same structure followed to show that the reciprocal max-type difference equation is eventually periodic (the proof was developed at the end of Sect. 5.1). In [16], the authors establish sufficient conditions on the periods pi of the sequences (Ai )n to guarantee the boundedness and persistent character of every positive solution of (26).

5.3 Powers in the Denominator A natural generalization of the reciprocal difference equation with maximum arises by considering powers of the delays,  xn+1 = max

 A0 A1 Ak , , , . . . , αk α1 xnα0 xn−1 xn−k

(27)

where Ai ≥ 0, and αi ≥ 0, for i = 0, . . . , k. A particular case of Eq. (27) appeared in [78], namely   1 A , xn+1 = max α , xn xn−1 where 0 < α < 1 and A > 0. The authors prove that every positive solution converges to x¯ = 1 if A ≤ 1, or is eventually periodic with period 4 if A > 1. Further research yields to more general results which concern the global stability of the solutions. In concrete, Stevi´c, [66], considers the difference equation  xn+1 = max

A1

,

A2

α1 α2 xn− p1 x n− p2

,...,

Ak

αk xn− pk

 ,

(28)

where k ∈ N, pi are natural numbers such that 1 ≤ p1 < · · · < pk , Ai > 0, αi ∈ (−1, 1), i = 1, .. . , k, and proves that every positive solution converges to 1 α

max1≤i≤k Ai 1

+1

.

It should be highlighted that the majority of the results related to the convergence of the positive solutions of Eq. (27) are based on the same technique: firstly, apply

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a change of variables that modifies the equation of interest into another max-type equation whose solutions are convergent to zero, and second, prove the convergence, based on the following lemma, see [62]. Lemma 1 Let (an ) be a sequence of non-negative numbers such that an+k ≤ q · max{an+k−1 , an+k−2 , . . . , an }, for n ∈ N, q ≥ 0 and k ∈ N fixed. Then, there is an M > 0 such that an ≤ √ M( k q)n , n ∈ N. Similar max-type difference inequalities are useful to see that the positive solution of certain equations converges to zero (see [67]). Furthermore, in [47], the authors study the nonautonomous max-type difference equation with two delays xn = max

fn

B

α , β xn−k xn−m

,

(29)

where k, m ∈ N, α, β ∈ R are fixed and ( f n ) is a positive sequence with a finite limit. They present some results which concern the global attractivity of positive solutions under some conditions. In concrete, they show that the positive solutions converge to max{(limn→∞ f n )1/(α+1) , B 1/(β+1) }. The reader interested in other particular cases of Eq. (27) is referred to [29–31, 69] or [65].

6 Other Classes of Max-Type Difference Equations In the present section we will deal with other max-type difference equations which are not directly related to the Lyness max-type difference equation or to the reciprocal difference equation with maximum. For example, different authors focus on the investigation of the behaviour of solutions of the difference equation  An = max , xn−m , xn−k 

xn+1

(30)

in terms of the sequence (An ) and k, m ∈ N ∪ {0}. For instance, in [22], we find Eq. (30) for k = 0, m = 1 and (An ) a two-periodic sequence with max{A0 , A1 } ≥ 0, that is,   An (31) xn+1 = max , xn−1 . xn

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Depending on the values of A0 , A1 , x−1 and x0 , three different cases are analyzed. The authors use the fact that solutions of Eq. (31) are eventually two-periodic whenever they are eventually positive or eventually negative. This allows them to prove that if A0 or A1 is equal to zero; or A0 = 0 = A1 and A0 > 0 or A1 > 0; or (An ) is a two periodic sequence such that A0 and A1 are not both negative, then every welldefined solution of Eq. (31) is eventually periodic with period two. However, when A0 < 0 and A1 < 0 we can find solutions of Eq. (31) which are not periodic. Moreover, they give explicit formulas from which the existence of unbounded solutions follows easily. Next, we deal with the boundedness character of the positive solutions of   xn−k+1 , xn+1 = max c, xn−m+1

(32)

where k, m ∈ N with gcd(k, m) = 1 and c is a positive constant. Berenhaut et al. prove in [6] that if c ≥ 1, then every solution of Eq. (32) is bounded, and if c ∈ (0, 1) and k is even, then there exist positive unbounded solutions. For the case c ∈ (0, 1) and k odd, they consider the related equation z n+1 = max{−1, z n−k+1 − z n−m+1 }

(33)

by making the change of variables xn = c1zn , and show that every integer solution is eventually periodic. Note that the case k = m is trivial, since in this case xn = max{c, 1}. Related to the boundedness character of the solutions of Eq. (33) with k odd, it has been conjectured that every solution is bounded. In [63], Stevi´c studies the boundedness and global attraction for the positive solutions of the difference equation  p  xn , xn+1 = max c, p xn−1 with p, c ∈ (0, ∞). It is shown that: there exist unbounded solutions whenever p ≥ 4; all positive solutions are bounded when p ∈ (0, 4); every positive solution is eventually equal to 1 when p ∈ (0, 4) and c ≥ 1; and all positive solutions converge to 1 whenever p, c ∈ (0, 1). Let us mention that, for the case p ≥ 4, the author transforms the equation by the change yn = ln(xn ) and studies the roots of the characteristic polynomial linked to the new equation. By considering more powers in the delays, [64], we find  p  xn , xn+1 = max A, r xn−2

(34)

where the parameters A and r are positive and p is a nonnegative real number. In the study of the boundedness character of positive solutions of Eq. (34), the author

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Table 4 Boundedness character of Eq. (34) 4 p3 27

0, generally speaking, two cases are considered: 3 3 4 p3 < r and 427p ≥ r . The quantity r − 427p arises as the value of the polynomial 27 p(λ) = λ3 − pλ2 + r at its local minimum λ = 23p . Such polynomial is the characteristic polynomial of the linear equation obtained from Eq. (34) with A = 0 when logarithms are taken. Table 4 summarizes the boundedness character of Eq. (34). Now, we consider Eq. (34) for a general delay k > 1. In this direction, [68], the study is concerned with the boundedness character of positive solutions of  xn+1 = max A,

p

xn

r xn−k+1

 ,

(35)

where k ∈ N \ {1}, the parameters A and r are positive and p is a nonnegative number. As in Eq. (34), the proofs are based on the Oachkatzlschwoif method, a proper change of variables and some inequalities verified by the characteristic polynomial associated to Eq. (35). For more particular cases of max-type difference equations different from Lyness and reciprocal types, see among others [21, 23, 29, 38, 66, 76, 77] or [79].

7 Rank-Type Difference Equations Let us consider a difference equation whose evolution rule is defined as the maximum of several first-order equations. In concrete, let p be a positive integer and let fi : R → R for i = 1, . . . , p be real-valued functions. Given the initial data x− p+1 , . . . , x0 , we define the max-type difference equation xn+1 = max{ f 1 (xn ), f 2 (xn−1 ), . . . , f p (xn− p+1 )}.

(36)

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Sauer, [58], proves that if the functions f i are contractive3 with fixed points ri , then Eq. (36) is globally convergent. In fact, he proves the result for a more general equation: Theorem 4 Consider p nonnegative integers q1 , . . . , q p and let 0 ≤ α < 1. For each i, j satisfying 1 ≤ i ≤ p, 1 ≤ j ≤ qi , assume that there exists a function f i j : R → R and a real number ri j satisfying | f i j (x) − ri j | ≤ α|x − ri j | for all x ∈ R. Then for any set (x− p+1 , . . . , x0 ) of initial values, the solution of the difference equation max { f i j (xn−i+1 )} (37) xn+1 = 1≤i≤ p;1≤ j≤qi

converges to maxi, j {ri j }. Notice that Eq. (36) is the particular case of Eq. (37) when q1 = · · · = q p = 1. The author presents several models where the result can be applied. For example, if Ai > 0 and −1 < αi < 1 for i = 1, . . . , p, the solution of α

xn+1 = max{A1 xnα1 , . . . , A p xn−p p+1 }  1  1−α is convergent to max1≤i≤ p Ai i for any initial conditions. In order to make conclusions about local convergence, it must be considered an extra hypothesis to control the contractivity between the individual fixed points. In fact, if the functions f i j : R → R are continuously differentiable with | f i j (x)| ≤ α for each 1 ≤ i ≤ p, 1 ≤ j ≤ q, 0 ≤ α < 1 and ri j ≤ x ≤ rim jm where rim jm = maxi j {ri j }, then the constant solution xn = rim jm of Eq. (37) is locally attractor, see [58]. It should be emphasized that the previous results hold for min-type difference equations, since it is enough to apply the max versions to − f i (x). Further research of the same author, [59], generalizes the previous results to ranktype equations. Definition 3 Let f i : R → R for i = 1, . . . , p be real-valued functions. A rank-type equation is the difference equation xn+1 = k-rank{ f 1 (xn ), f 2 (xn−1 ), . . . , f p (xn− p+1 )}

(38)

for initial conditions x− p+1 , . . . , x0 , where k-rank returns the kth-largest element of its argument. For instance, if we consider the equation A function f is contractive if there exists 0 ≤ α < 1 and a real number r such that | f (x) − r | ≤ α|x − r | for all x ∈ R. Notice that r is a globally attractor fixed point.

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xn+1 = 3-rank{ f 1 (xn ), f 2 (xn−1 ), f 3 (xn−2 ), f 4 (xn−3 )}, with initial conditions (x−3 , x−2 , x−1 , x0 ) and assume that f 3 (x−2 ) > f 1 (x0 ) > f 4 (x−3 ) > f 2 (x−1 ), then x1 = f 4 (x−3 ). Notice that max-type equations and min-type equations are particular cases of krank equations for k = 1 and k = p, respectively. So, rank-type equations generalize max-type difference equations. Sauer shows that, under the same assumptions as in Theorem 4, any set of initial conditions converges to k-rank1≤i≤ p; 1≤ j≤qi {ri j } through the difference equation xn+1 = k-rank1≤i≤ p; 1≤ j≤qi { f i j (xn−i+1 )}, where the ri j represents the kth-largest of the fixed points of the real valued functions f i j . Also, if the extra assumption of continuous differentiability is added, an analogous to the local convergence result of the maximum can be given for rank equations. In the case where the restriction α < 1 is relaxed, far less is known although in many cases the solution becomes eventually periodic. Proposition 1 ([59, Proposition 4.1]) Assume that b1 , . . . , b p in xn+1 = k-rank{−xn + b1 , . . . , −xn− p+1 + b p }

(39)

are ordered as bi1 ≥ . . . ≥ bik−1 > bik > bik+1 ≥ . . . ≥ bi p . That is, assume that bik , the kth-largest bi , is not repeated in the list. Then, there are uncountable many solutions with prime period 2i k . Now, by the change of variables, yn = e xn , we can consider the multiplicative version of the previous difference equation  xn+1 = k-rank

 Ap A1 . ,..., xn xn− p+1

(40)

Notice that for k = 1, the maximum case, we get the reciprocal difference equation studied in Sect. 5. The dynamics of Eq. (40) is not proven yet, but the following conjecture was established. Conjecture 2 Consider the difference Eq. (39) where the b1 , . . . , b p are ordered as bi1 > ... > bik−1 > bik > bik+1 > ... > bi p , that is, bik is the k-th largest of distinct bi . Then, all solutions are eventually periodic with period 2i k (the prime period may be a divisor of 2i k ). The requirement that bik is non-repeated is important, since if it is removed, although eventual periodicity is still expected, the formula for the period seems to be more complicated. Finally, Sauer and Berry, [5], study the convergence of rank-type equations where the functions involved are periodic with the discrete time variable n:

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Theorem 5 Let f 1 , . . . , f M : R × N → R be contractions with respect to the first variable, with common contraction α < 1, and P-periodic with respect to the second. Let 1 ≤ k ≤ M. Then, for any initial sequence, the solution of the difference equation xn+1 = k-rank1≤i≤M { f i (xn−i+1 , n)} is asymptotically periodic with period P. The conclusion fails if α ≥ 1, even if k = P = 1 as the difference equation xn = max{−xn−1 , −xn−2 } shows. It should be emphasized that Theorem 5 is independent of the memory length M, since the periodicity of the limit only depends on the periodicity of the forcing. The above theorem is also proved as a special case of a more general result that concerns sup-contractive4 difference equations. Moreover, it is proved that any solution converges to a unique P-periodic orbit. In some simple cases, the authors are able to identify explicit solutions. However, this task is significantly more complicated for larger period P and memory M and explicit formulas for the solutions remain to be found.

8 Applications The maximum function has been used to model a variety of phenomena, specially in the field of automatic control theory, optimal control theory or in biology. Nevertheless, since its first application appeared in the 1960s, every model was based on differential equations, but no application related to difference equations with maximum are known. For example, in automatic control theory, it often occurs that the law of regulation depends on the maximum values of some regulated state parameters over certain time interval. In [56], we can find a model of a system for regulating the voltage of a generator of constant current with parallel simulation and the regulated quantity was the voltage at the source electric current. The equation describing the work of the regulator involves the maximum of the unknown function and it has the form: T0 u (t) + u(t) + q · max u(s) = f (t), s∈[t−h,t]

where T0 and q are constants characterizing the object, u(t) is the regulated voltage and f (t) a perturbed effect. The discretization of the above differential equation yields to a difference equation with maximum. The reader interested in applications of max-type differential equations is referred to [3, 36, 53, 56]. Let G : Rn → Rm . Assume there exists L ≥ 0 such that ||G(x) − G(y)||∞ ≤ L||x − y||∞ for all x, y ∈ Rn , where || · ||∞ is the sup-norm. If L < 1, we will call G sup-contractive. Moreover, if L = 1, we call G sup-non-expansive.

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Nevertheless, in [13], the authors propose, for the sake of dialogue, an application for the nonautonomous reciprocal max-type difference equation  xn+1 = max

 A(0) A(1) A(k) n n n , ,..., , xn xn−1 xn−k

where the parameters are positive periodic sequences and the initial conditions are positive, as a phenomenological model of seizure activity as occurs in mesial temporal lobe epilepsy. In concrete, they propose the particular case  xn+1 = max

 An + Dn · H (xn ) Bn , , xn xn−1

where (An ), (Bn ) and (Dn ) are periodic parameters and H (x) is the Heaviside function  0, 0 ≤ x < ε, H (x) = 1, x ≥ ε, with ε sufficiently small. Motivated by numerical experiments, the authors call ε in the equation the seizure threshold. The state variable, xn , represents the density of activated neurons in the middle of the temporal lobe, the region of the brain where seizures occur in mesial temporal lobe epilepsy. The magnitudes of the three parameters (An ), (Bn ) and (Dn ) represent the degree to which neurons are inherently hyperexcitable. Numerical simulations indicate that the solutions of the equation are bounded and persist. Furthermore, it seems that, under additional conditions, the solutions are eventually periodic. In concrete, by assuming the following hypothesis: (H1) (An ), (Bn ) are both positive periodic sequences with period 2; (H2) (Dn ) is a positive periodic sequence with period 3; (H3) the inequalities A0 + D1 < B0 , B1 < A0 + D2 ; A1 + D1 < B0 , B1 < A1 + D2 hold; (H4) B0 , B1 > ε; (H5) ε ≤ 1; then, every positive solution is bounded and persists. Moreover, if the parameter (An + Dn · H (xn )) is eventually periodic, where (xn ) is a solution of the equation, then (xn ) is eventually periodic. Later on, C.M. Kent also proposed for the sake of dialogue a piecewise linear equation to model a bipolar disorder, [40]. In the same line of piecewise linear equations that can be transformed into max-type equation by the change of variables developed in Sect. 3, we can highlight the study of the convergence, the oscillatory character and chaos of a discrete model of combat analyzed by Sedaghat in [60]. Finally, let us mention the Hicks Equation as a economical model that emerged in the fifties. The analysis of its oscillatory character is treated, among other models of social sciences, in [61].

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9 Open Problems In the present section we collect some open problems and conjectures that may be of interest for the reader. The majority of them have been commented along the survey, but we find more useful to gather some of them in an specific section. Firstly, notice that in Sect. 4 we have analyzed Eq. (12) as a natural generalization of Lyness’ Equation proposed by Ladas in [43]. Nevertheless, according to the changes of variables developed in Sect. 3, a natural question that arises is to study the dynamics of Eq. (6) for a general value of m, and try to extend the results established in that section. Observe that it could be interesting to inquire into the relation between Eq. (6) and the generalized Lozi’s Equation. Moreover, related to the Lyness-max type difference equation, in Sect. 4.1 we have presented the equation xn+1 = max{xn−k+2 , xn−k+3 , . . . , xn , 0} − xn−k+1 .

(41)

In [46], a complete description of the set of periods for k = 4 is given. Nevertheless, a deep description of such set is not known for k ≥ 5. Moreover, the problem of studying the periodic structure of (41) when we change the constant 0 in the maximum function for a general positive number is open too. On the other hand, it is easy to check that the change of variables xn = ln(z n ), transforms Eq. (41) into z n+1 =

max{z n , z n−1 , . . . , z n−k+2 , 1} . z n−k+1

Related to this equation, Grove and Ladas proposed the following conjecture in [35, Conjecture 2.2]. Conjecture 3 Assume A ∈ (0, ∞). Show that no positive non-equilibrium solution n ,x n−1 } of the equation xn+1 = max{A,x has a limit. Extend and generalize. xn−2 Now, we focus on the reciprocal difference equation with maximum studied in Sect. 5,   A0 Ak , (42) xn+1 = max ,..., xn xn−k where the parameters A0 , . . . , Ak are real numbers and the initial conditions x−k , . . . , x0 are nonzero real numbers. Although a lot of specific and particular results are known related to the periodic and bounded character of the solutions of such equation, even its eventual periodicity, for the general case the determination of the period of their solutions is still an open problem. In concrete: Conjecture 4 ([44]) Assume that A0 , . . . , Ak−1 ∈ [0, ∞) and Ak ∈ (0, ∞). Show that every positive solution of Equation (42) is eventually periodic with period p ∈ {2, 3, . . . , 2(k + 1)}.

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In Table 3, we have collected the results known about eventual periodicity of the equation   1 An , (43) , xn+1 = max xn xn−1 where the coefficients (An ) are a periodic sequence of period k. Briden, Ladas and Nesemann, [12], published a collection of conjectures and open problems related to that equation. We state only some that, as far as we know, are still unsolved or that have been solved only partially. Open Problem 1 Let (An ) be a periodic sequence of positive real numbers with period k ≥ 3. Find necessary and sufficient conditions for each of the following statements related to Eq. (43) to be true: a) All positive solutions are bounded. b) All positive solutions are eventually periodic and determine all possible periods. Conjecture 5 Let (An ) be a periodic sequence of positive real numbers with prime period k ≥ 2. Assume that An ∈ (1, ∞) for n ≥ 0. Then every positive solution of Eq. (43) is eventually periodic with period 2k if k is even and 4k if k is odd. Finally, related to max-type difference equations presented in Sect. 6, we collect the coming open problems and conjectures. Conjecture 6 [21] Assume A ∈ R, k ∈ N. Show that every well-defined solution of the difference equation of order k 

xn+1

 A = max , xn−k , n ∈ N ∪ {0}, xn

is eventually periodic with period k + 1. Conjecture 7 [22] Every positive solution of xn = max{−1, xn−k − xn−m }, with k odd, is bounded. Conjecture 8 [64] Assume that r is a positive real number and p is a nonnegative 3 real number verifying the following inequalities: r ≤ 427p ; 1 < p < r + 1; r < 21 . Then, every solution of Equation  p  xn , xn+1 = max A, r xn−2 where the parameters A and r are positive and p is a nonnegative real number, is bounded. Open Problem 2 [22] It is known that every solution of equation

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 xn+1 = max

A xn−k

151

 , xn−l , n ∈ N ∪ {0},

where k, l ∈ N ∪ {0}, is eventually periodic. Find the prime periods of the equation.

10 Conclusions In the present work we have collected a wide variety of max-type difference equations. Apart from highlighting their properties, such as periodicity, boundedness, convergence, persistence..., we have focused on the different techniques and approaches that have been used over the years to attack this kind of equations. It is worth mentioning the richness of their dynamics and the variety of the techniques. Moreover, along the work and, specifically, in Sect. 9, we have gathered some open problems and conjectures. We hope that they will be of interest for the reader who may study and solve some of them. In view of the amount of problems still open and the recent contributions, we believe that this field could attract the interest of researchers. Finally, it is also significant to analyze the possible applications of the max-type difference equations presented in this survey. Some of them may be useful to model problem in automatic control theory or biology. Acknowledgements This work has been supported by the grant MTM2017-84079-P funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”, by the European Union.

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Catalan Numbers Recurrence as a Stationary State Equation of the Probabilistic Cellular Automaton Mariusz Białecki

Abstract A construction of a finite probabilistic cellular automaton is presented and the respective evolution rule is defined for a restricted space of states. The choice of variables describing the stationary state of the automaton—expected values weighted by the limiting distribution—is justified in terms of Markov chains framework. A system of equations for stationary state in mean-field approximation is derived and then reduced, by a special choice of dynamic parameters, to special case for which the system of equations is transformed into truncated Catalan numbers recurrence. Full recurrence appear in the limit of infinite size of the automaton. Keywords Catalan numbers · Cellular automata · Stationary stochastic processes · Random dynamical system · Markov chains · Finite systems · Integer sequences

1 Introduction There are many diversified research activities related to probabilistic cellular automata (PCA), including both mathematical and modeling issues [1]. Here, we focus on a construction of a specific 1D PCA leading to recurrence, that produces Catalan numbers, ubiquitous and important object in various branches of mathematics [2]. The automaton is a discrete one dimensional finite dynamical system, of a size L, which evolves in a discrete time, according to a given updating rule. The system may be understood also as a Markov process, for which stationary state is well defined, and we derive equations for stationary variables describing the system. For some equations, we use simplifying assumption, namely mean field approximation. Those equations are valid for a range of dynamic parameters (a constant and two functions of one variable), which may be specified in such a way, that equations take a form M. Białecki (B) Institute of Geophysics, Polish Academy of Sciences, Ks. Janusza 64, 01-452, Warszawa, Poland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_7

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of truncated Catalan numbers recurrence. The truncation is due to the finite size of the automaton. The full—infinite—recurrence is obtained in the limit L −→ ∞, i.e. for the size of the system enlarged up to infinity. The idea of construction a PCA with a stationary state equation in a form of a given recurrence originated from the paper [3], where an earthquake toy model in a form of PCA—Random Domino Automaton [4]—turn out to produce Motzkin numbers recurrence [5, 6]. In spite of simplicity of Random Domino Automaton, the model exhibits a complex behavior and, for example, can lead to so called bi-SOC-states [7]. The term SOC, which is an acronym for Self-Organizing Criticality, refers to the properties of dynamical systems that cause them to evolve spontaneously to a critical point, i.e. without external adjustment of a control parameter value (unlike for freezing water by externally forced cooling). Critical point is understood in physical manner, indicating various phase transition-like properties. Another stochastic cellular automaton leading to Catalan numbers—other one then proposed here—was described in the paper [8]. The main difference between these two automata is in the rule of evolution. Namely, they have different definitions of the shift (see Definition 6, point (b1) below), which differentiate their dynamics, which in consequence leads to various sets of equations for the stationary state. Moreover, the presentation of [8] was tailored for physically oriented audiences, i.e. with use of intuitive, physical terminology, but without rigorous definitions of states, evolution function, etc. This text is intended to be readable by mathematically oriented audiences, therefore the appropriate concepts are introduced and used to analyze the new cellular automaton associated with Catalan numbers. The plan of the paper is as follows. In Sect. 2 we introduce necessary definitions describing the system, and set the evolution rule. Section 3 contains a brief discussion of the automaton as a Markov chain and derivation of stationary state equations for general case. Next, in Sect. 4, dynamic parameters are chosen in a specific form, which imply that the equations take the form of Catalan numbers recurrence. Then, short analysis of the case and respective solution are also presented there. At the end, we conclude with comments in Sect. 5.

2 The Automaton First we introduce respective notion in order to define the finite one-dimensional cellular automaton as a dynamical system whose state evolves according to the given rule. Definition 1 (State function) Let Z L denotes the set of integers modulo L, and Q = {0, 1}. Then (1) Z L × N  (i, t) −→ s(i, t) ∈ Q defines the state of the site i ∈ Z L in time t ∈ N.

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Definition 2 (Occupied site, empty site) We say that site i at time t is occupied iff s(i, t) = 1. Otherwise, we say that the site i at time t is empty, iff s(i, t) = 0. Definition 3 (State of the automaton) State of the automaton at an instant t, S(t), is given by states of all sites at the time t S(t) := (s(0, t), s(1, t), . . . , s(L − 1, t)).

(2)

The set of all states is denoted by Q L . All site indexes are considered to be in Z L , i.e. we assume arithmetic modulo L. Moreover, in equations relating functions with the same value of time, the argument t may be omitted, and we may write, for example, s(i) = s( j), instead of s(i, t) = s( j, t). Definition 4 (Cluster of size m) Site i belongs to a cluster of size m iff for some kl ∈ Z L and kr ≡ kl + (m − 1) (mod L) the following conditions are satisfied: i ∈ {kl , kl + 1, . . . , kr }, s(kl − 1) = 0, s(kr + 1) = 0 and s( j) = 1 for all j ∈ {kl , kl + 1, . . . , kr }. Site kl is called the left end of the cluster, and site kr is the right end. State of the automaton may be described as consisting of clusters, separated by sequences of empty sites. The latter are called empty clusters—the respective definition is fully analogous to the definition of a cluster. The number of clusters at the instant t is denoted by n(t). The number of clusters of size m is denoted by n m (t). Analogously, for empty clusters we have n 0 (t) and n 0m (t). Obviously, n m (t) = 0 and n 0m (t) = 0 for m > L. Also, it follows n(t) =



n m (t), and n 0 (t) =

m≥1



n 0m (t).

(3)

m≥1

Moreover, because any cell is either occupied or empty, it follows 1 (n m (t) + n 0m (t))m = 1. L m≥1

(4)

Remark 1 Because the space variable (i.e. site index i) is defined in arithmetic modulo L, or, in other words, because of periodic boundary conditions, the number of clusters is equal to the number of empty clusters for all states, except for fully empty (0, 0, . . . , 0) and fully occupied (1, 1, . . . , 1). Definition 5 (Density) The density ρ(t) of the system, is the ratio of the number of occupied cells to the number of all sites L, i.e. ρ(t) =

1 n m (t)m. L m≥1

(5)

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In order to set the probabilistic rule of evolution for the automaton we define the following parameters: μ(m), δ(m), and ν, where μ(m) and δ(m) are [0, 1] ⊂ R valued functions of the size m of a cluster, satisfying μ(m) + δ(m) ≤ 1 for all values of m. Parameter ν ∈ [0, 1] is a constant. In the following, the short notation μm and δm will be used instead of μ(m), δ(m). Definition 6 (Evolution rule) Evolution rule of the automaton is given by f : Q L  S(t) −→ S(t + 1) ∈ Q L ,

(6)

where f is acting according to the following stochastic rule. For a time t choose randomly arbitrary site j ∈ Z L in such a way, that every site have the same chance. (a) If the site j is empty s( j, t) = 0 then: (a1) for s( j − 1, t) = 0 and s( j + 1, t) = 0 (both neighbours are empty) with probability ν set  s(i, t + 1) :=

1 for i = j, s(i, t) for i = j,

(7)

and, otherwise, with probability (1 − ν), nothing is changed, i.e. s(i, t + 1) := s(i, t) for all i;

(8)

(a2) for s( j − 1, t) = 1 or s( j + 1, t) = 1 (at least one neighbour is occupied) nothing is changed, i.e. s(i, t + 1) := s(i, t) for all i. (9) (b) If the site j is occupied s( j, t) = 1 then it belongs to a cluster of a certain size m. Its left and right ends are denoted by jl and jr , respectively. The left end of the other cluster next to the right (if exists) is denoted by kl . Then choose one from the three following excluding options (b1), (b2) and (b3), with respective probabilities δm , μm , and (1 − μm − δm ). (b1) For S(t) containing strictly less than two clusters s(i, t + 1) := s(i, t) for all i.

(10)

For S(t) containing at least two clusters s(i, t + 1) :=

⎧ ⎨

1 for i ∈ {kl − m, kl − m + 1, . . . , kl − 1}, 0 for i ∈ { jl , jl + 1, . . . , kl − m − 1}, ⎩ s(i, t) for i ∈ Z L − { jl , jl + 1, . . . , kl − 1}.

(11)

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(b2) For S(t) containing strictly less than two clusters s(i, t + 1) := s(i, t) for all i.

(12)

For S(t) containing at least two clusters  s(i, t + 1):=

0 for i ∈ { jl , jl + 1, . . . , jr }, s(i, t) for i ∈ Z L − { jl , jl + 1, . . . , jr }.

(13)

(b3) s(i, t + 1) := s(i, t) for all i.

(14)

The case (a1) refers to creation of a new cluster (of size 1). The case (b1) denotes the shift of the cluster, which results in merging it to the next right cluster. The case (b2) designates the removal of the cluster. The remaining two, (a2) and (b3) do not change the state of the automaton. The diagram below illustrates part (b) of Definition 6.

Definition 7 (Set of states of the automaton) Set of states of the automaton Q f is a set of all states S(t), which are accessible during evolution, given by the function f detailed in Definition 6, when starting from the following initial state S0 := (0, 0, 0, 0, 0, . . . , 0). Due to properties of the system described in the Sect. 3.1, the choice of the initial state S0 is arbitrary. Any other state communicated with S0 (explanation is given below) is equally good.

3 Equations for Expected Values We assume that size of the automaton is not too small, say L ≥ 10, in order to avoid incidental complications arising due to space limitation. This assumption comply with the aim to produce Catalan numbers recurrence, which is well approximated for large L.

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3.1 Stationary State The automaton is a Markov process, thus can be analyzed using relevant techniques, like in [9], where Finite Random Domino Automaton is investigated. This approach helps to solve the inverse problem, i.e. to reconstruct values of parameters (probabilities specifying the evolution) given a distribution of avalanches produced by the system (see [10] for more details), however, for computational purposes, such approach is not very effective. This is because of large number of states are involved for a big-sized system, which results in a large size of the transition matrix. From the other side, one can conclude on general properties of the automaton using basic theory of Markov processes (see for example [11]). Consider the space of states Q f of the automaton. The following properties can be deduced. Irreducibility. Every two states from Q f communicate, i.e. are accessible from each other. This means, for S1 and S2 , both S1 can evolve from S2 and S2 can evolve from S1 . It can be seen in the following way. Any state S ∈ Q f can be reduced to S0 (given by Definition 7) by removal of all clusters. Moreover, any state S ∈ Q f can be reconstructed by appropriate subsequent repetitions of creation and merging. This property ensures that the space of states Q f does not decompose into disjoint subsets of states (communicating classes), i.e. is irreducible. Recurrence. The same reasoning as above shows that every state S ∈ Q f is recurrent, i.e. any time the system changes its state from S, it will return to that state S in the future with probability one. Such a space of states, consisting of only recurrent states, is called recurrent. Aperiodicity. It follows directly from Definition 7, that each state S ∈ Q f has a selftransition (the system can remain in the same state S in the next time step), hence is aperiodic. Thus the space Q f is aperiodic. Proposition 1 The space of states Q f is irreducible, aperiodic and recurrent. It follows from theory of Markov processes (see for example [11]) that for processes with irreducible, aperiodic and recurrent space of states the limiting probability distribution of states is well defined. In physical terminology the limiting distribution is referred to as stationary state. Thus, it is legitimate to define stationary, time independent variables—ρ, n, n m —as expected values weighted by the limiting distribution, understood also as time average values. A stationary state does not change in time by definition, thus any increase of the number of occupied cells, any increase of the number of clusters, and any increase of the number of m-clusters are balanced by respective decrease during evolution. This property is also called “flow in = flow out” principle, and is the basis for deriving respective equations for stationary variables.

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3.2 Balance Equation for the Density The density ρ can be changed only by either creation of a new cluster (Definition 6 part (a1)) or by removal (Definition 6 part (b2)). Creation happens when both neighbors of the chosen site are empty sites, which may happen only for clusters of size m ≥ 3. The probability of choosing such site is  n 0 (m − 2) m

L

m≥3

·ν =ν

 1 (1 − ρ)L + n 01 − 2n , L

(15)

where the right hand side comes from inclusion m = 1 and m = 2 in the sum. Expected value of density’s drop in one time step caused by removal is  nm m m≥1

L

· μm · m.

(16)

Proposition 2 The balance equation for the density of the system is 

  μm n m m 2 = ν (1 − ρ)L + n 01 − 2n .

(17)

m≥1

3.3 Balance Equation for the Total Number of Clusters The number of clusters can only increase by one and it occurs with the probability given by formula (15). Also, the number of clusters can only decrease by one. This can only happen as a result of two processes: either because of a shift (Definition 6 part (b1)) or because of deletion (Definition 6 part (b2)). Their respective probabilities are  nm m m≥1

L

· δm , and

 nm m m≥1

L

· μm .

(18)

Proposition 3 The balance equation for the total number of clusters is 

  (δm + μm ) n m m = ν (1 − ρ)L + n 01 − 2n .

(19)

m≥1

Equations (17) and (19) have the same right-hand sides, thus  m≥1

μm n m m(m − 1) =

 m≥1

δm n m m.

(20)

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3.4 Balance Equation for the Number of M-Clusters The probability of increasing the number of 1-clusters—possible only by one—is given by formula (15). For 1 < m, a new m-cluster can appear as a result of merging of two smaller clusters of sizes k and (m − k), for all k ∈ {1, 2, . . . , m − 1}. The shift of a k-cluster leads to m-cluster with the probability given by m−1  k=1

nk k n m−k · δm · , N n

(21)

where no correlation between clusters is assumed. In particular, it may happen that (m − k) is equal to k, and then the fraction n m−kn −1 should be used instead of n m−k . To n avoid such corrections we assume also n k >> 1, which again comply with the limit L −→ ∞. This simplification is consistent with the structure of all equations—see Remarks 2 and 3 below. The number of m-clusters can be decreased by removals and by two kinds of shifts. The probability of the removal of the m-cluster is nm m · μm . N

(22)

The probability of the shift of the m-cluster and merging to another cluster is nm m · δm . N

(23)

Finally, for the shift of another cluster and merging to the m-cluster the probability is  nk k nm · δk · , (24) N n k≥1 where we assume that, there is no size correlation between clusters and n m >> 1. Note, contribution due to merging m-cluster to m-cluster—where the n m decreases by 2—is included in both of the above formulas. Proposition 4 The balance equation for the number of m-clusters is nm

1 δk n k k m (μm + δm ) + n k≥1



 =

ν[(1 − ρ)N + n 01 − 2n] for m = 1, (25) 1 m−1 k=1 kδk n k n m−k for m > 1, n

Remark 2 Sum of all equations for n m (25), for m = 1, 2, . . ., returns the Eq. (19), as would be expected from the definition of n (Eq. (3)). The quadratic term of Eq. (25) disappears due to the change of summation order, i.e. due to the following identities

Catalan Numbers Recurrence as a Stationary State Equation …

 m−1 

n k n m−k kδk =

m≥2 k=1

=

 

163

kδk n k n m−k =

k≥1 m≥k+1

(26)

   kδk n k n m−k = n kδk n k . k≥1

m≥k+1

k≥1

Remark 3 Sum of all equations for n m (25) multiplied by m, for m = 1, 2, . . ., returns the Eq. (17), as would be expected from the definition of ρ (Eq. (5)). Respective identities for the quadratic term are as follows   m−1 m≥2 k=1

=

 k≥1

mkδk n k n m−k =

 k≥1

⎛ kδk n k ⎝kn +





kδk n k ⎞

jn j ⎠ =

j≥1

mn m−k =

m≥k+1



nk 2 δk n k +

k≥1



kδk n k

k≥1



(27) mn m .

m≥1

Both above remarks above confirm, that obtained balance Eqs. (19), (17) and (25) are self-consistent, i.e. that assumptions are applied in a consistent way.

4 Catalan Numbers Recurrence To achieve the goal, we specify parameters μm and δm in the following way δm =

α β , and μm = , m m

(28)

with α and β constants, 0 < α, β, (α + β) < 1. For such choice, probabilities of the shift, and the removal, are the same for each cluster, independently of its size. In consequence of (28) balance equations are simplified. In particular the balance Eq. (17) for ρ is βρ L = ν[(1 − ρ)L − 2n + n 01 ], (29) the balance Eq. (19) for n (β + α)n = ν[(1 − ρ)L − 2n + n 01 ].

(30)

Corollary 1 The average size of a cluster for dynamic parameters in the form (28) is α+β ρL = . (31) < m >= n β

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The balance Eq. (25) for the number of clusters n m of a given size m is nm =

⎧ ⎨ ⎩





1 2α+β





α+β n 2α+β  i−1 α k=1 n k n i−k n

for

m = 1,

for

m ≥ 2.

(32)

Definition 8 Define a series cm , for m = 0, 1, . . ., by the following formula  cm =

α (2α + β)n

 n m+1 .

(33)

Proposition 5 The equation for the distribution of clusters for stationary state of the automaton can be written in the following form c0 = α(α + β)/(2α + β)2 , M  c M+1 = ck c M−k for M ≥ 0.

(34) (35)

k=0

Proof Setting M = m − 2, Eqs. (34) and (35) follows directly from Eq. (32). The Eq. (35) is the famous Catalan numbers recurrence, which gives Catalan numbers starting from initial value cˆ0 = 1. It can be solved, and Catalan numbers are given by exact formula cˆ M =

  2M 1 for M ≥ 0. M +1 M

(36)

The solution c M of Eqs. (34)–(35) is c M = cˆ M c0M+1 =

  2M M+1 1 c for M ≥ 0. M +1 M 0

(37)

The Eq. (34) shows that 0 < c0 < 1/4 for arbitrary α > 0 and β > 0, because c0 is decreasing function of β/α. Hence c M is a decreasing function of M and, due to relation (33), n m is a decreasing function of m. Finally, asymptotic behavior of Catalan numbers, obtained by application of the Stirling’s formula to the Eq. (36), is 1 4M  , cˆ M = √ π M(M + 1) 1 + O( M1 ) what assures no divergences in the limit L −→ ∞.

(38)

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5 Comments The paper shows how to define a probabilistic cellular automaton, for which the cluster size distribution for stationary state is given by rescaled Catalan numbers. Clearly, rescaling is necessary to avoid divergences in the limit L −→ ∞. So the result can not be improved with respect to this issue. However, there are other aspect to improve. Getting rid of the assumption of no correlation between clusters (aka mean-field approximation) is an obvious candidate. Possible usage of Markov chain framework may provide a solution, if a formula for transition matrix for a system of arbitrary size is discovered. Any progress in this topic will be a contribution to a more general program of constructing discrete dynamical systems, in particular probabilistic cellular automata, related with integer sequences. Acknowledgements The author has been supported by the National Science Center (Poland) under research grant No. 2017/27/B/ST10/02686 and by a subsidy from the Polish Ministry of Education and Science for the Institute of Geophysics PAS. The author thanks Toktam Zand for the comments on the text.

References 1. Fernández, R., Louis P.-Y., Nardi F.R.: Overview: PCA models and issues. In: Louis, P.-Y., Nardi F.R. (eds.) Probabilistic Cellular Automata. Emergence, Complexity and Computation, vol. 27, pp. 1–30. Springer, Berlin (2018). https://doi.org/10.1007/978-3-319-65558-1_1 2. Stanley, R.: Catalan Numbers. Cambridge University Press, Cambridge (2015) 3. Białecki, M.: Motzkin numbers out of a stochastic cellular automaton. Phys. Lett. A 376, 3098–3100 (2012). https://doi.org/10.1016/j.physleta.2012.09.022 4. Białecki, M., Czechowski, Z.: On one-to-one dependence of rebound parameters on statistics of clusters: exponential and inverse-power distributions out of random domino automaton. J. Phys. Soc. Jpn. 82, 014003 (2013). https://doi.org/10.7566/jpsj.82.014003 5. Aigner, M.: Catalan-like numbers and determinants. J. Combin. Theory Ser. A 87, 33–51 (1999). https://doi.org/10.1006/jcta.1998.2945 6. Aigner, M.: A Course in Enumeration. In: Graduate Text in Mathematics, vol. 238. Springer, Berlin (2007) 7. Czechowski, Z., Budek, A., Białecki, M.: Bi-SOC-states in one-dimensional random cellular automaton. Chaos 27, 103123 (2017). https://doi.org/10.1063/1.4997680 8. Białecki, M.: Catalan numbers out of random domino automaton. J. Math. Phys. 60, 012701 (2019). https://doi.org/10.1063/1.5027461 9. Białecki, M.: Properties of a finite stochastic cellular automaton toy model of earthquakes. Acta Geophys. 63, 923–956 (2015). https://doi.org/10.1515/acgeo-2015-0030 10. Białecki, M.: From statistics of avalanches to microscopic dynamics parameters in a toy model of earthquakes. Acta Geophys. 61, 1677–1689 (2013). https://doi.org/10.2478/s11600-0130111-7 11. Serfozo, R.: Basics of Applied Stochastic Processes. Springer, Berlin (2009)

Oscillation of Second Order Impulsive Neutral Difference Equations of Non-canonical Type G. N. Chhatria and A. K. Tripathy

Abstract In this article, necessary and sufficient conditions for the oscillation of a class of nonlinear second order neutral impulsive difference equations of the form: ⎧ ⎪ ⎨Δ[a(n)Δ(x(n) + p(n)x(n − τ ))] + q(n)F(x(n − σ)) = 0, n = m j Δ[a(m j − 1)Δ(x(m j − 1) + p(m j − 1)x(m j − τ − 1))] ⎪ ⎩ +r (m j − 1)F(x(m j − σ − 1)) = 0, j ∈ N have been discussed for p(n) ∈ (−1, 0] with fixed moments of impulsive effect. Here, we assume that the nonlinear function is either strongly sublinear or strongly superliner. Some examples are given to illustrate our main results. Keywords Oscillation · Nonoscillation · Neutral difference equation · Impulse · Lebesgue’s dominated convergence theorem

1 Introduction Consider a class of second order nonlinear neutral impulsive difference equations of the form: ⎧ Δ[a(n)Δ(x(n) + p(n)x(n − τ ))] + q(n)Ψ (x(n − σ)) = 0, n = m (1) j ⎨ (E) Δ[a(m j − 1)Δ(x(m j − 1) + p(m j − 1)x(m j − τ − 1))] ⎩ +r (m j − 1)Ψ (x(m j − σ − 1)) = 0, j ∈ N, (2)

G. N. Chhatria (B) · A. K. Tripathy Department of Mathematics, Sambalpur University, Sambalpur 768019, India e-mail: [email protected] A. K. Tripathy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_8

167

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where τ , σ are positive integers, a, p, q, r are real valued functions with discrete arguments such that q(n) > 0, r (n) > 0, a(n) > 0 and | p(n)| < ∞ for n ∈ N(n 0 ) = {n 0 , n 0 + 1, . . .}, n 0 ≥ 0, Ψ ∈ C(R, R) with the property xΨ (x) > 0 for x = 0, Δ is the forward difference operator defined by Δu(n) = u(n + 1) − u(n), Δ is another difference operator defined by Δu(m j − 1) = u(m j ) − u(m j − 1), and m 1 , m 2 , m 3 , . . . be the moments of impulsive effect such that 0 < m 1 < m 2 < · · · and lim j→∞ m j = +∞. A real valued function x(n) defined on N(n 0 − ρ) is said to be a solution of (E) if it satisfy (E) for n ≥ n 0 with the initial conditions x(i) = φ(i), i = n 0 − ρ, . . . , n 0 , where φ(i), i = n 0 − ρ, . . . , n 0 are given and ρ = max{τ , σ}. A nontrivial solution x(n) of (E) is said to be nonoscillatory, if it is either eventually positive or eventually negative. Otherwise, the solution is said to be oscillatory. The objective of this work is to establish the necessary and sufficient conditions for oscillation of all solutions of the impulsive system (E) under the assumption (C0 )

∞  1 < ∞. a(s) s=n 0

Actually, we say that (E) is of non-canonical type if (C0 ) holds; otherwise, it is of canonical type. About impulsive differential and difference equations, we refer to the monographs [1, 7, 11]. Indeed, this work is a continuation work of the authors’ earlier work [18], in which Tripathy and Chhatria studied the oscillation properties of (E) under the assumption (C00 )

∞  1 = ∞. a(s) s=n 0

In [3, 4, 10, 12, 13], the authors have studied the oscillatory and asymptotic behaviour of solutions of a class of second order nonlinear neutral difference equations of type (1) under a similar assumption to (C0 ), and this fact is very important while we go for the existence of positive bounded solutions of the respective difference equations. At the same time, we predict that the conditions necessary for the existence of a nonoscillatory solution may lead to an oscillatory solution subject to the influence of impulse (see for e.g. [8, 9, 19]). In contrast to these works and in our current state of the art, we enhance the above prediction a little bit farther in which x(n) is a solution of (1), and under the impact of impulse, x(m j ) is an impulsive solution satisfying another neutral difference Eq. (2). Of course, (E) is our required impulsive system and our state of art is our motivation. In this direction, we suggest readers to go for more information in the works [6, 14–18]. Definition 1 ([2]) A function Ψ is said to be strongly sublinear if there exists a constant α ∈ (0, 1) such that

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169

|Ψ (u)| |Ψ (v)| ≤ f or |u| ≥ |v|, uv > 0, α |u| |v|α

(3)

and it is said to be strongly superlinear if there exists a constant β > 1 such that |Ψ (u)| |Ψ (v)| ≥ f or |u| ≥ |v|, uv > 0. |u|β |v|β

(4)

Lemma 1 ([5]) If u and v are two nonnegative integers such that u < v, then v 1−γ − u 1−γ ≤ (1 − γ)v −γ (v − u), γ > 1.

2 Oscillation Criteria This section deals with the necessary and sufficient conditions for oscillation of all solutions of the nonlinear neutral impulsive system (E) under the key assumption (C0 ). Defining 

y(n) = x(n) + p(n)x(n − τ ) y(m j − 1) = x(m j − 1) + p(m j − 1)x(m j − τ − 1),

(5)

the impulsive system (E) becomes  (E 1 )

Δ[a(n)Δy(n)] + q(n)Ψ (x(n − σ)) = 0, n = m j Δ[a(m j − 1)Δy(m j − 1)] + r (m j − 1)Ψ (x(m j − σ − 1)) = 0, j ∈ N.

In the sequel, we use the following notion ˆ A(n) =

∞  1 . a(s) s=n

Theorem 1 Let −1 < p ≤ p(n) ≤ 0 and Ψ be strongly sublinear. In addition to (C0 ), let’s assume that (C1 ) Ψ (−u) = −Ψ (u), u ∈ R. Then every unbounded solution of (E) oscillates if and only if ⎤ ⎡ ∞ s−1 ∞    1 ⎣ q(t) + r (m j − 1)⎦ = ∞. (C2 ) a(s) s=n t=n ∗ j=1

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G. N. Chhatria and A. K. Tripathy

Proof Suppose that (C2 ) holds. On the contrary, let x(n) be an unbounded nonoscillatory solution of (E) for n ≥ n 0 > ρ + 1. Without loss of generality and due to (C1 ), we assume that x(n) > 0, x(n − τ ) > 0 and x(n − σ) > 0 for n ≥ n 1 > n 0 . From (E 1 ), it follows that a(n)Δy(n) is nonincreasing and y(n) is monotonic for n ≥ n 2 > n 1 . If y(n) < 0 for n ≥ n 2 , then x(n) < − p(n)x(n − τ ) ≤ x(n − τ ) ≤ x(n − 2τ ) ≤ x(n − 3τ ) · · · ≤ x(n 2 ) and x(m j − 1) ≤ x(m j − τ − 1) ≤ x(m j − 2τ − 1) ≤ x(m j − 3τ − 1) · · · ≤ x(n 2 ) due to nonimpulsive points m j − 1, m j − τ − 1, m j − 2τ − 1, . . . implies that x(n) is bounded for all non-impulsive point n and m j − 1 which is going against our hypothesis. Therefore, y(n) > 0 for n ≥ n 2 . Ultimately, we have the following possibilities: 1. a(n)Δy(n) > 0, y(n) > 0; 2. a(n)Δy(n) < 0, y(n) > 0. Case 1. In this case, we can find an n 3 > n 2 such that y(n) ≤ x(n) for n ≥ n 3 and (E 1 ) reduces to  Δ[a(n)Δy(n)] + q(n)Ψ (y(n − σ)) ≤ 0, n  = m j Δ[a(m j − 1)Δy(m j − 1)] + r (m j − 1)Ψ (y(m j − σ − 1)) ≤ 0, j ∈ N

(E 2 )

for n ≥ n 3 . Since y(n) is nondecreasing, then there exist an n 4 > n 3 + 1 and a constant C > 0 such that y(n) ≥ C for n ≥ n 4 and so also y(m j − 1) ≥ C. Hence, (E 2 ) is of the form  Δ[a(n)Δy(n)] + q(n)Ψ (C) ≤ 0, n = m j Δ[a(m j − 1)Δy(m j − 1)] + r (m j − 1)Ψ (C) ≤ 0, j ∈ N. Summing the last impulsive system from n 4 to n − 1, we get a(n)Δy(n) − a(n 4 )Δy(n 4 ) −

 n 4 ≤m j −1≤n−1

that is,

Δ[a(m j − 1)Δy(m j − 1)] +

n−1  s=n 4

q(s)Ψ (C) ≤ 0,

Oscillation of Second Order Impulsive Neutral Difference Equations …

⎡ a(n)Δy(n) ≤ a(n 4 )Δy(n 4 ) − Ψ (C) ⎣

n−1 



q(s) +

171

⎤ r (m j − 1)⎦

n 4 ≤m j −1≤n−1

s=n 4

→ −∞ as n → ∞, where we are using the fact that ∞ 



q(s) +

r (m j − 1) = ∞

n 4 ≤m j −1 0 and n 3 > n 2 such that y(n − σ) ≤ C and y(m j − σ − 1) ≤ C for n ≥ n 3 . Summing (E 2 ) from n 3 to n − 1, we obtain a(n)Δy(n) − a(n 3 )Δy(n 3 ) 

−

Δ[a(m j − 1)Δy(m j − 1)] +

n 3 ≤m j −1≤n−1

n−1 

q(s)Ψ (y(s − σ)) ≤ 0,

s=n 3

that is, n−1 



q(s)Ψ (y(s − σ)) +

r (m j − 1)Ψ (y(m j − σ − 1)) ≤ −a(n)Δy(n)

n 3 ≤m j −1 0, y(n) < 0, 4. a(n)Δy(n) < 0, y(n) < 0. The proofs for Case 1 and Case 2 are same as in Theorem 1. Case 3. In this case, limn→∞ y(n) exists. Therefore, 0 ≥ lim y(n) = lim sup(x(n) + p(n)x(n − τ )) ≥ lim sup(x(n) + px(n − τ )) n→∞

n→∞

n→∞

≥ lim sup x(n) + lim inf ( px(n − τ )) = (1 + p) lim sup x(n). n→∞

n→∞

n→∞

Because (1 + p) > 0, then lim supn→∞ x(n) = 0. Ultimately, limn→∞ x(n) = 0 for all nonimpulsive points n and m j − 1, j ∈ N. Due to m j − 1 < m j < n and an appli-

Oscillation of Second Order Impulsive Neutral Difference Equations …

175

cation of Sandwich theorem shows that lim j→∞ x(m j ) = 0. Hence, limn→∞ x(n) = 0 for all n. Case 4. Indeed, y(n) < 0 implies that x(n) < − p(n)x(n − τ ) < x(n − τ ) < x(n − 2τ ) < x(n − 3τ ) · · · < x(n 2 ) and x(m j − 1) < x(m j − τ − 1) < x(m j − 2τ − 1) < x(m j − 3τ − 1) · · · < x(n 2 ), that is, x(n) is bounded for all n and m j − 1, j ∈ N. We claim that x(m j ) is bounded for all j ∈ N. If not, let lim j→∞ x(m j ) = ∞. Now, y(m j ) = x(m j ) + p(m j )x(m j − τ ) ≥ x(m j ) − x(m j − τ ) ≥ x(m j ) − b shows that y(m j ) > 0 as j → ∞, a contradiction due to the fact that x(m j − τ ) ≤ b. So, our claim holds and hence y(n) is bounded for all n. The rest of this case follows from Case 3. Hence, (C2 ) is a sufficient condition. The necessary part is same as in the proof of Theorem 1. This completes the proof of the theorem. Theorem 3 Let −1 < p ≤ p(n) ≤ 0 and Ψ be strongly superlinear. Assume that (C0 ) and (C1 ) hold. Then every unbounded solution of (E) oscillates if and only if (C3 )

∞ 

ˆ − σ)) + q(s)Ψ (C A(s

s=n ∗

∞ 

ˆ j − σ − 1)) = ∞ r (m j − 1)Ψ (C A(m

j=1

for every constant C > 0. Proof On the contrary, we proceed as in Theorem 1 to obtain Case 1 and Case 2. Consider Case 1. Then for n ≥ n 2 , it follows that y(n) ≥ y(n 2 ) =

y(n 2 ) ˆ ˆ A(n) ≥ C A(n). ˆ A(n)

ˆ − σ) and y(m j − Hence, we can find n 3 > n 2 + σ + 1 such that y(n − σ) ≥ C A(n ˆ σ − 1) ≥ C A(m j − σ − 1) for n, m j ≥ n 3 . Consequently, (E 2 ) becomes ˆ − σ)) ≤ 0, n = m j , Δ[a(n)Δy(n)] + q(n)Ψ (C A(n ˆ j − σ − 1)) ≤ 0, j ∈ N. Δ[a(m j − 1)Δy(m j − 1)] + r (m j − 1)Ψ (C A(m Summing the last impulsive system from n 3 to n − 1, we get

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G. N. Chhatria and A. K. Tripathy

a(n)Δy(n) − a(n 3 )Δy(n 3 ) −



Δ[a(m j − 1)Δy(m j − 1) +

n 3 ≤m j −1≤n−1

n−1 

ˆ − σ)) ≤ 0, q(s)Ψ (C A(s

s=n 3

that is, n−1 

ˆ − σ)) + q(s)Ψ (C A(s



ˆ j − σ − 1)) r (m j − 1)Ψ (C A(m

n 3 ≤m j −1≤n−1

s=n 3

≤ a(n 3 )Δy(n 3 ) − a(n)Δy(n) ≤ a(n 3 )Δy(n 3 ) < ∞ as n → ∞, a contradiction to (C3 ). Therefore, Case 1 doesn’t arise. In Case 2, Δ[a(n)Δy(n)] ≤ 0 for n ≥ n 2 implies that a(l)Δy(l) ≤ a(n)Δy(n) for l ≥ n ≥ n 2 , that is, −Δy(l) ≥

−a(n)Δy(n) a(l)

which on summation from n to l gives y(n) − y(l + 1) ≥ −a(n)Δy(n)

l+1  1 . a(s) s=n

Letting l → ∞, we obtain y(n) ≥ −a(n)Δy(n)

∞  1 ˆ = −a(n)Δy(n) A(n). a(s) s=n

(7)

Since a(n)Δy(n) is nondecreasing and negative, then there exist an n 3 > n 2 + 1 and ˆ a constant C > 0 such that a(n)Δy(n) ≤ −C for n ≥ n 3 implies that y(n) ≥ C A(n) due to (7). Therefore, ˆ − σ) for n ≥ n 4 > n 3 + σ + 1, y(n − σ) ≥ C A(n and because of (4), Ψ (y(n − σ)) =

ˆ − σ)) β Ψ (y(n − σ)) β Ψ (C A(n y (n − σ) ≥ y (n − σ). y β (n − σ) C β Aˆ β (n − σ)

Summing (6) from n(≥ n 4 ) to l − 1, it yields that

Oscillation of Second Order Impulsive Neutral Difference Equations …

177

y(n) − y(l)

⎡ l−1 s−1  1 ⎣ ≥ q(t)Ψ (y(t − σ)) + a(s) t=n s=n n 4



⎤ r (m j − 1)Ψ (y(m j − σ − 1))⎦

4 ≤m j −1 0 Supported in part by Croatian Science Foundation under the project IP-2019-04-1239 and by the University of Rijeka under the projects uniri-prirod-18-9 and uniri-pr-prirod-19-16. D. Dragiˇcevi´c (B) Faculty of Mathematics, University of Rijeka, Radmile Matejˇci´c 2, 51000 Rijeka, Croatia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_9

183

184

D. Dragiˇcevi´c

such that supn∈Z Bn  ≤ c. Then, provided that c is sufficiently small, the perturbed linear difference equation xn+1 = (An + Bn )xn n ∈ Z,

(2)

also admits an exponential dichotomy. For the most relevant early contributions to the robustness problem of exponential dichotomies, we refer to the works of Coppel [13], Massera and Schäffer [17, 18] and Daleckij and Krein [14]. For some more recent contributions, we refer to [2, 19, 20, 22, 23, 25, 26] and references therein. Despite its importance, the notion of a (uniform) exponential dichotomy is somewhat restrictive and it is of interest to look for weaker concepts of exponential dichotomy. Recently, Barreira and Valls introduced the notion of a nonuniform exponential dichotomy, which allows for the rate of contraction (expansion) along the stable (unstable) space to depend on an initial time. The relevance of this notion stems from its ubiquity in the context of ergodic theory. We refer to [8] for details. The robustness property of nonuniform exponential dichotomies for discrete time dynamics was first discussed in [7]. It asserts the following: assume that (1) admits a nonuniform exponential dichotomy with exponents λ > 0 and  ≥ 0 (see Definition 1) and that (Bn )n∈Z is a sequence of operators with the property that there exists c > 0 such that (3) Bn  ≤ ce−2|n| , for n ∈ Z. Then, provided that c is sufficiently small, (2) also admits a nonuniform exponential dichotomy. We stress that in [7] it is required that coefficients An of (1) are invertible linear operators. This result was extended in [3], by considering the case when coefficients An in (1) are not necessarily invertible operators, and by replacing 2 by  in (3). For some other relevant contributions to the robustness property of nonuniform dichotomies (including those with not necessarily exponential growth rates), we refer to [4, 5, 10, 15, 16, 27–29] and references therein. Assume that (1) admits a nonuniform exponential dichotomy with constants λ > 0 and  ≥ 0. Provided that λ > 2 > 0 and that (3) holds with 3 instead of 2 (as well as with a sufficiently small c), Barreira and Valls proved in [9] that (2) admits a nonuniform exponential dichotomy with constants λ and 2. The main objective of this note is to generalize the aforementioned result of Barreira and Valls. More precisely, we show that if (1) admits a nonuniform exponential dichotomy with arbitrary constants λ > 0 and  ≥ 0, and if (3) holds with  + ρ, ρ > 0 instead of 2, then (provided that c is sufficiently small) equation (2) admits a nonuniform exponential dichotomy with the constants λ and . In fact, our result is even slightly more general; see Remark 1 for details. Thus, in comparison to [9], we remove any assumptions on the relationship between λ and . In addition, we consider a larger class of perturbations (Bn )n∈Z . Finally, despite weaker assumptions from those in [9], we obtain a sharper conclusion: the exponents related to nonuniform exponential dichotomies of (1) and (2) are the same. We emphasize that results similar to ours have been obtained in [1, 11] for uniform exponential dichotomies and for an invertible continuous time dynamics acting on a finite-dimensional space.

On the Robustness Property of Nonuniform Exponential Dichotomies

185

Our techniques rely on those developed in [9] (which are inspired by those in [23]). The novelty is that we show that those can be combined with the use of the so-called Lyapunov norms (see [3]), which in some sense eliminate the nonuniformity of our dynamics.

2 Robustness Property Let X = (X, | · |) be an arbitrary Banach space. By B(X ) we will denote the space of all bounded linear operators on X equipped with the operator norm  · . For a sequence (An )n∈Z ⊂ B(X ), we set  A(m, n) =

Am−1 · · · An for m > n; Id for m = n,

where Id denotes the identity operator on X . We recall the notion of a nonuniform exponential dichotomy (see [8]). Definition 1 We say that a sequence (An )n∈Z ⊂ B(X ) admits a nonuniform exponential dichotomy with exponents λ > 0 and  ≥ 0 if there exist a sequence of projections (Pn )n∈Z ⊂ B(X ) and a constant D > 0 such that: • for n ∈ Z,

Pn+1 An = An Pn ;

and An |KerPn : KerPn → KerPn+1 is invertible; • for m ≥ n, • for m ≤ n,

where

A(m, n)Pn  ≤ De−λ(m−n)+|n| ;

(4)

A(m, n)(Id − Pn ) ≤ De−λ(n−m)+|n| ,

(5)

−1  A(m, n) := A(n, m)|KerPm : KerPn → KerPm

for m ≤ n. The following is our robustness result for nonuniform exponential dichotomies. Theorem 1 Assume that a sequence (An )n∈Z ⊂ B(X ) admits a nonuniform exponential dichotomy with exponents λ > 0 and  ≥ 0. Moreover, let (Bn )n∈Z ⊂ B(X ) be a sequence with the property that there exists c > 0 such that

186

D. Dragiˇcevi´c



e|k| Bk  ≤ c.

(6)

k∈Z

Then, if c is sufficiently small, the sequence (An + Bn )n∈Z admits a nonuniform exponential dichotomy with the same exponents λ > 0 and  ≥ 0. Proof We start by introducing appropriate adapted norms. For n ∈ Z and x ∈ X , let |x|n := sup (|A(m, n)Pn x|eλ(m−n) ) + sup (|A(m, n)(Id − Pn )x|eλ(n−m) ). m≥n

m≤n

It follows from (4) and (5) that |x| ≤ |x|n ≤ 2De|n| |x|, for x ∈ X and n ∈ Z.

(7)

The proof of the following lemma is established in [3, Proposition 7]. Lemma 1 For m ≥ n and x ∈ X , |A(m, n)Pn x|m ≤ e−λ(m−n) |x|n .

(8)

Moreover, for m ≤ n and x ∈ X , |A(m, n)(Id − Pn )x|m ≤ e−λ(n−m) |x|n .

(9)

Next, we introduce some auxiliary sequence spaces. Let l

∞

  := x = (xn )n∈Z ⊂ X : x∞ := sup |xn |n < +∞ n∈Z

and

   |xn |n < +∞ . l 1 := x = (xn )n∈Z ⊂ X : x1 := n∈Z

Using (7), it is easy to verify that (l ∞ ,  · ∞ ) and (l 1 ,  · 1 ) are Banach spaces. Lemma 2 For each y = (yn )n∈Z ∈ l 1 , there exists a unique x = (xn )n∈Z ∈ l ∞ such that (10) xn+1 − An xn = yn+1 , n ∈ Z. Proof Take an arbitrary y = (yn )n∈Z ∈ l 1 and define x = (xn )n∈Z ⊂ X by xn =



A(n, k)Pk yk −

k≤n

By (8) and (9), we have that

 k>n

A(n, k)(Id − Pk )yk , n ∈ Z.

On the Robustness Property of Nonuniform Exponential Dichotomies

|xn |n ≤



|A(n, k)Pk yk |n +

k≤n

≤



≤

|A(n, k)(Id − Pk )yk |n

k>n

e−λ(n−k) |yk |k +

k≤n





187

|yk |k +

k≤n





e−λ(k−n) |yk |k

k>n

|yk |k

k>n

= y1 , for each n ∈ Z. Thus, x ∈ l ∞ . Furthermore, we have xn+1 − An xn =

n+1 

A(n + 1, k)Pk yk − An

k=−∞ ∞ 

−

n 

A(n, k)Pk yk

k=−∞

A(n + 1, k)(Id − Pk )yk

k=n+2 ∞ 

+ An

A(n, k)(Id − Pk )yk

k=n+1

=

n+1 

A(n + 1, k)Pk yk −

k=−∞ ∞ 

− +

k=n+2 ∞ 

n 

A(n + 1, k)Pk yk

k=−∞

A(n + 1, k)(Id − Pk )yk A(n + 1, k)(Id − Pk )yk

k=n+1

= Pn+1 yn+1 + (Id − Pn+1 )yn+1 = yn+1 , for every n ∈ Z and therefore (10) holds. Finally, in order to establish the uniqueness of x, it is sufficient to consider the case when y = 0. Thus, let x = (xn )n∈Z ∈ l ∞ be such that xn+1 = An xn , n ∈ Z. It follows that Pn xn = A(n, k)Pk xk for every k ≤ n, which together with (8) implies that |Pn xn |n ≤ e−λ(n−k) |xk | ≤ e−λ(n−k) x∞ , for k ≤ n. Letting k → −∞ yields that Pn xn = 0 for each n ∈ Z. Similarly, (Id − Pn )xn = 0 for n ∈ Z and thus x = 0. The proof of the lemma is completed.

188

D. Dragiˇcevi´c

We define a linear operator T : D(T ) ⊂ l ∞ → l 1 by (T x)n = xn − An−1 xn−1 for n ∈ Z, on the domain D(T ) that consists of all x = (xn )n∈Z ∈ l ∞ such that T x ∈ l 1 . Lemma 3 We have that T : D(T ) → l 1 is a closed operator. Proof Let (xk )k∈N be a sequence in D(T ) such that xk → x in l ∞ and yk := T xk → y in l 1 . It follows easily from (7) that xnk → xn and ynk → yn ,

(11)

for each n ∈ Z. Since yk = T xk , we have that k k − An xnk = yn+1 , for n ∈ Z and k ∈ N. xn+1

(12)

By passing to the limit when k → ∞ in (12) and using (11), we conclude that (10) holds. Therefore, x ∈ D(T ) and T x = y. The proof of the lemma is completed. For x ∈ D(T ), set

xT := x∞ + T x∞ .

By Lemma 3, we have that (D(T ),  · T ) is a Banach space. Moreover, Lemma 2 implies that T : (D(T ),  · T ) → l 1 is an invertible operator. We now introduce an operator T : D(T ) ⊂ l ∞ → l 1 by (T x)n = xn − (An−1 + Bn−1 )xn−1 for n ∈ Z, on the domain D(T ) that consists of all x = (xn )n∈Z ∈ l ∞ such that T x ∈ l 1 . Lemma 4 We have that D(T ) = D(T ). Moreover, provided that c is sufficiently small, T : (D(T ),  · T ) → l 1 is an invertible operator. Proof By (6) and (7), we have that T x − T x1 =

 n∈Z

≤ 2D

|Bn−1 xn−1 |n 

e|n| Bn−1  · |xn−1 |

n∈Z

≤ 2D



e|n| Bn−1  · |xn−1 |n−1

n∈Z

≤ 2De x∞

 n∈Z

≤ 2Dce x∞ ,

e|n| Bn 

On the Robustness Property of Nonuniform Exponential Dichotomies

189

and thus T x − T x1 ≤ 2Dce xT , for every x = (xn )n∈Z ∈ l ∞ . The above estimate readily implies that D(T ) = D(T ). In addition, since T is invertible, we have that T is also invertible when c is sufficiently small. This completes the proof of the lemma. 

Set

(Am−1 + Bm−1 ) · · · (An + Bn ) for m > n; ¯ A(m, n) = Id for m = n. Furthermore, for n ∈ Z set   ¯ S(n) := v ∈ X : sup |A(m, n)v|m < +∞ , m≥n

and let U(n) consists of all v ∈ X with the property that there exists a sequence ¯ (xm )m≤n ⊂ X such that xn = v, xm = A(m, m − 1)xm−1 for m ≤ n and supm≤n |xm |m < +∞. Clearly, S(n) and U(n) are subspaces of X for every n ∈ Z. Lemma 5 For n ∈ Z,

X = S(n) ⊕ U(n).

(13)

Proof Fix an arbitrary n ∈ Z and take v ∈ X . We define y = (ym )m∈Z ⊂ X by  v m = n; ym = 0 m = n.

(14)

Clearly, y ∈ l 1 . Since T is invertible, there exists x = (xm )m∈Z ∈ l ∞ such that xm+1 − (Am + Bm )xm = ym+1 , for m ∈ Z. Hence, xn − (An−1 + Bn−1 )xn−1 = v and xm = (Am−1 + Bm−1 )xm−1 , m = n. Thus, xn ∈ S(n) and (An−1 + Bn−1 )xn−1 ∈ U(n), which implies that v = xn − (An−1 + Bn−1 )xn−1 ∈ S(n) + U(n). Assume now that v ∈ S(n) ∩ U(n). Then, there exists a sequence (z m )m≤n ⊂ X such ¯ that z n = v, z m = A(m, m − 1)z m−1 for m ≤ n and supm≤n |z m |m < +∞. We define

190

D. Dragiˇcevi´c

x = (xm )m∈Z ⊂ X by

 xm =

¯ A(m, n)v m ≥ n; zm m < n.

Then, x ∈ l ∞ . Moreover, T x = 0. Since T is invertible, we conclude that x = 0 and therefore xn = v = 0. The proof of the lemma is completed. Lemma 6 For n ∈ Z, (An + Bn )S(n) ⊂ S(n + 1) and (An + Bn )U(n) = U(n + 1). Moreover, (An + Bn )|U (n) : U(n) → U(n + 1) is invertible. ¯ Proof Take an arbitrary n ∈ Z and v ∈ S(n). Then, supm≥n |A(m, n)v|m < +∞. Hence, ¯ ¯ ¯ n + 1)(An + Bn )v|m = sup |A(m, n)v|m ≤ sup |A(m, n)v|m , sup |A(m, m≥n+1

m≥n+1

m≥n

and therefore (An + Bn )v ∈ S(n + 1). Take now v ∈ U(n). Then, there exists a sequence (z m )m≤n ⊂ X such that z n = v, ¯ m − 1)z m−1 for m ≤ n and supm≤n |z m |m < +∞. We define a sequence z m = A(m, (¯z m )m≤n+1 ⊂ X by  zm m ≤ n; z¯ m = (An + Bn )v m = n + 1. ¯ Then, z¯ m = A(m, m − 1)¯z m−1 for m ≤ n + 1 and supm≤n+1 |¯z m |m < +∞. Consequently, (An + Bn )v ∈ U(n + 1). Let w ∈ U(n + 1). Then, there exists a sequence (z m )m≤n+1 ⊂ X such that z n+1 = ¯ m − 1)z m−1 for m ≤ n + 1 and supm≤n+1 |z m |m < +∞. Clearly, z n ∈ w, z m = A(m, ¯ + 1, n)z n = (An + Bn )z n , we conclude that U(n). Moreover, since w = z n+1 = A(n w ∈ (An + Bn )U(n). In order to complete the proof of the lemma, we need to show that (An + Bn )|U (n) : U(n) → U(n + 1) is injective. Take v ∈ U(n) and assume that (An + ¯ m− Bn )v = 0. There exists a sequence (z m )m≤n ⊂ X such that z n = v, z m = A(m, 1)z m−1 for m ≤ n and supm≤n |z m |m < +∞. We define a sequence x = (xm )m∈Z ⊂ X by  zm m ≤ n; xm = ¯ A(m, n)v m > n. Then, x ∈ l ∞ and T x = 0. Since T is invertible, x = 0 and consequently xn = v = 0. The proof of the lemma is completed.

On the Robustness Property of Nonuniform Exponential Dichotomies

191

For V ∈ B(X ) and n, m ∈ Z, set V n→m := sup |V x|m . |x|n ≤1

Observe that for V1 , V2 ∈ B(X ) and n, m, k ∈ Z, V1 V2 n→m ≤ V1 k→m · V2 n→k . Let I = {(m, n) ∈ Z × Z : m ≥ n}.   C = U : I → B(X ) : U C < +∞ ,

We consider

where

U C := sup{U (m, n)n→m eλ(m−n) : m ≥ n}.

Then, (C,  · C ) is a Banach space. Lemma 7 Provided that c is sufficiently small, there exists a unique U ∈ C such that U (m, n) = A(m, n)Pn +

m−1 

A(m, k + 1)Pk+1 Bk U (k, n)

k=n

−

∞ 

(15)

A(m, k + 1)Q k+1 Bk U (k, n),

k=m

for (m, n) ∈ I , where Q k = Id − Pk , k ∈ Z. Furthermore, for each ξ ∈ X , the sequence xm = U (m, n)ξ satisfies ¯ + 1, m)xm , m ≥ n. xm+1 = A(m

(16)

Proof For U ∈ C, let (LU )(m, n) := A(m, n)Pn +

m−1 

A(m, k + 1)Pk+1 Bk U (k, n)

k=n

−

∞ 

A(m, k + 1)Q k+1 Bk U (k, n).

k=m

We first claim that LU ∈ C. We begin by observing that (7) implies that Bk k→k+1 ≤ 2De e|k| Bk , k ∈ Z.

(17)

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By (8), (9) and (17) we have that (LU )(m, n)n→m ≤e

−λ(m−n)

+ 2De



m−1 

e−λ(m−k−1) e|k| Bk  · U (k, n)n→k

k=n

+ 2De

∞ 

e−λ(k+1−m) e|k| Bk  · U (k, n)n→k

k=m

≤ e−λ(m−n) + e−λ(m−n) 2De+λ U C

m−1 

(18)

e|k| Bk 

k=n

+ e−λ(m−n) 2De−λ U C

∞ 

e−2λ(k−m) e|k| Bk ,

k=m

for (m, n) ∈ I , which together with (6) yields that LU ∈ C. Take now U1 , U2 ∈ C. Proceeding as in (18), we have that (LU1 )(m, n) − (LU2 )(m, n)n→m ≤ 2De

m−1 

e−λ(m−k−1) e|k| Bk  · U1 (k, n) − U2 (k, n)n→k

k=n ∞   −λ(k+1−m) |k|

+ 2De

e

e

Bk  · U1 (k, n) − U2 (k, n)n→k

k=m

≤ e−λ(m−n) 2De+λ U1 − U2 C

m−1 

e|k| Bk 

k=n ∞ 

+ e−λ(m−n) 2De−λ U1 − U2 C

e−2λ(k−m) e|k| Bk ,

k=m

for (m, n) ∈ I , which together with (6) gives that LU1 − LU2 C ≤ c(2De+λ + 2De−λ )U1 − U2 C . Hence, provided that c is sufficiently small, we conclude that L is a contraction on C, which yields the first assertion of the lemma. The second assertion of the lemma can be established by a straightforward computation. Lemma 8 Provided that c is sufficiently small, U given by Lemma 7 satisfies that U (m, l)U (l, n) = U (m, n), m ≥ l ≥ n. In particular, U (n, n) is a projection for each n ∈ Z.

On the Robustness Property of Nonuniform Exponential Dichotomies

193

Proof Let us fix n ∈ Z and consider In := {(m, l) ∈ Z × Z : m ≥ l ≥ n}. Furthermore, set Cn := {H : In → B(X ) : H n < +∞}, where H n := sup{H (m, l)l→m : (m, l) ∈ In }. Then, (Cn ,  · n ) is a Banach space. We consider L 1 : Cn → Cn by (L 1 H )(m, l) =

m−1 

A(m, k + 1)Pk+1 Bk H (k, l)

k=l ∞ 

−

A(m, k + 1)Q k+1 Bk H (k, l),

k=m

for (m, l) ∈ In . We first claim that L 1 is well-defined. Indeed, for any H ∈ Cn , it follows from (8), (9) and (17) that (L 1 H )(m, l)l→m ≤

m−1 

e−λ(m−k−1) Bk k→k+1 · H (k, l)l→k

k=l ∞ 

+

e−λ(k+1−m) Bk k→k+1 · H (k, l)l→k

k=m

≤ 2De H n



e|k| Bk ,

k∈Z

for (m, l) ∈ In . By (6), we conclude that L 1 H ∈ Cn . In addition, a completely analogous argument gives that L 1 H1 − L 1 H2 n ≤ 2De H1 − H2 n



e|k| Bk  forH1 , H2 ∈ Cn .

k∈Z

Hence, it follows from (6) that L 1 is a contraction on Cn provided that c is sufficiently small. Therefore, L 1 has a unique fixed point. On the other hand, we observe that 0 ∈ Cn is a fixed point for L 1 . We define h(m, l) = U (m, l)U (l, n) − U (m, n), (m, l) ∈ In .

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Note that h(m, l)l→m ≤ U (m, l)l→m · U (l, n)n→l + U (m, n)n→m ≤ U n · U n + U n , for (m, l) ∈ In , and thus h ∈ Cn . A simple computation shows that h is a fixed point of L 1 . We conclude that h = 0, which yields the desired conclusion. Lemma 9 Let U be given by Lemma 7. Then, ImU (n, n) ⊂ S(n) for each n ∈ Z. Moreover, there exists K > 0 such that ¯ |A(m, n)U (n, n)ξ |m ≤ K e−λ(m−n) |ξ |n , for m ≥ n and ξ ∈ X.

(19)

Proof Take an arbitrary n ∈ Z and ξ ∈ X . It follows from Lemma 7 that the sequence (xm )m≥n given by xm = U (m, n)ξ satisfies (16). Hence, ¯ n)U (n, n)ξ, xm = A(m,

(20)

for m ≥ n. Moreover, |xm |m = |U (m, n)ξ |m ≤ U (m, n)n→m · |ξ |n ≤ U C e−λ(m−n) |ξ |n ,

(21)

for m ≥ n. Hence, supm≥n |x|m < +∞ and thus U (n, n)ξ ∈ S(n). We have proved that ImU (n, n) ⊂ S(n). Finally, (20) and (21) imply that (19) holds with K = U C > 0. Let J := {(m, n) ∈ Z × Z : m ≤ n}, and D := {V : J → B(X ) : V D < +∞}, where

V D := sup{V (m, n)n→m e−λ(m−n) : (m, n) ∈ J }.

By arguing as in the proof of Lemma 7, one can establish the following lemma. Lemma 10 Provided that c is sufficiently small, there exists a unique V ∈ D such that V (m, n) = A(m, n)Q n +

m−1 

A(m, k + 1)Pk+1 Bk V (k, n)

k=−∞

−

n−1  k=m

A(m, k + 1)Q k+1 Bk V (k, n).

On the Robustness Property of Nonuniform Exponential Dichotomies

195

Moreover, for each ξ ∈ X the sequence xm = V (m, n)ξ , m ≤ n satisfies ¯ + 1, m)xm , m < n. xm+1 = A(m

(22)

The proof of the following lemma can be obtained by repeating the arguments in the proof of Lemma 8. Lemma 11 Provided that c is sufficiently small, V given by Lemma 10 satisfies that V (m, l)V (l, n) = V (m, n), m ≤ l ≤ n. In particular, V (n, n) is a projection for each n ∈ Z. Lemma 12 Let V be given by Lemma 10. Then, ImV (n, n) ⊂ U(n) for each n ∈ Z. Moreover, there exists K > 0 such that ¯ |A(m, n)V (n, n)ξ |m ≤ K e−λ(n−m) |ξ |n , for m ≤ n and ξ ∈ X.

(23)

Proof Take an arbitrary n ∈ Z and ξ ∈ X . It follows from Lemma 10 that the ¯ m− sequence (xm )m≤n given by xm = V (m, n)ξ satisfies (22). Hence, xm = A(m, 1)xm−1 for m ≤ n. Moreover, |xm |m = |V (m, n)ξ |m ≤ V (m, n)n→m · |ξ |n ≤ V D e−λ(n−m) |ξ |n ,

(24)

for m ≤ n. Hence, supm≤n |x|m < +∞ and thus xn = V (n, n)ξ ∈ U(n). We have ¯ n)V (n, n)ξ , it folproved that ImV (n, n) ⊂ U(n). Finally, noting that xm = A(m, lows from (24) that (23) holds with K = V D > 0. Lemma 13 Provided that c is sufficiently small, U (n, n) + V (n, n) is invertible for each n ∈ Z. Proof Set Sn := U (n, n) + V (n, n), n ∈ Z. It follows from Lemmas 7 and 10 that Sn − Id = −

∞ 

A(n, k + 1)Q k+1 Bk U (k, n)

k=n

+

n−1 

A(n, k + 1)Pk+1 Bk V (k, n),

k=−∞

for n ∈ Z. By (8) and (9), we have that

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Sn − Idn→n ≤

∞ 

e−λ(k+1−n) Bk k→k+1 · U (k, n)n→k

k=n n−1 

+

e−λ(n−k−1) Bk k→k+1 · V (k, n)n→k

k=−∞

≤ e−λ U C

∞ 

e−2λ(k−n) Bk k→k+1

k=n λ

+ e V D

n−1 

e−2λ(n−k) Bk k→k+1

k=−∞ λ

≤ e (U C + V D )



Bk k→k+1 ,

k∈Z

which together with (17) implies that Sn − Idn→n ≤ 2De+λ (U C + V D )



e|k| Bk ,

k∈Z

for each n ∈ Z. Thus, (6) implies that Sn − Idn→n ≤ 2cDe+λ (U C + V D ), forn ∈ Z, which provided that c is sufficiently small readily implies the desired conclusion. Lemma 14 We have that ImU (n, n) = S(n) and ImV (n, n) = U(n). Proof We recall that ImU (n, n) ⊂ S(n) and ImV (n, n) ⊂ U(n). Take now v ∈ S(n). It follows from the previous lemma that there exists w ∈ X such that v = U (n, n)w + V (n, n)w. It follows that v − U (n, n)w = V (n, n)w ∈ S(n) ∩ U(n) and thus v = U (n, n)w ∈ ImU (n, n). We conclude that S(n) ⊂ Im U (n, n), and thus S(n) = ImU (n, n). Similarly, one prove that ImV (n, n) = U(n). In order to complete the proof of the theorem it is sufficient to note that (7) and (19) imply that ¯ |A(m, n)U (n, n)ξ | ≤ 2D K e−λ(m−n)+|n| |ξ |, for m ≥ n and ξ ∈ X. Moreover, (7) and (23) imply that ¯ |A(m, n)V (n, n)ξ |m ≤ 2D K e−λ(n−m)+|n| |ξ |n , for m ≤ n and ξ ∈ X.

On the Robustness Property of Nonuniform Exponential Dichotomies

197

We conclude that the sequence (An + Bn )n∈Z admits a nonuniform exponential dichotomy with exponents λ and . The proof of the theorem is completed. Remark 1 Let us compare Theorem 1 with [9, Theorem 1]. We first note that [9, Theorem 1] requires that  > 0 and λ > 2, while in Theorem 1 we don’t need these assumptions. Moreover, take any ρ > 0 and assume that there exists c > 0 such that Bk  ≤ ce−(+ρ)|k| , k ∈ Z. Then,

 k∈Z

e|k| Bk  ≤ c

(25)

1 + e−δ . 1 − e−δ

Hence, provided that c is sufficiently small, Theorem 1 is applicable. On the other hand, we note that [9, Theorem 1] requires that (25) holds with ρ = 2. Finally, we emphasize that in [9, Theorem 1] it is concluded that (An + Bn )n∈Z admits a nonuniform exponential dichotomy with exponents λ and 2, while we show (under milder assumptions) that it admits a nonuniform exponential dichotomy with exponents λ and .

References 1. Battelli, F., Feˇckan, M.: On the exponents of exponential dichotomies. Mathematics 8, 651 (2020) 2. Battelli, F., Franca, M., Palmer, K.J.: Exponential dichotomy for noninvertible linear difference equations. J. Differ. Equ. Appl. 27, 1657–1691 (2021) 3. Barreira, L., Dragiˇcevi´c, D., Valls, C.: Exponential dichotomies with respect to a sequence of norms and admissibility. Internat. J. Math. 25, 1450024 (2014), 20 pp 4. Barreira, L., Dragiˇcevi´c, D., Valls, C.: Tempered exponential dichotomies: admissibility and stability under perturbations. Dyn. Syst. 31, 525–545 (2016) 5. Barreira, L., Dragiˇcevi´c, D., Valls, C.: Nonuniform hyperbolicity and one-sided admissibility. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27, 235–247 (2016) 6. Barreira, L., Dragiˇcevi´c, D., Valls, C.: Admissibility and Hyperbolicity. SpringerBriefs in Mathematics, Springer, Cham (2018) 7. Barreira, L., Silva, C., Valls, C.: Nonuniform behavior and robustness. J. Differ. Equ. 246, 3579–3608 (2009) 8. Barreira, L., Valls, C.: Stability of nonautonomous differential equations. In: Lecture Notes in Mathematics, vol. 1926. Springer, Berlin (2008) 9. Barreira, L., Valls, C.: Nonuniform cocycles: robustness of exponential dichotomies. Discret. Contin. Dyn. Syst. 32, 4111–4131 (2012) 10. Bento, A.J., Silva, C.: Robustness of discrete nonuniform dichotomic behavior. arXiv:1308.6820 11. Calamai, A., Franca, M.: Mel’nikov methods and homoclinic orbits in discontinuous systems. J. Dyn. Differ. Equ. 25, 733–764 (2013)

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12. Chicone, C., Latushkin, Y.: Evolution semigroups in dynamical systems and differential equations. In: Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence, RI (1999) 13. Coppel, W.A.: Dichotomies in Stability Theory. In: Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978) 14. Daleckij, J.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence, RI (1974) 15. Dragiˇcevi´c, D.: Admissibility and nonuniform polynomial dichotomies. Math. Nachr. 293, 226–243 (2019) 16. Dragiˇcevi´c, D., Zhang, W.: Asymptotic stability of nonuniform behaviour. Proc. Am. Math. Soc. 147, 2437–2451 (2019) 17. Massera, J.L., Schäffer, J.J.: Linear differential equations and functional analysis I. Ann. Math. 67, 517–573 (1958) 18. Massera, J.L., Schäffer, J.J.: Linear Differential Equations and Function Spaces. Academic Press, New York (1966) 19. Naulin, R., Pinto, M.: Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal. 31, 559–571 (1998) 20. Palmer, K.: A perturbation theorem for exponential dichotomies. Proc. Roy. Soc. Edinburgh Sect. A 106, 25–37 (1987) 21. Perron, O.: Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32, 703–728 (1930) 22. Plis, V.A., Sell, G.R.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 11, 471–513 (1999) 23. Popescu, L.: Exponential dichotomy roughness on Banach spaces. J. Math. Anal. Appl. 314, 436–454 (2006) 24. Pötzsche, C.: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics, vol. 2002. Springer, Berlin (2010) 25. Sasu, A.L.: Exponential dichotomy and dichotomy radius for difference equations. J. Math. Anal. Appl. 344, 906–920 (2008) 26. Sasu, A.L., Sasu, B.: Strong exponential dichotomy of discrete nonautonomous systems: inputoutput criteria and strong dichotomy radius. J. Math. Anal. Appl. 504, 125373 (2021) 27. Silva, C.M.: Admissibility and generalized nonuniform dichotomies for discrete dynamics. Commun. Pure Appl. Anal. 20, 3419–3443 (2021) 28. Zhou, L., Lu, K., Zhang, W.: Roughness of tempered exponential dichotomies for infinitedimensional random difference equations. J. Differ. Equ. 254, 4024–4046 (2013) 29. Zhou, L., Zhang, W.: Admissibility and roughness of nonuniform exponential dichotomies for difference equations. J. Funct. Anal. 271, 1087–1129 (2016)

Implicit Linear First Order Difference Equations Over Commutative Rings Sergey Gefter, Anna Goncharuk, and Aleksey Piven’

Abstract In this paper we study the problem of existence and uniqueness of solutions of implicit linear difference equations over commutative rings. The cases where the ring is integrally closed, Noetherian, local are considered. The question of periodic and quasi-polynomial solutions of this equation is also studied. In the case of a local ring, we obtain a formula for the solution of the equation as a series converging in M-adic topology, where M is the maximal ideal of the ring in question. Keywords Implicit difference equation · Local ring · Noetherian ring · Periodical solution · Quasi-polynomial solution · M-adic topology MSC: 39A99 · 16P50

1 Introduction Let X and Y be real or complex vector spaces, A, B linear operators from X into Y , and f n (n = 0, 1, 2, 3, ...) a given sequence of elements from Y . Consider the following linear difference equation Bxn+1 + Axn = f n , n = 0, 1, 2, ....

(1)

If B be a non-invertible operator, then Eq. (1) is called implicit. Note that, in distinction from the classical theory of explicit difference equations (see, for example, S. Gefter · A. Goncharuk (B) · A. Piven’ School of Mathematics and Computer Science, V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv 61022, Ukraine e-mail: [email protected] S. Gefter e-mail: [email protected] A. Piven’ e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_10

199

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[1–3]), the implicit Eq. (1) can have no solution for some initial condition x0 ∈ X . In [4–8] and other works Eq. (1) in finite dimensional and Banach spaces was studied. In [9] the implicit linear difference equation in different classes of topological vector spaces was investigated. In the articles [10–12], in particular, a linear difference equation bxn+1 + axn = f n , n = 0, 1, 2, ....

(2)

over the ring of integers Z, which is analogous to Eq. (1), was studied. Here a, b, f n ∈ Z (n = 0, 1, 2, ...) and b = 0, ±1. The condition b = ±1 means that the element b is non-invertible in the ring of integers. Therefore Eq. (2) is also called implicit. As noted in [10], the implicit Eq. (2) does not always have a solution in integers, and in many cases it can not have more than one such solution. In Sect. 1 of this paper we study Eq. (2) over an arbitrary commutative ring R with identity in the case of a constant sequence f n ∈ R. As well as for R = Z, if b is a noninvertible element of the ring R, then Eq. (2) is called implicit. If a is not divisible by b in ring R, then Eq. (2) is called completely implicit (see Definitions 1 and 2). It is proved in this section that for a Noetherian local ring the implicit equation f , n = 0, 1, 2, . . . bxn+1 = xn + f for any f ∈ R has a unique solution xn = b−1 (Theorem 4). It is shown that for any subring of the ring Z of all integer algebraic numbers the implicit homogeneous equation bxn+1 = xn , n = 0, 1, 2, ... has only trivial solution (see Theorem 3). An example of such a subring of Z for which some completely implicit homogeneous equation bxn+1 = axn , n = 0, 1, 2, ... has a nontrivial solution is given (see Example 1). There is also an example of an integral domain for which some implicit homogeneous equation bxn+1 = xn , n = 0, 1, 2, ... has a non-trivial solution (Example 4). In Sect. 2 we study periodic solutions of the linear difference equation bn xn+1 = an xn + f n , n = 0, 1, 2, .... with periodic coefficients from a commutative ring. The main result of the section is Theorem 7 on existence and uniqueness of the periodic solution of this equation (see also Corollaries 9 and 10). In Sect. 3 we consider another generalization of the linear implicit equation with constant non-homogeneity, that is the equation with quasi-polynomial nonhomogeneity (see Eq. (18)). For this equation the theorem on existence and uniqueness of a quasi-polynomial solution is obtained (Theorem 8). In the final section of the paper, the general implicit non-homogeneous Eq. (2) is studied in the case when the commutative ring R is Noetherian and local. In this case, the solution of the equation is represented as a sum of series which converges in the M-adic topology of the ring R, where M is its maximal ideal (see Theorem 9). As a corollary, we obtain an explicit “ultrametric” formula for the quasi-polynomial solution of the implicit equation with quasi-polynomial non-homogeneity in the case of a local Noetherian domain (see Theorem 10 and Equality (24)).

Implicit Linear First Order Difference Equations …

201

The research was supported by the National Research Foundation of Ukraine funded by Ukrainian State budget in frames of project 2020.02/0096 “Operators in infinite-dimensionalspaces: the interplay between geometry, algebra and topology”.

2 Implicit Linear Difference Equation bxn+1 = axn + f Let R be a commutative ring with identity, a, b ∈ R, and { f n }∞ n=0 ⊂ R. Let us consider the following linear non-homogeneous difference equation over the ring R bxn+1 = axn + f n , n = 0, 1, 2, ....

(3)

Definition 1 The difference Eq. (3) is called implicit, if b is a non-invertible element of R. Definition 2 Let R be an integral domain. The difference Eq. (3) is called completely implicit, if a is not divisible by b in the ring R. In particular, in this case the element b is non-invertible. First study the uniqueness of the solution of Eq. (3). For this reason it is enough to study the homogeneous implicit difference equation bxn+1 = axn , n = 0, 1, 2, ....

(4)

Let R be an integral domain and F its quotient field. Let us remind that x ∈ F is almost integral over R if there exists 0 = d ∈ R such that d x n ∈ R for any n = 0, 1, 2, ... (see [13, Exercises to Sect. 9, p. 69]). Say that R is completely integrally closed if every x ∈ F, which is almost integral over R, belongs to R [13, Exercises to Sect. 9, p. 69] (see also [14, Chap. 5, Sect. 1.4, p. 312]). The following theorem is a corollary from [15, Proposition 2.3] but we reformulate and prove this proposition as a criterion of uniqueness of solution of Eq. (3). Denote by b R the principal ideal that is generated by b. Theorem 1 Let R be an integral domain. Equation (4) has only trivial solution over R for any a, b ∈ R such that a ∈ / b R if and only if the ring R is completely integrally closed. Proof Sufficiency. Let R be completely integrally closed. Consider Eq. (4) where over R, then it is also a solution a∈ / b R. If {xn }∞ n=0 is a solution of this equation  n over F. This solution has the form xn = ab x0 where ab ∈ F. Thus, ab is an almost integral element over R. Since the ring R is completely integrally closed, we have a ∈ R. This contradicts the condition a ∈ / b R. b Necessity. Suppose that Eq. (4) has only trivial solution over R for any a, b ∈ R such that a ∈ / b R. Let r = αβ ∈ F be an almost integral element over R, where α, β ∈ R. We prove that r ∈ R. Assume the contrary that r ∈ / R. Then α ∈ / β R. Since r is

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an almost integral element over R, there exists an nonzero element x0 ∈ R such that xn = r n x0 ∈ R, n ∈ N. The sequence xn is a non-trivial solution of the implicit equation βxn+1 = αxn , n = 0, 1, 2, .... This contradicts the assumption about the triviality of {xn }∞ n=0 . Corollary 1 Let R be a completely integrally closed ring and f ∈ R. The completely implicit difference equation bxn+1 = axn + f, n = 0, 1, 2, . . .

(5)

has a solution over R if and only if f ∈ (b − a)R. In this case the unique solution f , n = 0, 1, 2, . . . . of Eq. (5) is constant: xn = b−a Proof By Theorem 1 Eq. (4) has at most one solution over the ring R. Let {xn }∞ n=0 be a solution of Eq. (5). Then the sequence yn = xn+1 − xn is a solution of the homogeneous Eq. (4). Hence, yn = 0, n = 0, 1, 2, .... Therefore xn = xn+1 , n = 0, 1, 2, .... Then x0 is a solution of the equation (b − a)x0 = f , i.e. f ∈ (b − a)R. The statement is proved. Now consider the equation bxn+1 = xn , n = 0, 1, 2, ...,

(6)

which is a particular case of Eq. (4) with a = 1. ∞ 

Theorem 2 1. Let

(b R)n = {0}. Then Eq. (6) has only trivial solution over the

n=1

ring R. 2. If b is not a zero divisor and Eq. (6) has only trivial solution over R then ∞  (b R)n = {0}. n=1

Proof The part 1. Let {xn }∞ n=0 be a solution of Eq. (6). Then x j = bn xn+ j , Therefore, x j ∈

∞ 

(b R)n ,

j, n = 0, 1, 2, ....

j = 0, 1, 2, ..., i.e. x j = 0 for any j = 0, 1, 2, .... Let

n=1

us now prove the part 2. Let Eq. (6) have only trivial solution over R. Let 0 = ∞  (b R)n . Then for any n ∈ N there exists an element xn such that x0 = bn xn . x0 ∈ n=1

Therefore x0 = bn xn = bn+1 xn+1 , i.e. bn (xn − bxn+1 ) = 0, n = 0, 1, 2, .... Since b is not a zero divisor, we have bxn+1 − xn = 0, n = 0, 1, 2, .... This implies that Eq. (6) has a non-trivial solution. The theorem is proved. Corollary 2 Let R be a Noetherian domain and b is a non-invertible element of R. Then implicit Eq. (6) has only trivial solution over R.

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203

Proof The assertion of the Corollary follows from Theorem 2 and Krull’s Intersection Theorem (see [14, Chap. 3, Sect. 3.2, p. 200]). √ √ Example 1 Let d ∈ Z be a square-free integer and R = Z[ d] = {m + l d : m, l ∈ Z}. Then R is a Noetherian domain (see [14, Chap. 3, Sect. 2.10, p. 184]). Hence implicit Eq. (6) for any non-invertible element b ∈ R has only trivial solution over R. However, that would not be case for any d for the completely implicit √ Eq. (4). Indeed, if d = 4k + 1, k ∈ Z (for example, d = 3), then Z[ d] is completely integrally closed ring [14, Chap. √ 5, Sect. 1.4]. Hence completely implicit Eq. (4) has only trivial solution over Z[ d]. In the case d = 4k √ + 1 (for example, if d = 5) completely implicit Eq. (4) with b = 2 and a = 1 + d has the non-trivial solution 

√ n 1+ d xn = 2 2 √ √ over√Z[ d]. To prove that xn ∈ Z[ d] it is sufficient to note that x0 = 2, x1 = 1 + d and xn+2 = xn+1 + kxn , n = 0, 1, 2, .... The following theorem shows that the uniqueness of the solution of implicit Eq. (6) holds for any subring of the ring Z of all algebraic integers. Let us remind that an ∞  integral domain R is called Archimedean if (b R)n = {0} for any non-invertible element b ∈ R (see [15, Sect. 2]).

n=1

Theorem 3 Arbitrary subring R of Z is Archimedean. Thus Eq. (6) has only trivial solution over R for any non-invertible element b ∈ R. Proof First we prove that Z is Archimedean. Let b = 0 be a non-invertible element of ∞  Z. Let us show that (bZ)n = {0}. Assume to the contrary, that there exists element 0 = x0 ∈ ideal xn +

∞  n=1 n−1 

∞ 

n=1

(bZ) . We show that there exists a nonzero integer which belongs to the n

n=1

(bZ)n . Indeed, x0 is a root of some irreducible over Z polynomial p(x) = a j x j ∈ Z[x]. Then a0 ∈ Z and a0 = 0. By Vieta’s theorem there exists

j=0

a number α ∈ Z such that x0 α = a0 . Therefore a0 ∈

∞ 

(bZ)n . Since the element

n=1

b is algebraic over Q, by Proposition 1.4 in [16, Chap. 5, Sect. 1] there exists a finite extension S = {q(b) : q(x) ∈ Q[x]} of Q which contains b. Then the ring O S = S ∩ Z is the ring of algebraic integers from S. Moreover Z ⊂ O S and b ∈ O S . / Z the element b is non-invertible element of the ring O S . Hence bO S Since b1 ∈ is a proper ideal of O S . By Corollary 1 in [18, Chap. 12, Sect. 2] the ring O S is a Noetherian domain. By Krull’s Intersection Theorem

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(bO S )n = {0}.

(7)

n=1

Since a0 ∈

∞ 

(bZ)n then for any n ∈ N there exists an element z n ∈ Z such that

n=1

a0 = bn z n . Moreover, since S is a field, we have z n = ∞  (bO S )n . This contradicts to (7). Thus a0 ∈

a0 bn

∈ S and therefore z n ∈ O S .

n=1

Now let b = 0 be a non-invertible element of R. We show that

∞ 

(b R)n = {0}.

n=1

To prove this assertion it is sufficient to show that b is a non-invertible element of n−1  Z. Assume b1 ∈ Z. Then there exists a polynomial p(x) = x n + a j x j ∈ Z[x] j=0

such that p( b1 ) = 0. Now taking into account the inclusion Z ⊂ R we obtain n−1  1 = − a j bn−1− j ∈ R. This contradicts the assumption that b is a non-invertible b j=0

element of R. Since R is an integral domain, then Theorem 2 yields the uniqueness of the solution of Eq. (6). The theorem is proved. From our point of view in connection to Theorem 3 there are two following interesting problems. Problem 1. Is it true that an arbitrary subring of the field Q of all algebraic numbers is Archimedean? Problem 2. Does there exist a subring of the field C of complex numbers which is not Archimedean? In the sequel we consider following question: what kind of commutative rings give uniqueness of the solution of the implicit equation bxn+1 = xn + f, n = 0, 1, 2, . . .

(8)

for any element f . Proposition 1 If the implicit Eq. (8) for any f ∈ R has a unique solution then the ring R is local, i.e. R has a unique maximal ideal (see [14, Chap. 2, Sect. 3.1, p. 80]). Proof Similarly to the proof of Corollary 1 we obtain f ∈ (b − 1)R for any f ∈ R and any non-invertible b ∈ R. This means that for any non-invertible b the equation (b − 1)x = f has a solution for any f ∈ R, i.e. the element b − 1 is invertible for any non-invertible b. This is equivalent to R is local [14, Chap. 2, Sect. 3.1, p. 80]. Theorem 4 Let R be a local Noetherian ring. Then for any f ∈ R the implicit Eq. (8) f , n = 0, 1, 2, . . . . has the unique solution xn = b−1 Proof Since R is local, the element 1 − c is invertible for any non-invertible c ∈ R. f Hence the implicit Eq. (8) has the constant solution xn = b−1 , n = 0, 1, 2, . . .. By

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205

Krull’s Intersection Theorem [14, Chap. 3 Sect. 3.2, p. 200] and Theorem 2 we obtain that this solution is unique. Now from Proposition 1, Theorem 2 and from the proof of Theorem 4 we obtain the following result. Theorem 5 Let R be an integral domain. Every implicit equation (8) has a unique solution if and only if the ring R is local and Archimedean. Example 2 Let R = K [[ξ]] be the ring of formal power series over a Noetherian local ring K . Then K [[ξ]] also is a Noetherian local ring. Let b ∈ K [[ξ]] and b(0) = 0. Then b is a non-invertible element of K [[ξ]]. By Theorem 4 Eq. (8) for any f ∈ f (ξ) K [[ξ]] has the unique solution xn (ξ) = b(ξ)−1 , n = 0, 1, 2, . . .. Let us now give an example of a local ring for which the implicit homogeneous Eq. (6) has a non-trivial solution. Example 3 Let R be the ring of germs at t0 = 0 of real-valued functions continuous in a neighbourhood t0 , b be the germ of the function b(t) = t and x0 be the germ of 1 the function x0 (t) = e− t 2 , where x0 (0) = 0. Note that x0(n) (0) = 0, n = 0, 1, 2, .... Therefore x0 is divisible by bn in the ring R for any n ∈ N. Therefore, Eq. (6) over R has the non-trivial solution xn = bxn0 . Note that the ring R is local [14, Chap. 2 Sect. 3], but it is not Noetherian. The ring of germs from Example 3 is not an integral domain. The following example shows that the implicit homogeneous Eq. (6) with a non-invertible element b can have a non-trivial solution over an integral domain. Example 4 Let K be a field of zero characteristic. Let us consider the following

subring of the field of rational functions K (ξ, η): R = K ξ, η, ηξ , ηξ2 , ηξ3 , ... (see [14, Chap. 6, Sect. 8, Exercize 4]). Let b = η. Then b is a non-invertible element of R. Since xn = ηξn ∈ R, n = 0, 1, 2, .... this sequence is a non-trivial solution of Eq. (6) over R.

3 Periodic Solutions of Implicit Linear Difference Equations ∞ Let m ∈ N and {an }∞ n=0 , {bn }n=0 are m-periodic sequences in a commutative ring R. First consider the periodic solutions of non-autonomous homogeneous difference equation (9) bn xn+1 = an xn , n = 0, 1, 2, ....

Theorem 6 Equation (9) has only trivial m-periodic solution if and only if the element a0 · · · am−1 − b0 · · · bm−1 is not a zero-divisor.

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Proof If m = 1 then the existence of m-periodic (i.e. constant) solution of Eq. (9) is equivalent to the existence of the solution of the linear equation (a0 − b0 )x0 = 0 in variable x0 . This equation has only trivial solution if and only if element a0 − b0 is not a zero-divisor. Let now m ≥ 2. Let us consider the following matrix A with elements from R: ⎞ ⎛ a0 −b0 0 0 · · · 0 0 ⎜ 0 0 ⎟ a1 −b1 0 · · · 0 ⎟ ⎜ ⎜ 0 −b · · · 0 0 ⎟ 0 a 2 2 ⎟ ⎜ A=⎜ . .. .. .. .. .. .. ⎟ . ⎜ .. . . . . . . ⎟ ⎟ ⎜ ⎝ 0 0 0 0 · · · am−2 −bm−2 ⎠ −bm−1 0 0 0 · · · 0 am−1 The existence of m-periodic solution of Eq. (9) is equivalent to the existence of solution of the following linear system over R: Ax = 0, x = {xn }m−1 n=0 .

(10)

For the system (10) to have only trivial solution it is necessary and sufficient that det A is not a zero-divisor (see [17, Chap. 1, Corollary 1.30]). Using the last row decomposition we calculate the determinant of this matrix: det A = a0 · · · am−1 − b0 · · · bm−1 ([17, Chap. 1, Theorem 1.7]). The theorem is proved. Corollary 3 Let R be an integral domain. Equation (9) has only trivial m-periodic solution if and only if a0 · · · am−1 = b0 · · · bm−1 . Corollary 4 Let R be an integral domain, all elements a0 , ...., am−1 are invertible and at least one of the elements b0 , ..., bm−1 is non-invertible. Then Eq. (9) has only trivial m-periodic solution. Proof Since b0 · · · bm−1 is non-invertible, the condition a0 · · · am−1 = b0 · · · bm−1 holds. Example 5 Let R be an integral domain, m ≥ 2, b j = 0 ( j = 0, ..., m − 1) and a0 · · · am−1 = b0 · · · bm−1 . Consider Eq. (9) over R. Let F be a quotient field of , n= R. Introduce the following elements of F: x0 = b0 · · · bm−2 , xn = xn−1 abn−1 n−1 1, ...., m − 1 and xn+m = xn , n = 0, 1, 2, .... Then xn ∈ R and the sequence {xn }∞ n=0 is a m-periodic solution of Eq. (9). In particular, if a, b ∈ R, b = 0 and a m = bm , then the sequence xn = a n bm−n−1 , n = 0, 1, 2, ..., m − 1, xn+m = xn , n = 0, 1, 2, ... is a m-periodic solution of (4). Corollary 5 Let R be an integral domain, a, b ∈ R. The difference equation bxn+1 = axn has only trivial m-periodic solution if and only if a m = bm . In particular, if b is a non-invertible element and a is invertible element, then this equation has only trivial m-periodic solution.

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Corollary 6 If R is a subring of the field R, then the completely implicit difference equation bxn+1 = axn has not non-trivial periodic solutions. Proof Indeed, if R is a subring of R, then the assumption a m = bm yields the equality a = ±b. In this case the equation considered is not completely implicit. Corollary 7 The completely implicit difference equation bxn+1 = axn over an integral domain has not non-trivial 2-periodic solutions. Proof Indeed, the assumption a 2 = b2 implies that either a = b or a = −b. In both cases the equation considered is not completely implicit. Remark 1 The implicit difference equation bxn+1 = axn over a ring that is not an integral domain can have a non-trivial constant solution or a non-trivial 2periodic solution. For instance, let R = Z/6Z. Then the implicit difference equation 4xn+1 = xn , n = 0, 1, 2, .... has the constant solution xn = 2, n = 0, 1, 2, ..., and the equation 2xn+1 = xn has the 2-periodic solution  xn =

4, n is even, 2, n is odd.

Note that the completely implicit homogeneous equation bxn+1 = axn can have no non-trivial periodic solution, but still have a non-trivial solution. √ √ Example 6 Let R = Z[ 5] = {c + d 5 : c, d ∈ Z}. Then the completely implicit difference equation 2xn+1 = (1 +

√

5)xn , n = 0, 1, 2, ....

(11)

has the non-trivial solution 

√ n 1+ 5 , n = 0, 1, 2, ... xn = 2 2 (see Example 1). Since 2m = (1 + trivial periodic solutions.

√

5)m for any m ∈ N, then Eq. (11) has not non-

Now let us show that for any prime p = 2 the completely implicit equation bxn+1 = axn can have non-trivial p-periodic solutions (in Example 5 the equation can be explicit). Example 7 Let p = 2 be prime and let ω = e Consider the ring R=

⎧ ⎨ ⎩

c0 +

p−1  j=1

2πi p

be a primitive p-th root of unity.

2c j ω j : c0 , ..., c p−1

⎫ ⎬ ∈Z . ⎭

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The cyclotomic polynomial r (x) =

p−1 

x j is a minimal polynomial, that is irre-

j=0

ducible over Z and annulates the algebraic integer ω (see [18, Proposition 6.4.1]). / R. Show that ω ∈ / R. Since R is a subring of the ring of algebraic integers, then 21 ∈ Assume the contrary that ω ∈ R, i.e. there are integers c0 , ..., c p−1 ∈ Z such that p−1 p−1   2c j ω j . Then q(ω) = 0, where q(x) = c0 + 2c j x j − x ∈ Z[x]. It ω = c0 + j=1

j=1

follows from Proposition 6.1.7 in [18] and Exercise 5 in [18, Chap. 6] that the polynomial r (x) divides q(x), therefore there exists a constant c ∈ Z such that q(x) = cr (x). Equating in this equality the coefficients of x and x 2 , we get the contradictory in the relations 2c1 − 1 = c, 2c2 = c. Consider now the following elements of R: b = 2 and a = 2ω. Then a is not divisible by b and a p = b p . Hence difference equation 2xn+1 = 2ωxn , n = 0, 1, 2, ..... over the ring R is complitely implicit and by Corollary 5 it has non-trivial p-periodic solution (see Example 5). Now consider the non-homogeneous difference equation bn xn+1 = an xn − f n , n = 0, 1, 2, ...,

(12)

∞ ∞ where {an }∞ n=0 , {bn }n=0 , and { f n }n=0 are m-periodic sequences in a commutative ring R. Note that now we subtract f n for convenience.

Proposition 2 Suppose that the homogeneous Eq. (9) has only trivial solution. Then if the non-homogeneous Eq. (12) has a solution, then it is m-periodic. Proof Let {xn }∞ n=0 be a unique solution of Eq. (12). Let us prove that it is m-periodic. Note that the sequence yn = xn+m − xn , n = 0, 1, 2, .... satisfies the homogeneous Eq. (9). Since Eq. (9) has only trivial solution, then yn = 0, n = 0, 1, 2, ..... Therefore, xn+m = xn , n = 0, 1, 2, .... The proposition is proved. Now let us prove the existence and uniqueness of the m-periodic solution of Eq. (12). Theorem 7 Let R be an integral domain and m ≥ 2. Equation (12) has a unique m-periodic solution if and only if the element d = a0 · · · am−1 − b0 · · · bm−1 is not equal to zero and divides the elements f n an+1 · · · an+m−1 + bn f n+1 an+2 · · · an+m−1 + bn bn+1 f n+2 an+3 · · · an+m−1 + · · · + +bn · · · bn+m−3 f n+m−2 an+m−1 + bn · · · bn+m−2 f n+m−1 , n = 0, ..., m − 1. This solution has the form xn = ( f n an+1 · · · an+m−1 + bn f n+1 an+2 · · · an+m−1 +

Implicit Linear First Order Difference Equations …

209

+bn bn+1 f n+2 an+3 · · · an+m−1 + · · · + + bn · · · bn+m−3 f n+m−2 an+m−1 + bn · · · bn+m−2 f n+m−1 )/d, n = 0, ..., m − 1. (13) Proof Necessity. Note that m-periodicity implies d = an an+1 · · · an+m−1 − bn bn+1 · · · bn+m−1 , n = 0, ..., m − 1.

(14)

and d = det A (see the proof of Theorem 6). Since Eq. (9) has a unique m-periodic solution, then it follows from Theorem 6 that d = 0. Let {xn }∞ n=0 be an m-periodic solution of Eq. (12). Then b0 x1 = a0 x0 − f 0 and b1 x2 = a1 x1 − f 1 . Multiplying the last equation by b0 , we obtain b0 b1 x2 = b0 a1 x1 − b0 f 1 = a1 (a0 x0 − f 0 ) − b0 f 1 = a1 a0 x0 − a1 f 0 − b0 f 1 . (15) Multiplying b2 x3 = a2 x2 − f 2 by b0 b1 and taking into account (15), we get b0 b1 b2 x3 = a2 b0 b1 x2 − b0 b1 f 2 = a2 (a1 a0 x0 − a1 f 0 − b0 f 1 ) = = a2 a1 a0 x0 − f 0 a1 a2 − b0 f 1 a2 − b0 b1 f 2 . Continuing in the same way, we see that b0 b1 · · · bm−1 xm = a0 a1 · · · am−1 x0 − f 0 a1 · · · am−1 − b0 f 1 a2 · · · am−1 − −b0 b1 f 2 a3 · · · am−1 − · · · − b0 · · · bm−3 f m−2 am−1 − b0 · · · bm−2 f m−1 . Since x0 = xm , then the last equation implies that d divides the element f 0 a1 · · · am−1 + b0 f 1 a2 · · · am−1 + b0 b1 f 2 a3 · · · am−1 + · · · + +b0 · · · bm−3 f m−2 am−1 + b0 · · · bm−2 f m−1 . Similarly, by taking into account the equality (14) and m-periodicity of the solution {xn }∞ n=0 we get that d divides the elements f n an+1 · · · an+m−1 + bn f n+1 an+2 · · · an+m−1 + bn bn+1 f n+2 an+3 · · · an+m−1 + · · · + +bn · · · bn+m−3 f n+m−2 an+m−1 + bn · · · bn+m−2 f n+m−1 , n = 0, ..., m − 1. Sufficiency. Suppose that x0 , x1 are determined by formula (13). Let us check that they satisfy Eq. (12) as n = 0. Taking into account the m-periodicity of the sequences ∞ {an }∞ n=0 , {bn }n=0 we get

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b0 x1 − a0 x0 = (b0 f 1 a2 · · · am + b0 b1 f 2 a3 · · · am + +b0 b1 b2 f 3 a4 · · · am + · · · + b0 b1 · · · bm−2 f m−1 am + +b0 b1 · · · bm−1 f m − f 0 a0 · · · am−1 − b0 f 1 a0 a2 · · · am−1 − b0 b1 f 2 a0 a3 · · · am−1 − −b0 · · · bm−3 f m−2 a0 am−1 − b0 · · · bm−2 a0 f m−1 )/det A = = (b0 f 1 a0 a2 · · · am−1 + b0 b1 f 2 a0 a3 · · · am−1 + +b0 b1 b2 f 3 a0 a4 · · · am−1 + · · · + b0 b1 · · · bm−2 a0 f m−1 + +b0 b1 · · · bm−1 f m − f 0 a0 · · · am−1 − b0 f 1 a0 a2 · · · am−1 − b0 b1 f 2 a0 a3 · · · am−1 − −b0 · · · bm−3 f m−2 a0 am−1 − b0 · · · bm−2 a0 f m−1 )/det A = − f 0 . Thus, b0 x1 − a0 x0 = − f 0 . In the same way, if the sequence {xn }∞ n=0 is determined by formula (13), one can check that it is m-periodic and satisfies Eq. (12) for any n ≥ 1. Corollary 3 implies the uniqueness of the solution. The theorem is proved. Corollary 8 Let R be an integral domain, m ≥ 2, and d = a0 · · · am−1 − b0 · · · bm−1 an invertible element of R. Then Eq. (12) for any m-periodic sequence { f n }∞ n=0 has a unique m-periodic solution. This solution has the form xn = ( f n an+1 · · · an+m−1 + bn f n+1 an+2 · · · an+m−1 + +bn bn+1 f n+2 an+3 · · · an+m−1 + · · · + bn · · · bn+m−3 f n+m−2 an+m−1 + +bn · · · bn+m−2 f n+m−1 )d −1 , n = 0, ..., m − 1. Particularly, if R is a local integral domain and at least one of the elements b0 , ..., bn−1 is non-invertible, then the equation bn xn+1 = xn − f n , n = 0, 1, 2, ... has a unique m-periodic solution for any m-periodic sequence { f n }∞ n=0 . Corollary 9 Let R be an integral domain and a, b ∈ R. Then the equation bxn+1 = axn − f n , n = 0, 1, 2, ...

(16)

has a unique m-periodic solution if and only if a m = bm and the element a m − bm m−1  j m−1− j divides b a f n+ j for any n = 0, 1, ..., m − 1. This solution has the form j=0 m−1 

xn =

b j a m−1− j f n+ j

j=0

a m − bm

, n = 0, 1, ..., m − 1.

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Corollary 10 Let R be an integral domain and a, b ∈ R. Then Eq. (16) has a unique m m m-periodic solution for any m-periodic sequence { f n }∞ n=0 if and only if a − b is invertible in R. This solution has the form xn = (a m − bm )−1

m−1 

b j a m−1− j f n+ j , n = 0, 1, ..., m − 1.

(17)

j=0

Proof The sufficiency follows from Corollary 9. Let us prove the necessity. Let Eq. (16) have a unique m-periodic solution for any m-periodic sequence { f n }∞ n=0 . m−1  Then a m = bm and the element a m − bm divides the elements b j a m−1− j f j for any j=0

f 0 , ..., f m−1 ∈ R. Assuming f j = 1, j = 0, 1, 2, ..., m − 1, we obtain that a m − bm m−1  j m−1− j divides the element b a in the ring R, i.e. there exists an element c ∈ R such j=0

that

m−1  j=0

j

b j a m−1− j = (a m − bm )c. Assuming f j = (−1) j Cm−1 , j = 0, ..., m − 1,

we obtain that a m − bm divides the element (a − b)m−1 . On the other hand a m − bm = (a − b)

m−1 

b j a m−1− j = (a m − bm )c(a − b).

j=0

Since R is an integral domain, then this equality implies the invertibility of a − b. Since the element a m − bm divides the invertible element (a − b)m−1 , then it also is invertible.

4 Quasi-polynomial Solution of Implicit Linear Difference Equation Consider the following completely implicit difference equation with the quasipolynomial non-homogeneity over the ring R bxn+1 = axn + λn

s 

p j n j , n = 0, 1, 2, ...,

(18)

j=0

where λ, a, b, p j ∈ R ( j = 0, ..., s), λ = 0. Theorem 8 Let R be an integral domain with characteristics zero (see [19, 1.43]). Suppose (λb − a) j divides p j−1 , j = 1, ..., s + 1. Then there exists a unique quasipolynomial with the parameter λ solution of Eq. (18). It has a form

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x n = λn

s 

qjn j,

(19)

j=0

where q0 , ..., qs ∈ R. Proof Since Eq. (18) is completely implicit, then λb − a = 0. First let us prove the uniqueness. It is enough to prove that the homogeneous equation bxn+1 = axn has only trivial quasi-polynomial with the parameter λ solution. Substituting (19) for xn in the homogeneous equation, we get bλn+1

s 

qj

j 

C rj n r − aλn

s 

r =0

j=0

q j n j = 0, n = 0, 1, 2, ...

(20)

j=0

Since R is an integral domain, then (20) implies bλ

s 

qj

j=0

j 

C rj n r − a

r =0

s 

q j n j = 0, n = 0, 1, 2, ...

(21)

j=0

Since R is a ring of zero characteristics, then natural numbers are pairwise distinct and nonzero as elements of the ring R. Since R is an infinite integral domain, then equate coefficients of like powers of n in (21), we get bλ

s 

q j C rj − aqr = 0, r = 0, 1, 2, ..., s.

j=r

Therefore q j = 0, j = 0, ..., s. Now let us prove the existence of a quasi-polynomial with the parameter λ solution. Substituting (19) for xn in (18) in the same way for the coefficients of like powers we get s  q j C rj − aqr = pr , r = 0, 1, 2, ..., s. bλ j=r

Now the coefficients qs , qs−1 , ..., q0 could be found consequently. The theorem is proved.

5 The Case of a Local Ring Let R be a local Noetherian ring and M the maximal ideal of R. Let the arithmetic function w : R → N0 be given by the formula [20, Sect. 3.1]

Implicit Linear First Order Difference Equations …

 w(x) =

213

sup{k ∈ N0 : x ∈ Mk }, x = 0 ∞, x = 0

Assuming e−∞ = 0, we consider the distance d(x, y) = e−w(x−y) , x, y ∈ R. This distance induces the M-adic topology on the ring R [20, Sect. 3.1]. For b ∈ M consider a linear implicit difference equation over the ring R: bxn+1 = xn − f n , n = 0, 1, 2, ....

(22)

Note that now we subtract f n for convenience. Theorem 9 If Eq. (22) has a solution over R, then the series xn =

∞ 

bk f n+k , n = 0, 1, 2, ....

(23)

k=0

converge in R. In this case the sequence {xn }∞ n=0 is a unique solution of Eq. (22) over R. Proof Let {xn }∞ n=0 be a solution of Eq. (22) over R. Then for any m ∈ N xn =

m−1 

bk f n+k + bm xn+m .

k=0

Note that bm xn+m → 0, m → ∞ with respect to the topology on R for any n ∈ N0 . Taking the limit as m → ∞, we obtain the representation (23). Theorem 4 implies the uniqueness of the solution. Remark 2 It is easily shown that the converse to this theorem is also true: if all the series (23) converge in the ring R, then the sequence {xn }∞ n=0 is a unique solution of Eq. (22) in R. Example 8 Let p be a prime, Z p the ring of p-adic numbers, and M = pZ p . Then Z p is a local Noetherian domain, M is the maximal ideal of Z p and M-adic topology on Z p coincides with the standard p-adic topology (see [21, 7.6, Exercise 11 and 16.2]). Since the metric Z p space is complete, all the series (23) converge [22, Sect. 1.2]. Thus for any sequence { f n }∞ n=0 ⊂ Z p Eq. (22) has a unique solution over Z p and this solution has the form (23) (see also [10, Theorem 2.1]). Theorem 10 Let R be a local Noetherian domain with characteristics zero. Suppose a is invertible and b is non-invertible element of R. Then there exists a unique solution over R of Eq. (18). It has the form

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xn = −λn

∞ 

a −k−1 bk λk

k=0

s 

pj

j=0

j 

C rj k j−r n r , n = 0, 1, 2, ...,

(24)

r =0

where series (24) converge in the ring R with respect to the M-adic topology. Proof If a is invertible, then we can rewrite the equation as a −1 bxn+1 = xn − s  p j n j , n = 0, 1, 2, .... Also (λb − a) j = a j (a −1 λb − a −1 f n , where f n = −λn j=0

1) j . Then statement follows directly from Theorems 8 and 9. Note that the assumption of Theorem 8 holds as the ring R is local therefore (a −1 λb − 1) j , j = 1, ..., s + 1 are invertibles. The following example shows that if a is non-invertible, then Eq. (16) can have no solution at all. Suppose p is a prime. Denote by Z( p) the p-localization of Z. The ring Z( p) is local Noetherian domain. Example 9 Let R = Z(2) . The completely implicit equation 4xn+1 + 1 = 2xn has not solutions over Z(2) . It can be easily checked that Z(2) is completely integrally closed. Thus, by Corollary 1 if this equation has a unique solution, then it is constant. Thus the equation has not constant solutions, since its unique constant solution over / Z(2) . Q is equal to xn = − 21 ∈ Example 10 Let R be a local integral domain, let b be a non-invertible element in this domain. Then the element 1 − bm is invertible. The implicit Eq. (22) is a particular case of Eq. (16) as a = 1. By Corollary 10 Eq. (22) has a unique m-periodic solution for any m-periodic sequence { f n }∞ n=0 . This solution has the form xn = (1 − bm )−1

m−1 

b j f n+ j , n = 0, 1, ...m − 1

(25)

j=0

(see formula (17)). Note that the formula (25) determines a unique solution of Eq. (22) in the case of an arbitrary local ring R as well. The uniqueness of the solution follows from Theorem 6. Moreover, if R is Noetherian, then by Eq. (25) and Theorem 9 ∞  j=0

m −1

b f n+ j = (1 − b ) j

m−1 

b j f n+ j , n = 0, 1, 2, ..., m − 1.

j=0

It was shown that in the case of local Noetherian domain and periodic or quasipolynomial non-homogeneities the implicit Eq. (22) has a solution. The following example shows, that the implicit Eq. (22) can have no solution in the case of more complicated non-homogeneity.

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Example 11 Let R = Z( p) and let m ∈ N be co-prime with p, α = m1 and f n =   α = α(α−1)···(α−n+1) . Note that f n ∈ Z( p) [23, Chap. 4, Sect. 1], since α ∈ Z p . n! n Theorem 9 yields that if implicit Eq. (22) has a solution over Z( p) , then γ = ∞  f n bn ∈ Z( p) . Suppose p = 2, m = 2 and b = p 2 ; then (see [23, Chap. 4, Sect. 1]) n=0 2

/ Z( p) . Indeed, if γ ∈ Z( p) , then γ ∈ Q and as γ = p 2 + 1. We claim, that γ ∈ γ 2 = p 2 + 1, then γ ∈ Z. However, this equality doesn’t hold for any prime p and integer γ. Thus, Eq. (22) has not solutions over Z( p) . If p = 2, then take m = 3 and b = 4. Then γ 3 = 5 (see [23, Chap. 4, Sect. 1]), which does not hold for any γ ∈ Z(2) . Thus, in this case Eq. (22) also has not solutions over Z(2) .

References 1. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005) 2. Kelley, W.G.: Peterson, A.C., Difference Equation: An Introduction with Applications, 2nd ed., pp. 404. Academic Press (2001) 3. Halanay, A., Wexler, D.: Teoria Calitativa A Sistemelor Cu Impulsuri. Academiei Republicii Socialiste Romania, Bucuresti (1968) 4. Campbell, S.L.: Singular Systems of Differential Equations I, vol. 40. Pitman Publishing, Research Notes in Mathematics, San Francisko, London, Melbourne (1980) 5. Healton, J.W.: Discrete time systems, operator models and scattering theory. J. Funct. Anal. 16(1), 15–38 (1974) 6. Benabdallakh, M., Rutkas, A.G., Solov’ev, A.A.: Application of Asymptotic Expansions to the Investigation of an Infinite System of Equations, Axn+1 + Bxn = f n in a Banach Space. J. Soviet Math. 48 (1990). Iss. 2, 124–130. https://doi.org/10.1007/BF01095789 7. Bondarenko, M., Rutkas, A.: On a Class of Implicit Difference Equations. Dopov. Nac. Acad. Nauk Ukr., No. 7, 11–15 (1998) 8. Baskakov, A.G.: On the invertibility of linear difference operators with constant coefficients. Russ. Math. 45(5), 1–9 (2001) 9. Gefter, S.L., Piven, A.L.: Implicit linear nonhomogeneous difference equation in banach and locally convex spaces. J. Math. Phys. Anal. Geom. 15, No. 3, 336–353 (2019) 10. Gerasimov, V.A., Gefter, S.L., Goncharuk, A.B.: Application of the p-Adic topology on Z to the problem of finding solutions in integers of an implicit linear difference equation. J. Math. Sci. 235, 256–261 (2018). https://doi.org/10.1007/s10958-018-4072-x 11. Gefter, S., Goncharuk, A.: Generalized backward shift operators on the ring Z[[x]], Cramer’s rule for infinite linear systems, and p-adic integers. Oper. Theory: Adv. Appl. 268, 247–259 (2018). https://doi.org/10.1007/978-3-319-75996-8_13 12. Martseniuk, V., Gefter, S.L., Piven’, A.: Uniqueness criterion and cramer’s rule for implicit higher order linear difference equations over Z. In: Baigent, S., Bohner, M., Elaydi, S. (eds.), Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol. 341. Springer, Cham (2020). https://doi.org/10. 1007/978-3-030-60107-2_16 13. Matsumura, H.: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 2nd ed. Cambridge University Press (2006). https://doi.org/10.1017/CBO9781139171762 14. Bourbaki, N.: Commutative Algebra, Hermann (1972)

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15. Anderson, D.D., Anderson, D.F., Zaprullar, M.: Compeletely integrally closed Prufer vmultiplication Domains. Commun. Algebra 45(12), 5264–5282 (2017). https://doi.org/10. 1080/00927872.2017.1303502 16. Lang, S.: Algebra, Reviews, 3rd edn. Springer, New York (2002) 17. McDonald, B.R.: Linear Algebra over Commutative Rings, 1st ed. CRC Press (1984). https:// doi.org/10.1201/9781003065197 18. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, New York (1990) 19. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press (1996). https://doi.org/10. 1017/CBO9780511525926 20. Cahen, P.-J., Chabert, J.-L.: Integer-valued polynomials. Math. Surv. Monogr. (AMS) 48 (1997) 21. Dummit D.S., Foote R.M.: Abstract Algebra, 3rd ed. Wiley (2004) 22. Perez-Garcia, C., Schikhof, W. H.: Locally Convex Spaces over Non-Archimedean Valued Fields, pp. 472. Cambridge University Press (2010). https://doi.org/10.1017/ CBO9780511729959 23. Koblitz, N.: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer, New York, (1977)

Global Attraction and Repulsion of a Heteroclinic Limit Cycle in Three Dimensional Kolmogorov Maps Zhanyuan Hou

Abstract There is a recent development in the carrying simplex theory for competitive maps: under some weaker conditions a map has a modified carrying simplex (one of the author’s latest publications). In this paper, as one of the applications of the modified carrying simplex theory, a criterion is established for a three dimensional Kolmogorov map to have a globally repelling (attracting) heteroclinic limit cycle. As a concrete example, a discrete competitive model is investigated to illustrate the above criteria for global repulsion (attraction) of a hetericlinic limit cycle. Keywords Competitive maps · Carrying simplex · Hetericlinic limit cycle

1 Introduction We are concerned with the global asymptotic behaviour of the discrete dynamical system (1) x(n) = T n (x), x ∈ C, n ∈ IN, where C = IR+N = [0, +∞) N , IN = {0, 1, 2, . . .} and T : C → C is the Kolmogorov map having the form Ti (x) = xi f i (x), i ∈ I N = {1, 2, . . . , N }

(2)

and f ∈ C 1 (C, C) with f i (x) > 0 for all x ∈ C and i ∈ I N . It is known that system (1) with (2) is a typical mathematical model for the population dynamics of a community of N species, where each xi (n) represents the population size or density at time n (at the end of nth time period), and the function f i (x) denotes the per capita ∂ fi ≤ 0 for all i, j ∈ I N with i = j, then (1) models growth rate, of the ith species. If ∂x j the population dynamics of a community of competitive species. Z. Hou (B) SCDM, London Metropolitan University, London N7 8DB, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_11

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In this paper, we focus on a recent development on the carrying simplex theory for competitive maps. The main aim is to provide a criterion for a three dimensional system to have a globally attracting (repelling) heteroclinic limit cycle based on the existence of a modified carrying simplex. This demonstrates that the new development in the carrying simplex theory has the potential to have various applications for a broader class of systems. The paper is organised as follows: Sect. 2 is for the recent development on carrying simplex theory, Sect. 3 presents a criterion for global attraction (repulsion) of a heteroclinic limit cycle, Sect. 4 deals with a Ricker model as a concrete example illustrating the criterion given in Sect. 3 and Sect. 5 is for conclusion.

2 Carrying Simplex of Competitive Kolmogorov Maps Research on system (1) with (2) and its various particular instances as models has been flourishing in the last few decades. The carrying simplex theory and its various applications is one of the important and influential developments. This theory was originally established by Hirsch [9] (see [12, 13] for latest update) for competitive Kolmogorov systems of differential equations. Since then the idea of a carrying simplex for discrete systems gradually appeared in literature (see [5, 15, 20, 21] for example). But a more accepted theorem for existence and uniqueness of a carrying simplex was given by Hirsch [10] without proof. Then Ruiz-Herrera [19] presented a more general theorem covering Hirsch’s result with a complete proof. Definition 1 A nonempty set Σ ⊂ C is called a carrying simplex of (1) if it is a compact invariant hypersurface homeomorphic to Δ N −1 = {x ∈ C : x1 + · · · + x N = 1} such that every trajectory except the origin is asymptotic with a trajectory in Σ. If (1) has a carrying simplex Σ, then, since it attracts all the points x ∈ C\{0} so that the limit set ω(x) ⊂ Σ, the asymptotic dynamics of (1) on C is essentially described by the dynamics on Σ. Many applications of the carrying simplex theory actually utilise this attracting feature of Σ. For any x, y ∈ C, we write x ≤ y or y ≥ x if xi ≤ yi for all i ∈ I N ; x < y or y > x if x ≤ y but x = y; x y or y x if xi < yi for all i ∈ I N . Definition 2 The map T given by (2) is said to be retrotone (or competitive) in a subset X ⊂ C if for any x, y ∈ X , T (x) < T (y) implies xi < yi for all i ∈ I (y) = { j ∈ I N : y j = 0}. Let [0, r ] = {x ∈ C : 0 ≤ x ≤ r }. The theorem below is Theorem 6.1 in [19]. Theorem 1 Assume that T with T ([0, r ]) ⊂ [0, r ] for some r 0 satisfies the following conditions: (i) For each i ∈ I N , the map T restricted to the positive half xi -axis has a fixed point qi ei with qi > 0, ei the ith standard unit vector and q r .

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(ii) T is retrotone and locally one to one in [0, r ]. (iii) For any x, y ∈ [0, r ], if T (x) < T (y) then, for each j ∈ I N , either x j = 0 or f j (x) > f j (y). Then the map admits a carrying simplex Σ. Theorem 1 can be only applied to (1) restricted to the space [0, r ] ⊂ C if no condition for T on C\[0, r ] is provided. However, if for any compact set S ⊂ C there is a k ∈ IN such that T k (S) ⊂ [0, r ], then Theorem 1 can be applied directly to the system on C. When f on C is a C 1 map, T is also a C 1 map with Jacobian matrix DT (x) = diag( f 1 (x), . . . , f N (x))(I − M(x)),

(3)

where I is the identity matrix and   xi ∂ f i (x) . M(x) = (Mi j (x)) = − f i (x) ∂x j N ×N

(4)

Then, by Lemma 4.1, Corollary 6.1 and Remark 6.4 in [19], Theorem 1 has the following version with easily checkable conditions. Theorem 2 Assume that T satisfies the following conditions: (i) For each i ∈ I N , the map T restricted to the positive half xi -axis has a fixed point qi ei with qi > 0, ei the ith standard unit vector and q r for some r ∈ C. (ii) All entries of the Jacobian D f are negative. (iii) The spectral radius of M(x) satisfies ρ(M(x)) < 1 for all x ∈ [0, q]\{0}. Then the map admits a carrying simplex Σ. A more user-friendly variation of Theorem 2 given by Jiang and Niu [17, Theorem 3.1] is the above theorem with ρ(M(x)) < 1 replaced by an inequality involving ∂ fi (x). (See (7) or (8) below.) ∂x j We note that condition (ii) in Theorem 2 is very restrictive; it excludes the possibility of applying the theorem to systems with some zero entries of D f . But actually, condition (ii) is too strong and unnecessary, a compact invariant set attracting all the points of C\{0} with most of the features of a carrying simplex may still exist even if ∂ fi = 0 for some distinct i, j ∈ I N . The author’s recent work [14] provides a weaker ∂x j sufficient condition for (1) to have a modified carrying simplex. Let C˙ denote the ˙ interior of C and let B be either C or a positively invariant [0, r ] for some r ∈ C. Definition 3 A nonempty set Σ ⊂ B\{0} is called a modified carrying simplex of (1) if Σ meets the following requirements. (i) Σ is compact, invariant and homeomorphic to Δ N −1 by radial projection. (ii) Σ attracts all the points of B\{0}, i.e. ω(x) ⊂ Σ for each x ∈ B\{0}.

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Moreover, if x is below Σ with a nonempty support I (x) ⊂ I N , then there is a y ∈ Σ with I (y) = I (x) such that limn→+∞ (T n (x) − T n (y)) = 0. Note that the main difference between modified carrying simplex and the carrying simplex in literature is that the latter requires every trajectory in B\{0} to be asymptotic to one in Σ whereas the former requires every nontrivial trajectory below Σ to be asymptotic to one in Σ and Σ to attract all the points of B\{0}. Obviously, the concept of a modified carrying simplex is less restrictive and it includes carrying simplex as a particular class. Definition 4 The map T : C → C defined by (2) is said to be weakly retrotone (or weakly competitive) in a subset X ⊂ C if for x, y ∈ X with T (x) > T (y) and I = {i ∈ I N : Ti (x) > Ti (y)}, then x > y and xi > yi for all i ∈ I . Comparing this with the definition of retrotone we see that if T is retrotone then it is weakly retrotone, but not vice versa. The theorem below is Theorem 2.3 in [14]. Theorem 3 Assume that T defined by (2) with T ([0, r ]) ⊂ [0, r ] for some r ∈ C˙ satisfies the following conditions: (i) For each i ∈ I N , the map T restricted to the positive half xi -axis has a fixed point qi ei with qi > 0 and q r . (ii) T is weakly retrotone and locally one to one in [0, r ]. (iii) For any x, y ∈ [0, r ], if T (x) < T (y) with I = {i ∈ I N : Ti (x) < Ti (y)} then, for each j ∈ I , either x j = 0 or f j (x) > f j (y). Then 0 is a repeller with the basin of repulsion B(0) ⊂ [0, r ], (1) has a unique modified carrying simplex Σ and Σ = B(0)\({0} ∪ B(0)). Moreover, for each p ∈ Σ and every q ∈ [0, r ]\{0} with q < p, we have α(q) ⊂ πi (see its definition in the line before Theorem 5) provided qi < pi . Now utilising DT and D f , we obtain conditions which guarantee conditions (ii) and (iii) and the following version of Theorem 3 with easily checkable conditions. Consider the matrix M(x) given by (4) and   x j ∂ fi ˜ (x) . M(x) = ( M˜ i j (x)) = − f i (x) ∂x j N ×N

(5)

The theorem below is Theorem 2.4 in [14]. Note that (5) was used by some other authors in literature (e.g. [17]). Theorem 4 Assume that T given by (2) satisfies the following conditions: (i) For each i ∈ I N , the map T restricted to the positive half xi -axis has a fixed ˙ point qi ei with qi > 0 and q r for some r ∈ C. (ii) The entries of the Jacobian D f satisfy ∀x ∈ [0, r ], ∀i, j ∈ I N ,

∂ fi (x) ≤ 0, ∂x j

(6)

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and f i is strictly decreasing in xi ∈ [0, ri ] for x ∈ [0, r ]. ˜ (iii) For each x ∈ [0, q]\{0}, either ρ(M(x)) < 1 for M(x) given by (4) or ρ( M(x)) ˜ < 1 for M(x) given by (5). Then 0 is a repeller with the basin of repulsion B(0) ⊂ [0, r ], (1) has a unique modified carrying simplex Σ and Σ = B(0)\({0} ∪ B(0)). Moreover, for each p ∈ Σ and every q ∈ [0, r ]\{0} with q < p, we have α(q) ⊂ πi provided qi < pi . Note that f i (x) +

N 

xi

∂ fi (x) > 0, ∀i ∈ I N , ∂x j

(7)

xj

∂ fi (x) > 0, ∀i ∈ I N , ∂x j

(8)

j=1

implies ρ(M(x)) < 1 and f i (x) +

N  j=1

˜ implies ρ( M(x)) < 1 (see [14]). Then condition (iii) in Theorem 4 is satisfied if either (7) or (8) holds. Theorem 2 is a popular result in discrete competitive systems and has a large number of applications due to the important and interesting features of a carrying simplex: compact, invariant, unordered ( p ≤ q implies p = q for p, q ∈ Σ), homeomorphic to Δ N −1 by radial projection, and attracting all the points of C\{0}. The following are just a few examples. Ruiz-Herrera [19] investigated exclusion and dominance utilizing the existence of a carrying simplex. Jiang and Niu [16, 17] and Gyllenberg et al. [6, 8] dealt with some well known three-dimensional competitive models. Based on the existence of a carrying simplex, they classified the systems into 33 topologically equivalent classes and gave a phase portrait on Σ for each class. Jiang et al. [18] studied heteroclinic cycles via carrying simplex. Balreira et al. [4] and Gyllenberg et al. [7] provided criteria for global stability of an interior fixed point based on the existence of a carrying simplex. Baigent [1, 2] investigated the geometric feature of a carrying simplex and found conditions for Σ to be convex. Baigent and Hou [3] and Hou [11] provided split Lyapunov function method and geometric method for global stability. Although these methods were not based on the existence of a carrying simplex, comments and comparisons with those using carrying simplex were made there. With the introduction of a modified carrying simplex, we expect that Theorem 4 can be applied to a broader class of systems in various applications. In Sect. 3 of [14], geometric criteria for dominance and vanishing species was provided based on the existence of a modified carrying simple. In the next section, as an application of Theorem 4 we shall prove a theorem for (1) to have a globally attracting (repelling) heteroclinic limit cycle.

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3 Global Attraction and Repulsion of a Heteroclinic Limit Cycle In this section, we are going to find sufficient conditions for (1) to have a globally attracting (repelling) heteroclinic limit cycle. We assume that the conditions of Theorem 4 are satisfied so that (1) has a unique modified carrying simplex Σ. By a heteroclinic cycle we mean a closed curve that is topologically a circle consisting of fixed points p(i) for i ∈ I N , together with heteroclinic curves i connecting p(i) to p(i+1) (here p(N +1) = p(1) ). By a heteroclinic limit cycle Γ we mean a heteroclinic cycle Γ with an attracting (or repelling) neighbourhood N (Γ ) (restricted ˙ such that ω(x 0 ) = Γ (or α(x 0 ) = Γ ) for all x 0 ∈ N (Γ ). We first define the to C) concept of globally attracting or repelling heteroclinic limit cycle. Definition 5 We say that a heteroclinic cycle Γ0 of (1) is a • locally attracting (repelling) heteroclinic limit cycle if there is a neighbourhood V ⊂ C˙ (V ⊂ intΣ) of Γ0 such that ω(x 0 ) = Γ0 (α(x 0 ) = Γ0 ) for all x 0 ∈ V ; • globally attracting (repelling) heteroclinic limit cycle if ω(x 0 ) = Γ0 (α(x 0 ) = s ˙ ( p) (x 0 ∈ intΣ\{ p}), where W s ( p) is the one-dimensional Γ0 ) for all x 0 ∈ C\W ˙ stable manifold of a fixed point p ∈ C. For the case where N = 3 and system (1) admits a carrying simplex Σ, the three axial fixed points Q 1 , Q 2 , Q 3 are the only fixed points on ∂Σ, and ∂Σ is a heteroclinic cycle, Jiang et al. [18] gave a simple sufficient condition det(θ) < 0(> 0) for ∂Σ to be locally attracting (repelling), where θ = (ln f i (Q j ))3×3 . The main issue we address here is when the heteroclinic cycle is globally attracting (repelling) limit cycle. For convenience, let πi = {x ∈ C : xi = 0}, ∀i ∈ I N . Theorem 5 Assume that the following conditions hold for (1) with N = 3. (a) The conditions of Theorem 4 are met so (1) has a unique modified carrying simplex Σ ⊂ [0, r ]. (b) The three axial fixed points Q 1 , Q 2 , Q 3 are the only fixed points of (1) on ∂Σ = Σ ∩ ∂C and either the inequalities (9) or (10) hold: ∀i ∈ I3 , ∀ j ∈ I3 \{i, i + 1}, ∀i ∈ I3 , ∀ j ∈ I3 \{i, i + 1},

f i (Q i+1 ) < 1 < f j (Q i+1 ), f i+1 (Q i ) < 1 < f j (Q i ).

(9) (10)

(c) For all i ∈ I3 , ∂∂xfii (Q i ) < 0. (d) There is a unique fixed point p in intΣ = Σ\∂Σ that is hyperbolic with onedimensional stable manifold W s ( p) in C˙ and globally repelling on Σ. Then ∂Σ is a globally attracting heteroclinic limit cycle, so ω(x 0 ) = ∂Σ for all s ˙ x 0 ∈ C\W ( p).

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s ˙ Proof By conditions (a) and (d), we have ω(x 0 ) ⊂ ∂Σ for all x 0 ∈ C\W ( p). We 0 need only check that ∂Σ is a heteroclinic cycle and prove that ω(x ) = ∂Σ. Suppose the inequalities (9) hold. From (3), (4) and (5) we know that DT (Q 1 ) has eigenvalues f 3 (Q 1 ) < 1, f 2 (Q 1 ) > 1 and 1 + q1 ∂∂xf11 (Q 1 ). As ρ(M(Q 1 )) = ˜ 1 )) = −q1 ∂ f1 (Q 1 ), by (c) and condition (iii) of Theorem 4 we have 0 < ρ( M(Q ∂x1

−q1 ∂∂xf11 (Q 1 ) < 1 so 0 < 1 + q1 ∂∂xf11 (Q 1 ) < 1. Thus, Q 1 is a saddle point with stable manifold W s (Q 1 ) = π2 \π1 and unstable manifold W u (Q 1 ) = Σ ∩ (π3 \π1 ), which connects Q 1 to Q 2 . Similarly, Q 2 is a saddle point with stable manifold W s (Q 2 ) = π3 \π2 and unstable manifold W u (Q 2 ) = Σ ∩ (π1 \π2 ), which connects Q 2 to Q 3 ; Q 3 is a saddle point with stable manifold W s (Q 3 ) = π1 \π3 and unstable manifold W u (Q 3 ) = Σ ∩ (π2 \π3 ), which connects Q 3 to Q 1 . Therefore, ∂Σ is a heteroclinic cycle with the direction Q 1 → Q 2 → Q 3 → Q 1 . Next, we prove that ω(x 0 ) = ∂Σ. Since ω(x 0 ) is nonempty, closed, invariant and ω(x 0 ) ⊂ ∂Σ, there is a y ∈ ω(x 0 ) ∩ ∂Σ. We have either y ∈ {Q 1 , Q 2 , Q 3 } or both limn→∞ T n (y) and limn→∞ T −n (y) in {Q 1 , Q 2 , Q 3 }. Thus, ω(x 0 ) contains at least / W s (Q 1 ) one of the Q i . Without loss of generality, we suppose Q 1 ∈ ω(x 0 ). That x 0 ∈ 0 0 s / W (Q 1 ) ∪ W u (Q 1 ), we implies ω(x ) = {Q 1 }. Since Q 1 is a saddle point and x ∈ u s 0 u must have W (Q 3 ) = W (Q 1 ) ∩ ∂Σ ⊂ ω(x ) and W (Q 1 ) = W u (Q 1 ) ∩ ∂Σ ⊂ ω(x 0 ). So Q 2 , Q 3 ∈ ω(x 0 ). By the same reasoning as above, we also have W u (Q 2 ) ⊂ ω(x 0 ). As ∂Σ = {Q 1 , Q 2 , Q 3 } ∪ W u (Q 1 ) ∪ W u (Q 2 ) ∪ W u (Q 3 ), we have shown ∂Σ ⊂ ω(x 0 ). Hence, ω(x 0 ) = ∂Σ. If the inequalities (10) hold, the same reasoning as above is valid with the heteroclinic cycle ∂Σ having the direction Q 1 → Q 3 → Q 2 → Q 1 and ω(x 0 ) = ∂Σ. Theorem 6 Assume that (a)–(c) in Theorem 5 and the following condition hold for (1) with N = 3. ˙ (d) There is a unique interior fixed point p that is globally attracting in C. Then ∂Σ is a heteroclinic limit cycle globally repelling on Σ, so α(x 0 ) = Γ0 for all x 0 ∈ intΣ\{ p}. Proof In the proof of Theorem 5, replacing ω(x 0 ) by α(x 0 ), the argument is still valid.

4 An Example In this section, we are going to consider the three-dimensional competitive Ricker model where the map T of (1) is defined by ∀i ∈ I3 , Ti (x) = xi eu(1−xi −αxi+1 ) , α > 1, 0 < u < (1 + α)−1 , x4 = x1 .

(11)

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Since f i (x) = eu(1−xi −αxi+1 ) , we have ⎛ ∂f ⎜ D f (x) = ⎝

1 (x) ∂ f 1 (x) ∂ f 1 (x) ∂x1 ∂x2 ∂x3 ∂ f 2 (x) ∂ f 2 (x) ∂ f 2 (x) ∂x1 ∂x2 ∂x3 ∂ f 3 (x) ∂ f 3 (x) ∂ f 3 (x) ∂x1 ∂x2 ∂x3

Note that since

⎞

⎛ ⎞ −u f 1 (x) −uα f 1 (x) 0 ⎟ ⎝ 0 −u f 2 (x) −uα f 2 (x) ⎠ . (12) ⎠= 0 −u f 3 (x) −uα f 3 (x)

∂ f 2 (x) ∂ f 3 (x) ∂ f 1 (x) = = = 0, ∂x3 ∂x1 ∂x2

we cannot apply Theorem 2 to system (1) with T defined by (11) as its condition (ii) cannot be met. This example shows the situation where Theorem 4 is applicable but Theorem 2 is not. If we take q = (1, 1, 1) and any r q, the conditions of (i) and (ii) of Theorem 4 are satisfied. For x ∈ [0, q]\{0}, it follows from 0 < u < (1 + α)−1 and x1 ≤ 1 that f i (x) +

3  j=1

xi

∂ f i (x) = f i (x)[1 − xi u(1 + α)] > 0 ∂x j

for all i ∈ I3 . So (7) holds, which implies ρ(M(x)) < 1. Hence (iii) of Theorem 4 is met. Therefore, by Theorem 4, (1) with (11) has a unique modified carrying simplex Σ ⊂ [0, r ]. For x ∈ π1 , f 2 (x) = 1 if and only if x2 + αx3 = 1 and f 3 (x) = 1 if and only if x3 = 1. Thus, Q 2 = (0, 1, 0) and Q 3 = (0, 0, 1) are the only two fixed points of T on π1 \{0}. Similarly, Q 1 = (1, 0, 0) and Q 3 are the only two fixed points on π2 \{0}, Q 1 and Q 2 are the only two fixed points on π3 \{0}, so T has only three axial fixed points Q 1 , Q 2 , Q 3 on ∂Σ. As f 1 (Q 2 ) = f 2 (Q 3 ) = f 3 (Q 1 ) = eu(1−α) < 1, f 2 (Q 1 ) = f 3 (Q 2 ) = f 1 (Q 3 ) = eu > 1, the inequalities (9) hold. Thus, the conditions (a)–(c) of Theorem 5 are fulfilled. 1 1 1 , 1+α , 1+α ) with The map T has a unique interior fixed point p = ( 1+α ⎞ u uα 1 − 1+α − 1+α 0 u uα ⎠ − 1+α 0 1 − 1+α . DT ( p) = ⎝ uα u − 1+α 0 1 − 1+α ⎛

The matrix has three eigenvalues: λ1 = 1 − u, λ2,3

√ 3uα u(α − 2) ±i . =1+ 2(1 + α) 2(1 + α)

(13)

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Under the conditions given in (11), 0 < λ1 < 1. To determine whether |λ2,3 | > 1, we define the function F(α) =

α3 + α2 − 4α − 1 1 2 + α − α2 = − , α ≥ 0. 2 1+α 1−α+α (1 + α)(1 − α + α2 )

(14)

For system (1) with (11) we have the following result: Theorem 7 For system (1) with T defined by (11), the following conclusions hold. (a) There is an α0 ∈ (1, 2) such that α03 + α02 − 4α0 − 1 = 0 so F(α0 ) = 0, F(α) 2 < 0 for α ∈ [0, α0 ) and F(α) > 0 for α > α0 . For α > α0 and 2+α−α 1 so p is a repeller on Σ; for 0 ≤ α < α0 and 0 < u < 1+α , 1+α |λ2,3 | < 1 so p is asymptotically stable. (b) For α ≥ 2, p is a global repeller on Σ. (c) For α ∈ [0, α0 ) with 3u(1 − α + α2 ) < 2 + α − α2 ,

(15)

˙ p is globally asymptotically stable in C. 1 (d) For α ≥ 2 and 0 < u < 1+α , ∂Σ is a globally attracting heteroclinic limit cycle. 2

1 2+α−α (e) For α ∈ (1, α0 ) and 0 < u < min{ 1+α , 3(1−α+α 2 ) }, ∂Σ is a heteroclinic limit cycle globally repelling on Σ.

Proof (a) From (13) we have |λ2,3 | > 1 if and only if u>

(2 − α)(1 + α) 2 + α − α2 = . 2 1−α+α 1 − α + α2

1 . For α ∈ [0, 2), from (14) we see This holds obviously if α ≥ 2 since 0 < u < 1+α that the √ denominator of F(α) is positive and the numerator of F(α) has a minimum √ √ 13−1 13−1 at α = 13−1 , decreases for α ∈ [0, ) and increases for α > . As the 3 3 3 numerator is negative at α = 0 and α = 1 but positive at α = 2, there is an α0 ∈ (1, 2) such that α03 + α02 − 4α0 − 1 = 0 so F(α0 ) = 0, F(α) < 0 for α ∈ [0, α0 ) and F(α) > 0 for α > α0 . Then the conclusion (a) follows. (d) The conclusion follows from (b) and Theorem 5. (e) The conclusion follows from (c) and Theorem 6. (b) Let V (x) = x1 x2 x3 , W (x) = x1 + x2 + x3 , x ∈ C.

Then V (T (x)) = V (x)eu(3−(1+α)(x1 +x2 +x3 ))    3 − W (x) . = V (x) exp u(1 + α) 1+α

(16)

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˙ p}, W (x) > 3 implies V (T (x)) < V (x), W (x) < 3 Thus, for all x ∈ C\{ 1+α 1+α 3 implies V (T (x)) > V (x), and W (x) = 1+α implies V (T (x)) = V (x). For all x ∈ C that is not a fixed point of T , W (T (x)) = x1 eu(1−x1 −αx2 ) + x2 eu(1−x2 −αx3 ) + x3 eu(1−x3 −αx1 ) > x1 [1 + u(1 − x1 − αx2 )] + x2 [1 + u(1 − x2 − αx3 )] +x3 [1 + u(1 − x3 − αx1 )] = W (x)(1 + u) − u[x12 + x22 + x32 + αx1 x2 + αx2 x3 + αx1 x3 ] = W (x)(1 + u) − uW (x)2 − u(α − 2)(x1 x2 + x1 x3 + x2 x3 ). As x1 x2 + x1 x3 + x2 x3 ≤ x12 + x22 + x32 , we have 1 [2(x1 x2 + x1 x3 + x2 x3 ) + x12 + x22 + x32 ] 3 1 = W (x)2 . 3

x1 x2 + x1 x3 + x2 x3 ≤

Thus, if α ≥ 2 then u(1 + α) W (x)2 3   3 u(1 + α) W (x) − W (x) . = W (x) + 3 1+α

W (T (x)) > (1 + u)W (x) −

(17)

3 Therefore, for all x ∈ C which is not a fixed point such that W (x) ≤ 1+α , we have W (T (x)) > W (x). The conditions of Theorem 4 actually ensure that T from [0, r ] to T ([0, r ]) is a homeomorphism (see Remark (2.1) (b) in [14]). So, for each x ∈ T ([0, r ]), T −1 (x) exists in [0, r ]. Let

 S = x ∈ C : W (x) ≤

3 . 1+α

(18)

By (17) and continuity of T we have S ⊂ T (S). Since B(0) = Σ ∩ B(0) ⊂ [0, r ] is invariant under T so that T −1 (B(0)) = B(0), we have T −1 (S ∩ B(0)) ⊂ S ∩ B(0). Thus, for each x ∈ S ∩ B(0) which is not a fixed point and for all n ∈ IN, T −n (x) ∈ S ∩ B(0) and W (T −(n+1) (x)) < W (T −n (x)). Since S contains only two fixed points 0 and p, we must have limn→∞ W (T −n (x)) = 0 so limn→∞ T −n (x) = 0. Hence, for all x ∈ S ∩ B(0), if it is not a fixed point then x ∈ B(0). This shows that (Σ\{ p, Q 1 , Q 2 , Q 3 }) ∩ S = ∅. Now for each x ∈ intΣ\{ p} and all n ∈ IN, T n (x) ∈

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3 Σ\{ p} so W (T n (x)) > 1+α and V (T n+1 (x)) < V (T n (x)). Therefore, we must have n limn→∞ V (T (x)) = 0, so ω(x) ⊂ ∂Σ and p is a global repeller on Σ. (c) Now for α ∈ [0, α0 ) and x ∈ C not a fixed point, as 0 < u < 1, we have

W (T (x)) =
0 such that limn→∞ V (T n (x)) = c and V (ω(x)) = c. As ω(x) ⊂ 3 S\∂C, if there is a p1 ∈ ω(x)\{ p} then p1 is not a fixed point, so W (T ( p1 )) < 1+α and V (T 2 ( p1 )) < V (T ( p1 )) ≤ V ( p1 ) = c. But this contradicts V (T 2 ( p1 )) = c due to T 2 ( p1 ) ∈ ω(x) by the invariance of ω(x). Hence, ω(x)\{ p} = ∅ and ω(x) = { p}. If x ∈ / S but there is an m > 0 such that T m (x) ∈ S, then the above reasoning is valid / S for all n ∈ IN, then V (T n (x)) is for T m (x) so ω(x) = ω(T m (x)) = { p}. If T n (x) ∈ decreasing so limn→∞ V (T n (x)) = d for some d ≥ 0. If d = 0 then ω(x) ⊂ ∂Σ. By the same reasoning as that used in the proof of Theorem 5 we derive ω(x) = ∂Σ based on the fact that ∂Σ is a heteroclinic cycle. In particular, Q 1 = (1, 0, 0) ∈ ω(x). Note 3 3 . This contradicts the fact that W (y) ≥ 1+α for all y ∈ ω(x) that W (Q 1 ) = 1 < 1+α 3 n due to W (T (x)) > 1+α . This contradiction shows that d > 0. Then, by the same argument as above, we must have ω(x) = { p}. The proof of the theorem is complete.

5 Conclusion We have reviewed the modified carrying simplex theory as a recent development in this area. By comparing the concept of modified carry simplex and the sufficient conditions for system (1) with (2) to have a modified carrying simplex with those of carrying simplex, we note that the modified carrying simplex keeps most of the

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features for carrying simplex but requires much weaker conditions on the map T : it ∂ fi ∂ fi ≤ 0 rather than ∂x < 0 for all i, j ∈ I N and all x ∈ [0, r ]. This means requires ∂x j j that, like the theorems for carrying simplex, the theorems for modified carrying simplex can be applied to a broader class of systems in various applications. To demonstrate this point, we have found a criterion for a three dimensional system to have a globally attracting (repelling) heteroclinic limit cycle when it admits a modified carrying simplex. Then we have shown by a Ricker model as a concrete example that the theorem for modified carrying simplex is applicable but the theorem for carrying simplex is not. We expect that various applications of the theorems for modified carrying simplex will appear in future. Acknowledgement The author thanks the referee for providing positive comments and suggestions adopted in this version of the paper.

Appendix: Proof That H(x) < H( p0 ) for x ∈ S(θ)\{ p0 } For this purpose, we first show that H (x) < H ( p0 ) =

3θ 3θ = , x ∈ ∂ S(θ) = S(θ) ∩ ∂C. 1 − u + u(1 + α)θ c(θ)

As H (x) has the rotational symmetry about the components of x, we need only show the inequality for x ∈ S(θ) with x3 = 0 and x1 + x2 = 3θ. Clearly, for x = (3θ, 0, 0) 3θ < H ( p0 ) due to 3 > 1 + α. If x3 = 0 but x2 = 0, then we have H (x) = 1+u(3θ−1) H (x) − H ( p0 ) = =

=

= =

x2 3θ x1 + − 1 − u(1 − x1 − αx2 ) 1 − u(1 − x2 ) c(θ)  c(θ)x1 H ( p0 ) (1 − u)(x1 + x2 ) + 3uθ(x1 + αx2 ) c(θ)x2 −1 + (1 − u)(x1 + x2 ) + 3uθx2 )  c(θ)(x1 /x2 ) H ( p0 ) (1 + u(3θ − 1))(x1 /x2 ) + (1 + u(3θα − 1)) c(θ) −1 + (1 − u)(x1 /x2 ) + (1 + u(3θ − 1))  c(θ) c(θ)X + −1 H ( p0 ) AX + B CX + A H ( p0 )R(X ) , (AX + B)(C X + A)

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where A = 1 − u + 3uθ > c(θ) > 0, B = 1 − u + 3uθα > 0, C = 1 − u > 0, X = x1 /x2 ≥ 0 and R(X ) = C(c(θ) − A)X 2 − (A2 − 2c(θ)A + BC)X + B(c(θ) − A). As C(c(θ) − A) < 0 and B(c(θ) − A) < 0, we have H (x) − H ( p0 ) < 0 for all x ∈ ∂ S(θ) if either A2 − 2c(θ)A + BC ≥ 0 or [A2 − 2c(θ)A + BC]2 − 4BC(c(θ) − A)2 = (A2 − BC)[A2 − BC − 4c(θ)(A − c(θ))] < 0. The former is reduced to 3uθ(1 − 2α) ≥ (1 − u)(2 − α) > 0, which is impossible for α ≥ 0.5 or small uθ. Since A2 − BC = (1 − u + 3uθ)2 − (1 − u + 3uθα)(1 − u) = 3u(1 − u)θ(2 − α) + 9θ2 u 2 > 0, the latter becomes A2 − BC − 4c(θ)(A − c(θ)) < 0, which is reduced to uθ(2α − 1)2 < (2 − α)(1 − u).

(19)

1 1 , (19) is equivalent to (15). As 0 < θ ≤ 1+α , (19) holds When θ is replaced by 1+α under the condition (15). Therefore, H (x) < H ( p0 ) for all x ∈ ∂ S(θ). Next, we show that H (x) < H ( p0 ) for all x ∈ S(θ)\{ p0 }. Clearly, H (x) − H ( p0 ) = 0 for x = p0 . For any x0 ∈ intS(θ)\{ p0 }, x0 and p0 determine a unique line segment yz with y, z ∈ ∂ S(θ) such that x0 , p0 ∈ yz. The points on yz can be written as x(s) = y + s(z − y) for s ∈ [0, 1] such that x(0) = y, x(1) = z, x(s1 ) = x0 and x(s2 ) = p0 for some distinct s1 , s2 ∈ (0, 1). Let h(s) = H (x(s)) − H ( p0 ). From the P(x) , where definition of H (x) we can write it as H (x) = Q(x)

P(x) = x1 [1 − u(1 − x2 − αx3 )][1 − u(1 − x3 − αx1 )] +x2 [1 − u(1 − x1 − αx2 )][1 − u(1 − x3 − αx1 )] +x3 [1 − u(1 − x2 − αx3 )][1 − u(1 − x1 − αx2 )], Q(x) = [1 − u(1 − x2 − αx3 )][1 − u(1 − x2 − αx3 )][1 − u(1 − x3 − αx1 )]. ( p0 )Q(x(s)) h 1 (s) = Q(x(s)) with Q(x) > 0. As h(0) = H (y) − H ( p0 ) Then h(s) = P(x(s))−H Q(x(s)) < 0, h(1) = H (z) − H ( p0 ) < 0 and h(s2 ) = H ( p0 ) − H ( p0 ) = 0, the polynomial h 1 (s) of degree three satisfies h 1 (0) < 0, h 1 (1) < 0 and h 1 (s2 ) = 0. Also, from

d P(x(s)) |s=s2 = [c(θ)2 + 2uθc(θ)(1 + α)](W (z) − W (y)) = 0 ds and

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d Q(x(s)) |s=s2 = uc(θ)2 (1 + α)(W (z) − W (y)) = 0 ds we obtain h 1 (s2 ) = 0. Thus, h 1 (s) = (s − s2 )2 (Ds + E). As h 1 (0) = (s2 )2 E < 0 and h 1 (1) = (1 − s2 )2 (D + E) < 0, these imply that Ds + E < 0 for s ∈ [0, 1], so h 1 (s1 ) < 0 and h(s1 ) = H (x0 ) − H ( p0 ) < 0. Hence, H (x) < H ( p0 ) for all x ∈ S(θ)\{ p0 }.

References 1. Baigent, S.: Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems. J. Diff. Equ. Appl. 22(5), 609–620 (2016) 2. Baigent, S.: Convex geometry of the carrying simple for the May-Leonard map. Discret. Contin. Dyn. Syst. B 24(4), 1697–1723 (2019) 3. Baigent, S., Hou, Z.: Global stability of discrete-time competitive population models. J. Diff. Equ. Appl. 23, 1378–1396 (2017) 4. Cabral Balreira, E., Elaydi, S., Luís, R.: Global stability of higher dimensional monotone maps. J. Diff. Equ. Appl. 23, 2037–2071 (2017) 5. Diekmann, O., Wang, Y., Yan, P.: Carrying simplices in discrete competitive systems and age-structured semelparous populations. Discret. Contin. Dyn. Syst. 20, 37–52 (2008) 6. Gyllenberg, M., Jiang, J., Niu, L., Yan, P.: On the classification of generalised competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discret. Contin. Dyn. Syst. 38, 615–650 (2018) 7. Gyllenberg, M., Jiang, J., Niu, L.: A note on global stability of three-dimensional Ricker models. J. Diff. Equ. Appl. 25, 142–150 (2019) 8. Gyllenberg, M., Jiang, J., Niu, L., Yan, P.: On the dynamics of multi-species Ricker models admitting a carrying simplex. J. Diff. Equ. Appl. 25, 1489–1530 (2019) 9. Hirsch, M.W.: Systems of differential equations that are competitive or cooperative. III: competing species. Nonlinearity 1, 51–71 (1988) 10. Hirsch, M.W.: On existence and uniqueness of the carrying simplex for competitive dynamical systems. J. Biol. Dyn. 2(2), 169–179 (2008) 11. Hou, Z.: Geometric method for global stability of discrete population models. Discret. Contin. Dyn. Syst. B 25(9), 3305–3334 (2020) 12. Hou, Z.: On existence and uniqueness of a carrying simplex in Kolmogorov differential systems. Nonlinearity 33, 7067–7087 (2020). https://doi.org/10.1088/1361-6544/abb03c 13. Hou, Z.: Corrigendum: On existence and uniqueness of a carrying simplex in Kolmogorov differential systems (2020 Nonlinearity 33 7067). Nonlinearity 34, C5–C6 (2021). https://doi. org/10.1088/1361-6544/ac2a4f 14. Hou, Z.: On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems. J. Diff. Equ. Appl. 27 (2), 284–315 (2021). https://doi.org/10.1080/10236198.2021. 1894141 15. Jiang, J., Mierczy´nski, J., Wang, Y.: Smoothness of the carrying simplex for discrete-time competitive dynamical systems: a characterization of neat embedding. J. Diff. Equ. 246(4), 1623–1672 (2009) 16. Jiang, J., Niu, L.: On the equivalent classification of three-dimensional competitive AtkinsonAllen models relative to the boundary fixed points. Discret. Contin. Dyn. Syst. 36(1), 217–244 (2016) 17. Jiang, J., Niu, L.: On the equivalent classification of three-dimensional competitive LeslieGower models via the boundary dynamics on the carrying simplex. J. Math. Biol. 74, 1223– 1261 (2017)

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18. Jiang, J., Niu, L., Wang, Y.: On heteroclinic cycles of competitive maps via carrying simplices. J. Math. Biol. 72, 939–972 (2016) 19. Ruiz-Herrera, A.: Exclusion and dominance in discrete population models via the carrying simplex. J. Diff. Equ. Appl. 19(1), 96–113 (2013) 20. Wang, Y., Jiang, J.: The general properties of discrete-time competitive dynamical systems. J. Diff. Equ. 176, 470–493 (2001) 21. Wang, Y., Jiang, J.: Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems. J. Diff. Equ. 186, 611–632 (2002)

Bifurcation and Stability of a Ricker Host-Parasitoid Model with a Host Constant Refuge and General Escape Function Senada Kalabuši´c, Džana Drino, and Esmir Pilav

Abstract Motivated by the Ricker model and paper [7], in this paper, we investigate the dynamics of a class of Ricker host-parasitoid models, wherein for each generation, a constant number of hosts are safe from attack by parasitoids, and the Ricker equation governs the host population. For the escape function, we take a general probability function that satisfies specific conditions. We show that the system always possesses exclusion equilibrium. Furthermore, the unique interior equilibrium exists under certain conditions. We show that the exclusion equilibrium undergoes the transcritical and period-doubling bifurcations, while Neimark–Sacker and period-doubling bifurcations occur at the interior equilibrium point. We investigate the boundedness of solutions. We prove the global attractivity result for the interior equilibrium. Finally, we show the uniform persistence of the system, ensuring the long-term survival of both species. Then, using a few well-known escape functions, we provide numerical simulations to confirm our theoretical findings. Keywords Host–parasitoid model · Bifurcations · Persistence · Refuge · Ricker equation · Stability

1 Introduction Continuous-time models or discrete-time models usually describe the relationship between different species in the population dynamics. The continuous-time framework is applied when populations have overlapping generations and all-year-round reproduction. However, the discrete-time models are suited for populations whose generations do not overlap [23, 36]. Among these interactions, the important place occupies the host-parasitoid interactions because of the global prevalence of parasitoids and their influence in regulating their hosts. Host-parasitoid relationships are S. Kalabuši´c (B) · D. Drino · E. Pilav Department od Mathematics, University of Sarajevo, Zmaja od Bosne 33-35, Sarajevo, Bosnia and Herzegovina e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_12

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generally modeled using a system of difference equations. The first such models date back to the early works of Thompson [40] and Nicholson–Bailey [1]. A general model which describes host-parasitoid interaction has the following form. Hn+1 = r Hn f (Hn , Pn ) Pn+1 = eHn (1 − f (Hn , Pn ))

(1)

where Hn is the population size of the host, and Pn is the population size of the parasitoid at generation n. The parameter r > 0 is a reproductive rate of the host, i.e., the number of eggs laid by a host that survives through the larvae, pupae, and adult stages, and f (Hn , Pn ) denotes the proportion of host larvae that are safe from parasitism. The parameter e > 0 is the average number of viable eggs that a parasitoid lays on a single host. Assuming that the escape function is f (Hn , Pn ) = e−a Pn , leads to the classical discrete-time Nicholson–Bailey model, where a is searching efficiency of the parasitoid. However, there are several unrealistic assumptions in the Nicholson–Bailey model. For example, a constant host reproductive rate, constant search efficiency, and a homogeneous environment. These assumptions lead to the unstable behavior of the model. More precisely, at low densities of host and parasitoids, both populations start to grow. When the host population becomes large enough; then, the parasitoids start to overexploit the host population leading to a crash of the host population, followed by the crash of the parasitoid population [1, 23]. In 1975, Beddington et al. [2] modified the Nicholson–Bailey model by introducing density-dependent self-limitation of the host in the absence of parasitoids. As an implication of this assumption, the host population size does not grow exponentially without parasitism, i.e., the host net reproduction Hn+1 /Hn is density-dependent and follows the Ricker model: Hn+1 = (1 − γ)Hn er (1− K ) e−a Pn Hn

Pn+1 = γβ Hn (1 − e−a Pn )

(2)

where K is the carrying capacity and represents the maximum population size that can be supported by the available and potentially limited resources; r is the intrinsic growth rate of the host. It is assumed that the parameters a, γ, β, and r are positive real numbers, and 0 < γ < 1. Parameter β is the average number of offsprings that a parasitoid can reproduce from an infected host. Parameter γ is the host’s constant proportion available to the parasitoids in each generation. This additional nonlinearity (self-regulation of the host) significantly affects both populations’ stability and dynamics. See [7, 9, 14, 28, 42] and references therein for qualitative behavior of generalized Beddington model and generalized Beddington model with Allee effect. In natural ecosystems, prey populations often can take refuge in areas where it is safe from predators. A refuge plays an essential role in natural ecosystems, and the spatial distribution of these resources can have critical outcomes for individual and population capacity [7, 22]. Generally, there are two ways of introducing refuges in

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the model: a constant number of prey are safe from predation, and a constant portion of preys are invulnerable to predation [23]. The incorporation of refuges for the prey might stabilize predator-prey (host-parasitoid) interaction. There are a significant number of papers that include refuge for the host into the model. See for example [6–9, 12–14], and references within. In [3], the authors investigated a class of host-parasitoid models such that in each generation, a constant fraction of hosts are safe from parasitism, and the host population in the absence of parasitoids follows an exponential growth. More precisely, they investigated host-parasitoid interaction described by the following system Hn+1 = a Hn + bHn f (Pn ) Pn+1 = cHn (1 − f (Pn )) . where f is any real nonnegative, decreasing convex function, which is sufficiently smooth with f (0) = 1 and f (∞) = 0. They concluded that a constant proportion of the host population in refuge could stabilize both populations. In [4], authors considered a class of host-parasitoid models with a constant proportion of hosts free from infection, and the Beverton–Holt equation governs the host population in the absence of parasitoids. The following system of difference equations describes this type of host-parasitoid interaction. Hn+1 =

a Hn bHn + f (Pn ) 1 + Hn 1 + Hn

Pn+1 = cHn (1 − f (Pn )) , where f satisfies similar conditions like in [3]. The authors showed that the addition of a constant fraction of hosts in refuge does not change a qualitative behavior of positive equilibrium regarding the corresponding model without refuge. Papers [26, 27] deal with host-parasitoid models, with incorporated constant refuge for the host population. The authors showed that constant refuge might stabilize the behavior of the corresponding models. In [7], the authors modified the model of Hassell [21], by incorporating densitydependence into the host population. They assumed that in each generation, a constant fraction of the host is free from parasitism and that the Ricker equation governs the growth of the host population in the absence of the parasitoids. It is well-known that Ricker-type functions generate very complex dynamics, including chaos [32–35, 37, 43]. They incorporated the Allee term into the model. The authors wrote down the following model. Hn+1 = (1 − γ)Hn er (1− K ) + γ Hn er (1− K ) e−a Pn Hn

Pn+1 = γβ Hn (1 − e−a Pn )

Hn

(3)

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where γ, β, r are positive constants and 0 < γ < 1. Parameters K , a, r, β and γ have the same biological interpretations as those in model (2). They came up with interesting conclusions and discussions such as that the addition of the refuge can make the parasitoids extinct. At the same time, the hosts survive or may stabilize the host-parasitoid interaction, the addition of both refuge and strong Allee effects either has a negative or positive impact on the coexistence of both populations. Motivated by Ricker model and paper [7], in this paper we propose the following host-parasitoid model with constant host refuge Hn+1 = a + bHn er (1−Hn ) f (Pn )

(4)

Pn+1 = cHn (1 − f (Pn ))

which is generalisation of the Ricker host-parasitoid model (3) with refuge effects, where a, b, c > 0, r > 0 and for function f the condition (H1 ): (H1 ) : f ∈ C[0, ∞) ∩ C 3 (0, ∞), f (y) > 0, f  (y) < 0, f  (y) ≥ 0 for y > 0, and lim y→0+ y f  (y) = 0, f (0) = 1, f (∞) = 0 holds. Also, the system (4) can be reduced to the difference equation of second order:

Pn+2 =

  r Pn+1 ( f (Pn+1 ) − 1) f (Pn ) −ac f (Pn ) + ac + b Pn+1 f (Pn )e c( f (Pn )−1) +r f (Pn ) − 1

.

The rest of the paper is organized as follows. First, in Sect. 2, we study the existence of the equilibrium points of the system (4). Then in Sect. 3, we investigate the local behavior of the equilibrium points, including stable and unstable manifolds. Next, Sect. 4 provides an analysis of the local bifurcations of equilibrium points. In this section, we prove the occurrence of the transcritical, period-doubling, and Neimark– Sacker bifurcations. In Sect. 5, we study the boundedness of solutions of the model, while in Sect. 6, we prove the global attractivity result for the interior equilibrium. Finally, the uniform persistence of the model is proved in Sect. 7. Section 8 provides numerical examples for well-known escape functions. These examples illustrate theoretical results. The final section provides conclusions and discussions.

2 Equilibrium Points This section deals with the equilibrium points of system (4). In this regard, the following observation will be helpful. The host population satisfies the following a ≤ Hn ≤ a +

ber −1 r

for all n > 0.

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Indeed, the left-hand side of the inequality follows directly from the first equation of system (4). The right-hand side follows from the fact that the function g(x) = xer (1−x) attains its maximum at x = 1/r, equals to er −1 /r. Now, using the inequality for Hn , for the parasitoids population, we have Pn+1

  ber −1 for all n > 0. ≤c a+ r

To discuss the existence of the equilibrium points, we associate to system (4) a map T : IR2+ ∪ {(0, 0)} → [a, +∞) × [0, +∞),       H F(H, P) a + bH er (1−H ) f (P) T = = P G(H, P) cH (1 − f (P)) Equilibrium points (H ∗ , P ∗ ) satisfy the following equation  T

H∗ P∗



 =

H∗ P∗

 ,

which is equivalent to the following system of equations ∗

H ∗ = a + bH ∗ er (1−H ) f (P ∗ ) P ∗ = cH ∗ (1 − f (P ∗ ))

(5)

Using the previous inequalities for the host and parasitoids populations, we have that equilibrium point (H ∗ , P ∗ ) satisfies the following a < H∗ ≤ a + b

er −1 r

  er −1 and 0 ≤ P ∗ ≤ c a + b . r

Parasitoid-free equilibrium, that is the exclusion equilibrium (H0∗ , 0) always exists and satisfies the following equation ∗

H0∗ = a + bH0∗ er (1−H0 ) .

(6)

Since all parameters are greater than zero, then H ∗ = 0, which means that the host population cannot become extinct. The following theorem proves the uniqueness of the exclusion equilibrium point. Theorem 1 System (4) has a unique exclusion equilibrium point (H0∗ , 0), where   r −1 max a, 1 + lnr b < H0∗ ≤ a + b e r . Proof From the above discussion, it follows that H0∗ satisfies the right-hand side of ∗ ∗ the inequality. Since H0∗ (1 − ber (1−H0 ) ) = a and a > 0, it must be 1 − ber (1−H0 ) > ln b ∗ 0, i.e., H0 > 1 + r . These inequalities imply that the following holds

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  ln b er −1 < H0∗ ≤ a + b . max a, 1 + r r The Eq. (6) can be written as: H0∗ − a ∗ = ber (1−H0 ) . ∗ H0

(7)

Now, consider the following equation H −a = ber (1−H ) . H

(8)

Since relation (7) holds, then H0∗ is the solution of Eq. (8). We show that H0∗ is a unique solution to Eq. (8). For that purpose G 1 (H ) =

H −a H

and G 2 (H ) = ber (1−H ) . r −1

Obviously G 1 (H0∗ ) = G 2 (H0∗ ), where a < H0∗ ≤ a + b e r . It is straightforward to see that G 1 (H ) =

a >0 H2

and G 2 (H ) = −br er (1−H ) < 0.

 r −1 Thus, G 1 (H ) is increasing; while G 2 (H ) is decreasing on a, a + ber , hence they must intersect precisely in one point, i.e., Eq. (8) has exactly one solution in this interval.  In the sequel, we consider the existence of the coexistence equilibrium (H ∗ , P ∗ ), i.e., H ∗ , P ∗ = 0. Substituting f (P ∗ ) from the first equation of system (5) into the second equation, we obtain  P = cH 1 − ∗

∗

H∗ − a bH ∗ er (1−H ∗ )

 ,

where 0 < f (P ∗ ) < 1, a < H ∗ ≤ a + b

er −1 r

  er −1 and 0 < P ∗ ≤ c a + b . r ∗

Since 0 < f (P ∗ ) < 1, we have H0∗ = a + bH0∗ er (1−H0 ) > a + bH ∗ er (1−H f (P ∗ ) = H ∗ . That is, H0∗ > H ∗ . Indeed, consider the function Q(H ) =

H −a , H > a. bH er (1−H )

∗

)

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H −ar H +a We have Q  (H ) = r bH > 0 on (a, ∞) and Q(a) = 0, so Q(H ) is a positive 2 er (1−H ) ∗ ∗ increasing function on (a, ∞). Since P ∗ > 0 and Q(H ∗ ) = bH H∗ er −a (1−H ∗ ) = f (P ) < 2

f (0) = 1 =

H0∗ −a

∗

bH0∗ er (1−H0 )

= Q(H0∗ ), it follows that H ∗ < H0∗ .

The second order Maclaurin polynomial of the function f (P) is f (P) = f (0) +  f + (0)P + f 2!(ξ) P 2 for P > 0, ξ ∈ (0, P). Now, we have 1 − f (P) = − f + (0)P − f  (ξ) 2 P ≤ − f + (0)P, for P > 0. We obtain 2! P ∗ = cH ∗ (1 − f (P ∗ )) < −cH ∗ f + (0)P ∗ . Since H ∗ < H0∗ , then P ∗ ≤ −cH ∗ f + (0)P ∗ < −cH0∗ f + (0)P ∗ . Thus, if 1 + cH0∗ f + (0) ≥ 0 the positive equilibrium does not exist. More precisely, a necessary condition for the existence of interior equilibrium in the first quadrant is 1 + cH0∗ f + (0) < 0. The following theorem gives necessary and sufficient condition for the existence of the positive equilibrium point. Theorem 2 Assume that f satisfies conditions (H1), f + (0) exists and a unique positive equilibrium exists if and only if b1 (1 + 1 + cH0∗ f + (0)< 0. Then,  ac f + (0)) < e

r 1+ c f 1(0) +

.

Proof Consider the system F(H, P) = H G(H, P) = P Let HG (P) =

P , for 0 < P < ∞. c(1 − f (P))

Then the following holds. lim HG (P) = −

P→0+

1

> 0, lim HG (P) = ∞, HG (P) = c f + (0) P→∞

1 − f (P) + P f  (P) . c(1 − f (P))2

Let K (P) = 1 − f (P) + P f  (P). Since K  (P) = P f  (P) > 0 and lim+ K (P) = 0 then K (P) > 0 on (0, ∞). Therefore HG (P) > 0, ∀P > 0. Hence,

P→0

we have that HG (P) is a positive increasing function on (0, ∞). Now, the first equation of the system becomes: HG (P) = a + bHG (P)er (1−HG (P)) f (P) or equivalently

HG (P) − a = er (1−HG (P)) f (P) bHG (P)

240

Let F1 (P) =

S. Kalabuši´c et al. HG (P)−a bHG (P)

and F2 (P) = er (1−HG (P)) f (P). The following holds:

  a 1 1 1− = lim F1 (P) = lim+ 1− P→0+ P→0 b HG (P) b lim F1 (P) =

P→∞

F1 (P) =

−1 c f + (0)

=

 1 1 + ac f + (0) b

1 , b

a HG (P) > 0, b(HG (P))2

lim+ F2 (P) = lim+ er (1−HG (P)) f (P) = e

P→0



a

r (1+ c f 1(0) ) +

P→0

> 0,

lim F2 (P) = lim er (1−HG (P)) f (P) = 0,

P→∞

P→∞

F2 (P) = er (1−HG (P)) ( f  (P) − r f (P)HG (P)) < 0. Thus, in interval (0, ∞) function F1 (P) is increasing, while function F2 (P) is decreasing. Bearing in mind the limit behavior both functions at the point 0, we conclude that the equation F1 (P) = F2 (P) has a unique solutions on (0, ∞) if and only if the following condition holds  r 1 (1 + ac f + (0)) < e b

 1+ c f 1(0) +

. 

Remark 1 If f + (0) = −∞, then lim+ F1 (P) = −∞ and lim+ F2 (P) = er > 0. P→0

P→0

Hence, we conclude that in this case, a unique positive equilibrium always exists.

3 Linearized Stability Analysis In this section, we explore the local stability of the exclusion equilibrium > 0, and interior equilibrium E ∗ = (H ∗ , P ∗ ) = (H ∗ , E 0∗ = (H0∗ , 0), H0∗  H ∗ −a ∗ cH 1 − bH ∗ er (1−H ∗ ) , E ∗ ∈ IR2+ . Through the section, we assume that f + (0) exists.

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3.1 Local Stability of the Exclusion Equilibrium Point ∗

The exclusion equilibrium point satisfies the following a < H0∗ = a + bH0∗ er (1−H0 ) . The Jacobian matrix at (H0∗ , 0) is given by J (E 0∗ )

 =

∗

∗

ber (1−H0 ) (1 − H0∗r ) bH0∗ er (1−H0 ) f + (0) 0 −cH0∗ f + (0)

 (9)

∗

The eigenvalues are λ1 = ber (1−H0 ) (1 − H0∗r ), and λ2 = −cH0∗ f + (0). Obviously, ∗ ∗ λ2 > 0. From relation H0∗ = a + bH0∗ er (1−H0 ) , we obtain that ber (1−H0 ) = 1 − Ha∗ . 0   Now, for λ1 , we have λ1 = 1 − Ha∗ (1 − r H0∗ ) = 1 − H0∗r − Ha∗ + ar. Since 0 0 H0∗ > a, then λ1 < 1 − Ha∗ < 1. Thus, λ1 < 1. 0 The following theorem describes the local behavior of the exclusion equilibrium. Theorem 3 For equilibrium (H0∗ , 0), the following holds. ∗

(a) If −cH0∗ f + (0) < 1 and ber (1−H0 ) (1 − H0∗r ) > −1, then it is locally asymptotically stable. ∗ (b) If −cH0∗ f + (0) > 1 and ber (1−H0 ) (1 − H0∗r ) < −1, then it is a repeller. ∗ r (1−H0∗ ) (1 − H0 r ) > −1 and −cH0∗ f + (0) > 1, then (H0∗ , 0) is a saddle (c) If be point. Local stable manifold W1s and unstable manifold W1u are given by W1s = {(H, P) : a < H < ∞, P = 0}, W1u = {(H, P) : H = H0∗ + a1 P + a2 P 2 + O(P 3 ), H > a, P > 0} where a1 =

bH0∗ f  (0)  ∗ b H0∗r − 1 − cH0∗ f  (0)e H0 r −r 

and       ber −2a1 f  (0) H0∗r − 1 + a12 r H0∗r − 2 + H0∗ f  (0)   a2 = ∗ 2ber H0∗r − 1 + 2c2 (H0∗ )2 ( f  (0))2 e H0 r  ∗  a1 ce H0 r 2a1 f  (0) + H0∗ f  (0)   + ∗ 2ber H0∗r − 1 + 2c2 (H0∗ )2 ( f  (0))2 e H0 r ∗

If ber (1−H0 ) (1 − H0∗r ) < −1 and −cH0∗ f + (0) < 1 then (H0∗ , 0) is also a saddle point with stable manifold W2s = W1u and unstable manifold W2u = W1s . ∗ (d) If ber (1−H0 ) (1 − H0∗r ) = −1 or −cH0∗ f + (0) = 1, then it is a nonhyperbolic equilibrium. Proof The first two statements are straightforward. For the third statement, we find stable and unstable manifolds. To determine the stable and unstable manifold, we first

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shift the equilibrium to the origin by changing variables xn = Hn − H0∗ , yn = Pn . Then system (4) becomes   ∗ xn+1 = a + b H0∗ + xn f (yn ) er (−H0 −xn +1) − H0∗ (10) yn+1 = c (1 − f (yn )) H0∗ + xn . Let   ∗ f 1 (x, y) = a + b H0∗ + x f (y) er (−H0 −x ) − H0∗ ∗ f 2 (x, y) = c (1 − f (y)) H0 + x .

(11)

Taylor series expansion of functions f 1 (x, y) and f 2 (x, y) about (0, 0) is given by  ∗  ∗ ∗ f 1 (x, y) = −ber −H0 r H0∗ r − 1 x + bH0∗ v f  (0)er −H0 r y + 21 bH0∗ f  (0)er −H0 r y 2   ∗  ∗  + 21 br er −H0 r H0∗ r − 2 x 2 − b f  (0)er −H0 r H0∗ r − 1 x y + O((|x| + |y|)3 ) f 2 (x, y) = −cH0∗ f  (0)y − 21 cH0∗ f  (0)y 2 − c f  (0)x y + O((|x| + |y|)3 ).

(12) ∗ We consider the first case. That is ber (1−H0 ) (1 − H0∗r ) > −1 and −cH0∗ f + (0) > 1. Assume that unstable manifold is x = h(y) = a1 y + a2 y 2 + O(|y|3 ). From f 1 (h(y), y) − h( f 2 (h(y), y)) = 0, we obtain a1 =

bH0∗ f  (0)  ∗ b H0∗r − 1 − cH0∗ f  (0)e H0 r −r 

and       ber −2a1 f  (0) H0∗r − 1 + a12 r H0∗r − 2 + H0∗ f  (0)   a2 = ∗ 2ber H0∗r − 1 + 2c2 (H0∗ )2 ( f  (0))2 e H0 r  ∗  a1 ce H0 r 2a1 f  (0) + H0∗ f  (0)   + ∗ . 2ber H0∗r − 1 + 2c2 (H0∗ )2 ( f  (0))2 e H0 r The restriction of the mapping to the invariant manifold is given locally by   1 yn+1 = f 2 (h(yn ), yn ) = −cH0∗ f  (0)yn + −ca1 f  (0) − cH0∗ f  (0) yn2 + O(|yn |3 ). 2

Assume that stable manifold is y = h(x) = b1 x + b2 x 2 + O(|x|3 ). Then from h( f 1 (x, h(x))) − f 2 (x, h(x)) = 0, we obtain b1 = b2 = 0. Then the dynamics restricted to the invariant manifold is given locally by

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243

xn+1 = f 1 (xn , h(xn ))   1 ∗  ∗  = −ber −H0 r H0∗r − 1 xn + br er −H0 r H0∗r − 2 xn2 + O(|xn |3 ). 2 For the second case, the unstable manifold becomes stable and opposite. In the case when the equilibrium is nonhyperbolic, we show in Sect. 4 that period-doubling bifurcation and transcritical bifurcation appear, where the bifurcation parameter is c. 

3.2 Linearized Stability of the Coexistence Equilibrium ∗ ∗ Now, we investigate the local character of , P ∗)   the interior equilibrium E ∗ = (H H ∗ −a 2 ∗ ∗ ∗ ∈ IR+ , where P = cH 1 − bH ∗ er (1−H ∗ ) . The Jacobian matrix at (H , P ) is

∗

∗

J (H , P ) =

A1 B1



C 1 D1

=

∗

∗

b f (P ∗ )er (1−H ) (1 − H ∗r ) bH ∗ er (1−H ) f  (P ∗ ) c(1 − f (P ∗ ))

−cH ∗ f  (P ∗ )

 . (13)

The determinant and trace of the Jacobian matrix are ∗

det (J ) = ber (1−H ) cH ∗ (−1 + r H ∗ f (P ∗ )) f  (P ∗ ) =

c(H ∗ − a) (−1 + r H ∗ f (P ∗ )) f  (P ∗ ), f (P ∗ ) ∗

tr (J ) = ber (1−H ) (1 − r H ∗ ) f (P ∗ ) − cH ∗ f  (P ∗ ) =

H∗ − a (1 − r H ∗ ) − cH ∗ f  (P ∗ ). H∗

Observe that the determinant and trace can change the sign. It is obvious that B1 < 0, C1 > 0 and D1 > 0. We show that D1 < 1 and A1 < 1. For that purpose, let us consider D1 − 1 and A1 − 1. We have D1 − 1 = − cH ∗ f  (P ∗ ) − 1 = − =

P∗ f  (P ∗ ) − 1 1 − f (P ∗ )

P ∗ f  (P ∗ ) − f (P ∗ ) + 1 P∗ f  (P ∗ ) − 1 = . ∗ f (P ) − 1 f (P ∗ ) − 1

Let G(P) = P f  (P) − f (P) + 1. Then G  (P) = P f  (P) > 0. Since  lim+ P f (P) = 0, it follows G(P) > 0 for all P > 0. Thus, D1 < 1. For A1 − 1, P→0

we obtain

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A1 − 1 = b f (P ∗ )er (1−H ) (1 − H ∗r ) − 1 = −r (H ∗ )2 +ra H ∗ −a H∗

=

=

−r H ∗ (H ∗ −a)−a H∗

H ∗ −a (1 H∗

− H ∗r ) − 1

< 0,

(14)

that is A1 < 1. Since A1 < 1, D1 < 1, B1 < 0, C1 > 0, we have det (J ) + 1 − tr (J ) = A1 D1 − B1 C1 + 1 − A1 − D1 = (1 − A1 )(1 − D1 ) − B1 C1 > 0

and      ∗ det (J ) + tr (J ) + 1 = ber (1−H ) (cH ∗ f  P ∗ H ∗r f P ∗ − 1  ∗     + f P 1 − H ∗r ) − cH ∗ f  P ∗ + 1,

∗

det (J ) − 1 = cH ∗ ber (1−H ) f  (P ∗ ) (−1 + H ∗r f (P ∗ )) − 1 =

c(H ∗ −a) f  (P ∗ )(H ∗ r f (P ∗ )−1) f (P ∗ )

=

cr f (P ∗ ) f  (P ∗ )H ∗2 f (P ∗ )

+ ac f



(P ∗ )− f (P ∗ ) f (P ∗ )

+

−1

(−acr f (P ∗ ) f  (P ∗ )−c f  (P ∗ ))H ∗ f (P ∗ )

(15)

∗

= cH ∗ f  (P ∗ )(−ber (1−H ) + (H ∗ − a)r ) − 1.

From previous conclusions and Theorem 2.12 in [29], the following theorem follows. Theorem 4 The coexistence equilibrium (H ∗ , P ∗ ) is (a) locally asymptotically stable if      ∗  b f P ∗ er −H r H ∗r − 1 + cH ∗ f  P ∗ − 1      ∗ < bcH ∗ er −H r f  P ∗ H ∗r f P ∗ − 1 < 1, (b) a repeller if      ∗  b f P ∗ er −H r H ∗r − 1 + cH ∗ f  P ∗ − 1      ∗ < bcH ∗ er −H r f  P ∗ H ∗r f P ∗ − 1 and

     ∗ bcH ∗ er −H r f  P ∗ H ∗r f P ∗ − 1 > 1,

(c) a saddle point if      ∗ bcH ∗ er −H r f  P ∗ H ∗r f P ∗ − 1      ∗  < b f P ∗ er −H r H ∗r − 1 + cH ∗ f  P ∗ − 1,

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245

(d) a nonhyperbolic equilibrium if      ∗ bcH ∗ er −H r f  P ∗ H ∗r f P ∗ − 1      ∗  = b f P ∗ er −H r H ∗r − 1 + cH ∗ f  P ∗ − 1 or ∗

ber (1−H ) cH ∗ (−1 + r H ∗ f (P ∗ )) f  (P ∗ ) = 1 ∗

and ber (1−H ) (1 − r H ∗ ) f (P ∗ ) − cH ∗ f  (P ∗ ) ≥ −2. Remark 2 Since det (J ) + 1 + tr (J ) = (1 + A1 )(1 + D1 ) − B1 C1 , then to have the left-hand side greater than zero it is sufficient to have A1 + 1 > 0. That is, ∗ b f (P ∗ )er (1−H ) (1 − r H ∗ ) + 1 > 0.

4 Local Bifurcations of Equilibrium Points In this section, we study bifurcations of the exclusion and coexistence equilibrium points.

4.1 Period-Doubling Bifurcation of Exclusion Equilibrium Point In this subsection we study the period-doubling bifurcation of the exclusion equilibrium point (H0∗ (r ), 0). We assume that f + (0) exists and we take r as a bifurcation parameter. Theorem 3 proves that the exclusion equilibrium is nonhyperbolic if either λ1 = −1 or λ2 = 1. We show that in the first case, a period-doubling bifurcation occurs, taking r as a bifurcation parameter. In contrast, a transcritical bifurcation appears in the second case, taking c as a bifurcation parameter. The exclusion equi∗ librium satisfies following relation H0∗ (r ) = a + bH0∗ (r )er (1−H0 (r )) . Let r = r0 be such that the following holds r0 =

a − 2H0∗ (r0 )   a − H0∗ (r0 )

H0∗ (r0 )

(16)

and assume that −cH0∗ (r0 ) f + (0) < 1. One can show that (H0∗ ) (r )

      ∗ ber H0∗ (r ) − 1 H0∗ (r ) H0 (r ) − 1 H0∗ (r ) a − H0∗ (r )    =− r ∗ = ∗ be r H0 (r ) − 1 + er H0 (r ) r H0∗ (r ) H0∗ (r ) − a + a

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and (H0∗ ) (r0 ) =

1 ∗ (H (r0 ) − 1)(a − H0∗ (r0 )). 2 0

We shift the equilibrium to the origin by changing variables u n = Hn − H0∗ (r ), vn = Pn . The system becomes   ∗ u n+1 = a + b f (vn ) er (1−H0 (r )−u n ) H0∗ (r ) + u n − H0∗ (r ) (17)   vn+1 = c (1 − f (vn )) H0∗ (r ) + u n . Let Tr be the following map     

∗ a + b f (v) er (1−H0 (r )−u ) H0∗ (r ) + u − H0∗ (r ) u Tr =   v c (1 − f (v)) H0∗ (r ) + u .

(18)

Then the Jacobian matrix of Tr at point (0, 0) is

JTr (0, 0) =

   ∗ ∗ ber −r H0 (r ) 1 − r H0∗ (r ) ber −r H0 (r ) H0∗ (r ) f + (0) −cH0∗ (r ) f + (0)

0

.

(19)

It is straightforward that det (JTr0 ) + tr (JTr0 ) + 1 = 0. The eigenvalues of JTr0 are λ1 = −1 and λ2 = −cH0∗ (r0 ) f + (0). Obviously, λ2 > 0. The corresponding eigenvectors of JTr0 are,

   bH0∗ er0 f  (0) 1 H0∗ r0 ∗ ∗  r 0 v1 = , v2 = be ( H0 r0 −1)−cH0 f (0)e 0 1 respectively, where H0∗ = H0∗ (r0 ). Let r = r0 + η. Then, we have  JTr0 +η (0, 0) = JTr0 (0, 0) + where

and



J11 J12 J21 J22



J11 J12 J21 J22

 η + O(|η|2 ),

d(JTr0 +η (0, 0)) = dη

η=0

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247

    ∗ J11 := ber0 (1−H0 (r0 )) r0 r0 H0∗ (r0 ) − 2 (H0∗ ) (r0 )       ∗ + ber0 (1−H0 (r0 )) H0∗ (r0 ) r0 H0∗ (r0 ) − 1 − 2 + 1 ,    ∗ J12 := b f  (0)er0 (1−H0 (r0 )) (H0∗ ) (r0 ) − H0∗ (r0 ) r0 (H0∗ ) (r0 ) + H0∗ (r0 ) − 1 , J21 := 0 J22 := −c f  (0)(H0∗ ) (r0 ) . System (17) becomes 

u n+1 vn+1



 = Tr0 +η

un vn



 = JTr0

un vn



 +

 f 1 (u n , vn , η) . g1 (u n , vn , η)

where   1 ∗  ∗  br er −H0 r H0∗r − 2 u 2 − b f  (0)er −H0 r H0∗r − 1 uv 2 1 ∗ + bH0∗ f  (0)er −H0 r v 2 + J12 vη + J11 uη + O((|u| + |v| + |η|)3 ), 2 1 g1 (u n , vn , η) = −c f  (0)uv − cH0∗ f  (0)v 2 + J22 vη + O((|u| + |v| + |η|)3 ), 2 f 1 (u n , vn , η) =

and H0∗ = H0∗ (r0 ). We write f 1 (u n , vn , η) and g1 (u n , vn , η) as follows f 1 (u n , vn , η) = A1 u 2 + A2 uv + A3 v 2 + J12 vη + J11 uη + O((|u| + |v| + |η|)3 ) g1 (u n , vn , η) = B1 uv + B2 v 2 + J22 vη + O((|u| + |v| + |η|)3 )



Let A=

1 0

Then det (A) = 1. Let



bH0∗ er0 f  (0)

∗

ber0 ( H0∗ r0 −1)−cH0∗ f  (0)e H0 r0

1



un vn



 =A

u˜ n v˜n

 =

1 a1 0 1

 .



i.e. u n = u˜ n + a1 v˜n and vn = v˜n . Then, we have       u˜ n+1 f 1 (u˜ n + a1 v˜n , v˜n , η) u˜ n −1 −1 = A JTr0 A +A v˜n+1 v˜n g1 (u˜ n + a1 v˜n , v˜n , η)

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or equivalently 

u˜ n+1 v˜n+1



 =

−1 0 0 −det (JTr0 )



u˜ n v˜n



 +

 f 2 (u˜ n , v˜n , η) , g2 (u˜ n , v˜n , η)

(20)

where   f 2 (u, v, η) = A1 u 2 + v 2 a12 A1 + a1 A2 − a12 B1 − a1 B2 + A3 + uv (2a1 A1 − a1 B1 + A2 ) + J11 uη + (a1 J11 − a1 J22 + J12 ) ηv + O((|u| + |v| + |η|)3 ), g2 (u, v, η) = v 2 (a1 B1 + B2 ) + B1 uv + J22 vη + O((|u| + |v| + |η|)3 ).

Assume that the equation of the local central manifold is ˜ + h 3 η 2 + O((|u| ˜ + |η|)3 ). v˜ = h(u, ˜ η) = h 1 u˜ 2 + h 2 uη Then, ˜ h(u, ˜ η), η)) = −det (JTr0 )h(u, ˜ η) + g2 (u, ˜ h(u, ˜ η), η) h(−u˜ + f 2 (u, After equating similar terms, we obtain h 1 = h 2 = h 3 = 0. Hence, h(u, η) = O((|u| + |η|)3 ) and dynamics of (20) on central manifold is given by u n+1 = G(u n , η), where G(u, η) = −u + f 2 (u, h(u, η), η) + O((|u| + |η|)4 ) = −u + A1 u 2 + J11 uη + O((|u| + |η|)4 ). Now, we verify the following conditions from [44, p. 373]: (i)

∂G 2 ∂2G2 ∂G (0, 0) = −1, (ii) (0, 0) = 0, (iii) (0, 0) = 0, ∂u ∂η ∂u 2 (iv)

∂2 G2 ∂3G2 (0, 0) = 0, (v) (0, 0) = 0, ∂u∂η ∂u 3

where G 2 (u, η) = G(G(u, η), η). It is straightforward to see that (i), (ii) and (iii) hold. We have ∂2 G2 (0, 0) = −2J11 , ∂u∂η

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

and

249

∂3G2 (0, 0) = −12(A1 )2 . ∂u 3

Hence, if A1 = 0 and J11 = 0, then (0, 0) has period-doubling at η = 0 i.e. (H0 , 0) undergoes period-doubling bifurcation at r = r0 . If η = 0 then G(u, 0) = −u + A1 u 2 + O((|u|)4 ). Since Schwarzian derivative of the map G 1 (x) = −x + A1 x 2 + O((|x|)4 ) at x = 0 is −6(A1 )2 < 0, the origin is asymptotically stable by Theorem 1.16. [15]. Hence, the equilibrium point (H0∗ , 0) is asymptotically stable. It can be shown that ∂G ∂ 2 G ∂2 G = 2J11 , + 2 ∂η ∂u 2 ∂u∂η and 1 2



∂2 G ∂u 2

2 +

1 ∂3G = 2(A1 )2 . 3 ∂u 3

If J11 = 0, and A1 = 0, then the period-two solution is stable by Theorem 3.5.1. [17].

4.2 Transcritical Bifurcation of Exclusion Equilibrium Point In this subsection we take c as bifurcation parameter. Let c0 = − H ∗ f1 (0) . Then the 0 +  ∗  eigenvalues of Jacobian matrix are λ1 = ber −H0 r 1 − r H0∗ < 1 and λ2 = 1. By change of variables u n = Hn − H0∗ , vn = Pn , we obtain following system   ∗ u n+1 = a + b f (vn ) er (1−H0 −u n ) H0∗ + u n − H0∗ (21)   vn+1 = c (1 − f (vn )) H0∗ + u n . Let

  ∗ f 1 (u, v) = a + b f (v) er (1−H0 −u ) H0∗ + u − H0∗   f 2 (u, v) = c (1 − f (v)) H0∗ + u .

(22)

The equilibrium point of system (22) is (0, 0). Substituting c = − H ∗ f1 (0) + η for 0 + sufficiently small parameter η in (22), and expanding the system about (0, 0, 0), we obtain

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S. Kalabuši´c et al.

 ∗  ∗ f 1 (u, v, η) = −ber −H0 r H0∗r − 1 u + bH0∗ f  (0)er −H0 r v   1 ∗  ∗  + br er −H0 r H0∗r − 2 u 2 + b f  (0)er −H0 r 1 − H0∗r uv 2 1 ∗ + bH0∗ f  (0)er −H0 r v 2 + O((|u| + |v| + |η|)3 ), 2   1 f  (0) 2 ∗   − η f (0) uv + f 2 (u, v, η) = v − H0 f (0)vη + v H0∗ 2 f  (0) + O((|u| + |v| + |η|)3 ).



Let Q= 

and

ber H0∗ f + (0)

ber

un vn

(

)+e

r H0∗ −1

1



 =Q

1

r H0∗



0

u˜ n v˜n .



Then, the system becomes 

u˜ n+1 v˜n+1



 =

1 0 0 λ2



u˜ n v˜n



 +

 f˜1 (u˜ n , v˜n , η) , f˜2 (u˜ n , v˜n , η)

(23)

where

f˜1 (u, v, η) =

ber f  (0) f  (0)   + ∗ 2 f  (0) ber H0∗r − 1 + e H0 r

 u2 +

1 uv − H0∗ f  (0)uη H0∗

+ O((|u| + |v| + |η|)3 ), ∗  ∗   2 b f  (0)er −H0 r ber + H0∗r e H0 r  ∗ 1 r −H0∗ r ˜   uv H0 r − 2 v − f 2 (u, v, η) = br e ∗ 2 ber H0∗r − 1 + e H0 r b(H0∗ )2 er f  (0)2   ∗ uη ber H0∗r − 1 + e H0 r     b2 H ∗ e2r f  (0) + H ∗r 2 f  (0)2 − f  (0) + − 2  2 ber (H ∗r − 1) + e H ∗ r +

∗

+ +

b3 H ∗ e(H −3)(−r ) f  (0) (H ∗r − 1)2 2  2 ber (H ∗r − 1) + e H ∗ r b3 H ∗ e(H

∗

 f  (0)2 (−r H ∗ (H ∗r − 2) − 2) 2 u  2 2 ber (H ∗r − 1) + e H ∗ r

−3)(−r )

+ O((|u| + |v| + |η|)3 ).

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

251

Assuming that local central manifold is given by ˜ + h 3 η 2 + O((|u| ˜ + |η|)3 ) v˜ = h(u, ˜ η) = h 1 u˜ 2 + h 2 uη and using software package Mathematica, we obtain that          b2 H0∗ e2r −ber f  (0)2 H0∗r H0∗r − 2 + 2 − f  (0) H0∗r − 1 2    h1 = ∗  2 ber H0∗r − 1 + e H0 r 3    ∗  b2 H0∗ e2r −e H0 r f  (0) + H0∗r 2 f  (0)2 − f  (0)    − , ∗  2 ber H0∗r − 1 + e H0 r 3 ∗

b(H ∗ )2 f  (0)2 e H0 r +r h2 =  ∗ r 0 ∗  , bH0 e r + b (−er ) + e H0 r 2 h 3 = 0. Let

G(w, η) = w + w 2

ber



ber f  (0) f  (0)  ∗r + H ∗ 2 f  (0) H0 r − 1 + e 0

 − H0∗ f  (0)ηw + O((|w| + |η|)3 ).

We verify the following conditions from [44, p. 365] to prove that equilibrium undergoes transcritical bifurcation (i)

∂G ∂2 G ∂2 G ∂G (0, 0) = 1, (ii) (0, 0) = 0, (iii) (0, 0)  = 0, (iv) (0, 0)  = 0. ∂w ∂η ∂w∂η ∂w2

Conditions (i), (ii) and (iii) are obviously true since f + (0) < 0. Since f  (0) > 0, and    ∗   ∗  e H0 r H0∗r H0∗ − a + a r H0∗ r be H0 r − 1 + e = > 0, H0∗ we obtain

∂2 G 2ber f  (0) f  (0)  < 0. (0, 0) = r  ∗ +  ∗ 2 H r ∂w f (0) be H0 r − 1 + e 0

We conclude that (0, 0) undergoes transcritical bifurcation at η = 0, i.e., (H0∗ , 0) has transcritical bifurcation at c = c0 . For η = 0, the dynamics on center manifold is given by the map

u˜ n+1 = G(u˜ n , 0) = u˜ n + u˜

2

ber f  (0) f  (0)   ∗ + ∗ 2 f  (0) ber H0 r − 1 + e H0 r

 + O(|u| ˜ 3 ).

The second derivative of G(u˜ n , 0) at zero is negative. Hence, the equilibrium u = 0 of G(u, 0) is semi-stable from right [16].

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S. Kalabuši´c et al.

4.3 Local Bifurcations of Positive Equilibrium Point In this subsection, we study the conditions for the existence of the period-doubling bifurcation of the interior equilibrium (H ∗ (r ), P ∗ (r )) ∈ IR2+ , where P ∗ (r ) = c

∗ ∗ bH ∗ (r ) − (H ∗ (r ) − a)er (H (r )−1) (H ∗ (r ) − a)er (H (r )−1) and f (P ∗ (r )) = . b bH ∗ (r )

(24)

We take parameter r as a bifurcation parameter. By changing variables u n = Hn − H ∗ and vn = Pn − P ∗ , we shift the equilibrium (H ∗ (r ), P ∗ (r )) to the origin. System (4) becomes u n+1 = a + b(u n + H ∗ (r )) f (vn + P ∗ (r ))er (−H

∗

(r )−u n +1)

− H ∗ (r )

(25)

vn+1 = c(H ∗ (r ) + u n )(1 − f (vn + P ∗ (r ))) − P ∗ (r ). Let Fr be the following map Fr

 u v

=

a + b(u + H ∗ (r )) f (v + P ∗ (r ))er (−H

∗

(r )−u+1)

− H ∗ (r )

c(H ∗ (r ) + u)(1 − f (v + P ∗ (r ))) − P ∗ (r )

 .

(26)

Fr has a fixed point at (0, 0). The Jacobian matrix JFr (0, 0) is

JFr (0, 0) =

ber −r H

∗

(r )

f (P ∗ (r )) (1 − r H ∗ (r )) ber −r H

∗

(r )

H ∗ (r ) f  (P ∗ (r ))

−cH ∗ (r ) f  (P ∗ (r )) ⎛ (H ∗ (r )−a)(1−r H ∗ (r )) ⎞ ∗ ber −r H (r ) H ∗ (r ) f  (P ∗ (r )) H ∗ (r ) ⎠. = ⎝ r ( H ∗ (r )−1) ce (a−H ∗ (r )) ∗  ∗ + c −cH (r ) f (r )) (P bH ∗ (r )



c(1 − f (P ∗ (r ))

(27)

The Taylor series expansion of Fr about (0, 0) is       u u f˜(u, v, r ) Fr = JFr (0, 0) + , v v g(u, ˜ v, r )

(28)

where   r (H ∗ (r ) − a) (H ∗ (r )r − 2) 2 1 ∗ u + bH ∗ (r )er −H (r )r f  P ∗ (r ) v 2 f˜(u, v, r ) = ∗ 2H (r ) 2 −ber −H

∗

(r )r



   H ∗ (r )r − 1 f  P ∗ (r ) uv + O((|u| + |v|)3 ),

    1 g(u, ˜ v, r ) = −c f  P ∗ (r ) uv − cH ∗ (r ) f  P ∗ (r ) v 2 + O((|u| + |v|)3 ). 2

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

253

From (24), we obtain ∗ 

(H ) (r ) =

−er (H

∗

(r )−1)

  (H ∗ (r ) − 1)H ∗ (r )(H ∗ (r ) − a) cH ∗ (r ) f  (P ∗ (r )) + 1 , K1

where   ∗ K 1 = c(H ∗ (r ))2 er (H (r )−1) f  (P ∗ (r ))(1 − ar + r H ∗ (r )) − b   ∗ + er (H (r )−1) a − ar H ∗ (r ) + r (H ∗ (r ))2 and bc (H ∗ ) (r ) − cer (H

∗

(−ar + r H ∗ (r ) + 1) (H ∗ ) (r ) b   ∗ cer (H (r )−1) −(a + 1)H ∗ (r ) + a + H ∗ (r )2 − b   ∗ ∗ cer H (r ) (H ∗ (r ) − 1) (a − H ∗ (r )) er (H (r )−1) (a − H ∗ (r )) + bH ∗ (r ) , = bK 2

(P ∗ ) (r ) =

(r )−1)

where          H ∗ (r ) H ∗ (r ) c −ar + r H ∗ (r ) + 1 f  P ∗ (r ) + r − ar + a   − bcer H ∗ (r )2 f  P ∗ (r ) .

K 2 = er H

4.3.1

∗

(r )



Period-Doubling Bifurcation

Assume that a, b, c > 0 and r = r0 > 0, such that f  (P ∗ (r0 )) =

ar0 H ∗ (r0 ) − a − H ∗ (r0 ) (r0 H ∗ (r0 ) − 2)   cH ∗ (r0 ) 2 ar0 + ber0 −r0 H ∗ (r0 ) − r0 H ∗ (r0 ) + 1

(29)

and   ∗ H ∗ (r0 ) er0 (H (r0 )−1) (ar0 − r0 H ∗ (r0 ) + 1) + b  ∗ + (H ∗ (r0 ) (ar0 − r0 H ∗ (r0 ) + 2) − a) r0 er0 (H (r0 )−1) (a − H ∗ (r0 )) + b = 0. (30) Now, we obtain:

254

S. Kalabuši´c et al.

JFr0 = JFr (0, 0)|r =r0 ⎛ (a−H ∗ (r

0 ))(r0 H H ∗ (r0 )

∗

(r0 )−1)

⎜  = ⎝  er0 ( H ∗ (r0 )−1) (a−H ∗ (r0 )) c +1 bH ∗ (r0 )

ber0(H ∗ (r0 )(ar0 −H ∗ (r0 )r0 +2)−a)  ∗ cH ∗ (r0 ) ber0 +er0 H (r0 ) (ar0 −H ∗ (r0 )r0 +1) ∗ a+H (r0 )(−ar0 +H ∗ (r0 )r0 −2)   ∗ H ∗ (r0 ) ber0 −r0 H (r0 ) +ar0 −r0 H ∗ (r0 )+1

⎞ ⎟ ⎠

(31)

and      ∗ ∗ H (r0 ) − 1 a − H ∗ (r0 ) er0 H (r0 ) H ∗ (r0 ) −2ar0 + 2r0 H ∗ (r0 ) − 3 2I1    ∗   ∗ H (r0 ) − 1 a − H ∗ (r0 ) er0 H (r0 ) a − ber0 H ∗ (r0 ) − , 2I1

(H ∗ ) (r0 ) =

(P ∗ ) (r0 ) =

      ∗ c H ∗ (r0 ) − 1 a − H ∗ (r0 ) er0 ( H (r0 )−1) a − H ∗ (r0 ) + bH ∗ (r0 ) I2 −2bI1

,

where 

    H ∗ (r0 ) r0 H ∗ (r0 ) −2ar0 + r0 H ∗ (r0 ) − 1 + ar0 (ar0 + 2) − 1    ∗ − er0 H (r0 ) a 2 r0 + ber0 r0 H ∗ (r0 ) − 1 a − H ∗ (r0 )

I 1 = er 0 H

∗

(r0 )

and I 2 = er 0 H

∗

(r0 )

  ar0 − r0 H ∗ (r0 ) + 1 + ber0 .

Now, we have det (JFr0 ) + tr (JFr0 ) + 1 = 0. The eigenvalues of Jacobian matrix are λ1 = −1 and λ2 = −det (JFr0 )

  ∗ (H ∗ (r0 ) (ar0 − r0 H ∗ (r0 ) + 2) − a) r0 er0 H (r0 ) (a − H ∗ (r0 )) + ber0   =− . H ∗ (r0 ) er0 H ∗ (r0 ) (−ar0 + r0 H ∗ (r0 ) − 1) − ber0

Also, since (30) holds and det (JFr0 ) − tr (JFr0 ) + 1 > 0, we obtain: −1 = λ2 < 1. Let r = r0 + η. We obtain  JFr0 +η (0, 0) = JFr0 (0, 0) + where

and



J11 J12 J21 J22

 =

J11 J12 J21 J22

 η + O(|η|2 )

d(JFr0 +η (0, 0)) dη

η=0

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

J11 =

255

a(H ∗ ) (r0 ) + a − r0 (H ∗ ) (r0 ) − H ∗ (r0 ) H ∗ (r0 ) 2

J12 := ber0 (1−H (r0 )) H (r0 ) P  (r0 ) f  (P (r0 ))    + ber0 (1−H (r0 )) H  (r0 ) − H (r0 ) r0 H  (r0 ) + H (r0 ) − 1 f  (P (r0 )) J21 := −

∗

∗

(r0 )−1)

(r0 H ∗ (r0 ) (H ∗ (r0 ) − a) + a) (H ∗ ) (r0 ) bH ∗ (r0 ) 2

(H ∗ (r0 ) − 1) H ∗ (r0 ) (H ∗ (r0 ) − a) bH ∗ (r0 ) 2      := −c H ∗ (r0 ) (P ∗ ) (r0 ) f  P ∗ (r0 ) + (H ∗ ) (r0 ) f  P ∗ (r0 ) . −

J22

cer0 (H

cer0 (H

(r0 )−1)

Now, system (25) becomes 

u n+1 vn+1



 = Fr0 +η

un vn



 = JFr0

un vn



 +

 f˜1 (u n , vn , η) , g˜1 (u n , vn , η)

(32)

where r0 (H ∗ (r0 ) − a) (H ∗ (r0 )r0 − 2) 2 u f˜1 (u, v, η) = 2H ∗ (r0 )     ∗ − ber0 −H (r0 )r0 H ∗ (r0 )r0 − 1 f  P ∗ (r0 ) uv +

 bH ∗ (r0 ) r0 −H ∗ (r0 )r   ∗ e f P (r0 ) v 2 + J11 uη + J12 vη 2

+ O((|u| + |v| + |η|)3 ),     1 g˜1 (u, v, η) = −c f  P ∗ (r0 ) uv − cH ∗ (r0 ) f  P ∗ (r0 ) v 2 + J21 uη + J22 vη 2 + O((|u| + |v| + |η|)3 ). The functions f˜1 (u, v, η) and g˜1 (u, v, η) can be written in the following forms f˜1 (u, v, η) = A1 u 2 + A2 uv + A3 v 2 + J11 uη + J12 vη + O((|u| + |v| + |η|)3 ), g˜1 (u, v, η) = B1 uv + B2 v 2 + J21 uη + J22 vη + O((|u| + |v| + |η|)3 ). The eigenvectors of JFr0 corresponding to the eigenvalues −1 and −det (JFr0 ) are, respectively

v1 =



−

c e

1 ( H ∗ −1)r

0 (ar0 −H ∗ r0 +1)+b

b



 

, v2 =

b(H ∗ (ar0 −H ∗ r0 +2)−a) ∗ c(a−H ∗ )e( H −1)r0 +bcH ∗

1

 .

256

S. Kalabuši´c et al.

Let ⎛ A=⎝



−

c e

1 ( H ∗ −1)r

0 (ar0 −H ∗ r0 +1)+b

b(H ∗ (ar0 −H ∗ r0 +2)−a) ∗ )e( H ∗ −1)r0 +bcH ∗ c(a−H 

⎞ ⎠=



1

b

1 a1 a2 1

 ∗  c(H ∗ (ar0 −H ∗ r0 +2)−a) e( H −1)r0 (ar0 −H ∗ r0 +1)+b

We have det (A) = 1 − a1 a2 = ∗ c(a−H ∗ )e( H −1)r0 +bcH ∗ the relation (30) holds, it follows that 1 − a1 a2 = 0. Let 



un vn

 =A

u˜n v˜n

 .

+ 1. Since

 i.e. u n = u˜n + a1 v˜n

and vn = v˜n + a2 u˜n .

Now, we have 

u˜ n+1 v˜n+1

 =A

−1

 JFr0 A

u˜n v˜n

 +A

−1



f˜1 (u˜ n , v˜n , η) g˜1 (u˜ n , v˜n , η)



or equivalently 

u˜ n+1 v˜n+1



 =

−1 0 0 −det (JFr0 )



u˜ n v˜n



 +

 f˜2 (u˜ n , v˜n , η) , g˜2 (u˜ n , v˜n , η)

(33)

where −a 2 A −a A +a a 2 B +a a B −A u 2 f˜2 (u, v, η) = ( 2 3 2 2 a11 a22 −12 1 2 1 1 ) 1 a2 J22 −J11 )ηu + (−a2 J12 +a1 Ja211 a+a 2 −1

+(

−2a1 A1 −a2 a1 A2 −2a2 A3 +a2 a12 B1 +a1 B1 +2a2 a1 B2 −A2 )uv a1 a2 −1

+(

−a12 A1 −a1 A2 +a12 B1 +a1 B2 −A3 )v 2 a1 a2 −1

a12 J21 −a1 J11 +a1 J22 −J12 )ηv a1 a2 −1

+(

(34)

+ O((|u| + |v| + |η|)3 ),

a 3 A +a 2 A +a A −a 2 B −a B u 2 g˜2 (u, v, η) = ( 2 3 2 2 a12a21−1 2 2 2 1 ) a22 J12 +a2 J11 −a2 J22 −J21 )ηu a1 a2 −1

+(

a1 a22 A2 +2a22 A3 +2a1 a2 A1 +a2 A2 −a1 a2 B1 −2a2 B2 −B1 )uv a1 a2 −1

+(

a2 a12 A1 +a2 a1 A2 +a2 A3 −a1 B1 −B2 )v 2 a1 a2 −1

+(

1 J21 −J22 )ηv + (a1 a2 J11 +a2aJ112a2−a + O((|u| + |v| + |η|)3 ). −1

(35)

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

257

The central manifold is given by v˜ = h(u, ˜ η) = h 1 u˜ 2 + h 2 uη ˜ + h 3 η 2 + O((|u| + 3 |η|) ). The following holds h(−u˜ + f˜2 (u, ˜ h(u, ˜ η), η)) = −det (J Fr )(h 1 u˜ 2 + h 2 uη ˜ + h 3 η 2 ) + g˜ 2 (u, h(u, ˜ η), η). 0

We obtain a2 (a2 (a2 A3 + A2 − B2 ) + A1 − B1 ) a 2 J12 + a2 J11 − a2 J22 − J21 , h2 = 2 , (a1 a2 − 1) (det (JFr0 ) + 1) (a1 a2 − 1) (det (JFr0 ) − 1) h 3 = 0.

h1 =

Dynamics of (33) on the central manifold is given by the map w˜ n+1 = G(w˜ n , η), where G(w, η) = −w + f˜2 (w, h(w, η), η) + O((|w| + |η|)4 )  A1 +B1 )+h 1 (J22 −J11 )) = a1 (h 2 (−a2 (A2 −2B2a)−2 1 a2 −1 h (−2a2 A3 −A2 )+a12 (a2 B1 h 2 +h 1 J21 )−h 1 J12 w2 η + 2 a1 a2 −1 h −2a A −a a A −2a A +a2 a12 B1 +a1 B1 +2a2 a1 B2 −A2 ) 3 + 1 ( 1 1 2 1 2 2 a13 a2 −1 w

+(

(36)

−a22 A3 −a2 A2 +a1 a22 B2 +a1 a2 B1 −A1 ) 2 w a1 a2 −1

h a 2 J −a J +a J −J + 2 ( 1 21 a11 a112 −1 1 22 12 ) wη 2 1 a2 J22 −J11 + −a2 J12 +a1aJ121a2+a ηw − w + O((|w| + |η|)4 ). −1

We verify the following conditions from [44, p. 373] to prove that the system undergoes period-doubling bifurcation at r = r0 : (i)

∂G 2 ∂2G2 ∂G (0, 0) = −1, (ii) (0, 0) = 0, (iii) (0, 0) = 0, ∂u ∂η ∂u 2 (iv)

We have

∂2 G2 ∂3G2 (0, 0) = 0, (v) (0, 0) = 0. ∂u∂η ∂u 3

258

S. Kalabuši´c et al.

∂G 2 ∂2 G2 ∂G (0, 0) = −1 , (0, 0) = 0 , (0, 0) = 0, ∂w ∂η ∂w 2 ∂2 G2 2 (a2 J12 − a1 (a2 J22 + J21 ) + J11 ) (0, 0) = = 0, ∂w∂η a1 a2 − 1 12 (a2 (a2 A3 − a1 (a2 B2 + B1 ) + A2 ) + A1 ) 2 ∂3G2 (0, 0) = − ∂w 3 (a1 a2 − 1) 2 + +

12h 1 ((−a1 a2 − 1)A2 − 2a2 A3 ) 1 − a1 a2

12h 1 a1 (a1 a2 B1 + 2a2 B2 − 2 A1 + B1 ) = 0 (1 − a1 a2 )

If − (a1 a2 − 1) h 1 (a1 (a1 a2 B1 + 2a2 B2 − 2 A1 + B1 ) + (−a1 a2 − 1) A2 − 2a2 A3 ) − (a2 (a2 A3 − a1 (a2 B2 + B1 ) + A2 ) + A1 ) 2 = 0 and a2 J12 − a1 (a2 J22 + J21 ) + J11 = 0, then period-doubling bifurcation occurs at η = 0 i.e. at r = r0 .

4.3.2

Neimark–Sacker Bifurcation of the Interior Equilibrium

We prove that Neimark–Sacker bifurcation occurs for the interior equilibrium (H ∗ , P ∗ ), taking r as a bifurcation parameter. Let r0 > 0 be, such that f  (P ∗ ) =

cH ∗

1   ∗ r0 (H − a) − ber0 −H ∗ r0

where H ∗ = H ∗ (r0 ), P ∗ = P ∗ (r0 ). Let ⎛ JF = JFr (0, 0)|r =r0 =

(a−H ∗ )(r0 H ∗ −1) H∗ ⎝  r0 ( H ∗ −1) e (a−H ∗ ) c + bH ∗

∗

 1

ber0 −r0 H H ∗ f  (P ∗ ) −cH ∗ f  (P ∗ )

Then, it follows ⎛

(a−H ∗ )(r0 H ∗ −1) H∗

 JF = ⎝  er0 ( H ∗ −1) (a−H ∗ ) c +1 bH ∗

⎞

r0

− ber0 c+er0be H∗ r

0 (a−H

∗ )c

1 ∗ er0 −r0 H b+ar0 −r0 H ∗

⎠,

     ∗ ∗ det (JF ) = cH ∗ e−r0 H f  P ∗ r0 H ∗ − a e H r0 − ber0 = 1,

⎞ ⎠.

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

and

tr (JF ) =

1 ∗ ar0 +ber0 −H r0 −H ∗ r0

=−

eH

∗r

− H∗

0

−

a H∗

259

+ ar0 − H ∗r0 + 1

( H ∗ ( H ∗ r0 (−2ar0 +H ∗ r0 −1)+(ar0 +1)2 )−a 2 r0 ) ∗ ∗ H ∗ (−ar0 e H r0 +b(−er0 )+H ∗ r0 e H r0 ) ∗

∗

be (a−H )(H r0 −1) r0

(−ar0 e H ∗ r0 +b(−er0 )+H ∗ r0 e H ∗ r0 )

(37)

.

We will prove that Neimark–Sacker bifurcation occurs at r = r0 . Theorem 5 Assume that a, b, c, r > 0 and tr (JF ) = 0, tr (JF ) = −1. Let r0 be such that f  (P ∗ ) =

1   cH ∗ r0 (H ∗ − a) − ber0 −H ∗ r0

(38)

and eH

∗

r0



H ∗ (H ∗r0 (−2ar0 + H ∗r0 − 3) + ar0 (ar0 + 4) + 1) − a 2 r0



+ber0 (H ∗ (ar0 − H ∗r0 + 3) − a) > 0.

(39)

Then, the fixed point of the mapping Fr is (0, 0) and the Jacobian JFr (0, 0) has complex eigenvalues λ± (r0 ) with modulus 1, if parameter r is close to r0 :  tr (JF ) ± i 4 − (tr (JF ))2 , |λ± (r0 )| = 1. λ± (r0 ) = 2 d k Also, (λ± (r0 )) = 1 for k = 1, 2, 3, 4 and d(r0 ) = dr |λ± (r )| is given by (40). r =r0

Proof Using (39), and fact that f  (P ∗ ) given in (38) is negative, we obtain tr (JF ) + 2 = +

eH

∗

r0

  2 a r0 − H ∗ (H ∗r0 (−2ar0 + H ∗r0 − 3) + ar0 (ar0 + 4) + 1)   H ∗ r0 (H ∗ − a) e H ∗ r0 − ber0

ber0 (a − H ∗ (ar0 − H ∗r0 + 3))   >0 H ∗ r0 (H ∗ − a) e H ∗ r0 − ber0

Since det (JF ) − tr (JF ) + 1 > 0, and det (JF ) = 1, we have that tr (JF ) < 2, so −2 < tr (JF ) < 2. The complex eigenvalues λ± (r ) of JFr (0, 0) with modulus 1 exist at r = r0 and  tr (JF ) ± i 4 − (tr (JF ))2 . λ± (r0 ) = 2

260

S. Kalabuši´c et al.

If we assume that tr (JF ) = 0, and tr (JF ) = −1, then (λ± (r0 ))k = 1 for k = 1, 2, 3, 4. Let

∗

G 1 (H (r ), r ) := f

cer ( H

∗ (r )−1

) (a − H ∗ (r )) + bcH ∗ (r ) b

 −

er ( H

∗ (r )−1

) (H ∗ (r ) − a)

bH ∗ (r )

≡ 0.

We have ∂G 1

∂r (H ∗ ) (r ) = − ∂G

(H ∗ (r ), r ) (H ∗ (r ), r )

1

∂H∗

=−

er (H

∗

(r )−1)

  (H ∗ (r ) − 1) H ∗ (r ) (H ∗ (r ) − a) cH ∗ (r ) f  (P ∗ (r )) + 1 . I3

where       ∗ I3 = cH ∗ (r )2 er (H (r )−1) −ar + r H ∗ (r ) + 1 − b f  P ∗ (r )   ∗ + er (H (r )−1) −ar H ∗ (r ) + a + r H ∗ (r )2 Since f  (P ∗ (r0 )) =

1 ,  cH ∗ (r0 ) r0 (H ∗ (r0 ) − a) − ber0 −H ∗ (r0 )r0

we obtain   ∗ − (H ∗ − 1) H ∗ (H ∗ − a) er H (−ar0 + r0 H ∗ + 1) − be0r , (H ) (r0 ) = I4 ∗ 

where H ∗ = H ∗ (r0 ) and      H ∗ a 2 r02 + r0 H ∗ −2ar0 + r0 H ∗ + 1 + 1 − a 2 r0     +ber0 H ∗ ar0 − r0 H ∗ − 1 − a .

I 4 = er 0 H

∗



Since P ∗ (r ) = we have

cer (H

∗

(r )−1)

(a − H ∗ (r )) + bcH ∗ (r ) , b

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

261

  (a − H ∗ (r )) r (H ∗ ) (r ) + H ∗ (r ) − 1 (P ) (r ) = b   ∗ c b (H ∗ ) (r ) − er (H (r )−1) (H ∗ ) (r ) + b cer (H

∗ 

∗

(r )−1)

and (P ∗ ) (r0 ) =

   ∗ ∗ c (H ∗ − 1) (a − H ∗ ) r0 er0 (H −1) (a − H ∗ ) + b er0 H (a − H ∗ ) + ber0 H ∗ , − bI4 where H ∗ = H ∗ (r0 ).

d(r0 ) =



d |λ± (r )| dr r =r0

= 21 bcer0 −r0 H − 21 bcer0 −r0 H

∗

∗

=

d dr



det (JFr (0, 0))

r =r0

=



1 d(det (JFr (0,0))) 2 dr r =r0

∗ (r0 )+(r0 H ∗ (r0 )−1)(H ∗ ) (r0 ) (r0 ) (H ∗ (r0 )−1)H  ∗ ∗ cH (r0 ) −ar0 −ber0 −r0 H (r0 ) +r0 H ∗ (r0 )

(r0 )

(40)

H ∗ (r0 ) (P ∗ ) (r0 ) f  (P ∗ (r0 ))

∗ (r0 )(H∗ (r0 )−a)−r0 (a−2H ∗ (r0 ))(H ∗ ) (r0 ) ∗ cH ∗ (r0 ) −ar0 −ber0 −r0 H (r0 ) +r0 H ∗ (r0 )

+ 21 c H

+ 21 cr0 H ∗ (r0 ) (H ∗ (r0 ) − a) (P ∗ ) (r0 ) f  (P ∗ (r0 )) , Observe that the coefficient d(r0 ) can be positive or negative.



System (25) can be written using Taylor series expansion in the following form 

u n+1 vn+1



⎛ ⎜ =⎝

(a−H ∗ )(r0 H ∗ −1)

c

H∗  ∗ er0 ( H −1) (a−H ∗ ) bH ∗

 +1

−

ber0 ∗ ber0 c+er0 H r

⎞ 0

(a−H ∗ )c

1 ∗ er0 −r0 H b+ar0 −r0 H ∗

⎟ ⎠



un vn



 +

f (u n , vn ) g(u n , vn )

 ,

where f (u, v) =

r (H ∗ −a)(H ∗ r −2) 2 u 2H ∗

+

∗

ber (H ∗ r −1) uv cH ∗ (r (a−H ∗ )e H ∗ r +ber ) r 2 (a−H ∗ )(H ∗ r −3) 3 u 6H ∗ ber r (H ∗ r −2)  ∗ 2 f (P ) uv − 2cH ∗ r (a−H ∗ )e H ∗ r +ber u 2 v ( )

+ 21 bH ∗ er −H r f  (P ∗ ) v 2 + ∗

− 21 ber −H r (H ∗r − 1) ∗

(41)

+ 16 bH ∗ er −H r f (3) (P ∗ ) v 3 + O((|u| + |v|)4 ), g(u, v) =

uv H ∗ (ar +ber −H ∗ r −H ∗ r )

− 21 cH ∗ f  (P ∗ ) v 2 − 21 c f  (P ∗ ) uv 2

− 16 cH ∗ f (3) (P ∗ ) v 3 + O((|u| + |v|)4 ).

(42)

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S. Kalabuši´c et al.

We write f and g as follows f (u, v) = A20 u 2 + A11 uv + A02 v 2 + A30 u 3 + A21 u 2 v + A12 uv 2 + A03 v 3 +O((|u| + |v|)4 ),

(43)

g(u, v) = B11 uv + B02 v 2 + B12 uv 2 + B03 v 3 + O((|u| + |v|)4 ). Let ⎛

(a−H ∗ )(r0 H ∗ −1) H∗

⎞

r0

 JF = ⎝  er0 ( H ∗ −1) (a−H ∗ ) c + 1 ∗ bH

− ber0 c+er0be H∗ r

0 (a−H

∗ )c

1



⎠=

∗ er0 −r0 H b+ar0 −r0 H ∗

a11 a12 a21 a22

 .

The eigenvalues of JF are λ± = μ ± ωi. Let  L=

0 a12 μ − a11 −ω



Then −1 a12 ω

L −1 =



−ω 0 a11 − μ a12

 .

Using 

un vn



 =L

u˜ n v˜n





 ,

we obtain 

u˜ n+1 v˜n+1



 =

μ −ω ω μ

u˜ n v˜n

 +

 f 1 (u˜ n , v˜n ) , g1 (u˜ n , v˜n )

where     11 ) f 1 (u, + a u˜ 2 ˜ v) ˜ = (μ − a11 ) A02 (μ−a + A A 11 12 20 a12   2 11 −μ) + 2 A02 ω(a − A11 ω u˜ v˜ + Aa0212ω v˜ 2 a12 +

2 3 A03 (μ−a11 )3 +a12 A12 (μ−a11 )2 +a12 A21 (μ−a11 )+a12 A30 3 u˜ a12

+ ω2



3A03 (μ−a11 ) a12

 + A12 u˜ v˜ 2

2 ω 3A (μ−a11 )2 +2a12 A12 (μ−a11 )+a12 A21 ) − ( 03 v˜ u˜ 2 a12

−

A03 ω 3 3 v˜ a12

+ O((|u| ˜ + |v|) ˜ 4 ),

(44)

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

g1 (u, ˜ v) ˜ =



(μ−a11 )2 a12 (A11 −B02 ) a12 ω

263



+

2 (μ−a11 )(a12 (A20 −B11 )+A02 (μ−a11 )2 ) a12 ω

+

2 −a12 (μ−a11 )(A11 −2B02 )−2 A02 (μ−a11 )2 +a12 B11 u˜ v˜ a12

+ +



A02 ω(μ−a11 ) a12

u˜ 2

  3 12 −B03 ) − B02 ω v˜ 2 + (μ−a11 ) aa1212 (A ω

2 3 A30 ) (μ−a11 )(a12 (μ−a11 )(A21 −B12 )+A03 (μ−a11 )3 +a12 a12 ω

 u˜ 3

(45)

2 ω a (μ−a11 )(A12 −3B03 )+3A03 (μ−a11 )2 −a12 B12 ) + ( 12 u˜ v˜ 2 a12  2 A12 −3B03 ) − (μ−a11 ) a12a(2 12  2 (μ−a11 )(a12 (A21 −2B12 )+3A03 (μ−a11 )2 ) v˜ u˜ 2 + a12

+

ω 2 (A03 (a11 −μ)+a12 B03 ) 3 v˜ a12

+ O((|u| ˜ + |v|) ˜ 4 ).

The direction of Neimark–Sacker bifurcation is determined by following formula from Theorem 15.31 [20]

α(r0 ) = Re

ξ11 ξ20 (1 − 2λ+ (r0 )) (λ− (r0 ))2 1 − λ+ (r0 )

 +

|ξ11 |2 + |ξ02 |2 − Re (λ− (r0 )ξ21 ) , 2 (46)

where ξ20 = 18 {( f 1 )u˜ u˜ − ( f 1 )v˜ v˜ + 2(g1 )u˜ v˜ + i[(g1 )u˜ u˜ − (g1 )v˜ v˜ − 2( f 1 )u˜ v˜ ]} = i a 2 (A (λ −a )+B (a −λ ))+A (λ −a )(λ −a )2 = ( 12 20 − 11 11 114a12+ω 02 − 11 + 11 )

(47)

i A (a (a −2μ)+|λ+ |)−B02 (λ+ −a11 ) ) + ( 11 11 11 , 4ω 2

ξ11 = 41 {( f 1 )u˜ u˜ + ( f 1 )v˜ v˜ + i[(g1 )u˜ u˜ + (g1 )v˜ v˜ ]} i a 2 (A (λ −a )+B (a −μ))+A (λ −a )(λ −a )2 = ( 12 20 − 11 11 112a12 ω 02 + 11 − 11 )

(48)

11 )+B02 (a11 −λ+ )) + i(λ− −a11 )(A11 (μ−a , 2ω

ξ02 = 18 {( f 1 )u˜ u˜ − ( f 1 )v˜ v˜ − 2(g1 )u˜ v˜ + i[(g1 )u˜ u˜ − (g1 )v˜ v˜ + 2( f 1 )u˜ v˜ ]} = +

i(λ− −a11 )a12 (λ− −a11 )(A11 −B02 ) 4a12 ω

i(λ− −a11 )(

2 a12 (A20 −B11 )+A02 (λ− −a11 )2

4a12 ω

(49) ),

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S. Kalabuši´c et al.

ξ21 =

1 [( f 1 )u˜ u˜ u˜ 16

+ ( f 1 )u˜ v˜ v˜ + (g1 )u˜ u˜ v˜ + (g1 )v˜ v˜ v˜ ]

1 +[(g1 )u˜ u˜ u˜ + (g1 )u˜ v˜ v˜ − ( f 1 )u˜ u˜ v˜ − ( f 1 )v˜ v˜ v˜ ] 16 i 3 i 3A (a (a −2μ)+|λ+ |)2 +3a12 A30 (λ− −a11 )) = ( 03 11 11 8a12 ω

(50)

)(−3a11 +3μ+iω) + ia12 A21 (λ− −a118ω )(−3a11 +3μ−iω) + ia12 B12 (a11 −λ+8ω 11 −λ+ )+A12 (−3a11 +3μ−iω)) + i(a11 (a11 −2μ)+|λ+ |)(3B03 (a8ω ,

where λ± = λ± (r0 ). Since the expression for α(r0 ) is very complicated, we skip writing its form. However, in Sect. 8 we compute α(r0 ) for specific numerical values of parameters for all models that we consider. Now, we have the following theorem. Theorem 6 Assume that conditions of Theorem 5 are satisfied. The Neimark–Sacker bifurcation of the interior equilibrium (H ∗ , P ∗ ) occurs at r = r0 . If d(r0 ) > 0 and α(r0 ) > 0 (α(r0 ) < 0), there is an attracting (unstable) closed invariant curve when r is close to r0 and r > r0 (r < r0 ). If d(r0 ) < 0 and α(r0 ) > 0 (α(r0 ) < 0), there is an attracting (unstable) closed invariant curve when r is close to r0 and r < r0 (r > r0 ).

5 Boundedness of the Solutions In this section we study boundedness of the solutions of system (4). Obviously, Hn > a, Pn ≥ 0, ∀n ∈ IN. Since r = 0 the following holds:   ber −1 ber −1 , 0 ≤ Pn ≤ c a + a < Hn ≤ a + , ∀n > 0 r r r −1

because the maximum value of the function g(x) = xer (1−x) is e r . This means (4) are bounded and lie in the invariant interval  of system r −1 that all solutions r −1 × 0, c a + ber . a, a + ber r Also, if we assume that be < 1, we obtain: H1 ≤ a + ber H0 and then by induction it follows that Hn ≤ a(1 + ber + (ber )2 + · · · + (ber )n−1 ) + (ber )n H0 . This implies:

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

lim sup Hn ≤

a 1 − ber

lim sup Pn ≤

ca 1 − ber

n→∞

n→∞

265

    a ca so all solutions lie in the interval a, 1−be × 0, 1−be if ber < 1. r r Let z n+1 = h(z n ) where (H2 ) h ∈ C[0, ∞) ∩ C 2 (0, ∞), h(z) > 0, h  (z) > 0, h  (z) ≤ 0 h(0) = 0 and h(∞) < ∞. We will use the following lemma from [4].

(51)

for

z > 0,

Lemma 1 Assume that h satisfies (H 2). Then the following statements hold. (a) If h + (0) exists and h + (0) ≤ 1, then 0 is the only equilibrium point of (51) and all solutions of (51) converge to 0. (b) If h + (0) exists and h + (0) > 1 or h + (0) = ∞, then (51) has an unique positive equilibrium point z ∗ > 0 and lim z n = z ∗ for all z 0 > 0. n→∞

Now we will prove the following Lemma: Lemma 2 Assume that f + (0) exists. Then the following statements hold. (a) If ber < 1 and f + (0) > (b) If −c(a +

ber −1 , ac

then lim sup Hn ≤

r −1

n→∞

) f + (0) < 1, then lim Pn = 0. n→∞ > 1 or f + (0) = −∞, then lim inf n→∞ ∗ ∗

a 1−ber

and lim Pn = 0. n→∞

be r

(c) If −ac f + (0)

Pn ≥ z ∗ , where z ∗ is an unique

positive solution of z = ac(1 − f (z )). (d) If ber f (z ∗ ) < 1 and −ac f + (0) > 1 or f + (0) = −∞, then a < lim sup Hn ≤ n→∞

a . 1−ber f (z ∗ ) r ∗

(e) If be f (z ) < 1 and −ac f + (0) > 1 or f + (0) = −∞, then the set  a,

   a ac ∗ × z , 1 − ber f (z ∗ ) 1 − ber f (z ∗ )

is an invariant set for system (4). Proof (a) It follows from our assumptions that Hn+1 ≤ a + ber Hn . Let z n+1 = a + ber z n and H0 = z 0 . We have by induction Hn ≤ z n , ∀n > 0, and lim sup Hn ≤ n→∞

a . 1 − ber

a Let ε > 0. Then, there exists n 0 > 0, such that for n > n 0 holds Hn < 1−be r + ε. a  We can choose ε > 0 such that −c( 1−ber + ε) f + (0) < 1. Now we have

266

S. Kalabuši´c et al.

 Pn+1 = cHn (1 − f (Pn )) < c

 a + ε (1 − f (Pn )). 1 − ber

for all n > n 0 . Using Lemma 1, we obtain that lim Pn = 0. (b) We have already proved that Hn+1 ≤ a + polynomial of the function f (P) is

n→∞ ber −1 . The r

f (P) = f (0) + f + (0)P +

second order Maclaurin

f  (ξ) 2 P 2! 

for P > 0, ξ ∈ (0, P). We obtain 1 − f (P) = − f + (0)P − f 2!(ξ) P 2 ≤ − f + (0)P, for P > 0. If P = 0, the last inequality is obviously true. Now, we obtain:   ber −1 (− f + (0)Pn ). Pn+1 = cHn (1 − f (Pn )) ≤ c a + r r −1

Since − f + (0)c(a + ber ) < 1, it follows that lim Pn = 0. n→∞ (c) Since Hn > a for n > 0, the following holds Pn+1 > ac(1 − f (Pn )), ∀n > 0. We apply Lemma 1. Let z n+1 = ac(1 − f (z n )) and P0 = z 0 . It follows by induction that Pn > z n , ∀n > 0, and we conclude that lim inf Pn ≥ z ∗ > 0, where n→∞

z ∗ = lim z n . n→∞

(d) Since lim inf Pn ≥ z ∗ , it follows that for every ε > 0 there exists n 0 > 0 such that n→∞

for every n > n 0 holds: Pn > z ∗ − ε. Since f is decreasing, we have: f (Pn ) < f (z ∗ − ε). By our assumption, ber f (z ∗ ) < 1, so we can take ε > 0 such that ber f (z ∗ − ε) < 1. Now, it follows: Hn+1 = a + bHn er (1−Hn ) f (Pn ) < a + bHn er f (z ∗ − ε), ∀n > n 0 . From Lemma 1, we have lim sup Hn ≤ n→∞

a . 1 − ber f (z ∗ − ε)

Letting ε → 0+ , we obtain lim sup Hn ≤ n→∞

a . 1 − ber f (z ∗ )

  . We have (e) Let (H0 , P0 ) ∈ a, 1−bear f (z ∗ ) × z ∗ , 1−beac r f (z ∗ )

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

a < H1 = a + bH0 er (1−H0 ) f (P0 ) ≤ a + b =

267

a er f (z ∗ ) 1 − ber f (z ∗ )

a 1 − ber f (z ∗ )

and z ∗ = ca(1 − f (z ∗ )) < ca(1 − f (P0 )) ≤ cH0 (1 − f (P0 )) = P1 < cH0 ac ≤ . 1 − ber f (z ∗ ) 

The rest of the proof follows by induction. ∗

Lemma 3 If ber (1−H0 ) (1 − H0∗r ) > −1, then lim sup Hn ≤ H0∗ . n→∞

Proof The following holds: Hn+1 = a + bHn er (1−Hn ) f (Pn ) ≤ a + bHn er (1−Hn ) . Consider the difference equation z n+1 = a + bz n er (1−zn ) . Its unique equilibrium ∗ point H0∗ is locally asymptotically stable since −1 < ber (1−H0 ) (1 − H0∗r ) < 1. To prove that H0∗ is globally asymptotically stable, we apply Theorem 1.8.1 from [30]. r −1 r −1 Let h : [a, a + b e r ] → [a, a + b e r ] be defined as: h(x) = a + bxer (1−x) . This function has a unique maximum xm . Function is increasing for x ∈ (0, xm ) and decreasing for x ∈ (xm , ∞). One can prove that h has negative Schwarzian derivative S f (x) for all x, so Theorem 1.8.1. implies that H0∗ is globally asymptotically stable.  Now we obtain: lim sup Hn ≤ H0∗ . n→∞

Remark 3 Under assumptions of Lemma 3, the maximal host population size is H0∗ . If the host population is stabilized at this level, the expression −cH0∗ f + (0) can be interpreted as the maximal intrinsic growth rate of the parasitoid population [25]. If −cH0∗ f + (0) < 1, then Pn+1 = cHn (1 − f (Pn )) ≤ −cH0∗ f + (0)Pn , which implies that lim Pn = 0 i.e. the parasitoid population becomes extinct. n→∞

Remark 4 If P0 = 0, then Pn = 0, n > 0. System (4) reduces to an equation Hn+1 = a + bHn er (1−Hn ) , which has unique equilibrium point H0∗ . Then, it can be shown that if H0∗ < r1 , the equilibrium H0∗ is globally attractive. Let g(x) = a + r −1 bxer (1−x) . Then g obtains maximum value a + b e r at x = r1 . It is increasing function on (0, r1 ), and decreasing function on ( 1r , ∞). Also, g(x) > a, ∀x > 0. If we first assume that 0 < x0 < H0∗ , then x1 = g(x0 ) < g(H0∗ ) = H0∗ and x1 = g(x0 ) > x0 . It can be proved by induction that the sequence {xn } is increasing and bounded from above. Thus, we conclude that it must converge to H0∗ . Otherwise, if H0∗ < x0 < r1 , then x1 = g(x0 ) < x0 . Also, since H0∗ < x0 , then H0∗ = g(H0∗ ) < g(x0 ) = x1 . It can be shown by induction that sequence {xn } is decreasing and bounded from below. Hence, {xn } must converge to H0∗ . If x0 > r1 , then x1 = g(x0 ) ∈ (a, r1 ), and we can apply the procedure described above to conclude that H0∗ is globally attractive.

268

S. Kalabuši´c et al.

6 Global Attractivity In this section, we explore the global attractivity of the interior equilibrium (H ∗ , P ∗ ). To state the main result, we use the following theorem (Theorem 1.16 [18]). Theorem 7 Let I = [a, b] and J = [c, d] be intervals of real numbers, and let f 1 : I × J → I and f 2 : I × J → J be continuous functions. Consider the following system of difference equations xn+1 = f 1 (xn , yn ) yn+1 = f 2 (xn , yn )

(52)

with initial values (x0 , y0 ) ∈ I × J. Suppose that the following statements are true. (a) f 1 (x, y) is nondecreasing in x and is nonincreasing in y. (b) f 2 (x, y) is nondecreasing in both arguments. (c) If (m 1 , M1 , m 2 , M2 ) ∈ I 2 × J 2 is a solution of the system of equations m 1 = f 1 (m 1 , M2 ), M1 = f 2 (M1 , m 2 ), m 2 = f 2 (m 1 , m 2 ), M2 = f 2 (M1 , M2 ), then m 1 = M1 and m 2 = M2 . Then there exists exactly one equilibrium point (x ∗ , y ∗ ) of system (52) such that lim (xn , yn ) = (x ∗ , y ∗ ). n→∞

Theorem 8 Assume that ber f (z ∗ ) < 1, f + (0) exists and −ac f + (0) > 1. If ac f  (0)e1−ar (2+ber −1 )

< r1 and − +(1−ber f (z ∗ ))2 of system (4) is attractive in a 1−ber f (z ∗ )

 a,

< 1, an unique positive equilibrium (H ∗ , P ∗ )

   a ac ∗ × z , 1 − ber f (z ∗ ) 1 − ber f (z ∗ )

where z ∗ is an unique solution of the equation z ∗ = ac(1 − f (z ∗ )). Proof Let f 1 (x, y) = a + bxer (1−x) f(y) and f 2 (x, y) = cx(1 − f (y)). By a , then Lemma 2, if (x, y) ∈ a, 1−ber f (z ∗ ) × z ∗ , 1−beac r f (z ∗ )  f 1 (x, y) ∈ a, and

 f 2 (x, y) ∈ z ∗ ,

a 1 − ber f (z ∗ )



 ac . 1 − ber f (z ∗ )

It can be proved that f 1 (x, y) is nondecreasing in x and nonincreasing in y, and f 2 (x, y) is nondecreasing in both arguments. Let (m 1 , M1 , m 2 , M2 ) be a solution of the system

Bifurcation and Stability of a Ricker Host-Parasitoid Model …

269

m 1 = f 1 (m 1 , M2 ), M1 = f 2 (M1 , m 2 ), m 2 = f 2 (m 1 , m 2 ), M2 = f 2 (M1 , M2 ). Assume m 1 ≤ M1 and m 2 ≤ M2 . We have m 1 = a + bm 1 er (1−m 1 ) f (M2 ),

M1 = a + bM1 er (1−M1 ) f (m 2 )

(53)

M2 = cM1 (1 − f (M2 )).

(54)

M1 − a m1 − a and f (M2 ) = . r (1−M ) 1 bM1 e bm 1 er (1−m 1 )

(55)

and m 2 = cm 1 (1 − f (m 2 )), From (53), we have f (m 2 ) =

From (54), and using m 1 ≤ M1 , we obtain M2 − m 2 = cM1 (1 − f (M2 )) − cm 1 (1 − f (m 2 )) = c(M1 − m 1 ) + (cm 1 f (m 2 ) − cM1 f (M2 )) m 1 −a 1 −a = c(M1 − m 1 ) + cm 1 bMM r (1−M1 ) − cM1 bm er (1−m 1 ) 1e 1

≤ c(M1 − m 1 ) + = c(M1 − m 1 ) +

c ber c ber c ber

((M1 − a)er M1 − (m 1 − a)er m 1 ) (M1 − m 1 )er ξ (1 + r ξ − ar )

(56)

(M1 − m 1 )er ξ (1 + r ξ)  1  ≤ c(M1 − m 1 ) + bec r (M1 − m 1 )er r 1 + r · r1   1−r = (M1 − m 1 )c 1 + 2eb . ≤ c(M1 − m 1 ) +

where m 1 < ξ < M1 < r1 . From (55), we have 

M1 −a M1 e−r M1

1 ber

=

1 ber

≥

1 ber

=

M1 −a er m 1 − mm1 −a ber M1 1 er m 1 a(M1 −m 1 ) ber M1 m 1

= ≥

 

−

m 1 −a m 1 e−r m 1

f (m 2 ) − f (M2 ) =



(M1 −a)er M1 M1

−

(m 1 −a)er m 1 m1

(M1 −a)er m 1 M1

−

(m 1 −a)er m 1 m1





  (57)

er (a−1) a(M1 −m 1 ) . b M12

Since f  (x) ≥ 0, for x > 0, we have − f  (x) < − f + (0). From (56) and (57), we obtain

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S. Kalabuši´c et al.

M1 − m 1 ≤ =

M12 ber (1−a) ( a M12 ber (1−a) a

f (m 2 ) − f (M2 ))

f  (η)(m 2 − M2 )

=−

M12 ber (1−a) a

f  (η)(M2 − m 2 )

=−

M12 ber (1−a) a

f + (0)(M2 − m 2 )

2

r (1−a)

≤ − (1−bear f (z ∗ ))2 be a

(58)

f + (0)(M2 − m 2 )

r (1−a)

abe  ≤ − (1−be r f (z ∗ ))2 f + (0)(M2 − m 2 )

 aber (1−a)  ≤ − (1−be r f (z ∗ ))2 f + (0)(M1 − m 1 )c 1 + =−

ac f + (0)e1−ar (2+ber −1 ) (M1 (1−ber f (z ∗ ))2

2e1−r b



− m1)

ac f  (0)e1−ar (2+ber −1 )

where m 2 < η < M2 . Since − +(1−ber f (z ∗ ))2 < 1, it follows that M1 − m 1 ≤ 0.  Hence, M1 = m 1 . From (54), we have M2 = m 2 . Remark 5 Observe that we do not prove the global attractivity result because we cannot not apply  Theorem  1.16r −1[18]  on the whole invariant interval r −1 a, a + ber × 0, c a + ber , where the dissipative mapping T is defined, since the theorem requires the uniqueness of the equilibrium. However, in this interval, there are two equilibrium points, exclusion and extinction. Therefore, we find a subinterval where all conditions of Theorem 1.16 are satisfied, implying the existence of the unique interior equilibrium. All solutions that start in this interval converge to that Since we cannot  prove that attractivity result holds on the whole  equilibrium. ber −1 ber −1 × 0, c a + r , we cannot extend our attractivity result to the a, a + r global asymptotic stability result.

7 Uniform Persistence Persistence is a significant feature of ecological models because it addresses the longterm survival of one or all species that appear in the model [5, 24, 38]. System (4) is uniformly persistent, if there exists η > 0, such that lim inf Hn ≥ η and lim inf Pn ≥ n→∞

n→∞

η for all H0 > 0 and P0 > 0 [5, 24]. We apply Theorem 5.1 [5] to prove uniform persistence. The following result holds. ∗

Theorem 9 Assume ber (1−H0 ) (1 − H0∗r ) > −1. If f + (0) exists and 1 + cH0∗ f + (0) < 0 or f + (0) = −∞, then the system (4) is uniformly persistent. Proof Let Y = {(H, P) | H = 0 or P = 0} be the boundary of IR2+ . Then Y is forwardly invariant. Since

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ber −1 a < Hn ≤ a + r

271

  ber −1 and ≤ Pn ≤ c a + , ∀n > 0 r

all solutions of system are bounded and map T induced by system (4) is dissipative. We denote the restriction of T to Y by ∂T. We prove that the set M = {E 0 } = {(H0∗ , 0)} is an acyclic covering for (∂T ). ∗ Assume that f + (0) exists, ber (1−H0 ) (1 − H0∗r ) > −1 and 1 + cH0∗ f + (0) < 0. Now, we show that the stable set W + (E 0 ) of E 0 lies in Y. Assume the opposite i.e. there exists (H0 , P0 ) with H0 > 0, P0 > 0 such that lim (Hn , Pn ) = (H0∗ , 0). n→∞ Then for any ε > 0 there exists n 0 > 0 such that for all n ≥ n 0 the following holds: Pn ≤ ε and H0∗ − ε ≤ Hn ≤ H0∗ + ε. Since −cH0∗ f + (0) > 1, we can choose ε > 0 such that −c(H0∗ − ε) f + (0) > 1. Then Pn+1 = cHn (1 − f (Pn )) > c(H0∗ − ε)(1 − f (Pn )). Consider the difference equation z n+1 = c(H0∗ − ε)(1 − f (z n )).

(59)

Let z 0 = P0 . Then Pn ≥ z n , ∀n > 0. Using Lemma 3 [4], we conclude that Eq. (59) has a globally asymptotically stable equilibrium z > 0 and lim inf Pn ≥ z > 0. This n→∞

is contradiction. Hence, we conclude that the stable set W + (E 0 ) of E 0 lies in Y. Now, we prove that {E 0 } is isolated. If {E 0 } is not isolated, then for any ε > 0 there exists a compact invariant set K contained in B(E 0 , ε) ∩ IR2+ , such that {E 0 } ⊂ K . Let (H0 , P0 ) ∈ K , (H0 , P0 ) = E 0 and P0 > 0. If f + (0) exists, since −cH0∗ f + (0) > 1, we can choose ε > 0 such that −c(H0∗ − ε) f + (ε) > 1. Since K is invariant, we have (Hn , Pn ) ∈ K , ∀n > 0 i.e. H0∗ − ε < Hn < H0∗ + ε and 0 < Pn < ε, ∀n > 0. Hence, H0∗ − ε ≤ lim inf Hn . Now, for any n > 0 there exists ξn ∈ (0, Pn ) ⊂ (0, ε) n→∞

such that Pn+1 = cHn (1 − f (Pn )) ≥ c(H0∗ − ε) · ( f (0) − f (Pn )) = −c(H0∗ − ε) · f  (ξn )Pn > −c(H0∗ − ε) · f  (ε)Pn . Since −c(H0∗ − ε) · f  (ε) > 1, we conclude that Pn → ∞ as n → ∞, which is contradiction with Pn < ε. We conclude that {E 0 } is isolated and M is an acyclic covering for Y. Hence, the system (4) is uniformly persistent. If f + (0) = −∞, then the condition 1 + cH0∗ f + (0) < 0 obviously  holds. To prove the persistence, we proceed as in case when f + (0) exists. Remark 6 Since the system (4) is uniformly bounded and uniformly persistent, then it is permanent [19, 39].

8 Examples This section considers a few well-known probability functions from the paper [31]. Corresponding host-parasitoids models confirm the results that we obtained for the general probability function. Finally, we use numerical simulation to verify our

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theoretical findings, where numerical values of parameters are not taken from any study field. To produce the bifurcation plots, we use the code in the software package Mathematica from the paper [41]. Example 1 (Hassell–Varley host-parasitoid model (HV–model)) Let f (y) = m e−y , 0 < m ≤ 1. Then, we have the following HV–model r (1−Hn ) −Pn Hn+1 = a + bH  n e −P m e Pn+1 = cHn 1 − e n .

m

(60)

It is 1), i.e., f  (y) =  m−1 to see that the function f satisfies hypothesis (H  straightforward −y m  < 0 for y > 0, f (0) = 1, f (∞) = 0, lim+ y f (y) = 0, and y m −e y→0

f  (y) = m 2 e−y y 2m−2 − (m − 1)me−y y m−2 > 0 m

m

for y > 0.

If 0 < m < 1, then f + (0) = −∞. If m = 1, then f + (0) = −1. Hence, we conclude that all theoretical results hold. Figures 1 and 2 show that exclusion equilibrium may undergo transcritical bifurcation when c is a bifurcation parameter (other parameters are fixed e.g. a = 6.43, b = 1.24, r = 0.44, and m = 1). From previous results, we know that the exclusion equilibrium always exists, and using the software package Mathematica, we find that it is approximately equal to (7.04, 0). According to linearized stability analysis, the exclusion equilibrium is locally asymptotically stable for c < c0 = 0.142. Note that the interior equilibrium point does not exist for these values of c. At c = c0 , the stable interior equilibrium appears, while exclusion equilibrium becomes unstable. Figures 3 and 4 show the Neimark–Sacker bifurcation of positive equilibrium and period-doubling bifurcation of exclusion equilibrium. If we take a = 1.15, b = 2.21, c = 0.24 and m = 1, there exists an attracting invariant closed curve for small values of parameter r . As r increases, the size of the curve decreases, and at r ≈ 0.048, it disappears, leaving the stable positive equilibrium. The coefficients d(r0 ) and α(r0 ) are approximately equal to −2.86973 and 0.0343313, respectively. Using numerical computation in the software package Mathematica, we find that at r ≈ 0.35, the positive equilibrium disappears, and the exclusion equilibrium becomes

Fig. 1 Trajectories plotted for f (y) = e−y , a = 6.43, b = 1.24, r = 0.44, m = 1 a c = 0.04, b c = 0.13, c c = 0.15, d c = 0.17. Transcritical bifurcation of exclusion equilibrium occurs for c0 ≈ 0.142 m

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Fig. 2 Bifurcation diagram for f (y) = e−y , a = 6.43, b = 1.24, r = 0.44, m = 1 m

Fig. 3 f (y) = e−y , a = 1.15, b = 2.21, c = 0.24, m = 1, a r = 0.02, b r = 0.05, c r = 0.24, d r = 0.35, e r = 1.73, f r = 1.75 m

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Fig. 4 Bifurcation diagram m for f (y) = e−y , a = 1.15, b = 2.21, c = 0.24, m = 1

stable. It loses stability through period-doubling bifurcation at r ≈ 1.74. We can see that the period-2 solutions later merge into one stable fixed point on the bifurcation diagram. Besides Neimark–Sacker bifurcation, the interior equilibrium can also undergo period-doubling bifurcation, as shown on Figs. 5 and 6. For a = 0.35, b = 9.98, c = 4 and m = 0.49 period-doubling bifurcation occurs for r ≈ 2.66. Example 2 (May model (M–model)) If f (y) = (1 + my )−m , m > 0, then May’s model is given by  −m r (1−Hn ) 1 + Pmn Hn+1 = a + bH  ne   (61) −m . Pn+1 = cHn 1 − 1 + Pmn Straightforward calculations show that f satisfies all conditions in the hypoth −m−1 esis (H 1), i.e., f  (y) = − my + 1 < 0 for y > 0, f (0) = 1, f (∞) = 0, (m+1)( y +1)−m−2

m > 0 for y > 0. Figures 7 and 8 show that the f + (0) = −1. f  (y) = m positive equilibrium can undergo Neimark–Sacker bifurcation and period-doubling bifurcation. In this example, the bifurcation value for the Neimark–Sacker bifurcation is r0 ≈ 0.019 and the coefficients d(r0 ) and α(r0 ) are −0.691627 and 0.00503939, respectively. The value of parameter r at which the period-doubling bifurcation occurs is r ≈ 1.85. We see that the dynamics of M-model may vary from Neimark– Sacker bifurcation, stability to period-doubling bifurcation and chaos.

Example 3 (The parasitoid-parasitoid interaction model (PP–model)) Let f (y) = √ 1+y−1 e− m , m > 0. Then PP-model is given by following system √ 1+Pn −1 m

r (1−Hn ) − Hn+1 = a + bH  ne √ e 

Pn+1 = cHn 1 − e−

1+Pn −1 m

.

(62)

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Fig. 5 f (y) = e−y , a = 0.35, b = 9.98, c = 4, m = 0.49, a r = 0.01, b r = 0.76, c r = 2.67, d r = 5.46 m

One can show that f satisfies conditions in (H 1), i.e., √ 1− y+1

e √m < 0 for y > 0, f  (y) = f  (y) = − 2m y+1

f (0) = 1, f (∞) = 0,

f + (0)

=

−1 , 2m

√ 1− y+1

e m 4m 2 (y+1) 

+

√ 1− y+1

e m 4m(y+1)3/2

> 0 for y > 0,

and lim+ y f (y) = 0. y→0

The (PP) model also exhibits various dynamical behavior. As can be seen from Figs. 9 and 10, there exists stable invariant closed curve for small values of parameter r . When the bifurcation parameter r is close to r0 ≈ 0.059, the curve disappears, and

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Fig. 6 Bifurcation diagram in r − P plane: f (y) = e−y , a = 0.35, b = 9.98, c = 4, m = 0.49 m

Fig. 7 f (y) = (1 + my )−m , a = 0.69, b = 2.74, c = 0.85, m = 2.29, a r = 0.01, b r = 0.07, c r = 1.5, d r = 1.87, e r = 19.26

the coexistence equilibrium becomes stable (Neimark–Sacker bifurcation, where d(r0 ) ≈ −0.344831 and α(r0 ) ≈ 0.00217632). Finally, when r is near value 2.39, the positive equilibrium loses stability through period-doubling bifurcation, leading to the chaotic behavior of the system.

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Fig. 8 Bifurcation diagram for f (y) = (1 +

y −m , m)

277

a = 0.69, b = 2.74, c = 0.85, m = 2.29

√ 1− 1+y

Fig. 9 Trajectories plotted for f (y) = e m , a = 0.32, b = 2, c = 0.93, m = 0.35 and a r = 0.01, b r = 0.17, c r = 0.77, d r = 2.38, e r = 2.4, f r = 6.75

Numerical calculations show that the Neimark–Sacker bifurcation in all of these examples is supercritical, but in general, we cannot determine the sign of coefficients d(r0 ) and α(r0 ).

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Fig. 10 Bifurcation diagram √ 1− 1+y

for f (y) = e m , a = 0.32, b = 2, c = 0.93, m = 0.35

9 Conclusion and Further Discussion This paper investigated the dynamics of a class of host-parasitoid, where the constant number of hosts is safe from parasitoids in each generation, and Ricker’s equation governs the host population. We proved that system has two equilibria: the exclusion equilibrium, which exists for all values of parameters, and the coexistence equilibrium. The necessary condition for the existence of the exclusion equilibrium is 1 + cH0∗ f + (0) < 0, i.e., −cH0∗ f + (0) > 1. In Remark 3, we observed that under assumptions of Lemma 3, the maximal host population size is H0∗ . If the host population is stabilized at this level, the expression −cH0∗ f + (0) can be interpreted as the maximal intrinsic growth rate of the parasitoid population [25]. Hence, if −cH0∗ f + (0) < 1, then the parasitoid population becomes extinct. If the initial value of the parasitoid population is zero, then the parasitoid population will become extinct, while the host population will converge to its equilibrium, which will be the global attractor, Remark 4. According to the linearized stability analysis, transcritical bifurcation exists at the exclusion equilibrium taking c as the bifurcation parameter. Thus, for c < c0 , where c0 is the bifurcation value, the exclusion equilibrium is stable, meaning that the parasitoid population becomes extinct. Then, for c = c0 , the exclusion equilibrium becomes unstable, and at the same time, stable interior equilibrium is born. Thus, from the viewpoint of ecology, both populations will coexist for a long time. The appearance of the Neimark–Sacker bifurcation at the interior equilibrium indicates the oscillatory behavior of the solutions of the model. Period-doubling bifurcation appears for both exclusion and coexistence equilibrium, implying the possibility of chaotic behavior. Therefore, it is possible to use some methods of controlling chaos to stabilize solutions [10, 11, 13]. We proved the global attractivity result concerning the interior equilibrium, which means that both populations will coexist long. The global asymptotic stability result is tough to prove. However, we proved that the model is uniformly persistent, implying the long-term survival of both populations. For an illustration of theoretical results, we considered several well-known probability functions and provided numerical

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simulations. We observed that in all examples, the Neimark–Sacker bifurcation is supercritical. All simulations and bifurcation diagrams are generated in the software package Mathematica, where we chose numerical values of parameters to illustrate our theoretical findings. Hence, we did not take them from any study field. In the paper [28], the authors investigate the dynamics of the generalized Beddington host-parasitoid model. In that model, the probability function that regulates escape from parasitism of the host population is different for the host population concerning the parasitoid population, i.e., it differs in parameter. The model that we investigated is a general host-parasitoid model with constant host refuge and a general but the same probability function in both equations. We have a host-parasitoid model with a general probability function when we exclude the host refuge. However, the results obtained in the paper [28] are not a particular case of our results due to different probability functions. On the other hand, the following model Hn+1 = a Hn er (1−Hn ) f (b Pn ) Pn+1 = cHn (1 − f (Pn )), could be a generalization of the model investigated in the mentioned article, assuming that f satisfies hypothesis H1 . We did not investigate this model, but we believe that it would be challenging to prove the existence results for the interior equilibrium from the computational point of view and investigate the system’s dynamics in detail. Acknowledgements The authors thank the anonymous referee for the helpful comments.

References 1. Bailey, V.A., Nicholson, J.: The balance of animal populations. Proc. Zool. Soc. Lond. 3, 551–598 (1935) 2. Beddington, J.R., Free, C.A., Lawton, J.H.: Dynamic complexity in predator-prey models framed in difference equation. Nature 255, 58–60 (1975) 3. Bešo, E., Kalabuši´c, S., Muji´c, N., Pilav, E.: Stability of a certain class of a host-parasitoid models with a spatial refuge effect. J. Biol. Dyn. 14(1), 1–31 (2019) 4. Bešo, E., Kalabuši´c, S., Muji´c, N., Pilav, E.: Neimark-Sacker bifurcation and stability of a certain class of a host-parasitoid models with a host refuge effect. Int. J. Bifurc. Chaos 29(12) (2019). https://doi.org/10.1142/S0218127419501694 5. Butler, G., Waltman, P.: Persistence in dynamical systems. J. Diff. Equ. 63, 255–263 (1985) 6. Chow, Y., Jang, S.: Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete Contin. Dyn. Syst. - B 21(6), 1713–1728 (2016). https://doi.org/10.3934/ dcdsb.2016019, 2-16 7. Daiyong, W., Hongyong, Z.: Global qualitative analysis of a discrete host-parasitoid model with refuge and strong Allee effects. Math. Methods Appl. Sci. 41(1) (2018). https://doi.org/ 10.1002/mma.4731 8. Din, Q.: Global behavior of a host-parasitoid model under the constant refuge effect. Appl. Math. Modell. (2015). https://doi.org/10.1016/j.apm.2015.09.012 9. Din, Q.: Global stability of Beddington model. Qual. Theory Dyn. Syst. (2015). https://doi. org/10.1007/s12346-016-0197-9

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SageMath Tools for Stability and Bifurcation for Discrete Dynamical Systems Sinan Kapçak

Abstract This paper presents some interactive tools, created in the computer algebra system SageMath, dealing with the stability and bifurcation for one- and twodimensional discrete systems. The tools, geometrically, give information about the types of bifurcations (if any) and possible effects caused by the parameters. Using these tools, one can also obtain the basins of attraction of the fixed and periodic points, and the stability regions in parameter space. Keywords Parameter curves · Tr-det plane · Basin of attraction · Stability region · Discrete dynamical systems · Sagemath · Interactive tools

1 Introduction This work can be considered as a continuation of the paper [5], in which, the author introduces a collection of interactive tools for analyzing one- and two-dimensional discrete dynamical systems (DDS). The mentioned paper focuses on some of the very common and important tools in discrete dynamics, namely, algebraic stability, cobweb diagram, time-series, and bifurcation diagram for one-dimensional DDS; phase plane, time-series, bifurcation diagram, and parameter curve in tr-det plane for one-parameter case for two-dimensional DDS. In this paper, we will present algorithms/tools for parameter-derivative plane, basins of attraction, and stability regions for one-dimensional DDS; parameter curves in the tr-det plane, basins of attraction, and stability regions for two-dimensional DDS. This work is part of the book [7]. The tools presented in this paper are created in the computer algebra system SageMath. SageMath is an open and free software [9], whose mission is described in its official website www.sagemath.org as follows: “Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab”. For recent years, SageMath S. Kapçak (B) American University of the Middle East, Kuwait, Kuwait e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_13

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has been actively used in research mathematics, and it is particularly strong in number theory, algebraic combinatorics, and graph theory [4]. This paper is organized as follows: In Sect. 2, we introduce interactive tools dealing with stability and bifurcation for systems with parameters. Section 3 is devoted to an algorithm and the related tools generating the basins of attraction for one- and twodimensional discrete systems. Tools for the stability regions on parameter-parameter plane is presented in Sect. 4.

2 Parameter Curves In this section, we focus on two types of planar curves and some interactive tools generating them. Let us start with the one-dimensional case.

2.1 One-Dimensional Maps: Parameter-Derivative Plane Our first tool gives information, geometrically, about the stability and bifurcation of a given fixed point. User inputs a difference equation xn+1 = f (xn ; a), where a is a parameter, by entering a differentiable function f and one of the fixed points of it. Then the tool gives the graph of the function y = f  (x ∗ ; a), where x ∗ is the fixed point entered by the user, and the parameter a is represented by the horizontal axis. The graph shows the impact of the parameter a to the system. The horizontal lines f  (x ∗ ; a) = 1 and f  (x ∗ ; a) = −1 are also shown in the same coordinate plane so that the type of bifurcation (saddle node or period doubling), if any, can be seen clearly. As a concrete example, consider the following well-known population model, logistic map: xn+1 = axn (1 − xn ), where parameter a satisfies 0 ≤ a ≤ 4. In Fig. 1, the stability of two fixed points, x ∗ = 0 and x ∗ = 1 − a1 , is analyzed using the mentioned tool. It is clear from Fig. 1a that | f  (0, a)| < 1 if 0 < a < 1, which is, indeed, the stability condition for the fixed point x ∗ = 0. When a > 1, the fixed point x ∗ = 0 is unstable because | f  (0, a)| > 1. Similarly, we observe in Fig. 1b that the fixed point x ∗ = 1 − a1 is locally asymptotically stable if 1 < a < 3. Considering the two figures together, when a = 1, we have f  (0) = f  (1 − a1 ) = 1, which yields a saddle-node bifurcation. As a increases, x ∗ = 0 looses stability, and x ∗ = 1 − a1 becomes stable. Finally, when a = 3, we have f  (1 − a1 ) = −1, therefore a period-doubling bifurcation occurs.

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Fig. 1 Parameter-derivative plane for the logistic map xn+1 = axn (1 − xn ) Fig. 2 The interactive tool for the parameter-derivative plane

Figure 2 shows the interactive tool generating the parameter-derivative plane. In this case, the logistic map xn+1 = axn (1 − xn ) is entered with the fixed point x ∗ = 1 − a1 to obtain Fig. 1b. Intervals for horizontal and vertical axes are also adjustable. We may control the size and the aspect ratio of the figure as well.

2.2 Two-Dimensional Maps: tr-det Plane Having multiple parameters can make systems too complicated to understand. Here we present an interactive tool showing the tr-det plane along with some curves, which helps us analyze the impact of the parameters, and know how to reach (if possible) a particular type of bifurcation for a system with four parameters. Moreover, this will give us, geometrically, the dominant parameters and their sensitivity, which is important to analyze the qualitative behavior of the system. In [5], author introduce this tool for one-parameter systems. Consider the tr-det stability condition

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| tr J ∗ | − 1 < det J ∗ < 1,

(1)

where J ∗ is the Jacobian matrix evaluated at the fixed point, say P ∗ . The geometric representation of this condition, in the tr-det plane, is a triangular region. If a point A = (tr J ∗ , det J ∗ ) in the tr-det plane is in the triangular region, then the fixed point is asymptotically stable. If it is outside that region, then the fixed point is unstable. We have the non-hyperbolic cases when the point is on the sides or at the corners of the triangle. Now suppose that we have a parameter α ≤ t ≤ β in our system. Then, there will exist a corresponding parametric curve in the tr-det plane:   C(t) = tr J ∗ (t) , det J ∗ (t) , α ≤ t ≤ β. As an example, let us consider the following system, which has four parameters, namely a, b, c, d: xn+1 = xn exp(c − xn − ayn ), yn+1 = yn exp(d − yn − bxn ).

(2)

  One of the fixed points of system (2) is P ∗ = ad−c , bc−d . Now for this fixed point, ab−1 ab−1 we will analyze the potential behaviors of system (2), including the possible types of bifurcations based on the parameters. Figure 4 shows a screenshot of the tool, and an output of this tool is given in Fig. 3, in which we can see that for each parameter there exist a corresponding curve: a (red), b (green), c (blue), and d (orange). The

Fig. 3 Parametric parameter curves in the tr-det plane. Different colors represent different parameters. The position of the intersection point (black dot) is for the parameter values a = 2.30, b = 0.66, c = 2.05, d = 1.51

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Fig. 4 The interactive tool for the parametric curves in the tr-det plane

point where the curves intersect, when a = 2.30, b = 0.66, c = 2.05, d = 1.51, yields instability. Now with this parameter values, the red curve, which corresponds parameter a, crosses the sides for the flip and fold bifurcation. Clearly, moving along the red curve, it is possible to make the system stable for some values of parameter. We also have two types of bifurcation (flip and fold) for this case. On the other hand, no values of parameter b makes the system stable or gives any bifurcation. Similarly, parameter 0 ≤ c ≤ 3, can make system stable, and we have two possible bifurcation types, which are flip and Neimark-Sacker bifurcations. Finally, since orange curve crosses the all sides of the triangle, it is possible to have the three types of bifurcation. Note that when we move along a curve, we change the value of one parameter only. Therefore, the other curves may change and give us other possible cases. This tool is highly effective to have a quick understanding of the system with parameters. For creating examples with different bifurcation types, this tool can also be helpful. Furthermore, one may use this tool to control the instability or chaos in a

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system, and obtain parameters which yields a stable fixed point. Author also uses this tool to control the behavior, and construct discrete dynamical systems for generating algorithmic artworks [6].

3 Basins of Attraction Bifurcation diagrams and basins of attraction give a great deal of information on the dynamics of systems in one compact picture. Bifurcation diagrams deals with many parameter values for a single initial point, whereas basins of attraction deals with many initial points for a single parameter value. For systems with multi-stability, bifurcation diagrams may be misleading. Starting with two different initial points might give totally different pictures. However, basins of attraction is still effective and can be used for multi-stability. In this section, we introduce interactive tools for the basins of attraction for oneand two-dimensional discrete systems. Our algorithm uses a continuous version of Kronecker delta function, which we will start with.

3.1 A Closeness Function The well-known Kronecker delta function is defined as follows:  1 if i = j, δi j = 0 if i = j. In our algorithm, we will use a multivariate function which assigns a value from the interval [0, 1] to every couple of numbers/points based on their closeness. Function δi j can be considered as a very strict closeness function for i and j: when i and j are equal, it is 1; when i and j are not equal, it is zero. We now introduce a continuous and more flexible version of Kronecker delta function as follows: K α : R2 → R, K α (x, y) = e−α(x−y) , 2

(3)

where α > 0. The bigger the parameter α is, the stricter the behavior is. Therefore, large values of α mimics the function δi j better. For our case, we will take α = 1000. Clearly, the range of the function K α is (0, 1]. When x = y, we have K α (x, y) = 1. However, when x = y, the value of K α (x, y) depends on the closeness of the values x and y. If x and y are far from each other, then K α (x, y) is close to zero. We will call this function Closeness Function. More general version of Eq. (3) can be defined as follows:

SageMath Tools for Stability and Bifurcation for Discrete Dynamical Systems

K α (A, B) = e−α|AB| , 2

289

(4)

where |AB| is the Euclidean distance between the points A and B. For the twodimensional case, we will take A, B ∈ R2 .

3.2 An Algorithm We now introduce an algorithm to generate basins of attraction for one- and two dimensional systems for not only fixed points but also periodic points. First, consider the following straightforward procedure for two-dimensional DDS: Create a grid on the plane for a desired rectangular region. For every grid point, apply the following rule: Take the grid point as the initial point, and apply a large number of iterations, and obtain a terminal point. Then, paint the grid point with the color based on the location of the terminal point. Even though the above algorithm is good for visualizing the basins of attraction of fixed points, it is not suitable for periodic attractors. The reason is that two different initial points, although attracted by the same periodic attractor, might end up with different points of the cycles after some iterations, which results in different colors. For example, assume that we have a period-4 attractor, and we want to obtain the basin of attraction of it. The algorithm above would give us 4 different colored regions since different initial points ends up at points close to 4 different points after a certain number of iterations. In order to overcome this problem, we will use one common color for all points in the 4-cycle. In addition, we make the initial points visible if it is attracted by a period-4 cycle, and hide the initial points for all the other cases. We will do this by means of the opacity command (alpha) in SageMath, which takes values from zero (invisible) to one (opaque). Before giving the procedure in detail, we also define a way of addressing the points in the list of iterated points: P = ( p0 , p1 , p2 , . . . , pm−1 , pm ) We address the last point of this list, namely pm , by P[−1]. In general, we define: P[−i] = pm−i+1 , where m ≥ 0 and i > 0 are integers. Now for our example with period-4 attractors, we may follow the following procedure: • Focus on last 5 points after large number of iterations. • Find the number:

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K (P[−1], P[−5])

4 

(1 − K (P[−1], P[−i]))

(5)

i=2

• Assign this number as the opacity for the initial point. • Repeat the process for all points in the grid. Note that when a point is attracted by a period-4 attractor, then the number obtained in formula (5) will be close to 1, otherwise it will be close to zero. Assigning this value to the opacity, we show/hide each initial point in the grid plane. For the general version of the procedure for period-q, we deal with the last q + 1 points, and the opacity for the case is the following number: K (P[−1], P[−(q + 1)])

q 

(1 − K (P[−1], P[−i]))

i=2

3.3 One-Dimensional Maps We will present two examples with the following discrete dynamical system: xn+1 = xn + a sin (xn ).

(6)

When a = 3.5, system (6) has multiple period-4 attractors. Figure 5a shows, in the real line, two period-4 attactors and their basins of attractions. The set of four tiny circles in the dark green region is one of the period-4 attractors. Clearly, the other set of four tiny circles in the light green region is the other attractor. Starting with an initial point from the dark green region (light green region), the orbit will be attracted by the 4 points in the dark green region (light green region). To confirm this case geometrically, we use cobweb diagrams with different initial values, and show only several iterations after hundreds of them so that we are able see the period-4 clearly. See Fig. 5b, c. Initial points can be seen on the horizontal axes of the cobweb diagrams. As a second example, we have the parameter value a = 3.7266, which yields period-6 attractors. Similarly we give, in Fig. 6, basins of attraction along with two cobweb diagrams to confirm the output geometrically. Figure 7 shows the tool for basins of attraction for one-dimensional DDS.

3.4 Two-Dimensional Maps We use the same algorithm here for two-dimensional systems, however the closeness function will be for the points in the plane.

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Fig. 5 a Basins of attraction for system (6) when a = 3.5. The regions are confirmed by cobweb diagram in b and c with some initial values. Last several iterations are shown to observe the periodicity clearly

Fig. 6 a Basins of attraction for system (6) when a = 3.7266. The regions are confirmed by cobweb diagram in b and c with some initial values. Last several iterations are shown to observe the periodicity clearly

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Fig. 7 The interactive tool for the basins of attraction for one-dimensional maps

Consider the following system: xn+1 =

1 + xn + ayn2 , 1 + bxn2

yn+1 =

1 + yn + cxn2 . 1 + dyn2

The system, when a = 3, b = 5, c = −1.25, d = 5, has a period-3 attractor, whose basin of attraction is shown in Fig. 8. Even though the vector field of the system is shown in the figure, it is not easy, without the basin of attraction, to guess the dynamics. For example the point (−0.5, 0) is not attracted by the period-3 attractor, and we see that only by the help of basin of attraction. A screenshot of the related tool is given in Fig. 9.

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Fig. 8 Basin of attraction for a period-3 attractor when a = 3, b = 5, c = −1.25, d = 5

4 Stability Region in Parameter-Parameter Plane A stability region of a fixed point of a system with two parameters, say a and b, is nothing but the graphical representation of the stability condition in the a-b coordinate system. In this section, we present an interactive tool generating the stability regions for one- and two-dimensional systems.

4.1 One-Dimensional Maps In Fig. 10a, a screenshot of the tool for one-dimension is given. By using this tool, the stability region of the given fixed points of a difference equation xn+1 = f (xn ; a, b) can be obtained for parameters a and b. The regions are the graphs of the stability condition f  (x ∗ ; a, b) < 1 in a-b plane. User inputs a differentiable function f into the corresponding input box. The intervals of the diagram can also be entered by using the related input boxes. User enters fixed points in any order, and the corresponding label numbers and colors are assigned in the same order.

294 Fig. 9 Interactive tool for the basins of attraction for two-dimensional maps

S. Kapçak

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(a) Tool for stability regions

295

(b) Stability regions

Fig. 10 Stability regions for the map in (7)

As an example, we use the map xn+1 = xn + (xn − a)(xn − b)(xn − ab),

(7)

which has three fixed points, namely, a, b, and ab. Figure 10 shows the tool and its output for the stability regions.

4.2 Two-Dimensional Maps For two-dimensional maps, similarly, we will obtain the stability regions in the parameter plane. As an example, consider the following system: xn+1 = yn+1

xn + ayn2 1 + bxn2

yn + cxn2 = 1 + dyn2

(8)

Figure 11b shows the interactive tool along with the output, which is the stability regions for the fixed points (0, 0) and ab , ab for system (8). This tool is very powerful for two-parameter systems when the fixed points are known explicitly. It gives the stability regions for any number of fixed points by entering them in the input box. Note that, since we are talking about the local stability, it is possible that the stability regions for different fixed points overlap. In fact, in

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(a) x∗ = 0

(b) x∗ =

a−1 a

Fig. 11 Stability regions for the map in (8)

Fig. 11b, the region above the curve is such a region, where both fixed points are stable. In order to make it clear, we use transparency when coloring the regions. Furthermore, user may include/exclude the fixed points in the input box and obtain the regions for the fixed points of interest.

5 Conclusions Three main tools, namely parameter curves, basins of attraction, and stability regions were introduced. When a fixed point is entered, the tool for parameter curves helps us know all possible types of bifurcation. In two-dimensional systems, it is clearly more effective. Regarding the tool for the basins of attraction, we mostly focused on the algorithm of this tool. We defined a continuous function mimicking the Kronecker delta function, and used it in the coloring of the initial points to obtain the basins of attraction of periodic attractors. Finally, we introduced a tool for stability region in parameter plane for one- and two-dimensional systems. The codes have been written in the open free computer algebra system SageMath, Version 9.4, running on a PC with Linux (Ubuntu 21.10). As a further study, we will continue with stable, unstable, and center manifolds as well as the attracting closed invariant curve caused by the Neimark-Sacker bifurcation. Creating SageMath tools for stability and bifurcation in three-dimensional discrete systems is also a possible direction for research.

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References 1. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, vol. 13046. Addison-Wesley, Reading (1989) 2. Elaydi, S.: An Introduction to Difference Equations. Springer (2000) 3. Elaydi, S.: Discrete Chaos: With Applications in Science and Engineering, 2nd edn. Chapman & Hall/CRC (2008) 4. Eröcal, B., Stein, W.: The Sage Project: unifying free mathematical software to create a viable alternative to Magma, Maple, Mathematica and MATLAB. In: International Congress on Mathematical Software. Springer, Berlin, Heidelberg (2010) 5. Kapçak, S.: Discrete dynamical systems with SageMath. Electron. J. Math. Technol. 12(2), 292–308 (2018) 6. Kapçak, S.: Algorithmic art with discrete dynamical systems. In: Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture, pp. 237–242. Tessellations Publishing, Phoenix, Arizona (2020) 7. Kapçak, S., Elaydi S.: Discrete Dynamical Systems Using SageMath. (to be published) 8. Scheinerman, E.R.: Invitation to Dynamical Systems. Prentice-Hall, Upper Saddle River, NJ (1996) 9. Stein, W.: Sage: creating a viable free open source alternative to Magma, Maple, Mathematica, and MATLAB. In: Cucker, F., Krick, T., Pinkus, A., Szanto, A. (eds.), Foundations of Computational Mathematics, Budapest. London Mathematical Society Lecture Note Series, pp. 230–238. Cambridge University Press, Cambridge (2011). https://doi.org/10.1017/CBO9781139095402. 011

Pullback Attractors of Nonautonomous Lattice Difference Equations Peter E. Kloeden

Abstract Lattice difference equations are essentially difference equations on a Hilbert space of bi-infinite sequences. They are motivated by the discretisation of the spatial variable in integrodifference equations arising in theoretical ecology. It is shown in [8] that, under similar assumptions to those used for such integrodifference equations, autonomous lattice difference equations have a global attractor, to which the global attractors of finite dimensional approximations converge upper semi continuously. Here nonautonomous lattice difference equations are considered, both in two-parameter semi-group and skew-product forms, and the existence of nonautonomous pullback attractors established. Keywords Nonautonomous lattice difference equations · Discrete time semi-dynamical systems · Skew product flows · Pullback attractors Mathematics Subject Classification (2000): 37C70 · 37L60 · 39A60 · 92D40

1 Introduction Discrete time lattice models, i.e., lattice difference equations, arise from the spatial variable in integrodifference equations is discretised, and also when the temporal discretisation of lattice differential equations, see [5]. Integrodifference equations, themselves, are used as models in theoretical ecology describing the spatial dispersal and the temporal evolution of species with non-overlapping generations [7, 11]. They essentially involve the iteration of continuous functions on an ambient domain. In contrast, lattice difference equations involve the iteration of bi-infinite sequences. Autonomous lattice difference equations and their attractors were investigated in Kloeden [8].

P. E. Kloeden (B) Mathematisches Institut, Universität Tübingen, 72076 Tübingen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_14

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Here we consider nonautonomous lattice difference equations of the form u i (t + 1) = f i (t, u i (t)) +



κi,j gj (t, u j (t)),

i ∈ Zd , t ∈ Z,

(1.1)

j∈Zd

with weights κi,j , where i, j ∈ Zd . Remark 1.1 Representative functions f i and gi are a(t)ue−b(t)u ,

a(t)u , u ∈ R+ , t ∈ Z, b(t) + |u|

which arise in the spatial Ricker and Beverton-Holt equations with time dependent coefficients [7].

2 Set Up We consider a weighted space  of bi-infinite sequences with vectorial indices i = (i 1 , . . . , i d ) ∈ Zd with |i| = dj=1 i j . In particular, given a positive sequence of weights (ρi )i∈Zd , define the linear space    ρi u 2i < ∞ , 2ρ := u = (u i )i∈Zd : i∈Zd

which is a separable Hilbert space with the inner product and norm u, v :=



ρi u i vi , uρ :=

i∈Zd



ρi u 2i

 21

for u = (u i )i∈Zd , v = (vi )i∈Zd ∈ 2ρ ,

i∈Zd

respectively. Throughout this paper the weights ρi are assumed to satisfy  ρi < ∞. Assumption 1 ρi > 0 for all i ∈ Zd and ρ := i∈Zd

It will also be assumed that Assumption 2 The aggregate interconnection strength is reciprocal-weighted finite in the sense that 2  ki,j ≤ κ for some κ > 0; sup ρj i∈Zd d j∈Z

Assumption 3 for each i ∈ Zd the functions f i , gi : Z × R → R are globally Lipschitz continuous with

Pullback Attractors of Nonautonomous Lattice Difference Equations | f i (t, x) − f i (t, y)| ≤ L i (t)|x − y|, |gi (t, x) − gi (t, y)| ≤ Mi (t)|x − y|,

301 ∀ t ∈ Z, x, y ∈ R,

where sup L 2i (t) ≤ λ(t), for all t ∈ Z.

i∈Zd

and sup

 κ2i,j Mj2 (t) ρj

i∈Zd j∈Zd

= μ(t), for all t ∈ Z.

Assumption 4 for each i ∈ Zd the functions f i , gi : Z × R → R are bounded with | f i (t, x)|, |gi (t, x)| ≤ Bi (t) where



∀ t ∈ Z, x ∈ R,

ρi Bi2 (t) = β(t), for all t ∈ Z.

j∈Zd

Assumption 5 f i (t, 0) = gi (t, 0) = 0 for each i ∈ Zd , t ∈ Z.

2.1 Some Basic Estimates The proofs of the results presented in this subsection are very similar to those for autonomous lattice difference equations in Kloeden [8]. Define Ht (u) = {Hi (t, u)}i∈Zd for all t ∈ Z, u ∈ 2ρ componentwise by Hi (t, u) = f i (t, u i ) +



κi,j gj (t, u j ),

i ∈ Zd , u ∈ 2ρ , t ∈ Z.

j∈Zd

Lemma 2.1 H maps 2ρ to 2ρ and is uniformly bounded with Ht (u)ρ ≤



2β(t)(1 + κρ ),

∀u ∈ 2ρ , t ∈ Z.

Moreover Ht (0) = 0 for all t ∈ Z. Lemma 2.2 Ht : 2ρ → 2ρ is Lipschitz continuous for each t ∈ Z with Ht (u) − Ht (v)ρ ≤



2(λ(t) + μ(t)ρ )u − vρ , ∀u, v ∈ 2ρ , t ∈ Z.

A method adapted from Bates et al. [2] can then be uses show that the nonlinear operator Ht for each t ∈ Z satisfies a uniform tails condition. Lemma 2.3 For every ε > 0 there exist an M(ε) ∈ N such that

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ρi |Hi (t, u)|2 ≤ ε

|i|>M(ε)

for all u ∈ 2ρ and every t ∈ Z. From this follows Theorem 2.4 The nonlinear operator Ht on 2ρ is compact for each t ∈ Z.

3 Nonautonomous Discrete Time Lattice Dynamical System The lattice difference equation (1.1) can be written as a difference equation on 2ρ u(t + 1) = Ht (u(t)), t ∈ Z.

(3.1)

It generates a discrete time two-parameter semi-group or process φ by iterating the mapping H, i.e., φ(t, t0 , u) := Ht−1 ◦ · · · ◦ Ht0 (u), t ≥ t0 , t0 ∈ Z, with φ(t0 , t0 , u) = u for all u ∈ 2ρ and t0 ∈ Z. See [9, 10]. Moreover, the mapping u → φ(t, t0 , u) is Lipschitz continuous on 2ρ .

3.1 Existence of a Pullback Attractor A pullback attractor for a nonautonomous process is based on information of the system from the past and is defined using pullback convergence. See [9, 10]. Definition 3.1 A family A = {At , t ∈ Z} of nonempty compact subsets of 2ρ , which is φ-invariant, is called a (global) pullback attractor if it pullback attracts all families D = {Dt , t ∈ Z} of bounded subsets Dt of 2ρ , i.e.,

lim dist X φ(t, t0 , Dt0 ), At = 0,

t0 →−∞

(fixed t).

Define    Bt = u ∈ 2ρ : uρ ≤ 2β(t − 1)(1 + κρ ) , t ∈ Z, which is a closed and bounded subset of 2ρ for each t ∈ Z, and Define B = {Bt : t ∈ Z} .

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303

Since, by Lemma 2.1, Ht (2ρ ) ⊂ Bt+1 for each t ∈ Z, it follows that Ht (Bt ) ⊂ Bt+1 for each t ∈ Z. Hence B is positive invariant under φ. It is an absorbing set in both forward and pullback senses for the nonautonomous process φ. By Theorem 2.4, φ(t, t0 , Bt0 ) is a compact subset of 2ρ for each t ≥ t0 + 1 and all t0 ∈ Z and these sets form a nested sequence of compact subsets. It follows immediately the φ has a global attractor. Theorem 3.2 ([9, 10]) The discrete time process φ on 2ρ has a global pullback attractor A = {At : t ∈ Z} in 2ρ with nonempty compact component sets given by At =



φ(t, t0 , Bt0 ), t ∈ Z.

t0 ≤t

Note that 0 ∈ At for each t ∈ Z, since Ht (0) = 0, so 0 is a constant entire solution of the system and hence belongs to the pullback attractor. Remark 3.3 The above results also apply to lattice versions of integro-difference equations based on the Ricker and Beverton-Holt population models. Since the soluin 2ρ , tions are non-negative one restricts attention to the non-negative cone 2,+ ρ 2,+ which positive invariant under H, i.e., H(2,+ ρ ) ⊂ ρ . In this case the absorbing set + 2,+ is B = B ∩ ρ .

3.2 Existence of a Forward ω-Limit Sets The family pullback attractor A = {At : t ∈ Z} given by Theorem 3.2 need not be forward attracting, i.e., satisfy

lim dist X φ(t, t0 , Dt0 ), At = 0,

t→∞

fixed t0 ,

for all families D = {Dt , t ∈ Z} of bounded component subsets Dt of 2ρ . The problem is that there may be other forward ω-limit points starting in B than those starting within A. Recall that forward ω-limit set starting at time t0 in the component subset Bt0 of B is defined by ω Bt0 ,t0 :=



φ s, t0 , Bt0 , t≥t0

since the family B is positive invariant. It is a nonempty compact subset of 2ρ under the above assumptions. Note that



lim dist2ρ φ t, t0 , Bt0 , ω Bt0 ,t0 = 0.

t→∞

Since ω B,t0 ⊂ ω Bτ0 ,τ0 for t0 ≤ τ0 , the set

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ωB :=



ω Bt0 ,t0

t0 ∈Z

is nonempty and closed in 2ρ . Lemma 3.4 Suppose there is a nonempty compact subset K of 2ρ such that



lim dist2ρ φ t, t0 , Bt0 , K = 0, for all t0 ∈ Z.

t→∞

Then ωB is a compact subset of 2ρ . The proof follows by noting that ω Bt0 ,t0 ⊂ K for all t0 ∈ Z. The condition in Lemma 3.4 is a uniform asymptotic compactness condition. The compactness of the process mapping φ for each initial time t0 does not suffice to ensure the compactness of ωB . Note that ωB = t0 ≥T ∗ ω Bt0 ,t0 for any finite T ∗ ∈ Z. This is quite reasonable since the future asymptotic behaviour should not depend on that the system in the distant past. In fact the system itself need not be defined for all t ∈ Z, but just for t ≥ T ∗ for some T ∗ ∈ Z. This is more realistic in many biological and physical problems. The set ωB characterises the forward asymptotic behaviour of the nonautonomous system φ. It was called the forward attracting set of the nonautonomous system in Kloeden and Yang [10] and is closely related to the Haraux-Vishik uniform attractor, but it may be smaller and does not require the system to be defined for all time or the attraction to be uniform in the initial time.

4 Discrete Time Skew Product Lattice Systems A nonautonomous dfifference equation with a vector field that evolves according to an input driving system, i.e.,

xt+1 = f xt , θt ( p) , t ∈ Z,

(4.1)

where θ : P → P is a homeomorphism on a compact metric space (P, d P ). Note that the iterates {θt }t∈Z form a group on P under composition. Periodic difference equations can be formulated in this way. See [10]. The skew product counterpart of the nonautonomous difference equation is then written  κi,j gj (u j (t), θt ( p)), i ∈ Zd , t ∈ Z, p ∈ P. u i (t + 1) = f i (u i (t), θt ( p))) + j∈Zd

(4.2)

The discrete time lattice skew product system on 2ρ is then given by u(t + 1) = H(u(t), θt ( p), t ∈ Z, p ∈ P,

(4.3)

Pullback Attractors of Nonautonomous Lattice Difference Equations

305

where H(u, p) = {Hi (u, p)}i∈Zd for all p ∈ P, u ∈ 2ρ component-wise by Hi (u, p) = f i (u i , p) +



κi,j gj (u j , p), i ∈ Zd , p ∈ P, u ∈ 2ρ .

j∈Zd

In addition to Assumptions 1 and 2 it will be assumed that the coefficient functions satisfy Assumption 6 For each i ∈ Zd the functions f i , gi : R × P → R are continuous in both variables and globally Lipschitz continuous in the first variable with | f i (x, p) − f i (y, p| ≤ L i ( p)|x − y|, |gi (x, p) − gi (y, p)| ≤ Mi ( p)|x − y|, ∀ p ∈ P, x, y ∈ R,

where sup L 2i ( p) ≤ λ( p), for all p ∈ P,

i∈Zd

and sup

 κ2i,j Mj2 ( p ρj

i∈Zd j∈Zd

= μ( p), for all p ∈ P.

Assumption 7 for each i ∈ Zd the functions f i , gi : Z × P → R are bounded with | f i (x, p)|, |gi (x, p)| ≤ Bi ( p) where



∀ t ∈ Z, x ∈ R, p ∈ P,

ρi Bi2 ( p) = β( p), for all p ∈ P.

j∈Zd

Assumption 8 f i (0, p) = gi (0, p) = 0 for each i ∈ Zd , p ∈ P.

4.1 Some Basic Estimates The following properties of the mapping H are proved in the same way as those of its nonautonomous above and autonomous counterparts [8]. Lemma 4.1 H maps 2ρ × P to 2ρ and is uniformly bounded with H(u, p)ρ ≤



2β( p)(1 + κρ ),

∀u ∈ 2ρ , p ∈ P.

Moreover H(0, p) = 0 for all p ∈ P. Lemma 4.2 H : 2ρ × P → 2ρ is Lipschitz continuous it is first variable for each p ∈ P with

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H(u, p) − H(v, p)ρ ≤

 2(λ( p) + μ( p)ρ )u − vρ , ∀u, v ∈ 2ρ , p ∈ P.

Lemma 4.3 For every ε > 0 there exist an M(ε) ∈ N such that 

ρi |Hi (u, ω)|2 ≤ ε

|i|>M(ε)

for all u ∈ 2ρ and every ω ∈ Ω. From this follows Theorem 4.4 The nonlinear operator H(·, p) on 2ρ is compact for each p ∈ P.

4.2 Existence of a Random Attractor The lattice difference equation generates a discrete time skew product dynamical systems on 2ρ , i.e., with the dynamics in the state space 2ρ governed by a cocycle mapping φ by iterating the mapping H, i.e., ϕ(t, p, u) := H(θt−1 ( p)) ◦ · · · ◦ H(u, p), t ≥ 1, p ∈ P, with φ(0, p, u) = u for all u ∈ 2ρ and p ∈ P. Moreover, the mapping u → φ(t, p, u) is continuous on P × 2ρ . A pullback attractor for a discrete time skew product dynamical system is based on information of the system from the past and is defined using pullback convergence. Definition 4.5 A family A = {A( p) : p ∈ P} of nonempty compact subsets of 2ρ , which is ϕ-invariant in the sense that ϕ(t, p, A( p)) = A(θt ( p), is called a pullback attractor if it pullback attracts all families D = {D( p) : p ∈ p} of bounded subsets D( p) of 2ρ , i.e.,

lim dist X φ(t, θ−t ( p)), D(θ−t ( p))), A( p) = 0,

p ∈ P.

   B( p) = u ∈ 2ρ : uρ ≤ 2β( p)(1 + κρ ) ,

p ∈ P,

t→∞

Define

which is a closed and bounded subset of 2ρ for each p ∈ P, and write B = {B( p) : p ∈ P} . Since, by Lemma 4.1, H(2ρ , p) ⊂ B(θ( p)) for each p ∈ P, it follows that H(B( p)) ⊂ B(θ( p)) for each p ∈ P. Hence B is positive invariant under ϕ.

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It is an absorbing set in both forward and pullback senses for the random dynamical system ϕ. By Theorem 4.4, φ(t, θ−t ( p)), B(θ−t ( p))) is a compact subset of 2ρ for each t ≥ 0 and p ∈ P. Moreover, the sets ∪s≥t φ(s, θ−s ( p), B(θ−s ( p))) form a nested sequence of compact subsets. It follows immediately the ϕ has a pullback attractor. Theorem 4.6 The discrete time skew product lattice dynamical system ϕ on 2ρ has a global pullback attractor A = {A( p) : p ∈ P} in 2ρ with nonempty compact component sets given by A( p) =



φ(s, θ−s ( p)), B(θ−s ( p))),

p ∈ P.

t≥0 s≥t

References 1. Arnold, L.: Random Dynamical Systems. Springer, New York (1998) 2. Bates, P.W., Lu, K., Wang, B.: Attractors for lattice dynamical systems. Int. J. Bifur. Chaos App. Sci. Eng. 11, 143–153 (2001) 3. Crauel, H., Kloeden, P.E.: Nonautonomous and random attractors. Jahresbericht der Deutschen Mathematiker-Vereinigung 117, 173–206 (2015) 4. Cui, H., Kloeden, P.: Comparison of attractors of asymptotically equivalent difference equations. In: Elaydi, S. et al. (eds.), Difference Equations, Discrete Dynamical Systems and Applications. Springer Proceedings in Mathematics & Statistics, vol. 287, pp. 31–50. Springer Nature Switzerland (2019) 5. Han, X., Kloeden, P., Sonner, S.: Discretisation of the global attractor of a lattice system. J. Dyn. Diff. Equs. 32, 1457–1474 (2021) 6. Han, X., Kloeden, P., Usman, B.: Upper semi-continuous convergence of attractors for a Hopfield-type lattice model. Nonlinearity 33, 1881–1906 (2020) 7. Huynh, H., Kloeden, P.E., Pötzsche, C.: Forward and pullback dynamics of nonautonomous integrodifference equations: basic constructions. J. Dyn. Diff. Equ. (2020). https://doi.org/10. 1007/s10884-020-09887-8T 8. Kloeden, P.E.: Attractors of deterministic and random lattice difference equations. Stoch. Dyn. (2022). https://doi.org/10.1142/S0219493722400068 9. Kloeden, P., Rasmussen, M.: Nonautonomous Dynamical Systems. The American Mathematical Society, Providence (2011) 10. Kloeden, P.E., Yang, M.: Introduction to Nonautonomous Dynamical Systems and their Attractors. World Scientific Publishing Co. Inc, Singapore (2021) 11. Lutscher, F.: Integrodifference Equations in Spatial Ecology. Springer, Cham (2019)

Global Dynamics of Modified Discrete Lotka-Volterra Model M. R. S. Kulenovi´c and Sarah Van Beaver

Abstract In this paper, we will prove general results regarding the global stability of planar monotone systems without minimal period-two solutions on a rectangular region R. We will illustrate the general results by an example of a well known system used in mathematical biology, which is a modified Leslie-Gower system of the form cxn , a + cxn + yn dyn = βyn + (1 − β) , b + xn + dyn

xn+1 = αxn + (1 − α) yn+1

n = 0, 1, ..., where the parameters a, b, c, d are positive numbers, α, β ∈ (0, 1), and the initial conditions x0 , y0 are arbitrary nonnegative numbers. In most cases for different values of a, b, c, and d, there will either be one, two, three, or four equilibrium solutions present with at most one an interior equilibrium point. In the case when c = d = 1 and a = b there will exist an infinite number of interior equilibrium points, in which case we will find the basin of attraction for each of the equilibrium points. Keywords attractivity · Competitive · Cooperative · Difference equation · Invariant sets · Stable manifold · Unstable manifold

1 Introduction and Preliminaries In this paper, we will give global dynamic results for monotone systems with no minimal period-two solutions on a non-empty rectangular region R. These results will be found using the theory of global invariant manifolds developed by Kulenovi´c and Merino in [29–32]. M. R. S. Kulenovi´c (B) · S. Van Beaver University of Rhode Island, Kingston, RI 02881, USA e-mail: [email protected] URL: https://www.math.uri.edu/meet/mustafa-kulenovic/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_15

309

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We will illustrate the general results with an example of a system that has rational functions as transition functions, which has the form cxn xn+1 = αxn + (1 − α) a+cx , n +yn dyn yn+1 = βyn + (1 − β) b+xn +dyn , n = 0, 1, ...,

(1)

where the parameters a, b, c and d are positive numbers and α, β ∈ (0, 1). The initial conditions x0 , y0 are arbitrary nonnegative numbers. This system was originally outlined in [35] by Pakes and Maller as a way to model the application of plant growth. In particular, the system [35] came as a result of the study of the subterranean clover and its various strains found in the south-west of Western Australia. The motivation around this study of the clover and competing strains can be found in [40] by Rossiter. Rossiter et al. [37] and Pakes and Maller [35] formally derived the model from the experimental data. Their aim was to explore a binary mixture of two strains and observe the competition between desirable and undesirable strains of the clover to see which would endure. One trait that was considered is the hardness of the seeds, where seeds that soften begin to grow while hard seeds become part of the seed pool for the following year. The hardness has been studied by Taylor, Rossiter, and Palmer in [38, 39] as well as by others. It was found that in some strains the seeds soften at a faster rate as the years pass while for other strains the rate remains steady. The other quality observed is whether the seed has burrs or became a free seed, which will effect the rate of softening. It is assumed that the seed becomes free within a year once it softens. Some other early discrete models of competition and cooperation of two or more species can be found in [17, 21–23, 36]. Further models were given in [14–16, 27, 45, 46], where the conditions for two major dynamic scenarios in competitive systems were discussed. As it was shown in [5, 6] the global dynamics of discrete monotone systems can be derived from the corresponding local dynamics in the hyperbolic case as well as in some non-hyperbolic cases. See [7] for discussion when does local asymptotic stability imply global attractivity in rational equations. All the models in [14–17, 21–23, 36, 45, 46] have the transition functions which were either exponential (Ricker type models) or linear fractional (Beverton-Holt type models). The effect of quadratic terms in the transition functions of rational type was explored in [2, 5, 6, 13] where it was shown that the presence of such terms can cause the appearance of period-two solutions with substantial basins of attraction as well as the occurrence of the Allee effect. The first systematic analysis of the rational difference equations with quadratic terms is given in [11, 12, 24, 25, 41]. System (1) is a rational system with quadratic terms except when α = β = 0, in which case is original Leslie-Gower model [10]. System (1) is a modified Leslie-Gower system [10]. To understand the general Leslie-Gower system we first consider the system of uncoupled Beverton-Holt equations: cxn dyn , yn+1 = , n = 0, 1, . . . (2) xn+1 = 1 + xn 1 + yn

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where c, d > 0 and the initial conditions x0 and y0 are non negative. This system has been studied by many authors in [4, 8, 42]. The system (2) has an explicit solution of the form  xn =

1 1/(a−1)+(1/x0 −1/(a−1))1/a n 1 a=1 n+1/x0 ,

 yn =

if

1 1/(b−1)+(1/y0 −1/(b−1))1/bn 1 b=1 n+1/y0 ,

if

if a=1

n = 0, 1, . . .

if b=1

n = 0, 1, . . .

The following theorem summarizes the well known results regarding system (2). Theorem 1 The following statements are true for system (2). (1) All solutions (xn , yn ) are component-wise monotonic (xn and yn are increasing or decreasing sequences). Both axes are invariant sets. (2) If a ≤ 1, b ≤ 1, then E 0 (0, 0) is the only equilibrium and it is globally asymptotically stable. (3) If a ≤ 1, b > 1, then the equilibrium point E y (0, b − 1) is a global attractor of all positive solutions with x0 > 0, y0 ≥ 0. The basin of attraction of E 0 is the x-axis. (4) If a > 1, b ≤ 1, then the equilibrium point E x (a − 1, 0) is a global attractor of all positive solutions with y0 > 0, x0 ≥ 0. The basin of attraction of E 0 is the y-axis. (5) If a > 1, b > 1, then the equilibrium point E + (a − 1, b − 1) is a global attractor of all solutions with x0 , y0 > 0. The basin of attraction of E x (resp. E y ) is x-axis (resp. y-axis ) without E 0 . Remark 1 The global dynamic behaviors in (3) and (4) of Theorem 1 are known as global exclusion scenarios and the global dynamic behavior in (5) of Theorem 1 is known as global coexistence scenario. The variations of these two basic dynamic scenarios will appear in all modified versions of system (2), including system (1). The Beverton-Holt equations, system (2), can be modified to create a coupled system known as the Leslie-Gower model. This model is the two-species competition model of the form xn+1 =

cxn , 1 + a11 xn + a12 yn

yn+1 =

dyn , n = 0, 1, . . . , 1 + a21 xn + a22 yn

(3)

where c, d, ai j ≥ 0 and the initial conditions x0 , y0 are arbitrary nonnegative numbers, such that solution is defined for every n. As it was shown in [28] system (3) is semi implicit discretization of the classical Lotka-Volterra system of differential equations. The system (3) is a well known system that has been studied by numerous authors [2, 9, 10, 13, 17, 19, 20, 26, 29, 30, 34]. Note that the terms a12 and a21 are the constants added to couple the system as they represent the interspecific competition. System (1) is a modified version of system (3) where the linear factors αxn and

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βyn represent the stockings for two species in competition, see [14–16]. In this paper for system (1), we begin by finding the local stability results as well as proving both the (O+ ) condition and boundedness of solutions. Then, we use the global dynamic results from Sect. 3 to prove that solutions will converge to one of the equilibrium points in most cases. In one case, however, when c = d = 1 and a = b there will exist an infinite number of interior equilibrium solutions. In this case, we conclude that there is a global stable manifold which is the basin of attraction for each of the infinite equilibrium points. Our analysis will include all non-hyperbolic cases of system (1) which were not discussed so far. As we will show system (1) exhibits either global exclusion or global coexistence for all values of parameters, which is in agreement with the global dynamic scenarios of the Lotka-Volterra continuous case [1]:   dx 1 y(t) = r1 x(t) 1 − x(t)+a dt k 1   (4) dy y(t)+a2 x(t) , = r y(t) 1 − 2 dt k2 which semi-implicit discretization is:   xn+1 − xn xn + a1 yn , = r1 xn − xn+1 h k1   yn+1 − yn yn + a2 xn . = r2 yn − yn+1 h k2 Solving for xn+1 and yn+1 we have k1 (1 + hr1 )xn , k1 + hr1 xn + hr1 a1 yn k2 (1 + hr2 )yn = , k2 + hr2 yn + hr2 a2 xn

xn+1 = yn+1

which is exactly system (3). As it was shown in [10, 29, 30] the qualitative behavior of systems (4) and (3) is similar, so system (3) is acceptable linearization of LotkaVolterra system (4). In the rest of this section, we will give some basic definitions and results of monotone systems needed throughout the paper. In the second section, we will prove some general global dynamic results regarding monotone systems without minimal period-two solutions. In the third section, we will present the global dynamics for all cases of system (1). The obtained results will be new even in the non-hyperbolic case of system (3). A first order system of difference equations 

xn+1 = f (xn , yn ) , n = 0, 1, . . . yn+1 = g(xn , yn )

(5)

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where ( f, g) : S → S, S ⊂ R2 has nonempty interior, and both f , g are continuous functions is called competitive if f (x, y) is nondecreasing in x and nonincreasing in y, and g(x, y) is nonincreasing in x and nondecreasing in y. If both f and g are nondecreasing in x and y, system (5) is called cooperative. Competitive and cooperative maps are defined similarly. Strongly cooperative systems of difference equations or strongly competitive maps are those for which the functions f and g are coordinate–wise strictly monotone. Given a partial ordering  on R2 two points x, y are said to be related if x  y or y  x, and is said to be a strictly related if x ≺ y if x  y and x = y. A stronger inequality is defined as x = (x1 , x2 ) y = (y1 , y2 ) if x  y with x1 = y1 and x2 = y2 . We define a map T on a nonempty set R ⊂ R2 to be a continuous function T : R → R. The map, T , is monotone if x  y implies T (x)  T (y) for all x, y ∈ R, and furthermore is strongly monotone on R if x ≺ y implies that T (x) T (y) for all x, y ∈ R. The map is strictly monotone on R if x ≺ y implies that T (x) ≺ T (y) for all x, y ∈ R. This implies that being related is invariant under iteration for a strongly monotone map. A North-East ordering (NE) is an ordering for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x1 , y1 ) ne (x2 , y2 ) if x1 ≤ x2 and y1 ≤ y2 and the South-East (SE) ordering is defined as (x1 , y1 ) se (x2 , y2 ) if x1 ≤ x2 and y1 ≥ y2 . A map T on a nonempty set R ⊂ R2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive. The examples provided in this paper will be competitive and therefore follow a South-East ordering. Let S be a nonempty subset of R2 . A competitive map T : S → S is said to satisfy condition (O+) if for every x, y in S, T (x) ne T (y) implies x ne y, and T is said to satisfy condition (O−) if for every x, y in S, T (x) ne T (y) implies y ne x. For x ∈ R2 , define Q  (x) for  = 1, . . . , 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, Q 1 (x) = {y ∈ R 2 : ¯ y¯ ) of a map T , denoted x1 ≤ y1 , x2 ≤ y2 }. The basin of attraction of a fixed point (x, as B((x, ¯ y¯ )), is defined as the set of all initial points (x0 , y0 ) for which the sequence ¯ y¯ ). Similarly, we define a basin of attraction of iterates T n ((x0 , y0 )) converges to (x, of a periodic point of period p. The fixed point (x, ¯ y¯ ) is said to by non-hyperbolic if the Jacobian matrix has at least one eigenvalue on the unit circle (|λ| = 1). If the other eigenvalue is inside the unit circle (|λ| < 1) the fixed point is non-hyperbolic of stable type, and if the other other eigenvalue is outside of the unit circle (|λ| > 1) the fixed point is nonhyperbolic of unstable type. If both eigenvalues lie on the unit circle, the fixed point is non-hyperbolic of resonance type of either (1, 1), (1, −1), (−1, 1), or (−1, −1) depending on the values of the eigenvalues. It should be noticed that the eigenvalues of the Jacobian matrix of planar competitive or cooperative map are real numbers with largest eigenvalue which is positive. s u (x, y) and unstable manifold of Wloc (x, y) of an The local stable manifold Wloc equilibrium point (x, ¯ y¯ ) are defined as the sets

314

M. R. S. Kulenovi´c and S. Van Beaver s = {(x, y) : T n (x, y) ∈ U for all n ≥ 0, and T n (x, y) → ( x, Wloc ¯ y¯ ) as n → ∞},

u = {(x, y) : T −n (x, y) ∈ U for all n ≥ 0, and T −n (x, y) → ( x, ¯ y¯ ) as n → ∞} Wloc

where U is a neighborhood of the equilibrium point and T is the map. The global stable manifold W s and the global unstable manifold W u are then defined as the sets W s (E) =

∞  k=1

s T −k (Wloc (E)) and W u (E) =

∞ 

u T −k (Wloc (E)).

k=1

The following theorem is H. L. Smith generalization of deMottoni-Schiaffino result for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations [42–44]. Theorem 2 Let S be a nonempty subset of R2 . If T is a competitive map for which (O+) holds then for all x ∈ S, {T n (x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T . If instead (O−) holds, then for all x ∈ S, {T 2n (x)} is eventually componentwise monotone. If the orbit of x has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point. The next result from [30–32] is useful for determining basins of attraction of fixed points of competitive maps: have been obtained by H. L. Smith in [42, 43]. Theorem 3 Let T be a competitive map on a rectangular region R ⊂ R2 . Let x ∈ R be a fixed point of T such that Δ := R ∩ int (Q 1 (x) ∪ Q 3 (x)) is nonempty (i.e., x is not the NW or SE vertex of R), and T is strongly competitive on Δ. Suppose that the following statements are true. a. The map T has a C 1 extension to a neighborhood of x. b. The Jacobian JT (x) of T at x has real eigenvalues λ, μ such that 0 < |λ| < μ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis. Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T . The next two results are stated for order-preserving maps on Rn , but more general version is valid in ordered Banach spaces [18]. Theorem 4 Let T be a monotone map on a closed and bounded rectangular region R ⊂ R2 . Suppose that T has a unique fixed point x¯ in R. Then x¯ is a global attractor of T on R. Corollary 1 If the non-negative cone of  is a generalized quadrant in Rn , and if T has no fixed points in u 1 , u 2  other than u 1 and u 2 , then the interior of u 1 , u 2  is

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315

either a subset of the basin of attraction of u 1 or a subset of the basin of attraction of u 2 . The following result from [32] give conditions for the existence of the boundary curves of the basin of attraction. Theorem 5 Let R = (a1 , a2 ) × (b1 , b2 ), and let T : R → R be a strongly competitive map with a unique fixed point x¯ ∈ R, and such that T is twice continuously differentiable in a neighbourhood of x. ¯ Assume further that at the point x¯ the map T has associated characteristic values μ and ν satisfying 1 < μ and −μ < ν < μ, with ν = 0, and that no standard basis vector is an eigenvector associated to one of the characteristic values. Then there exists curves C1 , C2 in R and there exist p1 , p2 ∈ ∂R with p1 se x¯ se p2 such that i. For l = 1, 2, Cl is invariant, north-east strongly linearly ordered, such that x¯ ∈ Cl and Cl ⊂ Q3 (x) ¯ ∪ Q1 (x); ¯ the endpoints ql , rl of Cl , where ql ne rl , belong to the boundary of R. For l, j ∈ 1, 2 with l = j, Cl is a subset of the closure of one of the components of R\C j . Both C1 and C2 are tangential at x¯ to the eigenspace associated with ν. ii. For l = 1, 2, let Bl be the component of R\Cl whose closure contains pl . Then Bl is invariant. Also, for x ∈ B1 , T n (x) accumulates on Q2 ( p1 ) ∩ ∂R, and for x ∈ B2 , T n (x) accumulates on Q4 ( p2 ) ∩ ∂R. We will use the results from [30–32] to prove the general global dynamics results of competitive maps in the plane.

2 Global Dynamic Results We will prove some global dynamic results of the general monotone system (5). Theorem 6 Consider the map T generated by system (5) on a non-empty rectangular region R where the fixed point E 0 = (0, 0) is on the bottom left corner of R. Suppose that T is a strongly competitive map with no minimal period-two solutions on R. Furthermore, we will assume that conditions a and b of Theorem 3 holds for any saddle fixed point. (a) Assume the map T has the fixed points E x = (x, 0) which is a saddle point, E y = (0, y) which is locally asymptotically stable, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Then every solution which begins off the xaxis converges to E y , and every solution which begins on the x-axis without E 0 converges to E x . (b) Assume the map T has the fixed point E y = (0, y) which is a saddle point, the fixed point E x = (x, 0) which is locally asymptotically stable point, and the fixed point E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Then every

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(c)

(d)

(e)

(f)

(g)

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solution which begins off the y-axis converges to E x , and every solution which begins on y-axis without E 0 converges to E y . Assume the map T has the fixed points E x = (x, 0) and E y = (0, y) which are both saddle points, the fixed point E + = (x+ , y+ ), x+ > 0, y+ > 0 which is a locally asymptotically stable point, and the fixed point E 0 = (0, 0) which is a repeller. Assume E y se E 0 se E x and E y se E + se E x . Then every solution which begins off the x and y axes converges to E + . Every solution which begins on the x-axis without E 0 converges to E x and every solution which begins on the y-axis without E 0 converges to E y . Assume the map T has the fixed points E x = (x, 0) and E y = (0, y) which are both locally asymptotically stable points, the fixed point E + = (x+ , y+ ), x+ > 0, y+ > 0 which is a saddle point, and the fixed point E 0 = (0, 0) which is a repeller. Assume E y se E 0 se E x as well as E y se E + se E x . Then there will exist the continuous non-decreasing curve of the stable manifold W s (E + ) with an endpoint at E 0 . Every solution which begins to the right the stable manifold W s (E + ) converges to E x and every solution which begins to the left of the stable manifold W s (E + ) converges to E y . Every solution which begins on the stable manifold W s (E + ) converges to E + . Assume the map T has the fixed point E y = (0, y) which is locally asymptotically stable and the fixed point E 0 = (0, 0) which is a saddle point where E y se E 0 . Then every solution which begins off the x-axis converges to E y and every solution which begins on the x-axis converges to E 0 . Assume the map T has the fixed point E x = (x, 0) which is locally asymptotically stable and the fixed point E 0 = (0, 0) which is a saddle point where E 0 se E x . Then every solution which begins off the y-axis converges to E x and every solution which begins on the y-axis converges to E 0 . Assume the map T has one fixed point E 0 = (0, 0) which is locally asymptotically stable. Then every solution converges to E 0 .

Proof (a) As E x is a saddle point, there exists a global stable manifold W s (E x ), and global unstable manifold W u (E x ), by Theorems 2–5 in [31]. As there are no interior fixed points or minimal period-two solutions, the endpoints of both the stable and unstable manifolds will be on the boundary of R. For the unstable manifold, W u (E x ) the endpoint will be E y and for the stable manifold, W s (E x ) the endpoint will be the x-axis. Any point on the stable manifold, which in this case is the x-axis will converge to E x . As E y is locally asymptotically stable, points on the y-axis will converge to E y . We will consider the global dynamics in two cases based on the location of the initial point B = (x0 , y0 ) ∈ int R. For the first case, suppose the initial point B = (x0 , y0 ) is inside the rectangular region R0 = E y , E x  and below the unstable manifold W u (E x ) of E x . There will exists two projections of B onto the unstable manifold, W u (E x ), that is Px = (x, y0 ) and Py = (x0 , y) such that Py se B se Px . By monotonicity, T n (Py ) se T n (B) se T n (Px ).

(6)

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Taking the limit we obtain lim T n (Py ) se lim T n (B) se lim T n (Px ),

n→∞

n→∞

n→∞

(7)

which implies that E y se lim T n (B) se E y . n→∞

(8)

This yields that limn→∞ T n (B) = E y . Here we use the fact that the unstable manifold W u (E x ) has an endpoint at E y , limn→∞ T n (Px ) = limn→∞ T n (Py ) = E y . Next suppose the initial point B = (x0 , y0 ) is inside the rectangular region R0 = E y , E x  and above the unstable manifold W u (E x ) of E x . There will exists two projections of B onto the unstable manifold, W u (E x ), that is Px = (x, y0 ) and Py = (x0 , y) such that Px se B se Py . By monotonicity, T n (Px ) se T n (B) se T n (Py ).

(9)

Taking the limit we obtain lim T n (Px ) se lim T n (B) se lim T n (Py ),

n→∞

n→∞

n→∞

(10)

which implies that E x se lim T n (B) se E y . n→∞

(11)

This yields that limn→∞ T n (B) = E y . Here we use the fact that the unstable manifold W u (E x ) has an endpoint at E y , limn→∞ T n (Px ) = limn→∞ T n (Py ) = E y . Finally suppose that B = (x0 , y0 ) ∈ intR\R0 . There exists a projection Py = (0, y0 ) of B onto the y-axis and a projection Px = (x0 , 0) of B onto the x-axis such that Py se B se Px . By monotonicity this implies (6). As the x-axis is the stable manifold of E x , then limn→∞ T n (Px ) = E x . Furthermore, as E y is locally asymptotically stable, limn→∞ T n (Py ) = E y . So when the limit of the inequalities is taken we obtain (7) which implies E y se lim T n (B) se E x . n→∞

(12)

We can conclude that as n → ∞ then T n (B) → R0 . Once T n (B) enters the rectangular region R0 , the global behavior will follow from the previous cases. (b) This proof is analogous to the proof of case (a). The difference is that now we consider the stable and unstable manifolds of E y instead of E x as E y is a saddle point and E x is locally asymptotically stable. (c) By Theorems 2–5 in [31] as E x and E y are saddle points, there exist the global stable manifolds, W s (E x ) and W s (E y ), and global unstable manifolds, W u (E x ) and W u (E y ). As E + is the interior fixed point, it will be the endpoint of W u (E x ) and W u (E y ). The y-axis will be the stable manifold W s (E y ) of E y . Thus for

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any initial point that begins on the y-axis will converge to E y . The x-axis will be the stable manifold W s (E x ) of the E x . So we can conclude that any point that begins on the x-axis will converge to E x . We will consider the global dynamics in a few cases based on the location of the initial point B = (x0 , y0 ) ∈ int R. First suppose that B = (x0 , y0 ) is inside the rectangular region R0 = E y , E x  and below both the unstable manifolds W u (E x ) and W u (E y ) of E x and E y respectively. There will exists two projections Px = (x, y0 ) and Py = (x0 , y) of B, which will either be on the unstable manifold of E x (W u (E x )) or E y (W u (E y )) depending on the initial location of the point B. As the proof holds regardless of whether the projections are onto W u (E x ), W u (E y ), or both, without loss of generality we can suppose that Px is on W u (E x ) and Py is on W u (E y ) such that Py se B se Px . By monotonicity, we obtain (6). In view of limn→∞ T n (Px ) = E + and limn→∞ T n (Py ) = E + we obtain (7), which implies that E + se lim T n (B) se E + , n→∞

and so limn→∞ T n (B) = E + . Next suppose that B = (x0 , y0 ) is inside the rectangular region R0 = E y , E x  and above both the unstable manifolds W u (E x ) and W u (E y ) of E x and E y respectively. There will exists two projections Px = (x, y0 ) and Py = (x0 , y) of B, which will either be on the unstable manifold of E x (W u (E x )) or E y (W u (E y )) depending on the initial location of the point B. Without loss of generality suppose that Px is on W u (E x ) and Py is on W u (E y ) such that Px se B se Py . By monotonicity, we obtain (9). In view of limn→∞ T n (Px ) = E + and limn→∞ T n (Py ) = E + we obtain (10) which implies that E + se lim T n (B) se E + , n→∞

and so limn→∞ T n (B) = E + . Finally suppose that B = (x0 , y0 ) ∈ int R\R0 . There exists a projection Py = (0, y0 ) of B onto the y-axis and a projection Px = (x0 , 0) of B onto the x-axis such that Py se B se Px . By monotonicity this implies (6). As the x-axis is the stable manifold of E x , then limn→∞ T n (Px ) = E x . Furthermore, as the y-axis is the stable manifold of E y , limn→∞ T n (Py ) = E y . So when the limit of the inequalities is taken we obtain (7), which implies (12). We can conclude that as n → ∞ then T n (B) → R0 . Once T n (B) enters the rectangular region R0 , the global behavior will follow from the previous cases. (d) As E + is a saddle point, by Theorems 2–5 in [31], there exists the global stable manifold W s (E + ), and the global unstable manifold W u (E + ). As there are no other interior fixed points besides E + , the endpoints of W u (E + ) will be E x and E y on the boundary of the region. The endpoint of W s (E + ) will be E 0 . As E y is locally asymptotically stable, solutions on the y-axis will converge to E y and as E x is locally asymptotically stable, solutions on the x-axis will converge to E x . Any point that begins on the stable manifold W s (E + ) of E + will converge to

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E + . We will describe the global dynamics in a few cases based on the location of the initial point B = (x0 , y0 ) ∈ int R. First suppose that B = (x0 , y0 ) is inside the rectangular region R0 = E y , E x  and is both to the left of the stable manifold W s (E + ) and below the unstable manifold W u (E + ) of E + . There will exist a projection Py = (x0 , y) of B onto the unstable manifold W u (E + ) of E + as well as another projection Px = (x, y0 ) of B such that Py se B se Px . The projection Px will either be on the unstable manifold W u (E + ) or on the stable manifold W s (E + ) depending on the initial point B. We will first suppose that the projection Px is on the unstable manifold W u (E + ). By monotonicity, (6) holds. Once the limit of the inequalities is taken we obtain (7) which implies that E y se lim T n (B) se E y n→∞

as both Px and Py are on the unstable manifold so that limn→∞ T n (Px ) = limn→∞ T n (Py ) = E y . We can conclude that limn→∞ T n (B) = E y . Next suppose that the projection Px is on the stable manifold W s (E + ). By monotonicity, (6) still holds. Once the limit of the inequalities is taken we have (7). As Py is on the unstable manifold, limn→∞ T n (Py ) = E y , and as Px is on the stable manifold, then limn→∞ T n (Px ) = E + . This implies that E y se lim T n (B) se E + . n→∞

We can conclude that limn→∞ T n (B) = E y as B does not begin on the stable manifold of E + . The proof in other cases when the initial point is in R0 is similar. Finally, suppose that B = (x0 , y0 ) ∈ intR\R0 . There exists a projection Py = (0, y0 ) of B onto the y-axis and a projection Px = (x0 , 0) of B onto the xaxis such that Py se B se Px . By monotonicity this implies (6). As E x is locally asymptotically stable, then limn→∞ T n (Px ) = E x . Furthermore, as E y is locally asymptotically stable, limn→∞ T n (Py ) = E y . So when the limit of the inequalities is taken we obtain (7), which implies (12). We can conclude that as n → ∞, then T n (B) → R0 . Once T n (B) enters the rectangular region, R0 , the global behavior will follow from one of the previous cases. (e) As E 0 is a saddle point, there exists a global stable manifold W s (E 0 ), and global unstable manifold W u (E 0 ), by Theorems 2–5 in [31]. As there are no interior fixed points or minimal period-two solutions, the endpoints of both the stable and unstable manifolds will be on the boundary of R. The unstable manifold W u (E 0 ) will be the y-axis with the endpoint of E y . So if a point begins on the y-axis, it will converge to E y . The stable manifold W s (E 0 ) will be the x-axis. Thus, if a point begins on the x-axis, it will converge to E 0 .

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Suppose that B = (x0 , y0 ) ∈ int R. There will exists two projections Px = (x0 , 0) and Py = (0, y0 ) of B onto the x and y axis such that Py se B se Px . By monotonicity, (6). As E y is locally asymptotically stable, limn→∞ T n (Py ) = E y . Additionally, as Px is on the stable manifold of E 0 , limn→∞ T n (Px ) = E 0 . Taking the limit of the inequalities we obtain (7), which implies E y se lim T n (B) se E 0 . n→∞

As B does not begin on the stable manifold and by the monotone system theory [31] as the stable manifold is unique, we can conclude that lim n→∞ T n (B) = E y . (f) This proof is analogous to the proof of case (e). The difference is that we consider the existence of E x instead of E y , where E x will be locally asymptotically stable. (g) Let the map T contain one fixed point E 0 = (0, 0) that is locally asymptotically stable. As E 0 is the only fixed point on the rectangular region R and there are no minimal period-two points, all solutions must converge to E 0 by Theorem 4, that is limn→∞ T n (B) = E 0 for an initial point B = (x0 , y0 ).  The next result gives global dynamics in some non-hyperbolic cases. Theorem 7 Consider the map T generated by system (5) on a rectangular region R where the fixed point E 0 = (0, 0) is on the bottom left corner of R. Suppose that T is a strongly competitive map with no minimal period-two solutions on R. Furthermore, we will assume that conditions a and b of Theorem 3 holds for any saddle fixed point. (a) Assume that the map T has the fixed points E y = (0, y) which is non-hyperbolic of stable type, E x = (x, 0) which is locally asymptotically stable, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the y-axis converges to E x and every solution which begins on the y-axis without E 0 converges to E y . (b) Assume that the map T has the fixed points E y = (0, y) which is locally asymptotically stable, E x = (x, 0) which is non-hyperbolic of stable type, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the x-axis converges to E y and every solution which begins on the x-axis without E 0 converges to E x . (c) Assume that the map T has the fixed points E y = (0, y) which is non-hyperbolic of stable type, E x = (x, 0) which is a saddle point, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the x-axis converges to E y and every solution which begins on the x-axis without E 0 converges to E x . (d) Assume that the map T has the fixed points E y = (0, y) which is a saddle point, E x = (x, 0) which is non-hyperbolic of stable type, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the y-axis converges to E x and every solution which begins on the y-axis without E 0 converges to E y .

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(e) Assume that the map T has the fixed points E y = (0, y) and E x (x, 0) which are both non-hyperbolic of stable type and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution on the x-axis without E 0 will converge to E x and every solution on the y-axis without E 0 will converge to E y . Every solution which begins off the x and y axis will converge to exactly one of E x or E y . (f) Assume that the map T has the fixed points E y = (0, y) which is locally asymptotically stable and E 0 = (0, 0) which is non-hyperbolic of unstable type where E y se E 0 . Then there will exists two curves, C1 and C2 , C2 se C1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C1 and C2 coincide with each other or C2 ∈ / R, every solution which begins off the x-axis will converge to E y . Every solution which begins on the x-axis will converge to E 0 . If there exists both C1 , C2 ∈ R then every solution to the left of C2 will converge to E y and every solution to the right of C2 will converge to E 0 . (g) Assume that the map T has the fixed points E x = (x, 0) which is locally asymptotically stable and E 0 = (0, 0) which is non-hyperbolic of unstable type where E 0 se E x .Then there will exists two curves, C1 and C2 , C2 se C1 , that are continuous and non-decreasing with an endpoint at E 0 . If the curves C1 and C2 coincide with each other or C2 ∈ / R, every solution which begins off the y-axis will converge to E x . Every solution which begins on the y-axis will converge to E 0 . If there exists both C1 , C2 ∈ R then every solution to the left of C1 will converge to E 0 and every solution to the right of C1 will converge to E x . (h) Assume that the map T has one fixed point E 0 = (0, 0) which is non-hyperbolic. Then every solution converges to E 0 . (i) Assume the map T has the fixed points E x = (x, 0) and E y = (0, y) both of which are non-hyperbolic of stable type, E 0 = (0, 0) which is a repeller, and an infinite number of equilibrium points which are non-hyperbolic of stable type. Suppose the map T is strongly competitive and has a C 1 extension. For each of the infinite equilibrium points suppose that conditions a and b of Theorem 3 are satisfied. Then for each of the infinite equilibrium points, there is a stable manifold as its basin of attraction. Each stable manifold will have an end point at E 0 and they are graphs of continuous and non-decreasing functions. The points will depend continuously on the initial point (x0 , y0 ). Proof (a) As E y is non-hyperbolic of stable type, there exists a stable manifold W s (E y ) by Theorems 3 and 2 in [31], which in this case will be the y-axis. Any point on the y-axis will converge to E y . As E x is locally asymptotically stable, an initial point on the x-axis will converge to E x . Suppose there exists an initial point B = (x0 , y0 ) ∈ int R. There will exist two projections of B, Px = (x0 , 0) onto the x-axis and Py = (0, y0 ) onto the y-axis such that Py se B se Px . By monotonicity this implies (6). Furthermore, taking the limits we have (7), which implies that E y se lim T n (B) se E x . n→∞

This step was obtained using the fact that E x is locally asymptotically stable so limn→∞ T n (Px ) = E x and the y-axis is the stable manifold W s (E y ) so that

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limn→∞ T n (Py ) = E y . We can conclude that limn→∞ T n (B) = E x as the stable manifold of E y is unique and B does not begin on it. This case is analogous to case (a) where in this case we use the stable manifold of E x , and E y is now locally asymptotically stable. This proof is analogous to case (a) of Theorem 6 where instead of claiming E y is locally asymptotically stable, E y is now non-hyperbolic of stable type. We can instead use the fact that the y-axis is the stable manifold W s (E y ) of E y to proceed with proof using the same technique. This proof is analogous to the proof of case (b) of Theorem 6 where instead of claiming E x is locally asymptotically stable, E x is now non-hyperbolic of stable type. We can instead use the fact that the x-axis is the stable manifold W s (E x ) of E x to proceed with proof using the same technique. As E x and E y are non-hyperbolic of stable type, there exist stable manifold W s (E x ), which is the x-axis, and W s (E y ), which is the y axis respectively in this case by Theorems 2 and 3 in [31] Any point on the y-axis without E 0 will converge to E y and any point on the x-axis without E 0 will converge to the E x . For both E x and E y there will exist a center manifold. This manifold can be used to show all solutions in the interior of R will either converge to E x or E y . Alternatively, one can use Theorem 5 to show that if there are points in the interior of R which converge to E x and E y , there must exist the points which will be attracted to E 0 , which is contradiction. By Theorem 5, there exist two curves C1 and C2 , where C2 se C1 that are continuous and non-decreasing with an endpoint at E 0 . The curve C1 is the boundary of the basin of attraction of a point at infinity, and the curve C2 is the boundary of the basin of attraction of E y . This proof is analogous to case (e) of / R where instead Theorem 6 when the two curves C1 and C2 coincide or C2 ∈ of claiming E 0 is a saddle point, E 0 is now non-hyperbolic of unstable type. We can instead use the curve C1 , that is a stable manifold W s (E 0 ) of E 0 to proceed with proof using the same technique. If both C1 , C2 ∈ R, then, any points on C1 and C2 converge to E 0 . Suppose there exists a point B0 = (x0 , y0 ) ∈ R to the right of C2 . There will exist two projections of B, P2 onto the curve C2 and P1 onto the curve C1 such that P2 se B0 se P1 . By monotonicity this implies T n (P2 ) se T n (B0 ) se T n (P1 ), which once the limits are taken becomes lim T n (P2 ) se lim T n (B0 ) se lim T n (P1 ),

n→∞

n→∞

n→∞

and furthermore E 0 se T n (B0 ) se E 0 .

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The last inequalities were obtained using the fact that P1 is on C1 so that limn→∞ T n (P1 ) = E 0 and P2 is on C2 so that limn→∞ T n (P2 ) = E 0 . Therefore, limn→∞ T n (B0 ) = E 0 . Next suppose that there exists a point B0 = (x0 , y0 ) ∈ R and to the left of C2 . Then B0 is in the region of the basin of attraction to E y . As B0 does not begin on C1 or C2 , then the point will converge to E y . (g) This proof is analogous to case ( f ) of Theorem 6 when the two curves C1 and / R where instead of claiming E 0 is a saddle point, E 0 is C2 coincide or if C2 ∈ now non-hyperbolic of unstable type. We can instead use the curve C1 , that is the stable manifold W s (E 0 ) of E 0 to proceed with proof using the same technique. When the two curves C1 and C2 do not coincide, the proof will be analogous to case ( f ) given above. (h) Assume that the map T has one fixed point of E 0 = (0, 0) which is nonhyperbolic. As E 0 is the only fixed point on R and there are no minimal periodtwo all solutions must converge to E 0 by Theorem 4, that is lim n→∞ T n (B) = E 0 for an initial point B = (x0 , y0 ). (i) For the fixed points suppose both E x = (x, 0) and E y = (0, y) are non hyperbolic of stable type, E 0 = (0, 0) is a repeller or singular point, and there exist an infinite number of interior equilibrium points on the rectangular region R. As the map T is strongly competitive, has a C 1 extension, and for each interior equilibrium point conditions a and b of Theorem 3 are satisfied, then by Theorem 3 there is a stable manifold as its basin of attraction for each of the equilibrium points. Each stable manifold will have an endpoint at E 0 and they are graphs of continuous and non-decreasing functions where the points depend continuously on the initial point (x0 , y0 ).  Remark 2 Theorems 6 and 7 can be generalized. Instead of considering E 0 = (0, 0), E x = (x, 0), and E y = (0, y), one can instead consider a rectangular region R where E 0 is a fixed point on the bottom left corner of the boundary, E x is a fixed point on the bottom boundary, and E y is a fixed point on the left boundary of R where the points are not necessarily on the axes.

3 Global Dynamics of System (1) We will investigate the global dynamics of system (1), where the parameters a, b, c and d are positive numbers and 0 < α, β < 1.

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3.1 Local Stability Results First we perform local stability analysis of system (1). Additionally, we will prove that the (O+ ) condition is satisfied as well as the fact that system (1) is bounded, which will help in proving the global results. Lemma 1 The following holds for system (1) where α, β ∈ (0, 1): (a) (b) (c) (d)

point. E 0 = (0, 0) is always an equilibrium is an equilibrium point. If d > b, then E y = 0, d−b d  If c > a, then E x = c−a , 0 is an equilibrium point. c If cd > 1 and both d(1 − c + a) < b and c(1 + b − d) < a hold or cd < 1 and both d(1 − c + a) > b and c(1 + b − d) > a hold, then E + =  d(1−c+a)−b c(1+b−d)−a is an equilibrium point. , 1−cd 1−cd

Proof The equilibrium points satisfy cx x = αx + (1 − α) a+cx+y

and

y = β y + (1 − β)

dy . b + x + dy

Clearly, one of the equilibrium points is always E 0 = (0, 0). Suppose that x = 0 , and so we get the equilibrium point E y = (0, d−b ) when and y = 0. Then y = d−b d d , which shows d > b. Next suppose that x = 0 and y = 0. It follows that x = c−a c , 0) providing that c > a. Finally, that there is the equilibrium point of E x = ( c−a c and assume that x, y = 0. Straightforward calculation yields that x = d(1−c+a)−b 1−cd c(1+b−d)−a d(1−c+a)−b c(1+b−d)−a y= . Therefore, the interior point of E + = ( 1−cd , 1−cd ) exist 1−cd d(1−c+a)−b > 0 and c(1+b−d)−a > 0 holds.  when both 1−cd 1−cd To find the local stability of each equilibrium point we find the Jacobian matrix. The map corresponding to system (1) is T (u, v) = ( f (u, v), g(u, v)) where cu dv f (u, v) = αu + (1 − α) a+cu+v and g(u, v) = βv + (1 − β) b+u+dv . The Jacobian matrix of T is

 2 u cu (α − 1) (a+cu+v) α + (1 − α) (a+cu+v)c−c 2 (a+cu+v)2 J (u, v) = . 2 v dv (β − 1) (b+u+dv) β + (1 − β) (b+u+dv)d−d 2 (b+u+dv)2 First, the Jacobian matrix evaluated at the equilibrium point E 0 = (0, 0) gives J (0, 0) = with the eigenvalues λ1 =

c+α(a−c) a

c+α(a−c) a

0 and λ2 =

0

d+β(b−d) b d+β(b−d) . b

,

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Lemma 2 The equilibrium point E 0 = (0, 0) is (a) (b) (c) (d)

locally asymptotically stable if a > c and b > d. repeller if a < c and b < d. saddle point if a < c and b > d or a > c and b < d. non-hyperbolic if a = c or b = d.

Proof The results follow from the eigenvalues. When both eigenvalues lie within the unit circle, the equilibrium point will be locally asymptotically stable. Note that |λ1 | < 1 and |λ2 | < 1 when −a < c + α(a − c) < a ⇔ −(a + c) < α(a − c) < (a − c), −b < d + β(b − d) < b ⇔ −(b + d) < β(b − d) < (b − d). This will hold true when a > c and b > d. When both eigenvalues lie outside the unit circle, the equilibrium point will be a repeller. So |λ1 | > 1 and |λ2 | > 1 when −a > c + α(a − c) > a ⇔ −(a + c) > α(a − c) > a − c, −b > d + β(b − d) > b ⇔ −(b + d) > β(b − d) > b − d. This will hold true when a < c and b < d. Note that in this case λ1 > 1 and λ2 > 1, as α, β < 1. It cannot hold that −(a + c) > α(a − c) and −(b + d) > β(b − d). When one eigenvalue lies outside the unit circle and the other within the unit circle the equilibrium point will be a saddle point. Based on the previous calculations this will occur when a < c and b > d or a > c and b < d. Finally, when at least one eigenvalue lies on the unit circle, the equilibrium point is non-hyperbolic. So |λ1 | = 1 or |λ2 | = 1 respectively when −a = c + α(a − c) = a ⇔ −(a + c) = α(a − c) = (a − c), −b = d + β(b − d) = b ⇔ −(b + d) = β(b − d) = (b − d). This will hold true when a = c or b = d. Note that as a, b, c, d > 0 then λ1 = 1 or λ2 = 1. It cannot happen that either is equal to −1 as −(a + c) = α(a − c) and −(b + d) = β(b − d) cannot hold true.  Next we evaluate the Jacobian matrix at the point of E y = (0, d−b ): d  cd+α(−b+d+ad−cd)  d −b −b+d+ad = J 0, −(1 − β) (d−b) d d2 The eigenvalues of the matrix are λ1 =

0 b+β(d−b) d

.

and λ2 =  Lemma 3 When d > b the equilibrium point of E y = 0, d−b is d cd+α(−b+d+ad−cd) −b+d+ad

(a) locally asymptotically stable if b < d(a + 1 − c).

b+β(d−b) . d

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(b) saddle point if b > d(a + 1 − c). (c) non-hyperbolic of stable type if b = d(a + 1 − c). Proof The results follow from the eigenvalues. As d > b, then we can conclude 0 < λ2 < 1. Indeed −d < b + β(d − b) < d ⇔ −(d + b) < β(d − b) < (d − b) ⇔ d > b. When both eigenvalues lie within the unit circle, the equilibrium point will be locally asymptotically stable. Note that |λ1 | < 1 when −(−b + d + ad) < cd + α(−b + d + ad − cd) < (−b + d + ad) ⇔ −(−b + d + ad + cd) < α(−b + d + ad − cd) < (−b + d + ad − cd). This will hold true when d(a + 1 − c) > b. When one eigenvalue lies outside the unit circle and the other within the unit circle the equilibrium point will be a saddle point. Note that |λ1 | > 1 when −(−b + d + ad) > cd + α(−b + d + ad − cd) > (−b + d + ad) ⇔ −(−b + d + ad + cd) > α(−b + d + ad − cd) > (−b + d + ad − cd). This will hold true when d(a + 1 − c) < b. Note again that as α < 1, then −(−b + d + ad + cd) > α(−b + d + ad − cd) cannot hold resulting in the fact that λ1 > 1. Finally, when at least one eigenvalue (in this case |λ1 | = 1) lies on the unit circle, the equilibrium point is non-hyperbolic. This will happen when −(−b + d + ad) = cd + α(−b + d + ad − cd) = (−b + d + ad) ⇔ −(−b + d + ad + cd) = α(−b + d + ad − cd) = (−b + d + ad − cd). This holds true when d(a + 1 − c) = b. As a, b, c, d > 0, then −(−b + d + ad + cd) = α(−b + d + ad − cd) does not hold true resulting in the fact that λ1 cannot be −1.  Next we evaluate Jacobian matrix at the equilibrium point E x = ( c−a , 0). c  J

 a+α(c−a) c−a −(1 − α) (c−a) c c2 ,0 = cd+β(bc+c−a−cd) . 0 c bc+c−a

It follows that the eigenvalues will be λ1 =

and λ2 = cd+β(bc+c−a−cd) . bc+c−a  c−a Lemma 4 When c > a, the equilibrium point of E x = c , 0 will be a+α(c−a) c

(a) locally asymptotically stable if a < c(b + 1 − d). (b) saddle point if a > c(b + 1 − d). (c) non-hyperbolic of stable type if a = c(b + 1 − d).

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Proof The proof is similar to the proof of Lemma 3.



Next we prove that system (1) is bounded and furthermore, is component-wise decreasing when c ≤ a and d ≤ b. Lemma 5 The solutions of system (1) are bounded. In addition, {xn } is decreasing when c ≤ a and {yn } is decreasing when d ≤ b. Proof Note that cxn ≤ αxn + (1 − α) a + cxn + yn

(13)

dyn ≤ βyn + (1 − β). b + xn + dyn

(14)

xn+1 = αxn + (1 − α)

yn+1 = βyn + (1 − β)

Let xn ≤ u n and yn ≤ vn where u n+1 = αu n + 1 − α and vn+1 = βvn + 1 − β. Assume the initial conditions of x0 ≤ u 0 and y0 ≤ v0 hold. When iterated u n and vn become u n = (u 0 − 1)α n + 1 ≤ (u 0 − 1) + 1 = u 0 vn = (v0 − 1)β n + 1 ≤ (v0 − 1) + 1 = v0 . Therefore, we have that xn ≤ u n ≤ u 0 and yn ≤ vn ≤ v0 . Next, we want to see when xn+1 ≤ xn and when yn+1 ≤ yn holds. By reexamining (13) we see that xn+1 = αxn + (1 − α)

c cxn cxn = (α + (1 − α) )xn . ≤ αxn + (1 − α) a + cxn + yn a a

Thus xn+1 ≤ xn when (α + (1 − α) ac ) ≤ 1. This will happen when c ≤ a. Similarly, by rewriting (14) we see that yn+1 = βyn + (1 − β)

c a

≤ 1 that is

dyn d ≤ (β − (1 − β) )yn . b + xn + dyn b

Then yn+1 ≤ yn when (β − (1 − β) db ) ≤ 1. This will hold true when d ≤ b.



Next, we will use an alternative  method to prove thelocal stability in two cases for the equilibrium point E + = d(1−c+a)−b . , c(1+b−d)−a 1−cd 1−cd Lemma 6 Let b < d(1 + a − c), a < c(1 + b − d), c > a, d > b, and cd < 1. Furthermore, suppose that E x and E y are locally asymptotically stable while E 0 is a repeller. Then E y se E + se E x and E + will either be non-hyperbolic of unstable type or a saddle point.

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Proof First note that d −b c(1 + b − d) − a > ⇔ (d − b)(1 − cd) > d(c(1 + b − d) − a) ⇔ d(1 + a − c) > b, d 1 − cd

and c−a d(1 − c + a) − b > ⇔ (c − a)(1 − cd) > c(d(1 − c + a) − b) ⇔ c(1 + b − d) > a, c 1 − cd

which implies that E y se E + se E x . By Corollary 1, intE y , E +  is a subset of the basin of attraction of either E y or E + . Since the equilibrium point E y is locally asymptotically stable, the interior of E y , E +  is a subset of the basin of attraction of E y . This means that E + cannot be locally asymptotically stable. By Theorem 5, there exists an invariant, north-east strongly linearly ordered curve Cx that is the boundary of the basin of attraction for E x with an endpoint of E 0 that passes through E + , and an invariant, north-east strongly linearly ordered curve C y that is the boundary of the basin of attraction for E y with an endpoint of E 0 that passes through E + where Cx may coincide with C y . Any points on the curves Cx and C y will be attracted to E + as they are on the boundary, E 0 is a repeller, and by the theorem cannot cross over to the other boundary. If C y and Cx do not coincide, we can suppose that there exists a point B0 = (x, y) that is in the region between Cx and C y . Additionally, there exists the points B1 ∈ C y and B2 ∈ Cx such that B1 se B0 se B2 . This implies that T n (B1 ) se T n (B0 ) se T n (B2 ), and furthermore lim T n (B1 ) se lim T n (B0 ) se lim T n (B2 ).

n→∞

n→∞

n→∞

However, as limn→∞ T n (B1 ) = limn→∞ T n (B2 ) = E + , we conclude that lim T n (B0 ) = E + .

n→∞

As E + attracts some points, E + cannot be a repeller. Thus, E + is either a saddle point or a non-hyperbolic point. Based on Mathematica calculations the eigenvalues of E + cannot be 1, however, one eigenvalue can potentially be −1. The other eigenvalue will always be greater than 1. If E + exists, it will be non-hyperbolic of unstable type or a saddle point.  The following lemma will give results regarding the spectral radius based on the slopes of the tangent lines at E + . Let ρ(J ) be the spectral radius of J (E + ). The result was suggested by O. Merino and the related result is given in [3, 33]. Lemma 7 Suppose the tangent lines to f (x, y) = x and g(x, y) = y at E + are not parallel to one of the axes. Let m 1 and m 2 be the slopes of the tangent lines respectively. The following holds true: (i) If m 1 − m 2 > 0, then ρ(J ) > 1. (ii) If m 1 − m 2 = 0, then ρ(J ) = 1. (iii) If m 1 − m 2 < 0, then ρ(J ) < 1.

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Proof For the proof let the equilibrium E + be represented as (x, ¯ y¯ ). Note that the Jacobian matrix of J (E + ) can be rewritten as J (E + ) =

fx f y . gx g y

Using the Taylor expansion of the map T (x, y) we obtain  (x, y) = T (x, y) = (x, ¯ y¯ ) + J (E + )

x − x¯ y − y¯

 + ....

We will consider the linear part to find the slope of the tangent lines. Rewritten this becomes ¯ y¯ )(x − x) ¯ + f y (x, ¯ y¯ )(y − y¯ ) x − x¯ = f x (x, y − y¯ = gx (x, ¯ y¯ )(x − x) ¯ + g y (x, ¯ y¯ )(y − y¯ ). Let Δx = x − x¯ and Δy = y − y¯ . Substituting this in we have ¯ y¯ )Δx + f y (x, ¯ y¯ )Δy Δx = f x (x, Δy = gx (x, ¯ y¯ )Δx + g y (x, ¯ y¯ )Δy. From the two equations we have that 1 − fx Δy = Δx fy gx Δy = m2 = . Δx 1 − gy m1 =

As the map is competitive, then f x , g y > 0, f y , gx < 0, as well as m 1 , m 2 < 0. This implies that f x , g y < 1. The term m 1 − m 2 is equal to m1 − m2 =

1 − g y − f x + f x g y − gx f y p(1) 1 − fx gx = . − = fy 1 − gy f y (1 − g y ) f y (1 − g y )

Note the characteristic polynomial is p(λ) = λ2 − ( f x + g y )λ + ( f x g y − f y gx ), and p(1) = 1 − ( f x + g y ) + f x g y − gx f y . As f y (1 − g y ) < 0, then m 1 − m 2 will either be less than, greater than, or equal to zero based on p(1). The characteristic polynomial at 1 is equivalent to p(1) = 1 − tr(J ) + det(J ). Then we have that p(1) > 0 when ρ(J ) < 1, p(1) < 0 when ρ(J ) > 1, and p(1) = 0 when ρ(J ) = 1.  For system (1), x = f (x, y) means that x = αx + (1 − α)

cx ⇔ y = −cx + (c − a) a + cx + y

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and y = g(x, y) means that y = βy + (1 − β)

dy 1 b ⇔ y = − x + (1 − ). b + x + dy d d

= −c + d1 . This results that The criteria of m 1 − m 2 is equivalent here to −c − −1 d m 1 − m 2 < 0 when 1 < cd, m 1 − m 2 > 0 when 1 > cd, and m 1 − m 2 = 0 when 1 = cd. Lemma 8 Let b > d(1 + a − c), a > c(1 + b − d), c > a, d > b, and cd > 1. Furthermore, suppose that E x and E y are both saddle points while E 0 is a repeller. Then E y se E + se E x and E + will be locally asymptotically stable. Proof First note that d −b a − c(1 + b − d) > ⇔ (d − b)(cd − 1) > d(a − c(1 + b − d)) ⇔ d(1 + a − c) < b, d cd − 1

and c−a b − d(1 − c + a) > ⇔ (c − a)(cd − 1) > c(b − d(1 − c + a)) ⇔ c(1 + b − d) < a, c cd − 1

which implies that E y se E + se E x . As E x and E y are both saddle points, there exists global stable manifolds, W s (E x ) and W s (E y ), and global unstable manifolds, W u (E x ) and W u (E y ), by Theorems 2–5 in [31]. The endpoint of the unstable manifolds of W u (E x ) and W u (E y ) will be E + . The stable manifold of W s (E y ) will be the y-axis and the stable manifold of W s (E x ) will be the x-axis. So E + is clearly not a repeller. Suppose that E + is a saddle point and let Rsp = E y , E + . As E + is a saddle point, by Theorems 2–5 in [31] there exists an unstable manifold W u (E + ). However, this causes a contradiction as the curve conflicts with the unstable manifold W u (E y ). So E + cannot be a saddle point. By Lemma 7 ρ(J ) < 1 and therefore, E + will not be non-hyperbolic and furthermore, must be locally asymptotically stable.  Next we will prove that the (O+ ) condition holds. The (O+ ) condition implies that there will be no minimal period-two solutions. Lemma 9 System (1) satisfies the (O+ ) condition. Proof It suffices to prove that if T (x1 , y1 ) ne T (x2 , y2 ), then x1 ≤ x2 and y1 ≤ y2 , where T (x, y) is a map associated with system (1). The condition T (x1 , y1 ) ne T (x2 , y2 ) is equivalent to cx1 cx2 ≤ αx2 + (1 − α) a + cx1 + y1 a + cx2 + y2 dy1 dy2 βy1 + (1 − β) ≤ βy2 + (1 − β) , b + x1 + dy1 b + x2 + dy2 αx1 + (1 − α)

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331

which can be reduced to α(x1 − x2 )(a + cx1 + y1 )(a + cx2 + y2 ) ≤ (1 − α)(ac(x2 − x1 ) + c(x2 y1 − x1 y2 )) (15)

β(y1 − y2 )(b + x1 + dy1 )(b + x2 + dy2 ) ≤ (1 − β)(bd(y2 − y1 ) + d(x1 y2 − x2 y1 )). (16)

We know that either x1 ≤ x2 or x1 > x2 . Suppose that x1 > x2 . By (15), as α(x1 − x2 )(a + cx1 + y1 )(a + cx2 + y2 ) > 0 then ac(x2 − x1 ) + c(x2 y1 − x1 y2 ) > 0. This implies that c(x2 y1 − x1 y2 ) > ac(x1 − x2 ) > 0. As c > 0, then x2 y1 − x1 y2 > 0. Since x1 > x2 , then y1 > y2 . Using the fact that y1 > y2 and (16), (1 − β)(bd(y2 − y1 ) + d(x1 y2 − x2 y1 )) > 0. So this implies that (bd(y2 − y1 ) + d(x1 y2 − x2 y1 )) > 0. By reducing this we see that (x1 y2 − x2 y1 ) > b(y1 − y2 ) > 0. This is a contradiction as we stated x2 y1 − x1 y2 > 0 so that x1 y2 − x2 y1 > 0 cannot hold true. Therefore, x1 ≤ x2 must hold. Next we know either y1 ≤ y2 or y1 > y2 . Suppose that y1 > y2 . By (16) as β(y1 − y2 )(b + x1 + dy1 )(b + x2 +dy2 ) > 0, then (1 − β)(bd(y2 − y1 ) + d(x1 y2 − x2 y1 )) > 0. As β < 1, then (bd(y2 − y1 ) + d(x1 y2 − x2 y1 )) > 0. With some reduction this implies (x1 y2 − x2 y1 )) > b(y1 − y2 ) > 0. Therefore, (x1 y2 − x2 y1 ) > 0 and moreover x1 > x2 . Using (15) and the fact that x1 > x2 , then c(x2 y1 − x1 y2 ) > ac(x1 − x2 ) > 0. This implies that x2 y1 − x1 y2 > 0. However, this is a contradiction  as we already stated that x1 y2 − x2 y1 > 0. Therefore, y1 ≤ y2 .

3.2 Global Stability Results In this section, we will compile the local stability results and use both Theorems 6 and 7 to give conclusions regarding the global dynamics of system (1). Lemma 10 The following hold: (a) If c > a > 0, d > b > 0, b < d(1 + a − c), and a < c(1 + b − d), then cd < 1. (b) If c > a > 0, d > b > 0, b > d(1 + a − c), and a > c(1 + b − d), then cd > 1. Proof (a) Rewriting the two inequalities give us d(c − a) < d − b and c(d − b) < c − a. This implies that cd(c − a) < c − a. As c > a this inequality can be reduced to cd < 1. (b) Rewriting the two inequalities give us d(c − a) > d − b and c(d − b) > c − a

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This implies that cd(c − a) > c − a. As c > a this inequality can be reduced to cd > 1.  Lemma 11 The equilibrium point E + will not exist when (a) (b) (c) (d) (e) (f) (g) (h) (i)

c > a > 0, d > b > 0, b > d(1 + a − c), and a < c(1 + b − d). c > a > 0, d > b > 0, b < d(1 + a − c), and a > c(1 + b − d). c > a > 0, d > b > 0, and a = c(1 + b − d). c > a > 0, d > b > 0, and b = d(1 + a − c). c > a > 0 and b > d > 0. a > c > 0 and d > b > 0. 0 < c < a and 0 < d < b. a = c. d = b.

Proof (a) Clearly E + cannot exist by Lemma 1 as either d(1 − c + a) < b and c(1 + b − d) < a or d(1 − c + a) > b and c(1 + b − d) > a must hold. exist. (b) For the same reasons as case (a) clearly E + cannot   , 0 . However, (c) As a = c(1 + b − d), then E + can be reduced to d(1−c+a)−b 1−cd d(1−c+a)−b 1−cd

=

c−a . c

This results in E + = E x .

  . However, (d) As b = d(1 + a − c), then E + can be reduced to 0, c(1+b−d)−a 1−cd = d−b , which results in E + = E y . d > 0 holds when c(1 + b − d) < (e) First suppose that cd > 1. Then c(1+b−d)−a 1−cd a. This can be rewritten as c(b − d) < a − c, which is not true in this case > 0 when as c > a and b > d. Next suppose that cd < 1. Then, d(1+a−c)−b 1−cd d(1 + a − c) > b. Note that the inequality can be rewritten as d(a − c) > b − d, which is false and therefore, E + does not exist. > 0 when d(1 + a − c) < b. This (f) Suppose that cd > 1. Then d(1+a−c)−b 1−cd inequality does not hold true as a > c and d > b since the inequality can be >0 rewritten as d(a − c) < b − d. Next suppose that cd < 1. Then, c(1+b−d)−a 1−cd when c(1 + b − d) − a > 0. This inequality can be rewritten as c(b − d) > a − c, and so will not hold. Therefore, E + does not exist. (g) Let cd > 1. This implies that c(1+b−d)−a 1−cd

d(1 + a − c) − b < 0 ⇔ d(a − c) < b − d, and c(1 + b − d) − a < 0 ⇔ c(b − d) < a − c. This will provide a contradiction as these inequalities imply cd(a − c) < c(b − d) < a − c. However, this can be reduced to cd < 1 since a > c. Next suppose that cd < 1. This gives us that

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333

d(1 + a − c) − b > 0 ⇔ d(a − c) > b − d, and c(1 + b − d) − a > 0 ⇔ c(b − d) > a − c. Again combining the two inequalities, cd(a − c) > c(b − d) > a − c, which in turn can be reduced to cd > 1. This again provides a contradiction, and therefore E + cannot exist.   d−b c(b−d) (h) As a = c then E + can be reduced to 1−cd , 1−cd . Note that one coordinate will be negative or if d = b, then E + =E 0 . Therefore,  E + cannot exist. d(a−c) c−a (i) Similar to case (h) we can reduce E + to 1−cd , 1−cd . This point will therefore not exist as one coordinate will be negative or if c = a, then E + = E 0 .  Theorem 8 Consider system (1), and let a, b, c, d > 0, 0 < α, β < 1, and x0 , y0 ≥ 0. (a) If c > a > 0, d > b > 0, b < d(1 + a − c), a < c(1 + b − d) and E + is a saddle point, it follows that E 0 is a repeller, and both E x and E y are locally asymptotically stable such that E y se E + se E x . Every solution which begins off the stable manifold W s (E + ) to the right of the manifold converges to E x , and to the left of the manifold converges to E y . Every solution which begins on the stable manifold W s (E + ) converges to E + . Every solution which begins on the x-axis without E 0 converges to E x , and every solution which begins on the y-axis without E 0 converges to E y . (b) If c > a > 0, d > b > 0, b > d(1 + a − c), a > c(1 + b − d), and E + is locally asymptotically stable, then E 0 is a repeller, and both E x and E y are saddle points such that E y se E + se E x . Every solution which begins off the x and y axes converges to E + . Every solution which begins on the x-axis without E 0 converges to E x , and every solution which begins on the y-axis without E 0 converges to Ey. (c) If c > a > 0, d > b > 0, b > d(1 + a − c), and a < c(1 + b − d), then E y is a saddle point, E x is locally asymptotically stable, E 0 is a repeller, and E + does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis without E 0 converges to E y . (d) If c > a > 0, d > b > 0, b < d(1 + a − c), and a > c(1 + b − d), then E y is locally asymptotically stable, E x is a saddle point, E 0 is a repeller, and E + does not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis without E 0 , converges to E x . (e) Suppose that c > a > 0 and b > d > 0. The equilibrium point E 0 will be a saddle point, while E x will be locally asymptotically stable,and both E y and E + will not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis converges to E 0 .

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(f) Suppose that a > c > 0 and d > b > 0. The equilibrium point E 0 will be a saddle point, while E y will be locally asymptotically stable, and both E x and E + will not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis converges to E 0 . (g) Suppose that 0 < c < a and 0 < d < b. The equilibrium points of E x , E y , and E + do not exist and E 0 is globally asymptotically stable. Proof For each case the existence of the equilibrium points is given by Lemma 1. The local stability results of E 0 are given in Lemma 2, of E y are given in Lemma 3, and of E x are given in Lemma 4. For cases (a) and (b), Lemma 10 establishes that cd < 1 or cd > 1 respectively, conditions necessary to give the existence of E + . Additionally, for case (a) the local behavior of E + was proved in Lemma 6 and for case (b) the local behavior of E + was proved in Lemma 8. In all other cases E + will not exist by Lemma 11. Using Lemma 9 the (O+ ) condition holds resulting in the fact that there are no minimal period-two solutions. As the solutions are bounded by Lemma 5, then Theorem 6 will give us the global results of all the cases (a)–(g).  Theorem 9 Consider system (1), and let a, b, c, d > 0, 0 < α, β < 1, and x0 , y0 ≥ 0. (a) If c > a > 0, d > b > 0, b = d(1 + a − c) and a < c(1 + b − d), then E y is non-hyperbolic of stable type, E x is locally asymptotically stable, E 0 is a repeller, and E + does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis without E 0 converges to E y . (b) If c > a > 0, d > b > 0, b < d(1 + a − c) and a = c(1 + b − d), then E y is a saddle point, E x is non-hyperbolic point of stable type, E 0 is a repeller, and E + does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis without E 0 converges to E y . (c) If c > a > 0, d > b > 0, b = d(1 + a − c) and a > c(1 + b − d), then E y is non-hyperbolic of stable type, E x is a saddle point, E 0 is a repeller, and E + does not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis without E 0 , converges to E x . (d) If c > a > 0, d > b > 0, b > d(1 + a − c) and a = c(1 + b − d), then E y is locally asymptotically stable, E x is non-hyperbolic of stable type, E 0 is a repeller, and E + does not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis without E 0 converges to E x . (e) If c > a > 0, d > b > 0, b = d(1 + a − c) and a = c(1 + b − d), then both E y and E x are non-hyperbolic of stable type, E 0 is a repeller, and E + does not exist. Every solution on the x-axis without E 0 will converge to E x and every solution on the y-axis without E 0 will converge to E y . Every solution which begins off the x and y axis will converge to exactly one of E x or E y . (f) Suppose that c = d = 1 and a = b. Then c > a > 0 and d > b > 0, E 0 is a repeller, E x and E y are non-hyperbolic of stable type, and there exist an infinite number of solutions of the form E K = {(x, 1 − K − x)|0 < x < 1 − K and K = a = b}. For each of the equilibrium points of the form E K , there is

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(g)

(h)

(i)

(j)

335

a stable manifold W s (E K ) as its basins of attraction. All W s (E K ) have an end point at E 0 and they are graphs of continuous and non-decreasing functions. The equilibrium points E K depends continuously on the initial point (x0 , y0 ). Suppose that c = a and d > b > 0. The equilibrium point E 0 is non-hyperbolic of unstable type, while E y is locally asymptotically stable, and both E x and E + do not exist. There exist two curves, C1 and C2 , C2 se C1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C1 and C2 coincide with each other or C2 is not in the region, every solution which begins off the x-axis will converge to E y . Every solution which begins on the x-axis will converge to E 0 . If there exists both C1 , C2 in the region then every solution to the left of C2 will converge to E y and every solution to the right of C2 will converge to E 0 . Suppose that c = a and b > d > 0 or b = d and a > c > 0. The equilibrium point E 0 will be non-hyperbolic of stable type, while E y , E x , and E + will not exist. Every solution will converge to E 0 . Suppose that b = d and c > a > 0. The equilibrium point E 0 is non-hyperbolic of unstable type, while E x is locally asymptotically stable, and both E y and E + do not exist. Then there exist two curves, C1 and C2 , C2 se C1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C1 and C2 coincide with each other or C2 is not in the region, every solution which begins off the y-axis will converge to E x . Every solution which begins on the y-axis will converge to E 0 . If there exists both C1 , C2 in the region then every solution to the left of C1 will converge to E 0 and every solution to the right of C1 will converge to E x . Suppose that c = a and b = d. Then E 0 is non-hyperbolic of resonance type (1, 1), and E x , E y , and E + will not exist. Every solution will converge to E 0 .

Proof For each case the existence of the equilibrium points is given by Lemma 1. The local stability results of E 0 are given in Lemma 2, of E y are given in Lemma 3, and of E x are given in Lemma 4. Additionally, E + will not exist in all cases by Lemma 11. For case ( f ), we need to check the eigenvalues of the Jacobian matrix for each of the infinite equilibrium points of the form E K . As c = d = 1 and a = b substituting in E K into the Jacobian matrix will yield J (x, 1 − K − x) =

α + (1 − α)(1 − x) (α − 1)x , (β − 1)(1 − K − x) β + (1 − β)(k + x)

The eigenvalues of this matrix will be λ1 = 1 and λ2 = K + αx + (1 − K − x)β. Note that as |λ2 | < 1, then λ1 > |λ2 | > 0. Using Lemma 9 the (O+ ) condition holds resulting in the fact that there are no minimal period-two solutions. As the solutions are bounded by Lemma 5, then Theorem 7 will give us the global results of all the cases (a)–(j).  Remark 3 Theorems 8 and 9 give global dynamics of system (1) in both hyperbolic and non-hyperbolic cases showing that in all case except case (b) of Theorem 8 and

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( f ) of Theorem 8 system (1) will exhibit competitive exclusion and only the case (b) will exhibit competitive coexistence scenario. Remark 4 In terms of population dynamics we conclude the following: (1) If the interspecific competition is not too strong the two populations can cooexist stably, but at lower populations than their respective carrying capacities—Case (b) of Theorem 8. (2) If the interspecific competition is strong and ultimately one population wins, while the other is driven to extinction. The winner depends upon which has the starting advantage—Case (a) of Theorem 8. (3) If the interspecific competition of one species dominates the other and, then the species with the strongest competition always drives the other to extinction— Cases (c)–(f) of Theorem 8. (4) If the birth rates of both species are critically low both species go to extinction— Case (g) of Theorem 8. Some non-hyperbolic cases described in Theorem 9 are either of the competitive exclusion type or of new type with an infinite number of the equilibrium points.

References 1. Allen, L.J.S.: An Introduction to Mathematical Biology. Prentice Hall (2006) 2. Assas, L., Livadiotis, G., Elaydi, S., Kwessi, E., Ribble, D.: Competition models with Allee effects. J. Differ. Equ. Appl. 20, 1127–1151 (2014) 3. Basu, S., Merino, O.: On the global behavior of solutions to a planar system of difference equations. Commun. Appl. Nonlinear Anal. 16, 89–101 (2009) 4. Bilgin, A., Kulenovi´c, M.R.S.: Global asymptotic stability for discrete single species population models. Disc. Dyn. Nat. Soc. Art. ID 5963594, 15 pp (2017) 5. Bilgin, A., Brett, A., Kulenovi´c, M.R.S., Pilav, E.: Global dynamics of a cooperative discrete system in the plane. Int. J. Bifurc. Chaos 28, 17p (2018) 6. Brett, A., Kulenovi´c, M.R.S.: Two species competitive model with the Allee effect. Adv. Differ. Equ. 2014, 307, 28 p (2014) 7. Camouzis, E., Ladas, G.: When does local asymptotic stability imply global attractivity in rational equations? J. Differ. Equ. Appl. 12, 863–885 (2006) 8. Clark, D., Kulenovi´c, M.R.S.: A coupled system of rational difference equations. Comput. Math. Appl. 43, 849–867 (2002) 9. Cushing, J.M.: The Allee effect in age–structured population dynamics. Hallam, T.G., Gross, L.J., Levin, S.A. eds., Mathematical Ecology, pp. 479–505. World Scientific Publishing Co., Singapore (1988) 10. Cushing, J.M., Levarge, S., Chitnis, N., Henson, S.M.: Some discrete competition models and the competitive exclusion principle. J. Differ. Equ. Appl. 10, 1139–1152 (2002) 11. Dehghan, M., Kent, C.M., Mazrooei-Sebdani, R., Ortiz, N.L., Sedaghat, H.: Dynamics of rational difference equations containing quadratic terms. J. Differ. Equ. Appl. 14, 191–208 (2008) 12. Dehghan, M., Kent, C.M., Mazrooei-Sebdani, R., Ortiz, N.L., Sedaghat, H.: Monotone and oscillatory solutions of a rational difference equation containing quadratic terms. J. Differ. Equ. Appl. 14, 1045–1058 (2008)

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13. Elaydi, S., Livadiotis, G.: General Allee effect in two-species population biology. J. Biol. Dyn. 6, 959–973 (2012) 14. Franke, J.E., Yakubu, A.-A.: Mutual exclusion versus coexistence for discrete competitive systems. J. Math. Biol. 30, 161–168 (1991) 15. Franke, J.E., Yakubu, A.-A.: Global attractors in competitive systems. Nonlinear Anal. TMA 16, 111–129 (1991) 16. Franke, J.E., Yakubu, A.-A.: Geometry of exclusion principles in discrete systems. J. Math. Anal. Appl. 168, 385–400 (1992) 17. Hassell, M.P., Comins, H.N.: Discrete time models for two-species competition. Theor. Popul. Biol. 9, 202–221 (1976) 18. Hess, P.: Periodic-parabolic boundary value problems and positivity. Pitman Research Notes in Mathematics Series, vol. 247. Longman Scientific and Technical, Harlow (1991) 19. Hirsch, M., Smith, H.L.: Monotone dynamical systems. Handbook of Differential Equations, Ordinary Differential Equations (second volume), pp. 239–357. Elsevier B.V., Amsterdam (2005) 20. Hirsch, M., Smith, H.L.: Monotone maps: a review. J. Differ. Equ. Appl. 11, 379–398 (2005) 21. Jones, F., Perry, J.: Modelling populations of cyst-nematodes (nematoda: heteroderidae). J. Appl. Ecol. 15, 349–371 (1978) 22. Jiang, H., Rogers, T.D.: The discrete dynamics of symmetric competition in the plane. J. Math. Biol. 25, 573–596 (1987) 23. Krawcewicz, W., Rogers, T.D.: Perfect harmony: the discrete dynamics of cooperation. J. Math. Biol. 28, 383–410 (1990) 24. Kent, C.M., Sedaghat, H.: Global attractivity in a quadratic-linear rational difference equation with delay. J. Differ. Equ. Appl. 15, 913–925 (2009) 25. Kent, C.M., Sedaghat, H.: Global attractivity in a rational delay difference equation with quadratic terms. J. Differ. Equ. Appl. 17, 457–466 (2011) 26. Kulenovi´c, M.R.S., McArdle, D.: Global dynamics of Leslie-Gower competitive systems in the plane. Mathematics 7(1), 76, 18 (2019) 27. Kulenovi´c, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, Boca Raton, London (2002) 28. Kulenovi´c, M.R.S., Merino, O.: A global attractivity result for maps with invariant boxes. Disc. Contin. Dyn. Sys. B. 6, 97–110 (2006) 29. Kulenovi´c, M.R.S., Merino, O.: Competitive-exclusion versus competitive-coexistence for systems in the plane. Disc. Contin. Dyn. Sys. Ser. B. 6, 1141–1156 (2006) 30. Kulenovi´c, M.R.S., Merino, O.: Global bifurcations for competitive system in the plane. Disc. Contin. Dyn. Syst. Ser. B. 12, 133–149 (2009) 31. Kulenovi´c, M.R.S., Merino, O.: Invariant manifolds for competitive discrete systems in the plane. Int. J. Bifur. Chaos 20, 2471–2486 (2010) 32. Kulenovi´c, M.R.S., Merino, O.: Invariant curves for planar competitive and cooperative maps. J. Differ. Equ. Appl. 24, 898–915 (2018) 33. Kulenovi´c, M.R.S., Merino, O., Nurkanovi´c, M.: Global dynamics of certain competitive system in the plane. J. Differ. Equ. Appl. 18, 1951–1966 (2012) 34. Kulenovi´c, M.R.S., Nurkanovi´c: Global behavior of a two-dimensional competitive system of difference equations with stocking. Math. Comput. Model. 55, 1998–2011 (2012) 35. Maller, R.A., Pakes, A.: Mathematical Ecology of Plant Species Competition. Cambridge University Press, Cambridge (1990) 36. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ (2001) 37. Maller, R.A., Rossiter, R.C., Pakes, A.: model of changes in the composition of binary mixtures of subterranean clover strains. Aust. J. Agric. Res. 36, 119–143 (1985) 38. Palmer, M.J., Rossiter, R.C.: An analysis of seed yield in some strains of subterranean clover (Trifolium subterraneum L.) when grown in binary mixtures. Aust. J. Agric. Res. (32), 445–452 (1981)

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Nonwandering Sets and Special α-limit Sets of Monotone Maps on Regular Curves Habib Marzougui and Aymen Daghar

Abstract Let X be a regular curve and let f : X → X be a monotone map. We show that AP( f ) = R( f ) = Ω( f ), where AP( f ), R( f ) and Ω( f ) are the sets of almost periodic points, recurrent points and nonwandering points of f , respectively. On the other hand, we show that for every x ∈ X \P( f ), the special α-limit set sα f (x) is a minimal set, where P( f ) is the set of periodic points of f and that sα f (x) is always closed, for every x ∈ X . In addition, we prove that SA( f ) = R( f ), where SA( f ) denotes the special α-limit set of f . Further results related to the continuity of the limit maps are also obtained, we prove that the map ω f (resp. α f , resp. sα f ) is continuous on X \P( f ) (resp. X ∞ \P( f )), where X ∞ = ∩ f n (X ). n≥0

Keywords Minimal set · Regular curve · ω-limit set · Local dendrite · Monotone map

1 Introduction In the last two decades, a wide literature on the dynamical properties of maps on some one-dimensional continua has developed. Examples of continua become increasingly studied include, graphs, dendrites and local dendrites (see for instance [1–3, 6, 7, 23, 30, 32, 36]). Recently a large class of continua called regular curves has given a special attention (see for example, [5, 14–17, 25, 26, 34, 35, 37]). These form a large class of continua which includes local dendrites. The Sierpi´nski triangle is a well known example of a regular curve which is not a local dendrite. Regular curves appear in continuum theory and also in other branches of Mathematics such as complex dynamics; for instance, the Sierpi´nski triangle can be realized as the H. Marzougui (B) · A. Daghar Faculty of Science of Bizerte, Dynamical Systems and their Applications (UR17ES21), University of Carthage, 7021 Jarzouna, Bizerte, Tunisia e-mail: [email protected] A. Daghar e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_16

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Julia set of the map p(z) = z 2 + λz where λ ≈ −0.59257 (see [11, p. 109]). Also the Julia set of the complex polynomial p(z) = z 2 + i is a dendrite (see [18], p. 198). Seidler [37] proved that every homeomorphism of a regular curve has zero topological entropy (later, this result was extended by Kato in [25] to monotone maps). In [34, 35], Naghmouchi proved that any ω-limit set (resp. α-limit set) of a homeomorphism f on a regular curve is a minimal set. Moreover he established the equality between the set of nonwandering points and the set of almost periodic points. In [14], the first author gave a full characterization of minimal sets for homeomorphisms without periodic points on regular curves. In [17] it was shown that the set of periodic points is either empty or dense in the set of non-wandering points, for homeomorphisms on regular curves. In the present paper, we deal with several questions/problems. First, we address the question of the equality between the set of nonwandering points and the set of almost periodic points. This was proved in two cases: − for homeomorphisms on regular curves (Naghmouchi [35]). − for monotone maps on local dendrites (Abdelli et al. [3]). In Theorem 2, we prove a more general result by showing that this is true for monotone maps on regular curves. Nevertheless, the later result is false in general for monotone maps on dendroids as it is shown in Example 1. Moreover, we show in Theorem 3 that the set of nonwandering points coincides with the set of points belonging to their special α-limit sets (Proposition 3). Second, beside the usual limits sets ω-limit and α-limit, we are interested in the study of another kind of limit sets, called special α -limit set (see Section 6). We ask the question whether every special α-limit set is a minimal set? We show that for a monotone map f on a regular curve, every special α-limit set sα f (x) of a non periodic point x is a minimal set. However, we built an example of a monotone map f on an infinite star for which sα f (x) is not a minimal set for some periodic point x. Moreover we show that the inclusion sα f (x) ⊂ α f (x) is strict. In addition, we prove that SA( f ) = R( f ), where SA( f ) (called the special α-limit set of f ) and R( f ) denote the union of all special α-limit sets and the set of recurrent points of f , respectively. Notice that it is shown recently that, for mixing graph maps f : G → G, every special sα-limit set is the ω-limit set of some point from G and moreover, for graph maps f : G → G with zero topological entropy, every special sα-limit set is a minimal set (cf. [19, Theorem 3.8]). On the other hand, Kolyada et al. showed in [27, Theorem 3.3 and Corollary 3.11], that sα f (x) is closed whenever f is either an interval map for which the set of all periodic points is closed or f is transitive. Recently, Hantáková and Roth [21] showed that a special α-limit set is closed for a piecewise monotone interval map. It is previously known that sα f (x) is closed for homeomorphisms on any compact metric space. In Corollary 4, we extend the later result to monotone maps on regular curves by showing that for any x ∈ X , sα f (x) is closed. In fact we show more precisely that for every x ∈ X ∞ , α f (x) ∩ Ω( f ) = sα f (x) (cf. Theorem 6).

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2 Preliminaries Let X be a compact metric space with metric d and let f : X → X be a continuous map. The pair (X, f ) is called a dynamical system. Let Z, Z+ and N be the sets of integers, non-negative integers and positive integers, respectively. For n ∈ Z+ , denote by f n the n-th iterate of f ; that is, f 0 = identity and f n = f ◦ f n−1 if n ∈ N. For any x ∈ X , the subset Orb f (x) = { f n (x) : n ∈ Z+ } is called the orbit of x (under f ). A subset A ⊂ X is called f − invariant (resp. strongly f −invariant) if f (A) ⊂ A (resp., f (A) = A); it is further called a minimal set (under f ) if it is closed, non-empty and does not contain any f -invariant, closed proper non-empty subset of X . We define the ω -limit set of a point x ∈ X to be the set: ω f (x) = {y ∈ X : lim inf d( f n (x), y) = 0} n→+∞  k = { f (x) : k ≥ n}. n∈N

The set α f (x) =



f −n (x) is called the α-limit set of x. Equivalently a point

k≥0 n≥k

y ∈ α f (x) if and only if there exist an increasing sequence of positive integers (n k )k∈N and a sequence of points (xk )k≥0 such that f n k (xk ) = x and lim xk = y. It is much k→+∞

harder to deal with α-limit sets since there are many choices for points in a backward orbit. When f is a homeomorphism, α f (x) = ω f −1 (x). Balibrea et al. [8] considered exactly one branch of the backward orbit as follows. Definition 1 Let x ∈ X . A negative orbit of x is a sequence (xn )n of points in X such that x0 = x and f (xn+1 ) = xn , for every n ≥ 0. The α-limit set of (xn )n≥0 denoted by α f ((xn )n ) is the set of all limit points of (xn )n≥0 . It is clear that α f ((xn )n ) ⊂ α f (x). The inclusion can be strict; even for onto monotone maps on regular curves (see Example 2). Notice that we have the following equivalence: (i) α f (x) = ∅, (ii) x ∈ X ∞ . In particular, if f is onto, then α f (x) = ∅ for every x ∈ X. Proposition 1 ([16, Corollary 2.2]) Let f : X −→ X be a continuous self mapping of a compact space X . Let x ∈ X ∞ . Then: (i) α f (x) is non-empty, closed and f -invariant. In addition, it is strongly f -invariant whenever lim diam( f −n (x)) = 0. n→+∞

(ii) α f ((xn )n≥0 ) is non-empty, closed and strongly f -invariant, for any negative orbit (xn )n≥0 of x.

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A point x ∈ X is called: − Periodic of period n ∈ N if f n (x) = x and f i (x) = x for 1 ≤ i ≤ n − 1; if n = 1, x is called a fixed point of f i.e. f (x) = x; − Almost periodic if for any neighborhood U of x, there is N ∈ N such that { f i+k (x) : 0 ≤ i ≤ N } ∩ U = ∅, for all k ∈ N. −Recurrent if x ∈ ω f (x). − Nonwandering if for any neighborhood U of x there is n ∈ N such that f n (U ) ∩ U = ∅. We denote by P( f ), AP( f ), R( f ), Ω( f ) and Λ( f ) the sets of periodic points, almost periodic points, recurrent points, nonwandering points and the union of all ω-limit sets of f , respectively. Define the space f n (X ). From the definition, we have the following inclusions: X∞ = n≥0

P( f ) ⊆ AP( f ) ⊆ R( f ) ⊆ Λ( f ) ⊆ Ω( f ) ⊆ X ∞ . In the definitions below, we use the terminology from Nadler [33] and Kuratowski [28]. Definition 2 ([28, p. 131]) Let X, Y be two topological spaces. A continuous map f : X −→ Y is said to be monotone if for any connected subset C of Y , f −1 (C) is connected. When f is closed and onto, Definition 2 is equivalent to that the preimage of any point by f is connected (cf. [28, p. 131]). In particular, this holds if X is compact, Y is Hausdorff and f is onto. Notice that f n is monotone for every n ∈ N when f itself is monotone. A continuum is a compact connected metric space. An arc I (resp. a circle) is any space homeomorphic to the compact interval [0, 1] (resp. to the unit circle S1 = {z ∈ C : |z| = 1}). A space is called degenerate if it is a single point, otherwise; it is non-degenerate. By a graph G, we mean a continuum which can be written as the union of finitely many arcs such that any two of them are either disjoint or intersect only in one or both of their endpoints. A dendrite is a locally connected continuum which contains no circle. Every subcontinuum of a dendrite is a dendrite [33, Theorem 10.10] and every connected subset of D is arcwise connected. A local dendrite is a continuum every point of which has a dendrite neighborhood. By [28, Theorem 4, p. 303]), a local dendrite is a locally connected continuum containing only a finite number of circles. As a consequence every subcontinuum of a local dendrite is a local dendrite [28, Theorems 1 and 4, p. 303]. Every graph and every dendrite is a local dendrite. A regular curve is a continuum X with the property that for each point x ∈ X and each open neighborhood V of x in X , there exists an open neighborhood U of x in V such that the boundary set ∂U of U is finite. Each regular curve is a 1-dimensional locally connected continuum. It follows that each regular curve is locally arcwise

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connected (see [28, 33], for more details). In particular every local dendrite is a regular curve (cf. [28, p. 303]). A continuum X is said to be finitely suslinean continuum provided that each infinite family of pairwise disjoint continua is null (i.e. for any disjoint open sets U, V , only a finite number of elements of the family meet both U and V ). Note that each regular curve is finitely suslinean (see [29]). A continuum is called hereditarily locally connected continuum, written hlc, provided that every subcontinuum of X is locally connected. In particular, finitely suslinean continua and (hence regular curves) are hlc (see [28]). A continuum X is said to be rational curve provided that each point x of X and each open neighborhood V of x in X , there exists an open neighborhood U of x in V such that the boundary set ∂U of U is at most countable. Clearly, regular curve are rational. Let X be a compact metric space. We denote by 2 X (resp. C(X )) the set of all non-empty compact subsets (resp. compact connected subsets) of X . The HausC(X )) is defined as follows: d H (A, B) = dorffmetric d H on 2 X (respectively  max supa∈A d(a, B), supb∈B d(b, A) , where A, B ∈ 2 X (resp. C(X )). For x ∈ X and M ∈ 2 X , d(x, M) = inf y∈M d(x, y). With this distance, (C(X ), d H ) and (2 X , d H ) are compact metric spaces. Moreover if X is a continuum, then so are 2 X and C(X ) (see [33], for more details). Let f : X → X be a continuous map of X . We denote by 2 f : 2 X → 2 X , A → f (A), called the induced map. Then 2 f is also a continuous self mapping of (2 X , d H ) (cf. [33]). For a subset A of X , we denote by diam(A) = supx,y∈A d(x, y) and card(A) the cardinality of A. We let Mesh(A) = sup{diam(C) : C is a connected component of A}. A family {Ai : i ∈ I } of subsets of X is said to be a null family if for any ε > 0 there exists a finite subset J of I such that for any i ∈ I \J , we have diam(Ai ) < ε. We recall some results which are needed for the sequel. Proposition 2 ([29]) Let X be a regular curve. Then for any ε > 0 and for any family of pairwise disjoint subcontinua (Ai )i∈I of X , the set {i ∈ I : diam(Ai ) ≥ ε} is finite. In particular if (An )n≥0 is a sequence of pairwise disjoint continua, then (An )n≥0 is a null family. Definition 3 (Weak incompressibility) A set A ⊂ X is said to have the weak incompressibility property if for any proper closed subset F  A (i.e. F is nonempty and distinct from A), we have that F ∩ f (A\F) = ∅. Notice that the term weak incompressibility seems to have appeared first in [9]. Lemma 1 ([12, Lemma 3, p. 71]) For any x ∈ X , ω f (x) has the weak incompressibility property. Lemma 2 Let A ⊂ P( f ) be a closed invariant subset of X with the weak incompressibility property. If some a ∈ A is an isolated point of A, then A = O f (a).

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Proof Assume that for some a ∈ A, {a} is an open subset of A and O f (a)  A. Since f |A is an homeomorphism, O f (a) is a finite open subset of A, thus also a proper closed subset of A. The subset A has the weak incompressibility property and O f (a) is a closed subset of A, we get O f (a) ∩ f (A\O f (a)) = ∅. Therefore O f (a) ∩ (A\O f (a)) = ∅, which contradict the fact that f is one to one on A. Let f be a monotone map on a regular curve X . If M is an infinite minimal set of f , we call B(M) = {x ∈ X : ω f (x) = M} the basin of attraction of M. Theorem 1 ([16]) Let f : X −→ X be a regular curve monotone map. Then the following assertions hold: (1) ω f (x) is a minimal set, for all x ∈ X . (2) α f ((xn )n≥0 ) is a minimal set any negative orbit (xn )n≥0 of x ∈ X ∞ . Moreover if x ∈ X ∞ \P( f ), then α f (x) is a minimal set and α f (x) = α f ((xn )n≥0 ), for any negative orbit (xn )n≥0 of x. (3) For every infinite minimal set M, B(M) is a closed subset of X . (4) For every x ∈ X ∞ , if ω f (x) is infinite, then α f (x) = ω f (x). (5) AP( f ) = Λ( f ) = R( f ).

3 Nonwandering Sets of Monotone Maps on Regular Curves The aim of this section is to prove the following theorem which extends [3, Theorem 1.1] and [35, Theorem 2.1]. Theorem 2 Let X be a regular curve and f a monotone self mapping of X . Then Ω( f ) = R( f ) = Λ( f ) = AP( f ). Proof Following Theorem 1, (5), it suffices to prove that Ω( f ) ⊂ R( f ). Assume that there exists x ∈ Ω( f )\R( f ). We distinguish two cases: Case 1: x ∈ / P( f ). In this case, we follow similarly the proof given in [35, Theorem 2.1]. Let V be an open neighborhood of x such that V ∩ P( f ) = ∅. As x ∈ Ω( f ), there is a sequence (xk )k≥0 in X converging to x and a sequence of positive integers (n k )k≥0 such that f n k (xk ) converges to x. Since x ∈ / R( f ), x ∈ / ω f (x). Then we can find an open neighborhood U1 ⊂ V of x with finite boundary and an open neighborhood U2 of ω f (x) such that U1 ∩ U2 = ∅. Since X is locally connected, so by [28, Theorem 4, p. 257], for k large enough, we can find a sequence of arcs (Ik )k≥0 ⊂ U1 joining xk and x such that lim Ik = {x} (with respect to the Hausdorff metric). k→+∞

Thus f n k (Ik ) will meets U1 and U2 , for k large enough and so it meets the boundary ∂U1 in a point bk , for k large enough. As ∂U1 is finite, one can assume that bk = b for infinitely many k. Therefore f −n k (b) ∩ Ik = ∅, for infinitely many k. It follows

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that x ∈ α f (b). As b ∈ / P( f ), then α f (b) is a minimal set of f (Theorem 1). So ω f (x) = α f (b) and hence x ∈ ω f (x). A contradiction. Case 2: x ∈ P( f ). Let (xn )n≥0 ⊂ P( f ) be a sequence converging to x and set pn = Per(xn ) the period of xn , n ≥ 0. Then ( pn )n≥0 in unbounded: otherwise, x ∈ P( f ) ⊂ R( f ), a contradiction. So we can assume that ( pn )n≥0 goes to infinity. Since α f (x) is a minimal set, x ∈ / α f (x) (because otherwise, x ∈ ω f (x), a contradiction). Now as in Case 1, let U2 be an open neighborhood of α f (x) and U1 an open neighborhood of x with finite boundary and disjoint from U2 . Let (Ik )k≥0 be a sequence of arcs joining x and xk such that lim Ik = {x} (with respect to the Hausdorff metric). k→+∞

Thus f − pk (Ik ) meets ∂U1 infinitely many times and thus there exists b ∈ ∂U1 such that f pk (b) ∈ Ik . This implies that x ∈ ω f (b). As ω f (b) is a minimal, so x ∈ R( f ). A contradiction. Corollary 1 If Ω( f ) is finite or countable, then Ω( f ) = P( f ). Proof Since Ω( f ) = AP( f ) (Theorem 2), so AP( f ) is at most countable and hence every minimal set for (X, f ) is a periodic orbit. Therefore AP( f ) = P( f ) and then Corollary 1 follows. In [13], Coven and Nitecki have shown that for continuous maps f of the closed interval [0, 1], Ω( f ) = {x ∈ [0, 1] : x ∈ α f (x)}. Later, it is extended to graph maps in [31, Corollary 1]. However, for dendrite maps, it does not holds (see [38]). The following theorem extend the above result to monotone maps on regular curves. Theorem 3 Let X be a regular curve and f a self monotone mapping of X . Then Ω( f ) = {x ∈ X : x ∈ α f (x)}. Proof Let x ∈ Ω( f ). By Theorem 2, x ∈ ω f (x) and hence there exists a negative orbit (xn )n≥0 of x such that (xn )n≥0 ⊂ ω f (x). By minimality of ω f (x), we have x ∈ ω f (x) = α f ((xn )n≥0 ) ⊂ α f (x). Conversely, let x ∈ X such that x ∈ α f (x). We can assume that x ∈ / P( f ) (if x ∈ P( f ) then x ∈ Ω( f ) and we are done). So by Theorem 1, α f (x) is a minimal set and x ∈ α f (x). Therefore ω f (x) = α f (x). Hence x ∈ ω f (x) and so x ∈ R( f ) = Ω( f ). Corollary 2 Let X be a regular curve and f a self monotone mapping of X . If x ∈ Ω( f ), then : (i) ω f (x) ⊂ α f (x). (ii) If x ∈ / P( f ), then ω f (x) = α f ((xn )n≥0 ) = α f (x), for any negative orbit (xn )n≥0 of x. Proof (i) Since x ∈ Ω( f ), so x ∈ α f (x) (Theorem 3) and hence ω f (x) ⊂ α f (x) (since by Proposition 1, α f (x) is closed and f -invariant). (ii) If x ∈ / P( f ), then α f ((xn )n ) = α f (x) (Theorem 1). Again by Theorem 1 and from (i), we get α f (x) = ω f (x).

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In the following example, we show that Theorem 2 cannot be extended to rational curves; we construct a self monotone map f on a rational curve D (it is in fact a dendroid) such that the inclusion is strict: R( f )  Ω( f ). Notice also that Theorem 2 does not hold even for continuous maps on intervals; for instance; we may extend the map g : L → L (defined in Example 1) into a / R(h). continuous map h of [0, 1] so that T1 ∈ Λ(h) ⊂ Ω(h) but T1 ∈ Example 1 The idea of the construction consists to define a continuous map g on a countable compact set L satisfying R(g)  Ω(g) and then extends it to a monotone map f defined on a dendroid D. • Let σ be the shift map on the Cantor set {0, 1}N endowed with the metric d defined as follows: For x, y ∈ {0, 1}N , d(x, y) = 2−N (x,y) , where N (x, y) = min{n ∈ N : xn = yn } with N (x, x) = +∞, ∀x ∈ {0, 1}N . We denote by: We let T0 = 0 and Ti = (000000 . . .

1 

0 . . . ), for i > 0. Set

i th −position

Z = ( 10  100 1000 . . . 0 1 . . . ),   . . . 10   n−zer o

O− (Z) = {T−i = (000 . . .

Z ) : i ≥ 1}, 

i th −position

L = Oσ (Z) ∪ {Ti : i ≥ 0} ∪ O− (Z). Observe that (T−i )i≥1 converges show that ωσ (Z ) = {Ti : i ≥ 0}.

to T0 . Let usn(n−1) k + n = + n. It is clear that (kn )n∈N is For each n ∈ N, we let kn = n−1 k=1 2 an increasing sequence. Observe that σ kn (Z) gives the first apparition of the number ¯ < 2−n and so 0¯ ∈ ωσ (Z). Let one after n zeros. Then for each n ∈ N, d(σ kn (Z), 0) i ≥ 2 and suppose that n ≥ i. Then σkn +(n−i+1) (Z) gives i − 1 zero before the first apparition of one. Then d(σ kn +(n−i+1) (Z), Ti ) < 2−(n+1) , so Ti ∈ ωσ (Z ), ∀i ≥ 2. ¯ ∪ {Ti , i ∈ N} = For i = 1, we have T1 = (100 . . . ) = σ (T2 ) ∈ ωσ (Z). In result, {0} {Ti : i ≥ 0} ⊂ ωσ (Z). Conversely, let y = (yn )n∈N ∈ ωσ (Z) and suppose that there exist i, j ∈ N such that i < j and yi = y j = 1, let (n s )s≥0 be an increasing sequence of integer be such that lims→+∞ σ n s (Z) = y. Then there exists s1 ∈ N such that ∀s ≥ s1 , we have n s > k j+1 , so d(σ n s (Z), y) ≥ 2− j , ∀s ≥ s1 ( we have only a single one in the first block with length j in σ n s (Z) but, we have two ones in the first block with length j in y). So every point of ωσ (Z) contains at most only one. Consequently, ¯ = {Ti : i ≥ 0}. ωσ (Z) = {Ti , i ∈ N} ∪ {0} We conclude that L is a countable compact set. We let g = σ|L . Then g(Ti ) = Ti−1 , for any i ≥ 1. Pick a homeomorphic copy of L in [0, 1] × {1}. For i ∈ Z, let Ii be an arc joining Ti and T0 so that S := Ii is an infinite star centered at T0 . in particular i∈Z

I0 = {T0 }. For each k ∈ Z, let i k ∈ Z such that d(g k (Z), S) = d(g k (Z), Iik ) and let Jk be an arc joining T0 and g k (Z) which is included in the closed ball B F (Iik , d(g k (Z), Iik )) of the plan. We may assume that for any k ∈ Z, S ∩ Jk = {T0 } and for any k =

Nonwandering Sets and Special α-limit Sets of Monotone Maps …

l, Jk ∩ Jl = {T0 }. Let D = S

 

347

Jk . Clearly D is a countable union of arcs, so

k≥0

by [28, Theorem 6, p. 286] D is a rational curve which is a dendroid. • We extend g into a map f on D so that f (Jk ) = Jk+1 and f |Jk is affine, for each k ∈ Z and f (Ii ) = Ii−1 for i ≥ 1. Then f (I1 ) = {T0 } and f |L = g. It is clear that f is a pointwise monotone map on D and onto, thus monotone. Moreover R( f )  Ω( f ), / R( f ) (Fig. 1). since for instance T1 ∈ Λ( f ) ⊂ Ω( f ) but T1 ∈

Fig. 1 The map f on the dendroid D

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4 The Space of Minimal Sets with Respect to the Hausdorff Metric The main result of this section is to prove the following theorem. Theorem 4 Let X be a regular curve and f a monotone self mapping of X . Then any limit (with respect to the Hausdorff metric) of a sequence of minimal sets (Mn )n≥0 is a minimal set. Proof Let (Mn )n≥0 be a sequence of minimal sets converging to some M ⊂ X . Then f (M) = M. Since each Mn ⊂ R( f ), so by Theorem 2, M ⊂ R( f ) = AP( f ). Therefore M = Mi , where Mi is a minimal set for any i ∈ I . Assume that M is not a i∈I

minimal set. Then card(I ) > 1. Claim: M ⊂ P( f ). Suppose that M  P( f ), so there exists i 0 ∈ I such that Mi0 is infinite. Set P = Mi0 and N = M j0 , where j0 = i 0 . Then P and N are disjoint minimal sets and so does B(P) and N . By Theorem 1, B(P) is a closed set. So let U an open set with finite boundary ∂U of cardinality k such that B(P) ⊂ U and N ∩ U = ∅. Recall that P ∪ N ⊂ M = lim Mn , so fix some y ∈ P, z ∈ N and let yn , z n ∈ n→+∞

Mn such that (yn )n≥0 (resp. (z n )n≥0 ) converges to y (resp. z). Let (In )n≥0 be a sequence of arcs joining y and yn so that {y} = lim In . n→+∞

Since Mn is a minimal set, we can find sn ≥ 0 such that d( f −sn (yn ), z n ) ≤ n1 , for every n ≥ 1. Hence for n large enough, f −sn (yn )  U and f −sn (In )  U . On the other hand, f −sn (In ) meets P at some point of f −sn (y). As f is monotone, thus for n large enough, f −sn (In ) ∩ ∂U = ∅. As ∂U is finite, then there exists z ∈ ∂U such that z ∈ f −sn (In ), for infinitely many n. Hence y ∈ ω f (z) (since d( f sn (z), y) ≤ diam(In ) + d(yn , y)). By minimality of ω f (z) (see Theorem 1), we get ω f (z) = P. Hence z ∈ B(P) ∩ ∂U . A contradiction with B(P) ⊂ U . The proof is complete. Proof of Theorem 4. As proven by the claim above, we have M ⊂ P( f ). Since any minimal set Mn has the weakly incompressibility property (Lemma 1), so does its limit M by [4, Proposition 3.1]. Since M ⊂ P( f ), it is uncountable, otherwise, M will have an isolated point a, so by Lemma 2, M = O f (a) is a periodic orbit. A contradiction. From M = ∪n∈N Fix( f n ) ∩ M, there exist N > 0 and an infinite sequence (z n )n≥0 in Fix( f N ) ∩ M with disjoint orbits converging to some z ∈ Fix( f N ) ∩ M. As O f (z 0 ) ∩ O f (z) = ∅, there exists an open set Vz with finite boundary such that O f (z) ⊂ Vz and O f (z 0 ) ∩ Vz = ∅. Since (z n )n≥0 converges to z, there is some p ∈ N such that for any n ≥ p, O f (z n ) ⊂ Vz . For each n ≥ p, let (In,q )q≥0 be a sequence of arcs joining z n and z n,q ∈ Mq so that (In,q )q≥0 converges to {z n } as done in the proof of the claim above. As Mq is minimal and Mq  U , we can find sq > 0 such that f −sq (z n,q )  U . Recall that O f (z n ) ⊂ U , so f −sq (In,q ) ∩ ∂U = ∅, for any q ≥ 0.

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By the same argument as in the proof of the claim above, we can find, for each n ≥ p, bn ∈ ∂U such that ω f (bn ) = O f (z n ). This leads to a contradiction since {O f (z n ) : n ≥ p} is an infinite family of disjoint periodic orbits.

5 On the Continuity of Limit Maps ω f and α f In this section, we shall investigate the continuity of the limit maps: ω f : X → 2 X ; x → ω f (x) and α f : X ∞ → 2 X ; x → α f (x). Theorem 5 The maps ω f and α f are continuous everywhere except may be at the periodic points of f . We use the following lemmas. Lemma 3 ([15, Lemma 4.2]) Let X be a hereditarily locally connected continuum, F X a closed subset and (On )n≥0 a sequence of open subsets of X such that F= On . Then lim Mesh(On \F) = 0. n≥0

n→+∞

Lemma 4 The restriction map (ω f )|Ω( f ) is continuous. Proof Let (xn )n≥0 a sequence of Ω( f ) converging to some x ∈ Ω( f ). By Theorem 2, xn ∈ ω f (xn ) and by Theorem 1, (ω f (xn ))n≥0 is a sequence of minimal set, thus by Theorem 4, any limit point of that sequence should be a minimal set. Observe that x belongs to any limit point of the sequence (ω f (xn ))n≥0 , therefore any limit point of the sequence (ω f (xn ))n≥0 is a minimal set that contains x, therefore it has to be ω f (x). Let (X, f ) be a dynamical system. For x ∈ X , we denote by Sx = f −i (x). i≥0

Lemma 5 Let x, y ∈ X such that O f (x) and O f (y) are two disjoint orbits. Then Sx and S y are disjoint. Proof Assume that Sx ∩ S y = ∅. Then we can find two positive integers m ≤ n and some z ∈ f −m (x) ∩ f −n (y). Therefore f n (z) ∈ O f (x) ∩ O f (y), which will lead to a contradiction. Lemma 6 Let X be a regular curve and f a monotone self mapping of X . The limit maps ω f and α f are continuous at any point of X \ Ω( f ). Proof First note that X \ R( f ) = X \ Ω( f ) is a open set disjoint from P( f ). Thus for any point a ∈ X \ Ω( f ), α f (a) and ω f (a) are minimal sets. Let x ∈ X \ R( f ):

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• Continuity of ω f : Assume that ω f is not continuous at x and set M = ω f (x), then we can find a sequence (xn )n≥0 that converges to x such that ω f (xn ) converges to A = M. By Theorem 4, A is a minimal set, thus A ∩ M = ∅. Let m ≥ 0 and V A,m ⊂ B(A, m1 ) be an open set with finite boundary such that M ∩ V A,m = ∅. Let (In )n≥0 be a sequence of arcs joining x and xn , where (In )n≥0 converge to {x} with respect to the Hausdorff metric. Recall that A = lim ω f (xn ), so for n large enough n→+∞

we can find some pn such that f pn (xn ) ∈ V A,m and f pn (x) ∈ / V A,m . Then f pn (In ) / Ω( f ), we will meets ∂ V A,m and so x ∈ α f (bm ) for some bm ∈ ∂ V A,m . Since x ∈ conclude that α f (bm ) is not a minimal set, thus by Theorem 1, bm ∈ P( f ). Claim: The sequence (bm )m≥0 satisfies the following properties: (i) For any m ≥ 0, bm ∈ P( f ) and x ∈ α f (bm ). (ii) The sequence (O f (bm ))m≥0 converges to A. (iii) By passing to a subsequence if needed, the family (O f (bm ))m≥0 is pairwise disjoint. Proof of the claim: (i) is already proven above, for the proof of (ii) observe that any limit point of (bm )m≥0 is a point of A and since {bm : m ≥ 0} ∪ A ⊂ Ω( f ), so Lemma 4 ends the proof of (ii). (iii) Clearly for any m ≥ 0, O f (bm ) ∩ A = ∅. Let bm 0 = b0 and O1 be an open neighborhood of A such that O f (bm 0 ) ∩ O1 = ∅. As (bm )m≥0 converges to A we can find m 1 > 0 such that bm 1 ∈ O1 . Thus O f (b0 ) and O f (b1 ) are disjoint. For N ≥ 0, (O f (bm i ))0≤i≤N is defined and pairwise disjoint. Let O N +1 an open neighborhood of A such that  O f (bm i ) ∩ O N +1 = ∅. We can find m N +1 > m N such that bm N +1 ∈ O N +1 and 0≤i≤N

then (O f (bni ))0≤i≤N +1 is pairwise disjoint. We have defined by induction the subsequence (bm i )i≥0 so that (O f (bm i ))i≥0 are pairwise disjoint. This ends the proof of the claim. Now x ∈ / R( f ), so x ∈ / A. Let V be an open neighborhood of A such that x∈ / V and set k = card(∂ V ). By (iii) there exists N ≥ 0 such that for any m ≥ N , ω f (bm ) = O f (bm ) ⊂ V . By (i), we may find for each m ≥ 0 some sm ≥ 0 such that f −sm (bm )  V . Fix N ≤ m 0 < m 2 < . . . m k . By Lemma 5 and monotonicity of f , the family ( f −smi (bm i ))0≤i≤k is a pairwise disjoint family of connected subset of X . Observe that for each 0 ≤ i ≤ N , f −smi (bm i )  V and f −smi (bm i ) meets V at some point of O f (bm i ). Therefore for each 0 ≤ i ≤ N , f −smi (bm i ) meets ∂ V . This leads to a contradiction. • Continuity of α f : Set M = α f (x). By Theorem 1, M is a minimal set, assume that (α f (xn ))n≥0 converges to L = M, then L  M, so let t ∈ L\M. Let Vt be an open neighborhood of t with finite boundary such that M ∩ Vt = ∅ and (In )n≥0 a sequence of arcs joining x and xn converging to {x} with respect to the Hausdorff metric. Since t ∈ L = lim α f (xn ), for n large enough, we can find n→+∞

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351

some pn ≥ 0 such that f − pn (xn ) ⊂ Vt and f − pn (x)  Vt . Then by monotonicity of f , f − pn (In ) will meets ∂ Vt and thus x ∈ ω f (b) for some b ∈ ∂ Vt . By Theorem 2, ω f (b) ⊂ R( f ), hence x ∈ R( f ). A contradiction. Proof of Theorem 5. Let x ∈ X \ P( f ). If x ∈ / Ω( f ), then by Lemma 6, ω f is continuous at x. So assume that x ∈ Ω( f )\ P( f ) and set M = ω f (x). Then x ∈ M and M is an infinite minimal set. • Continuity of α f on X \P( f ): By Theorem 1, α f (x) = M. Assume that there exists some sequence (xn )n≥0 of X converging to x such that α f (xn ) converges to L and L = M. By minimality of M, L  M. Assume that there exists y ∈ L\M. We distinguish two cases: Case 1: For infinitely many n ≥ 0, xn ∈ P( f ). Since x in non periodic point, so per((xn ))n is unbounded. By choosing (xn ))n≥0 with pairwise distinct periods per((xn ))n , one can assume that (O f (xn ))n≥0 is pairwise disjoint. Therefore by Lemma 5, the family (Sxn )n≥0 is pairwise disjoint. Let now Vy , VM be two disjoint open neighborhoods of y, M with finite boundary. By Lemma 4, the restriction map (ω f )|Ω( f ) is continuous. Therefore as x ∈ R( f )\ P( f ), we conclude that the sequence O f (xn ))n≥0 converges to ω f (x) = M. So one can assume that for any n ≥ 0, we have O f (xn ) ⊂ VM . As y ∈ lim α f (xn ), then for each n ≥ 0, one can n→+∞

find m n > 0 such that f −m n (xn ) ∩ Vy = ∅. So let Sm n ,n be the connected component of Sn containing f −m n (xn ). Hence (Sm n ,n )n≥0 is a family of pairwise disjoint connected sets each of which meets VM and Vy (since xn is a periodic point such that O f (xn ) ⊂ VM ). This leads to a contradiction since ∂(VM ) is finite. / P( f ) . In this case, by Theorem 1, α f (xn ) is a Case 2: For n large enough, xn ∈ minimal set. Thus by Theorem 4, L is also a minimal set and then M ∩ L = ∅. So let VM and VL be two disjoint neighborhoods of M, L with finite boundary and set k = card(∂(VM )). As (xn )n≥0 converges to x, then we can find k + 1 pairwise disjoint arcs I0 , I1 , . . . , Ik , each I j joins f j (x) and f j (x N ), for some N large enough and satisfying α f (x N ) ⊂ VL . Then we can find s ≥ 0 such that for any i ≥ s, f −i (x N ) ⊂ VL . Consider the family ( f −(s+k) (I j ))0≤ j≤k , which is a family of pairwise disjoint connected sets, each of which meets VL at f −(s+k) (x N ) and meets VM at some point of f −(s+k) ( f j (x)), this will lead to a nonempty intersection with ∂(VM ) for each 0 ≤ j ≤ k. A contradiction. This ends the proof of the continuity of α f . • Continuity of ω f : Assume that there is some sequence (xn )n≥0 of X converging to x such that ω f (xn ) converges to L and L = M. As ω f (xn ) is a minimal set, for any n ≥ 0, thus by Theorem 4, L is also a minimal set and then M ∩ L = ∅. Claim 1. For any neighborhood VL of L, there exists z ∈ VL ∩ P( f ) such that M ⊂ α f (z).

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Proof Fix VL some neighborhood of L. Let U M and U L be two disjoint neighborhoods of M, L with finite boundary such that U L ⊂ VL . Set k = card(∂(U L )). As (xn )n≥0 converges to x and O f (x) is infinite, then for each 0 ≤ j ≤ k, we can find a sequence of arcs (In, j )n≥0 joining f j (xn ) and f j (x) converging to { f j (x)} such that for any n, m ≥ 0 and for any 0 ≤ i < j ≤ k, In,i ∩ Im, j = ∅. Let η = min{d( f p (x), f q (x)), 0 ≤ p < q ≤ k}. Clearly η > 0 and we may assume that ε = inf n≥0 {d(In, p , In,q ) : 0 ≤ p < q ≤ k} > 0. We can assume that for any n ≥ 0, ω f (xn ) ⊂ U L . Thus for each n ≥ 0, we can find m n > 0 such that { f m n ( f j (xn )) : 0 ≤ j ≤ k} ⊂ U L . Therefore for each n ≥ 0, 0 ≤ j ≤ k, f m n (In, j ) ∩ ∂(U L ) = ∅. Then for each n ≥ 0, there exists z n ∈ ∂(U L ) and 0 ≤ j1,n < j2,n ≤ k such that z n ∈ f m n (In, j1,n ) ∩ f m n (In, j2,n ). Hence f −m n (z n ) ∩ In, j p,n = ∅, for p ∈ {0, 1}. Recall that z n ∈ ∂(O L ) which is finite and 0 ≤ jn,1 < jn,2 ≤ k. Hence there exists z ∈ ∂(U L ) and 0 ≤ j1 < j2 ≤ k such that f −m n (z) ∩ In, j p = ∅, for infinitely many n, p ∈ {0, 1}. Therefore for infinitely many n, we have that diam( f −m n (z)) ≥ ε. Hence z ∈ P( f ). Moreover as f −m n (z) ∩ In, j p = ∅, then O f (x) ∩ α f (z) = ∅ and hence M ⊂ α f (z). This ends the proof of Claim 1. Now by Claim 1, we may find a sequence (z n )n≥0 of periodic points with disjoint orbits converging to some point l ∈ L such that M ⊂ α f (z n ). By Lemma 4, (O f (z n ))n≥1 converges to L. Moreover for each n ≥ 1, we can find sn > 0 such that d( f −sn (z n ), M) ≤ n1 . As O f (z n ) ⊂ P( f ), then O f (z n ) ∩ f −sn (z n ) = ∅. Therefore . we can find N ≥ 0 such that for any n ≥ N , we have diam( f −sn (z n )) ≥ d(M,L) 2 Recall that (z n )n≥0 is a sequence of periodic points with disjoint orbits, hence ( f −sn (z n ))n≥N is a non null family of pairwise disjoint subcontinua. A contradiction with the fact that X is a regular curve and hence finitely Suslinean. This ends the  proof of the continuity of ω f .

6 On Special α-limit Sets In [22], Hero introduced another kind of limit sets, called the special α-limit sets. He considered the union of the α-limit sets over all backward orbits of x. Definition 4 Let X be a metric compact space, f : X → X a continuous map and  x ∈ X . The special α-limit set of x, denoted sα f (x), is the union sα f (x) = α f ((xn )n≥0 ) taken over all negative orbits (xn )n≥0 of x. Notice that we have the following equivalence: (i) α f (x) = ∅, (ii) sα f (x) = ∅, (iii) x ∈ X ∞ . In particular, if f is onto, then sα f (x) = ∅ for every x ∈ X . It is clear that sα f (x) ⊂ α f (x). The inclusion can be strict. Hero [22] provided an example of a continuous map on the interval for which the inclusion sα f (x) ⊂ α f (x) is strict. Even, one can provide an onto monotone interval map. Indeed, consider the monotone map g : [0, 1] −→ [0, 1] : x −→ max{0, 2x − 1}. We see that sαg (0) = {0, 1}  αg (0) = [0, 1].

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In [27, Theorem 3.3 and Corollary 3.11], Kolyada et al. provided an example of a map on a subset of R2 where a special α-limit set is not closed. Recently, Hantáková and Roth proved in [21, Theorem 37] that a special α-limit set for interval map is always Borel, and in fact both Fσ and G δ . Furthermore, they provided a counterexample of an interval map with a special α-limit set which is not closed, this disproves the conjecture 1 in [27]. However they showed that a special α-limit set is closed for a piecewise monotone interval map. Jackson et al. [24] proved that a special α-limit set is always analytic (i.e. a continuous image of a Polish space) and provide an example of a map of the unit square with special α-limit set not a Borel set. Here we show that, for a monotone map on a regular curve, the special α-limit set is always closed.

6.1 Relation Between Nonwandering Sets, α-limit Sets and Special α-limit sets The aim of this paragraph is to prove the following theorem. Theorem 6 Let X be a regular curve and f a monotone self mapping of X . Then for every x ∈ X ∞ , we have that sα f (x) = α f (x) ∩ Ω( f ). First, we derive from Theorem 1 the following corollary. Corollary 3 Let X be a regular curve and f a monotone self mapping of X . Then for any x ∈ X ∞ : (i) sα f (x) ⊂ Ω( f ), (ii) sα f (x) is a union of minimal sets. (iii) if x ∈ X ∞ \P( f ), then sα f (x) = α f (x) is a minimal set. Proof Let x ∈ X ∞ . By Theorem 1, (2), α f ((xn )n≥0 ) is a minimal set for every negative orbit (xn )n≥0 of x, thus (i) and (ii) follow. Assume now that x ∈ X ∞ \P( f ), again by Theorem 1, (2), we have α f ((xn )n≥0 ) = α f (x) is a minimal set for any negative orbit (xn )n≥0 of x, thus (iii) follows. The Proof of Theorem 6 needs the following lemma. Lemma 7 Let X be a regular curve and f a monotone self mapping of X . If x ∈ P( f ), then every minimal set M ⊂ α f (x) is a periodic orbit. Proof Let x ∈ P( f ) and M an infinite minimal set such that M ⊂ α f (x). Fix some y ∈ M ⊂ α f (x), we can find an increasing sequence of integers (m n )n≥0 and xn ∈ f −m n (x) such that (xn )n≥0 converges to y. Clearly for any n ≥ 0, we have ω f (xn ) = O f (x). By Theorem 5, ω f is continuous at y, therefore we get ω f (y) = M = O f (x), which is finite. A contradiction.

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Recall that given a subset A of a topological space X , the arc connected component C of a ∈ A is defined as C = {y ∈ A : there exists an arc I ⊂ A joining a and y}. Lemma 8 Let X be a regular curve and A a subset of X . Then the following hold: (i) Every arc connected component of A is closed in A. (ii) If C is arcwise connected, then it is locally arcwise connected. Proof Since a regular curve is finitely suslinean continuum, so the proof of (i) results from [41, Corollary 2.2] and the proof of (ii) results from [20, Corollary 5.5]. Lemma 9 Let X be a regular curve and f a monotone self mapping of X . If x ∈ P( f ) and O f (x)  α f (x), then Ω( f ) = X . Proof Assume that Ω( f ) = X , by Theorem 2, Ω( f ) = R( f ), thus R( f ) = X . Observe that for some n ≥ 0, f −n (x)  O f (x), if not O f (x) = α f (x). So let / O f (x). t ∈ X \O f (x) and n > 0 such that f n (t) = x, then ω f (t) = O f (x) and t ∈ This will lead to a contradiction since t ∈ R( f ). Proof (Proof of Theorem 6) If x ∈ X ∞ \P( f ), then by Corollary 3, sα f (x) = α f (x) ⊂ Ω( f ) and so sα f (x) = α f (x) ∩ Ω( f ). Now assume that x ∈ P( f ). The inclusion sα f (x) ⊂ α f (x) ∩ Ω( f ) follows from Corollary 3. Now let y ∈ α f (x) ∩ Ω( f ). By Lemma 7, ω f (y) is a periodic orbit and so y ∈ P( f ). If O f (y) ∩ O f (x) = ∅, then y ∈ O f (x) ⊂ sα f (x). Now assume that O f (y) ∩ O f (x) = ∅, then η = d(O f (x), O f (y)) > 0 and O f (x)  α f (x). Therefore by Lemma 9, Ω( f )  X . By Lemma 3, we can find an open set U such that Ω( f ) ⊂ U and Mesh(U \Ω( f )) < η2 .  Define A = f −n (x) ∪ O f (x) ∪ O f (y). We observe that f (A) ⊂ A. n≥0

Claim: A has a finite number of arc connected components. Denote by q the period of x. We have  

q−1

A=

i=0

f −i ( f −qn (x)) ∪ O f (x) ∪ O f (y).

n≥0

Since f is monotone so for each n ≥ 0, f −qn (x) is a subcontinuum of the regular  curve X , thus arcwise connected, moreover it contains x. We conclude that ( f −qn (x)) is an arcwise connected subsets of X . Again since f is monotone and 

n≥0

( f −qn (x)) is arcwise connected so for each 0 ≤ i ≤ q − 1,

n≥0

is also arcwise connected. Indeed for a, b ∈ arc I joining f i (a) and f i (b) in

 n≥0





f −i ( f −qn (x))

n≥0

f −i ( f −qn (x)), we can find an

n≥0

( f −qn (x)), thus we can find some arc J in

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f −i (I ) ⊂



355

f −i ( f −qn (x)) joining a and b. We conclude that A can be written as

n≥0

a finite union of arcwise connected subsets of X , this end the proof of the claim. Recall that y ∈ α f (x) then we can find an increasing sequence of integer (m n )n≥0 and xn ∈ f −m n (x) so that (xn )n≥0 converges to y. For each n ≥ 0, xn ∈ A which has a finite number of arc connected components by the claim above, namely C1 , . . . Cm , so we may find 1 ≤ j ≤ m such that for each n ≥ 0, xn ∈ C j so that Therefore by Lemma y ∈ C j . By Lemma 8, (i), C j ∪ {y} is arcwise connected.

8, (ii), it is locally arcwise connected. Next let In ⊂ C j ∪ {y} ∩ B(y, n1 ) be an arc in C j ∪ {y} joining y and xn ∈ f −m n (x) so that (In )n≥0 converges to {y}. We

have for each n ≥ 0 {x, f m n (y)} ⊂ f m n (In ) ⊂ A ⊂ X \Ω( f ) ∪ {O f (x), O f (y)}. Therefore for each n ≥ 0, f m n (In ) is an arcwise connected set of X meeting Ω( f ) only at O f (x), O f (y). Therefore f m n (In )  U . Thus for each n ≥ 0, there exists z n ∈ f m n (In ) ∩ ∂U ⊂ X \Ω( f ). As ∂U is finite there exists z ∈ ∂U such that z n = z, for infinitely many n. Hence f −m n (z) ∩ In = ∅ for infinitely many n ≥ 0. Since z ∈ ∂U ⊂ X \P( f ), so by Theorem 1, α f (z) is a minimal set containing y. Thus z ∈ X ∞ and α f (z) = O f (y). Recall that for each n ≥ 0, z n ∈ A and hence z ∈ A. Then for some p > 0, we have f p (z) = x, therefore any negative orbit of z can be completed into a negative orbit of x hence sα f (z) ⊂ sα f (x). By Theorem 1, we have sα f (z) = α f (z) = O f (y) and so y ∈ sα f (x). Corollary 4 Let X be a regular curve and f a monotone self mapping of X . Then for every x ∈ X , sα f (x) is a closed set. Proof The proof follows from Theorem 6 (since Ω( f ) and α f (x) are closed). The following proposition improves Theorem 3. Proposition 3 Let X be a regular curve and f a monotone self mapping of X . Then Ω( f ) = {x ∈ X : x ∈ sα f (x)}. Proof Let x ∈ X such that x ∈ sα f (x). By Corollary 3, sα f (x) is a union of minimal sets, thus x ∈ Ω( f ). Conversely, let x ∈ Ω( f ). Then by Theorem 2, x ∈ ω f (x). Thus x has a negative orbit (xn )n≥0 ⊂ ω f (x). Therefore by minimality of ω f (x) (cf. Theorem 1), we have α f ((xn )n≥0 ) = ω f (x). Hence x ∈ α f ((xn )n≥0 ) ⊂ sα f (x). The proof is complete.

6.2 Further Results on Special α-limit Sets Theorem 7 (Continuity of special α-limit maps) Let X be a regular curve and f a monotone self mapping of X . Then the special α-limit map sα f is continuous everywhere except may be at the periodic points of f . Proof Let x ∈ X ∞ \P( f ) and (xn )n≥0 be a sequence of X ∞ converging to x. By Theorem 5, the sequence α f (xn )n≥0 converges to α f (x). Let F be any limit point

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of sα f (xn )n≥0 with respect to the Hausdorff metric. Clearly F ⊂ α f (x), moreover by Corollary 4, F is an invariant closed subset of X . By Theorem 1, (2), α f (x) is a minimal set, therefore F = α f (x). It turn out that α f (x) is the unique limit point of the sequence sα f (xn )n≥0 . We conclude that the sequence sα f (xn )n≥0 converges to α f (x) (since 2 X is compact). Again by Theorem 1, (2), α f (x) = sα f (x) and the result follows. Note that it may happened that the maps ω f , α f and sα f (x) are discontinuous at some periodic point: For example consider the map f : X → X given in [16, Example 5.10]. Example 2 ([16, Example 5.10]) There exists a monotone map f on an infinite star X centered at a point z 0 ∈ R2 and beam In , n ≥ 0 with endpoints z 0 and z n satisfying the following properties. (i) For any x ∈ X \{z n : n ≥ 0} we have ω f (x) = {z 0 } and α f (x) = {z n x }, where n x ≥ 1 such that x ∈ In x . (ii) For any n ≥ 1 we have α f (z n ) = ω f (z n ) = {z n }. (iii) ω f (z 0 ) = {z 0 }, α f (z 0 ) = X . We have ω f is discontinuous at z n , for every n ≥ 0 and α f (resp. sα f ) is discontinuous at z 0 . Denote by SA( f ) (resp. A( f )) the special α-limit set (resp. the α-limit set) of a map f ; that is SA( f ) (resp. A( f )) is the union of all special α-limit sets (resp. all α-limit sets) of f . In [19, Corollary 2.2], it was shown that R( f ) ⊂ SA( f ) holds for general dynamical system (X, f ). For continuous map f on graphs, it was shown that SA( f ) ⊂ R( f ) ⊂ Λ( f ) (see [39], cf. [10, 22]) and that there are continuous maps f on dendrites with SA( f )  R( f ) (cf. [40]). In the following theorem, we extend the above results to regular curves when we restricted to monotone maps. Theorem 8 Let X be a regular curve and f a monotone self mapping of X , the following hold: (i) SA( f ) = R( f ). (ii) SA( f ) and A( f ) are closed. (iii) If P( f ) = ∅, then SA( f ) = R( f ) = A( f ). Proof Recall that by Theorem 2, Ω( f ) = R( f ). Assertion (i) follows immediately from Corollary 3 and Proposition 3. (iii): Assume now that P( f ) = ∅. Then by Corollary 3, sα f (x) = α f (x) is a minimal set, for any x ∈ X ∞ , so assertion (iii) follows. (ii): By (i), Ω( f ) = R( f ) = SA( f ) and so SA( f ) is closed. Let us show that A( f ) is closed. Let (xn )n≥0 be a sequence of A( f ) converging to some x ∈ X . We can assume that x ∈ / Ω( f ) since otherwise x ∈ Ω( f ) ⊂ A( f ) and we are done. For / A( f ), then by each n ≥ 0, let bn ∈ X ∞ such that xn ∈ α f (bn ). Suppose that x ∈ passing to a subsequence if needed, one can assume that the sequence (bn )n≥0 is infinite (since otherwise x ∈ A( f )).

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Since x ∈ / Ω( f ), so for n large enough, xn ∈ / Ω( f ) and therefore α f (bn ) is not a minimal set. Therefore by Theorem 1, bn ∈ P( f ). So by passing to a subsequence if needed, the family (O f (bn ))n≥0 is pairwise disjoint. Let U be an open neighborhood of x with finite boundary such that U ∩ Ω( f ) = ∅. As xn ∈ α f (bn ) and the sequence (xn )n≥0 converges to x, there exists N ≥ 0 such that for any n ≥ N , we can find sn ≥ 0 such that f −sn (bn ) ∩ U = ∅. Since bn ∈ P( f ), we have f −sn (bn )  U . Since f is monotone, it follows that f −sn (bn ) ∩ ∂U = ∅, for any n ≥ N . By Lemma 5, the family ( f −sn (bn ))n≥N is pairwise disjoint and as proven above f −sn (bn ) ∩ ∂U = ∅, for each n ≥ N , this will lead to a contradiction since ∂U is finite. Remark 1 If P( f ) = ∅, we have the inclusion SA( f ) ⊂ A( f ) which can be strict for monotone maps on regular curves. Indeed, consider the map f ∞ : S∞ → S∞ of Example 3. We have that SA( f ∞ ) = R( f ∞ ) = P( f ∞ )  A( f ∞ ) = S∞ . Theorem 9 Let X be a regular curve and f a monotone self mapping of X . Then for any x ∈ P( f ), sα f (x) is either a finite union of periodic orbits or an infinite sequence of periodic orbits converging to O f (x). Proof Let x ∈ P( f ). By Corollary 3 and Lemma 7, sα f (x) is a union of periodic orbits. Assume that sα f (x) is infinite and that there is an accumulation point a of sα f (x) such that a ∈ / O f (x). We can find an infinite sequence (an )n≥0 of sα f (x) converging to a. By Theorem 4, (O f (an ))n≥0 converges to a minimal set L. As an ∈ P( f ), then a ∈ L and hence L = O f (a). Then for i large enough, say i ≥ p,

d O f (ai ), O f (x) ≥

  d O f (x), O f (a) 2

.

Since Ω( f )  X (Lemma 9), so by Lemma 3, there is an open set O such that card(∂ O) = k and

d O f (x), O f (a) Mesh(O\Ω( f )) < . 2 As done in the Proof of Theorem 6, the set   A= f −n (x) {O f (ai ), p ≤ i ≤ k + p} n≥0

is f -invariant and has a finite number of arc connected components. For each p ≤ i ≤ k + p, let Ci be the arc connected component of A containing ai . Since ai ∈ α f (x), for each p ≤ i ≤ k + p, we can find an increasing sequence of integers (m n,i )n≥0 and a sequence (xn,i )n≥0 which converges to ai , where xn,i ∈ f −m n,i (x). By the same arguments as in the Proof of Theorem 6, we can assume that xn,i ∈ Ci for infinitely many n ≥ 0. Now as the ai are pairwise distinct and the Ci are locally arcwise connected, p ≤ i ≤ k + p, so for n large enough, we may find a sequence

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of pairwise disjoint arcs (In,i )n≥0 in Ci joining xn,i and ai which converges to {ai }. Choose N large enough and set Ii = I N ,i , p ≤ i ≤ k + p. So (Ii ) p≤i≤k+ p is a family of k + 1 pairwise disjoint arcs in A joining ai and x N ,i ∈ f −m N ,i (x). Set η := min{d(Ii , I j ) : p ≤ i < j ≤ k + p} > 0 and m > max{m N ,i : p ≤ i ≤ k + p}. Since A is f -invariant, f m (Ii ) ⊂ A. Moreover f m (Ii ) is a subcontinuum of X which meets the orbits of x and ai . As A ∩ Ω( f ) = O f (x) ∪ {O f (ai ) : p ≤ i ≤ k + p}, then we can find an arc J ⊂ f m (Ii ) joining x  ∈ O f (x) and bi ∈ O f (ai ) such that J \{x  , bi } ⊂ A\Ω( f ). Obviously J \{x  , bi } is connected and diam(J \{x  , bi }) ≥ d ( O f (x), O f (a)) . Therefore J  O and J ∩ O = ∅. Thus J ∩ ∂ O = ∅ and therefore 2 f m (Ii ) ∩ ∂ O = ∅, for each p ≤ i ≤ k + p. It follows that there exists z m ∈ ∂ O such that z m ∈ f m (Ii ) ∩ f m (I j ), for some p ≤ i < j ≤ k + p. Hence diam( f −m (z m )) > η. As ∂ O is finite, there exists z ∈ ∂ O such that z = z m for infinitely many m, thus lim sup diam( f −s (z)) > 0. This implies that the family ( f −n (z))n≥0 is not pairs→+∞

wise disjoint (otherwise it will be a null family), and hence for some q1 < q2 , we have f −q1 (z) ∩ f −q2 (z) = ∅. Thus z = f q2 −q1 (z) and so z ∈ P( f ). As z ∈ ∂ O and P( f ) ⊂ Ω( f ) ⊂ O, then z ∈ X \P( f ). A contradiction. We conclude that any accumulation point of sα f (x) is a point of O f (x). It turns out that sα f (x) is a compact space having a finite set of accumulation points, hence it is countable. The following corollary is in contrast to the properties of ω-limit sets. Corollary 5 Let X be a regular curve and f a monotone self mapping of X and x ∈ X ∞ . If sα f (x) contains an isolated point, then sα f (x) ⊂ P( f ). Proof If x ∈ / P( f ), then sα f (x) is a minimal set and hence it is a periodic orbit since it contains z which is an isolated point in it. If x ∈ P( f ), then from Lemma 7, we have sα f (x) ⊂ P( f ). Corollary 6 A countable sα-limit set for a monotone map on a regular curve is a union of periodic orbits. Proof Let x ∈ X ∞ such that sα f (x) is countable. We may that sα f (x) is not minimal; otherwise sα f (x) will be a finite minimal set and we are done. Therefore by Theorem 1, (2), x ∈ P( f ) and by Theorem 9, the result follows. Remark 2 Notice that Corollaries 5 and 6 extend those of [21, Theorem 20 and Corollary 21] to monotone maps on regular curves.

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Remark 3 (1) A special α-limit set can be totally periodic and infinite for monotone map on a regular curve; for example, in Example 2, we have sα f (z 0 ) is infinite and composed of fixed points. This is in contrast to some properties of ω-limit sets (cf. [16, Theorem 2.4]). (2) In [5], it was constructed a continuous self-mapping (not monotone) of a dendrite having totally periodic ω-limit set with unbounded periods. In Example 3 below, we construct analogously, a monotone map on a dendrite having a totally periodic special α-limit set with unbounded periods. Example 3 The idea is to slightly change the map f in Example 2 so that sα f (z 0 ) is infinite and composed of periodic orbits with unbounded period. Let z 0 some point of the plan R2 , N ≥ 1 and denote by S N , the N -star centered at z 0 . For N = 1, we define f 1 : S1 −→ S1 identified as the map f 1 = g : [0, 1] −→ [0, 1] : x −→ max{0, 2x − 1}, where z 0 and S1 play the role of 0 and [0, 1], respectively. Let now N −1 N ≥ 1, S N = Ik,N , each Ik,N is an arc of R2 , where z 0 is one of its endpoints, k=0

we denote by z k,N the other endpoint of Ik,N distinct from z 0 . Let h N : S N −→ S N be the homeomorphism of S N defined as h N (Ik,N ) = Ik+1,N and h N (I N −1,N ) = I0,N in an affine way. We let f N = h N ◦ f 1 . Clearly f N : S N −→ S N is monotone and continuous. Observe that End(S N ) = {z k,N : 0 ≤ k ≤ N − 1} is a periodic orbit of ≤ k ≤ N − 1} and α f N (z 0 ) = S N . We may pick period N , sα f N (z 0 ) = {z 0 , z k,N : 0 S N is an infinite star centered at z 0 . Define the (S N ) N ≥1 in such a way that S∞ = n≥1

map f ∞ : S∞ → S∞ given by its restriction ( f ∞ )|SN = f N . Clearly f ∞ is monotone and continuous. Moreover α f∞ (z 0 ) = S∞ and sα f∞ (z 0 ) = {z 0 , z k,N : N ≥ 1, 0 ≤ k ≤ N − 1} is totally periodic which contains periodic orbits of period N , for any N ≥ 1 (Figs. 2 and 3). Fig. 2 The map g

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Fig. 3 The map f ∞

Acknowledgements This work was supported by the research unit: “Dynamical systems and their applications”, (UR17ES21), Ministry of Higher Education and Scientific Research, Faculty of Science of Bizerte, Bizerte, Tunisia.

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Asymptotic Stability, Bifurcation Analysis and Chaos Control in a Discrete Evolutionary Ricker Population Model with Immigration Karima Mokni and Mohamed Ch-Chaoui

Abstract This research deals with the derivation and dynamical analysis of a discrete-time evolutionary Ricker population model. The model is built using Evolutionary Game Theory and takes into consideration the effect of immigration. The positive fixed point’s existence and local asymptotic stability are examined. Further, it is shown that the evolutionary model experiences Neimark–Sacker bifurcation (NSB) and period doubling bifurcation (PDB) in a small neighborhood of the positive fixed point under certain conditions. To make the chaotic behavior predictable and stable, three different chaos control strategies are applied. Detailed numerical simulations are carried out to not only verify our theoretical results but also exhibit the rich dynamics of the derived system. Keywords Evolutionary game theory · Asymptotic stability · Bifurcation analysis · Chaos control

1 Introduction Discrete dynamical systems using difference equations are largely used to produce the dynamics of population models [7, 14, 23]. The dynamical behavior is characterized by rich and complex dynamics, including bifurcations and chaos, see e.g. [14, 18, 19, 25, 31, 32]. Controlling chaos constitutes a challenging topic in studying dynamical systems [1, 17, 24, 26, 28, 29]. The behavior of species interaction in such populations shows rich and complex dynamics when evolution is taken into account [4, 11]. The derivation of Darwinian models using Evolutionary Game Theory (EGT) methodology means that the model’s parameters are assumed to depend K. Mokni (B) · M. Ch-Chaoui MRI Laboratory, Faculté Polydisciplinaire Khouribga, Sultan My Slimane University, BP 145 Khouribga principale, 25000 Khouribga, Morocco e-mail: [email protected] M. Ch-Chaoui e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_17

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on some phenotypic traits subject to Darwinian evolution. For a single species population, we consider the following difference equation [13] x(t + 1) = x(t)r (x(t)).

(1)

In Eq. (1), x(t) models the population size or the density at season t, and r is the fitness function which models the change per capita in population density from season t to the next season t + 1. Following EGT approach, the fitness r will depend on a phenotypic trait of an individual (labeled v) and of the main trait of population labeled latter by u. Thus (1) will formulated on two equations, one describing the populations dynamics x(t) and the other one describing the dynamics on the mean trait of the population u(t) as follows:   x(t + 1) = x(t)r (x(t), v, u(t))

v=u(t)

,

  u(t + 1) = u(t) + σ + ∂v (ln r (x, v, u(t))) 

(2)

2

  where ∂v (ln r (x, v, u(t))) 

v=u(t)

(3) v=u(t)

is the fitness gradient with respect to the trait v

and the constant of proportionality σ 2 models the speed of evolution for the trait (strategy). In the recent work [22], some global stability results are established for a special class of discrete time evolutionary models for both single species and multi-species dynamics, that are derived according to the evolutionary game theory methodology. The key of the analysis is the decoupling of the mean trait dynamics from the population dynamics. In [4] the following evolutionary Ricker model is derived  x(t + 1) = b0 x(t) exp(−x(t)) exp

 −u 2 (t) , 2

u(t + 1) = −c1 σ 2 x(t) + (1 − σ 2 )u(t).

(4) (5)

In [4], the local dynamic is investigated, with the focus on how evolution select against or promote complex dynamics. The author shows that the answer relies on the speed of evolution and how the intra-specific competition coefficient c depends on the evolving trait. Motivated by the previous cited works, we extend the model derived in [4], by adding the effect of immigration. The novelty of this study, is a rigorous bifurcation analysis by using bifurcation theory and center manifold theory [27]. In particular, we prove the existence of the period doubling bifurcation (PDB) and a Neimark–Sacker bifurcation (NSB) in a small neighborhood of the positive fixed point. Moreover, we employ three different methods to control chaos and stabilize chaotic orbits. The reader interested in such evolutionary discrete time models is

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addressed to, among others [5–9]. To sum up, here’s how the document is structured: In Sect. 2, the evolutionary discrete Ricker model with immigration is presented. The existence and stability of the positive fixed point and the existence of bifurcations are discussed in Sect. 3. The bifurcation analysis is studied through NSB and PDB in Sect. 4. In Sect. 5 the chaos is controlled. The evolutionary discrete Ricker model is supported with the help of numerical simulations in Sect. 6. Finally, Sect. 7 makes the conclusion of this study.

2 An Evolutionary Discrete Model Thorough this study, the population dynamic is represented by the Ricker model, and assumed to have immigration proportional to their density, and we write x(t + 1) = b exp(−cx(t))x(t) + h x(t) = r (x(t))x(t),

(6)

where the two nonnegative parameters b and c stand for per capita fertility rate, and a competition coefficient respectively. The coefficient h is related to the effect of immigration 0 ≤ h < 1 [3]. If h = 0, Eq. (6) is the famous Ricker map, which has been thoroughly studied in the literature e.g. [2, 10, 13–15, 20, 21]. The density dependant fertility rate is r (x) = b exp(−cx) + h.

(7)

Now, in order to derive the corresponding evolutionary Ricker model, r becomes a function of the phenotypic trait of an individual, (labeled v), and the mean trait of the population, (labeled u). Letting r = r (x, v, u), the single-species model is then given by the coupled population Eq. (2) and the trait (3). To specify the evolutionary model, we follow the same assumption made in [4, 30]. That is, b and c are functions of a phenotypic trait of the individual, denoted by v, that is subject to evolutionary change over time. We assume that the birth rate b is a function of v alone, on the other hand, the competition coefficient c is assumed dependent on the individual’s trait v and that of other individuals with whom it competes, as represented by the mean trait u. Thus we assume b = b(v), c = c(v, u).

(8)

The density dependent birth rate (7) is then r (x, v, u) = b(v) exp(−c(v, u)x) + h.

(9)

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To simplify the model, we choose the reference point and scale as vm = 0 and 2 w = 1. That is b(v) = b0 exp(− v2 ). The competition coefficient c is a function of the difference v − u [4], that is (c = c(v − u)) where the function is continuously differentiable for all values of z = v − u which means the competition that an individual experiences depends on how different it is from other individual of the population (represented by the mean trait u). (10) c(v − u) = c0 exp(−c1 (v − u)), c1 > 0. Note that function c(z) is continuously differentiable for all value z and Eq. (10) models a hierarchical situation in which individuals with larger traits feel less competitive pressure [4]. We take c(0) = 1, then the evolutionary discrete model (2)–(3) becomes   u(t)2 exp(−c(u(t), u(t))x(t))x(t) + h x(t), x(t + 1) = b0 exp − 2

(11)

exp(−c(u(t),u(t))x(t))∂v c(v,u(t)) u(t + 1) = u(t) + σ 2 −u(t)b(u(t)) exp(−c(u(t),u(t))x(t))−b(u(t))x |v=u(t) . b(u(t)) exp(−c(u(t),u(t))x(t))+h

(12) Then, the system (11)–(12) becomes   u(t)2 exp(−x(t))x(t) + h x(t), x(t + 1) = b0 exp − 2   2 u(t + 1) = u(t) + σ 2 b0 exp − u(t) exp(−x(t)) 2

 −u(t)+c  1 x(t) . 2 b0 exp − u(t) exp(−x(t))+h 2

(13) (14)

3 Local Dynamics In this section, we study the existence and stability properties of the evolutionary Ricker equations (13)–(14) using b0 as a bifurcation parameter. The equations for the fixed point pair are  2 u exp(−x) + h, 1 = b0 exp − 2

(15)

0 = −u + c1 x.

(16)

If b0 < 1 − h, then from the first Eq. (15) there is no positive fixed point (x, u). However, if b0 > 1 − h then there exists a unique positive fixed point obtained from (15) and (16) which are simplified to

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

 2 2 c x exp(−x) + h, 1 = b0 exp − 1 2 and

367

u = c1 x,

b0 c12 x 2 + x = ln , 2 1−h

(17)

(18)

gives the formulas for positive fixed points: 

 x(b0 ), u(b0 ) =

⎧ ⎪ ⎪ ⎪ ⎨ ln



b0 ,0 , 1−h     b0 ⎪ −1+ 1+2c12 ln 1−h

⎪ ⎪ ⎩

c12

,

   b0 −1+ 1+2c12 ln 1−h

if c1 = 0, (19)

, if c1 = 0.

c1

The Jacobian matrix of (13)–(14) is  J (x, u) =

 ,

(20)

 2 u exp(−x)(1 − x) + h, j11 = b0 exp − 2

where

j12

j21 =

j11 j12 j21 j22

 2 u exp(−x)xu, = −b0 exp − 2

   2  2 σ 2 b0 exp − u2 exp(−x) c1 (b0 exp − u2 exp(−x) + h) − (c1 x − u)h 

 b0 exp

j22 = 1 − σ 2

2 − u2



exp(−x) + h

2

   2  2 b0 exp − u2 exp(−x) b0 exp − u2 exp(−x) + h + u(c1 x − u) 2

(b0 exp(− u2 ) exp(−x) + h)2

,

.

The Jacobian matrix (20) evaluated at the positive fixed point becomes, when (17) are used,   1 − x(b0 )(1 − h) −c1 (1 − h)x 2 (b0 ) J (x(b0 ), u(b0 )) = , (21) 1 − σ 2 (1 − h) σ 2 c1 (1 − h) when (18) is utilized the matrix (21) becomes  J (x(b0 ), u(b0 )) =

b0

 − x(b0 ) 1 − x(b0 )(1 − h) − c21 (1 − h) ln 1−h . 1 − σ 2 (1 − h) σ 2 c1 (1 − h)

(22)

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For the case c1 = 0 the eigenvalues of this Jacobian are 

b0 λ1 = 1 − ln 1−h

 (1 − h),

λ2 = 1 − σ 2 (1 − h).

(23)

2 , There exists positive fixed points for Theorem 1 Assume c1 = 0, and σ 2 < 1−h and only for b0 > 1 − h They are locally asymptotically stable if 1 − h < b0 < 2 2 2 (1 − h)e 1−h , and unstable if b0 > (1 − h)e 1−h . When b0 = (1 − h)e 1−h the Jacobian has eigenvalue value −1.

Consider now the case c1 = 0. To study the eigenvalues of the Jacobian we employ the trace and determinant criteria which imply both eigenvalues have magnitude less than 1 and the positive fixed point is locally asymptotically stable if and only if the three inequalities are verified:[13] tr J (x, u) < 1 + det J (x, u),

(24)

− 1 − det J (x, u) < tr J (x, u),

(25)

det J (x, u) < 1.

(26)

If inequality (24) or (25) become equalities, then the Jacobian has an eigenvalue equal to +1 or −1 respectively. If inequality (26) becomes an equality, then the Jacobian has a complex eigenvalue whose absolute value equals 1. The characteristic equation of Jacobian matrix (22) can be written as λ2 − tr J (x(b0 ), u(b0 )) + det(J (x(b0 ), u(b0 ))) = 0,

(27)

tr J (x(b0 ), u(b0 ))) = 2 − x(b0 )(1 − h) − σ 2 (1 − h),

(28)

where

and     b0 − x(b0 ) . det J (x(b0 ), u(b0 ) = (1 − x(b0 )(1 − h))(1 − σ 2 (1 − h)) + 2σ 2 (1 − h)2 ln 1−h

(29) The discriminant of (27) is Δ = tr J (x(b0 ), u(b0 ))2 − 4 det J (x(b0 ), u(b0 )),

(30)

Lemma 1 Assume c1 = 0 in the evolutionary Ricker model (13)–(14). The inequality (24) is true for all σ 2 and b0 > 1 − h. Proof Using (28) and (29), it is easy to show that inequality (24) can be reduced to

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

 x(b0 ) < 2 ln

b0 1−h

369

 ,

(31)

replacing the value of x(b0 ) in (31), one gets 

 1 + 2c12 ln

b0 1−h



 < 1 + 2c12 ln

b0 1−h

 ,

(32)

which is true and completes the proof. Next, we turn attention to inequality (25). Lemma 2 Assume c1 = 0 and σ 2 < • If σ2
(1 − h) exp 1−h such that inequality (25) holds for b0 satisfying 1 < b0 < κ. Inequality (25) is reversed if b0 is greater than but near κ The Jacobian J (x(κ), u(κ)) has eigenvalue −1. • If 2 σ2 > , (34) (1 − h) + 8c12 then inequality (25) holds for all b0 > 1 − h. Proof Using (28) and (29) together with the fixed point formulas (19), one can re arrange inequality (25) to the inequality  (1 − h)(2 + σ 2 (1 − h)) 1 + 2c12 y < (1 − h)(2 + σ 2 (1 − h)) + 2σ 2 c12 (1 − h)2 y + 2c12 (2 − σ 2 (1 − h)), (35) where we have defined y = ln

b0 . 1−h

Since both sides of (35) are positive, we can retain the inequality by squaring both sides, after which we re-arrange the result into an equivalent inequality p1 (y) > 0 where p1 (y) is the quadratic polynomial p1 (y) = Ay 2 − By + C. where

(36)

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K. Mokni and M. Ch-Chaoui

A = σ 4 c14 (1 − h)4 , B = 2(c12 (1 − h)2 (2 − σ 2 (1 − h))(2 + σ 2 ((1 − h) − 4c12 ))), and C = (2c12 (2 − σ 2 (1 − h))(2 + σ 2 (1 − h))(1 − h) + c12 (2 − σ 2 (1 − h))). The quadratic p1 (y) has a global minimum p1 (yc ) =

  1 2 2 2 2 2 2 , (2 − σ (1 − h))(2 + σ (1 − h)) ((1 − h) + 8c ) σ − 1 8σ 4 (1 − h) + 8c12

(37)

attained at the critical point yc = If σ 2
0. It follows that the two roots are both negative or both positive, depending on whether yc < 0 or yc > 0 respectively. Clearly yc > 0 if (1 − h) − 4c12 > 0 . Suppose, on the other hand, that (1 − h) − 4c12 < 0 , Then yc > 0 if and only if 2 2 −1 > σ 2 < 2(4c12 − (1 − h))−1 which holds by σ 2 < (1−h)+8c 2 since (4c1 − (1 − h)) 1

((1 − h) + 8c12 )−1 Thus, in this case, p1 (y) has two positive roots. If we denote the smaller by y2 then p1 (y) > 0 for 0 < y < y2 . p1 (y) changes sign as y increases through y2 . Since p1 (y2 ) = 0 inequality (25) becomes an equality which means the 2 . One way Jacobian has an eigenvalue of −1. Finally we need to show that y2 > 1−h 2 2 to do this is to show p1 ( 1−h ) = 2c14 (2 + σ 2 (1 − h))2 > 0 and p1 ( 1−h ) = c12 (σ 2 (1 − h) + 2)(σ 2 (1 − h) + 4c12 σ 2 − 2) < − 21 c12 (σ 2 (1 − h) + 2)(2 − σ 2 (1 − h)) < 0 b2 the real 1−h is equal to exp(y2 ) where y2 is the smaller of the positive roots of p1 (y). Yields √ (2−σ 2 (1−h)(2+σ 2 (1−h)−4σ 2 c12 )−(σ 2 (1−h)+2) (2−σ 2 (1−h))(2−σ 2 (1−h)−σ 2 8c12 ) . (39) y2 = 2 4 2 4σ c (1−h) 1

Therefore  κ = (1 − h) exp

  (2−σ 2 (1−h)(2+σ 2 (1−h)−4σ 2 c12 )−(σ 2 (1−h)+2) (2−σ 2 (1−h))(2−σ 2 (1−h)−σ 2 8c12 )

,

4σ 4 c12 (1−h)2

(40) 

and κ > (1 − h) exp

2 1−h

 .

(41)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

371

2 Lemma 3 Assume c1 = 0 and σ 2 < 1−h in the evolutionary Eqs. (13)–(14). There 1 exists a real γ > (1 − h) exp( 2(1−h) ) such that inequality (26) holds for 1 − h < b0 < γ . The Jacobian J (x(γ ), u(γ )) has a complex eigenvalue of absolute value 1.

Proof Inequality (26) can be re-arranged as  σ

2



b0 2 ln 1−h



 (1 − h) − 1 < x(b0 )(σ 2 (1 − h) + 1),

(42)

1 1 which is true for 1 < b0 < (1 − h) exp( 2(1−h) ). For b0 > (1 − h) exp( 2(1−h) ) we use the formula (19) for x(b0 ) and re-arrange the inequality as

1 + (2y(1 − h) − 1)

c12 σ 2 < σ 2 (1 − h) + 1

 1 + 2c12 y,

(43)

b0 1 where y = ln 1−h > 2(1−h) . Since both sides are positive, we can square them and re-arrange the inequality to obtain an equivalent inequality





0< p2 (y)=−4c12 σ 4 (1−h)2 y 2 + 2(1−σ 2 (1−h))(σ 2 (1−h)+1)y+4(1−h)c12 σ 4

+σ 2 (2(σ 2 (1−h)+1)−c12 σ 2 ).

(44) 1 ) > 0 ,this quadratic polynomial has a unique positive root y3 > 21 Since p2 ( 2(1−h) and p2 (y) > 0, for 21 < y < y3 . Since p2 (y3 ) = 0, then inequality (26) becomes an equality, which implies the Jacobian has a complex eigenvalue of absolute value 1. Now the real γ is equal to (1 − h) exp(y3 ) where y3 is the unique positive root greater than 21 of p2 (y) . A formula for y3 is y3 =

√

1−σ 4 (1−h)2 +2(1−h)c12 σ 4 +(σ 2 (1−h)+1) (σ 2 (1−h)+1)2 +4(1−h)c12 σ 4 . 4c12 σ 4 (1−h)2

(45)

Hence  γ = (1 − h) exp

  1−σ 4 (1−h)2 +2(1−h)c12 σ 4 +(σ 2 (1−h)+1) (σ 2 (1−h)+1)2 +4(1−h)c12 σ 4 4c12 σ 4 (1−h)2

> exp 21 .

(46) The three Trace-Determinant stability inequalities (24)–(25)–(26) for local stability, together with the three Lemmas 1,2 and 3, yields the following theorem. 2 Theorem 2 Assume c1 = 0 and σ 2 < 1−h in the evolutionary discrete model (13)– (14), and let κ and γ are defined by (40)–(46).

• Assume σ2
, (48) (1 − h) + 8c12 then the positive fixed point (19) is locally asymptotically stable for 1 − h < b0 < γ and unstable for b0 greater than, but near γ . The Jacobian has a complex eigenvalue of absolute value 1 when b0 = γ .

4 Bifurcations Analysis Bifurcation is a nonlinear phenomenon in a dynamical system that implies a qualitative change in system attributes when a system parameter crosses a boundary. In this section, Neimark–Sacker bifurcation (NSB) and period doubling bifurcation (PDB) are studied rigorously. NSB is the analogous of the Hopf bifurcation that occurs in continuous dynamical systems. NSB is the primary tool for determining if a map has quasi periodic orbits. It introduces closed invariant curves into the system, which shows more complex behaviors. A PDB happens when a new limit cycle emerges from an existing one, with the new limit cycle’s period being twice that of the existing one. Sometimes periodic doubling bifurcation is called “flip bifurcation”.

4.1 Existence of Bifurcations About the Positive Fixed Point of the Model Based on Theorem 2, the existence of bifurcations about the positive fixed point (x(b0 ), u(b0 )) can be summarized as follows: • From Theorem 2, we see that if b0 = κ (κ is defined in (40) )holds, then one of the eigenvalues is −1 . So a period-doubling bifurcation exists by the variation of parameter in a small neighborhood of b0 = κ. More precisely we can also represent the parameters satisfying b0 = κ as Pd = {(b0 , h, c1 , σ 2 ) : tr J (x(b0 ), u(b0 ))2 > 4 det J (x(b0 ), u(b0 )), b0 = κ}, (49) • From Theorem 2, we see that if b0 = γ (γ is defined in (46) holds, then one of the eigenvalues of J are a pair of complex conjugate with modulus 1. So a Neimark– Sacker bifurcation exists by the variation of parameter in a small neighborhood of b0 = γ . More precisely we can also represent the parameters satisfying b0 = γ as

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

373

Ns = {(b0 , h, c1 , σ 2 ) : tr J (x(b0 ), u(b0 ))2 < 4 det J (x(b0 ), u(b0 )), b0 = γ }. (50)

4.2 Neimark–Sacker Bifurcation About (x(b0 ), u(b0 )) The roots of the characteristic Eq. (27) at (x(b0 ), u(b0 )) are a pair of complex conjugate numbers λ1 , λ2 given by λ1,2

 tr J (x(b0 ), u(b0 )) ± i 4 det J (x(b0 ), u(b0 )) − (tr J (x(b0 ), u(b0 ))2 , (51) = 2

with tr J (x(b0 ), u(b0 )) and det J (x(b0 ), u(b0 )) are given in (28) and (29) respectively. Now NSB occurs when one of the roots of the above equation are complex conjugates with unit modulus. If we vary b0 in the neighborhood of b0 = γ and keeping other parameters constant. Then (x(b0 ), u(b0 )) undergoes NSB. Taking a perturbation b0∗ where (b0∗  1) of the parameter b0 in the neighborhood of b0 = γ in the system (13)–(14), we have   2 + h x(t), x(t + 1) = (b0 + b0∗ )x(t) exp(−x(t)) exp − u(t) 2 (52a) 2

u(t + 1) = u(t) + σ 2 (b0 + b0∗ ) exp(− u(t) ) exp(−x(t)) 2

−u(t)+c1 x(t) . 2 (b0 +b0∗ ) exp(− u(t) 2 ) exp(−x(t))+h

(52b) Let v(t) = x(t) − x(b0 ) , w(t) = u(t) − u(b0 ), then from (52) we set v(t + 1) = (b0 +

b0∗ )(v(t)



(w(t) + u(b0 ))2 + x(b0 )) exp − 2



× e−(v(t)+x(b0 )) + h(v(t) + x(b0 )) − x(b0 ),  −

(w(t)+u(b0 ))2 2

(53)



w(t + 1) = w(t) + σ 2 (b0 + b0∗ )e −(w(t) + u(b0 )) + c1 (v(t) + x(b0 )) . × e−(v(t)+x(b0 )) (w(t)+u(b0 ))2 2 (b0 + b0∗ )e− ev(t)+x(b0 ) + h

(54)

Expanding the above in Taylor series at (v(t), w(t)) = (0, 0) considering the terms up to second order, we have

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K. Mokni and M. Ch-Chaoui

v(t + 1) = α1 v(t) + α2 w(t) + α12 v(t)w(t) + α11 v 2 (t) + α22 w 2 (t)   2 + O (| v(t) | + | w(t) |) , (55a) w(t + 1) = β1 v(t) + β2 w(t) + β12 v(t)w(t) + β11 v 2 (t) + β22 w 2 (t)   2 (55b) + O (| v(t) | + | w(t) |) , where α1 = f x (x(b0 ), u(b0 ), 0) = 1 − (1 − h)x(b0 ), α2 = f u (x(b0 ), u(b0 ), 0) = −c1 (1 − h)x 2 (b0 ), α12 = f xu (x(b0 ), u(b0 ), 0) = −(1 − h)c1 x(b0 )(1 + x(b0 )),   x(b0 ) , α11 = f x x (x(b0 ), u(b0 ), 0) = (1 − h) −1 + 2   x(b0 ) c12 x 3 (b0 ) + (1 − h), α22 = f uu (x(b0 ), u(b0 ), 0) = − 2 2 β1 = gx (x(b0 ), u(b0 ), 0) = σ 2 (1 − h)c1 , β2 = gu (x(b0 ), u(b0 ), 0) = 1 − σ 2 (1 − h), β12 = gxu (x(b0 ), u(b0 ), 0) = σ 2 h(1 − h)(1 − c12 x(b0 )), β11 = gx x (x(b0 ), u(b0 ), 0) = −σ 2 h(1 − h)c1 , β22 = guu (x(b0 ), u(b0 ), 0) = σ 2 h(1 − h)c1 x(b0 ). The characteristic equation associated with the linearization of the model (55) at (v(t), w(t)) = (0, 0) is λ2 − tr J (b0∗ )λ + det J (b0∗ ) = 0.

(56)

The roots of (56) are λ1,2 (b0∗ )

 tr J (b0∗ ) ± i 4 det J (b0∗ ) − (tr (b0∗ ))2 , = 2

with |λ1,2 (b0∗ )| =



det J (b0∗ ),

d|λ1,2 | |b0∗ =0 = 0 db0∗

(57)

(58)

Additionally, we required that when b0∗ = 0 , λm 1,2  = 1, m = 1, 2, 3, 4. This is equivalent to tr J (0) = −2, −1, 1, 2. the invertible Let η = Re(λ1,2  ) and ξ = I m(λ1,2 ), and define     matrix 0 α2 v(t) X (t) P= , and using the transformation =P . Then η − α1 −ξ w(t) U (t) the system (55) reduces to the following form

Asymptotic Stability, Bifurcation Analysis and Chaos Control …



X (t + 1) U (t + 1)



 =

η −ξ −ξ η

where F(v(t), w(t)) =  G(v(t), w(t)) =

(η−α1 )α12 ξ α2





X (t) U (t)

α12 v(t)w(t) α2

−

β12 ξ

+



 +

375

 F(v(t), w(t)) , G(v(t), w(t))

α11 2 v (t) α2

+

(59)

α22 2 w (t), α2

  1 )α11 v(t)w(t) + (η−α − ξ α2

β11 ξ

(60)  v 2 (t)+

 (η − α1 )α22 β22 w 2 (t). − ξ α2 ξ

Writing v(t) = α2 X (t), and w(t) = (η − α1 )X (t) − ξU (t), we obtain F(X (t), U (t)) =

α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 2 X (t)+ α2

α22 ξ 2 2 −(ξ α12 α2 + 2α22 ξ(η − α1 )) X (t)U (t) + U (t), α2 α2 G(X (t), U (t)) =  1 (α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 ) η−α − ξ α2 

β12 α2 (η−α1 )+β11 α22 +β22 ξ 2 ξ

 X 2 (t)−

 η − α1 (α12 α2 + 2α22 (η − α1 )) − (β12 α2 + 2β22 (η − α1 )) X (t)U (t)+ α2   η − α1 α22 ξ − β22 ξ U 2 (t). α2

In order for (13)–(14) to undergo a NSB, the following discriminatory quantity must be nonzero, (i.e, L = 0 [27]),   2 1 (1 − 2λ)λ τ11 τ20 − | τ11 |2 − | τ02 |2 +(λτ21 ), L = − 1−λ 2

(61)

where τ02 =

1 [FX (t)X (t) − FU (t)U (t) − 2G X (t)U (t) + i(G X (t)X (t) − G U (t)U (t) − 2FX (t)U (t) )](0,0) , 8

τ11 =

1 [FX (t)X (t) + FU (t)U (t) + i(G X (t)X (t) + G U (t)U (t) )](0,0) , 4

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K. Mokni and M. Ch-Chaoui

τ20 =

1 [FX (t)X (t) − FU (t)U (t) + 2G X (t)U (t) + i(G X (t)X (t) − G U (t)U (t) − 2FX (t)U (t) )](0,0) , 8

τ21 =

1 [FX (t)X (t)X (t) + FX (t)U (t)U (t) + G X (t)X (t)U (t) + G U (t)U (t)U (t) + 16

i(G X (t)X (t)X (t) + G X (t)U (t)U (t) − FX (t)X (t)U (t) − FU (t)U (t)U (t) )](0,0) .  τ02 =

1 4

α12 α2 (η−α1 )+α11 α22 +α22 (η−α1 )2 α2

(ξβ12 α2 + 2β22 ξ(η − α1 )) ξ



 −

α22 ξ 2 α2

 +i

(α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 )

β12 α2 (η−α1 )+β11 α22 +β22 (η−α1 )2 ξ



τ11 =

i

1 2





η − α1 + ξ α2

  1 − α22 ξ η−α − β ξ + 22 α2

(ξ α12 α2 + 2α22 ξ(η − α1 )) α2

 ,

α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 α2

 +

α22 ξ 2 + α2

 η − α1 (α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 ) ξ α2

β12 α2 (η − α1 ) + β11 α22 + β22 (η − α1 )2 + ξ τ20 =

 1 + (ξ α12 α2 + 2α22 ξ(η − α1 )) η−α − ξ α2

1 4





  η − α1 + α22 ξ − β22 ξ , α2

α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 α2

 (ξ α12 α2 + 2α22 ξ(η − α1 ))

−

α22 ξ 2 − α2

 η − α1 (ξβ12 α2 + 2β22 ξ(η − α1 )) + − ξ α2 ξ

 1 i (α12 α2 (η − α1 ) + α11 α22 + α22 (η − α1 )2 ) η−α + ξ α2  1 α22 ξ 2 η−α − ξ α2



β22 ξ 2 ξ



 +

β12 α2 (η−α1 )+β11 α22 +β22 (η−α1 )2 ξ

(ξ α12 α2 +2α22 ξ(η−α1 )) α2

 ,

τ21 = 0. Based on the above analysis, we arrive to the following result on NSB.

 −

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

377

Theorem 3 If the condition (58) holds and L defined in (61) is nonzero, then the evolutionary Ricker model (13)–(14) undergoes a NSB at the positive fixed point (x(b0 ), u(b0 )) when b0∗ changes in the small neighborhood of b0 = γ and (b0 , h, c1 , σ 2 ) ∈ 50 Moreover, if L < 0 (L > 0) then an attracting (respectively repelling) invariant closed curve bifurcates from the fixed point (x(b0 ), u(b0 )) for b0 > γ (respectively, b0 < γ ).

4.3 Period-Doubling Bifurcation About (x(b0 ), u(b0 )) For the fixed point (x(b0 ), u(b0 )) of the system (13)–(14), if (b0 , h, c1 , σ 2 ) ∈ Pd where Pd is defined in (49), one of the eigenvalues of J (x(b0 ), u(b0 )) is −1 and the other is neither 1 nor −1. Therefore the system (13)–(14) admits a PDB at the positive fixed point (x(b0 ), u(b0 )) if b0 varies in the small neighborhood of b0 = κ and (b0 , h, c1 , σ 2 ) ∈ Pd . Giving a perturbation b0∗ (where b0∗  1) of the parameter b0 in the neighborhood of b0 = κ to the system (13)–(14) we have x(t + 1) = (b0 + b0∗ )x(t) exp(−x(t)) exp(− u(t + 1) = u(t) + σ 2 (b0 + b0∗ )e−

u(t)2 2

e−x(t)

u(t)2 ) + h x(t) = f (x(t), u(t), b0∗ ) 2 (62) −u(t)+c1 x(t)

= g(x(t), u(t), b0∗ ).

2

(b0 +b0∗ ) exp(− u(t) 2 ) exp(−x(t))+h

Let v(t) = x(t) − x(b0 ) , w(t) = u(t) − u(b0 ), the system (62)–(63) becomes v(t + 1) = (b0 + b0∗ )(v(t) + x(b0 ))e−

(w(t)+u(b0 ))2 2

w(t + 1) = w(t) + σ 2 (b0 + b0∗ )e−

(63)

exp(−(v(t) + x(b0 ))) + h(v(t) + x(b0 )) − x(b0 ),

(64) (w(t)+u(b0 ))2 2

e−(v(t)+x(b0 ))

−(w(t)+u(b0 ))+c1 (v(t)+x(b0 )) (b0 +b0∗ )e−

(w(t)+u(b0 ))2 2

ev(t)+x(b0 ) +h

.

(65) Expanding the above in Taylor series at (v(t), w(t), b0∗ ) = (0, 0) considering the terms up to second order, we have v(t + 1) = α1 v(t) + α2 w(t) + α12 v(t)w(t) + α11 v 2 (t) + α22 w 2 (t)+ α13 b0∗ v(t)

+

α23 b0∗ w(t)

+

w(t + 1) = β1 v(t) + β2 w(t) + β13 b0∗ v(t)

+

β23 b0∗ w(t)

+

+ α113 b0∗ v 2 (t) + α223 b0∗ w 2 (t), β12 v(t)w(t) + β11 v 2 (t) + β22 w 2 (t)+

(66a)

α123 b0∗ v(t)w(t)

β123 b0∗ v(t)w(t)

+

β113 b0∗ v 2 (t)

+

β223 b0∗ w 2 (t),

where α1 = f x (x(b0 ), u(b0 ), 0) = 1 − x(b0 )(1 − h), α2 = f u (x(b0 ), u(b0 ), 0) = −c1 (1 − h)x 2 (b0 ),

(66b)

378

K. Mokni and M. Ch-Chaoui

α12 = f xu (x(b0 ), u(b0 ), 0) = −(1 − h)c1 x(b0 )(1 + x(b0 )), α11

  x(b0 ) , = f x x (x(b0 ), u(b0 ), 0) = (1 − h) −1 + 2

 x(b0 ) c12 x 3 (b0 ) + (1 − h), = f uu (x(b0 ), u(b0 ), 0) = − 2 2 

α22

  u(b0 )2 α13 = f xb0∗ (x(b0 ), u(b0 ), 0) = exp − exp(−x(b0 ))(1 − x(b0 )) + h, 2 α23

  u(b0 )2 exp(−x(b0 ))c1 x 2 (b0 ), = f ub0∗ (x(b0 ), u(b0 ), 0) = − exp − 2

  u(b0 )2 exp(−x(b0 ))c1 x(b0 )(1 + x(b0 )), α123 = f xub0∗ (x(b0 ), u(b0 ), 0) = − exp − 2

α113

    u(b0 )2 x(b0 ) exp(−x(b0 )) −1 + , = f x xb0∗ (x(b0 ), u(b0 ), 0) = exp − 2 2

   c2 x 3 (b0 ) u(b0 )2 x(b0 ) α223 = f uub0∗ (x(b0 ), u(b0 ), 0) = exp − − ) exp(−x(b0 ) + 1 , 2 2 2

β1 = gx (x(b0 ), u(b0 ), 0) = σ 2 (1 − h)c1 , β2 = gu (x(b0 ), u(b0 ), 0) = 1 − σ 2 (1 − h), β12 = gxu (x(b0 ), u(b0 ), 0) = σ 2 h(1 − h)(1 − c12 x(b0 )), β11 = gx x (x(b0 ), u(b0 ), 0) = −σ 2 h(1 − h)c1 , β22 = guu (x(b0 ), u(b0 ), 0) = σ 2 h(1 − h)c1 x(b0 ), β13 = g

xb0∗

  2 u (b0 ) exp(−x(b0 ))σ 2 c1 h, (x(b0 ), u(b0 ), 0) = exp − 2

  2 u (b0 ) exp(−x(b0 )), β23 = gub0∗ (x(b0 ), u(b0 ), 0) = −σ 2 h exp − 2

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

379

β123 = gxub0∗ (x(b0 ), u(b0 ), 0)   u(b0 )2 2 = σ exp − exp(−x(b0 ))(2h 2 − h + c1 x 2 (b0 )(2 + h − 2h 2 )), 2   u(b0 )2 exp(−x(b0 ))c1 h(1 − 2h), β113 = gx xb0∗ (x(b0 ), u(b0 ), 0) = σ 2 exp − 2

β223 = guub0∗ (x(b0 ), u(b0 ), 0)   u(b0 )2 2 = σ exp − exp(−x(b0 ))c1 x(b0 )(3h − 4 + 2c1 x(b0 )(1 − h)). 2 The model can be written us        v(t + 1) α1 α2 v(t) f 1 (v(t), w(t), b0∗ ) . = + β1 β2 g1 (v(t), w(t), b0∗ ) w(t + 1) w(t) Where f 1 (v(t), w(t), b0∗ ) = α12 v(t)w(t) + α11 v 2 (t) + α22 w 2 (t) + α13 b0∗ v(t)+ α23 b0∗ w(t) + α113 b0∗ v 2 (t) + α223 b0∗ w 2 (t) + α123 b0∗ w(t)v(t), g1 (v(t), w(t), b0∗ ) = β12 v(t)w(t) + β11 v 2 (t) + β22 w 2 (t) + β13 b0∗ v(t)+ β23 b0∗ w(t) + β123 b0∗ v(t)w(t) + β113 b0∗ v 2 (t) + β223 b0∗ w 2 (t). Now, we consider the invertible matrix P associated to the real eigenvalues −1 and λ2   α2 α2 , (67) P= −1 − α1 λ2 − α1 

and use the translation

v(t) w(t)



 =P

 X (t) , U (t)

one gets  P where

X (t + 1) U (t + 1)



 =

α1 α2 β1 β2



 P

X (t) U (t)



 +

 f 1 (X (t), U (t), b0∗ ) , g1 (X (t), U (t), b0∗ )

380

K. Mokni and M. Ch-Chaoui

f 1 (X (t), U (t), b0∗ ) =



  − α12 α2 (1 + α1 ) + α11 α22 + α22 (1 + α1 )2 X 2 (t) + α113 α22 +

 α223 (1 + α1 ) − α123 α2 (1 + α1 ) X 2

2

 + α12 α2 (λ2 − α1 ) + α11 α22 +

(t)b0∗

   2 2 2 α22 (λ2 − α1 ) U (t) + α223 α2 + α223 (λ2 − α1 ) + α123 α2 (λ2 − α1 ) U 2 (t)b0∗ + 2

  α12 α2 (λ2 − α1 ) − α12 α2 (1 + α1 ) + 2α11 α22 − 2α22 (1 + α1 )(λ2 − α1 ) X (t)U (t)+   2α113 α22 + 2α223 (1 + α1 )(λ2 − α1 ) + α123 α2 (λ2 − α1 ) − α123 α2 (1 + α1 ) X (t)U (t)b0∗ +

    ∗ α13 α2 − α23 (1 + α1 ) X (t)b0 + α13 α2 + α23 (λ2 − α1 ) U (t)b0∗ , and g1 (X (t), U (t), b0∗ )

 =

 − β12 α2 (1 + α1 ) +



β11 α22

 β113 α22

+ β223 (1 + α1 ) − β123 α2 (1 + α1 ) X 2

2

+ β22 (1 + α1 )

(t)b0∗

2

X 2 (t)+

 + β12 α2 (λ2 − α1 )+

   β11 α22 + β22 (λ2 − α1 )2 U 2 (t) + β223 α22 + β223 (λ2 − α1 )2 + β123 α2 (λ2 − α1 ) U 2 (t)b0∗ +   β12 α2 (λ2 − α1 ) − β12 α2 (1 + α1 ) + 2β11 α22 − 2β22 (1 + α1 )(λ2 − α1 ) X (t)U (t)+   2β113 α22 + 2β223 (1 + α1 )(λ2 − α1 ) + β123 α2 (λ2 − α1 ) − β123 α2 (1 + α1 ) X (t)U (t)b0∗ +

    ∗ β13 α2 − β23 (1 + α1 ) X (t)b0 + β13 α2 + β23 (λ2 − α1 ) U (t)b0∗ . Therefore, one obtains the system 

X (t + 1) U (t + 1)



 =

−1 0 0 λ2



X (t) U (t)



+ P −1



 f 1 (X (t), U (t), b0∗ ) , g1 (X (t), U (t), b0∗ )

Asymptotic Stability, Bifurcation Analysis and Chaos Control …



where P 

Thus

X (t + 1) U (t + 1)



 =

−1

=

−1 0 0 λ2

1 −λ2 − α1+λ − λ21+1 2

1+α1 1+λ2



X (t) U (t)

 .

1 λ2 +1



 +

381

(68)

 F1 (X (t), U (t), b0∗ ) , G 1 (X (t), U (t), b0∗ )

(69)

where F1 (X (t), U (t), b0∗ ) =

 1 (λ2 − α1 )(−α12 α2 (1 + α1 ) + α11 α22 + α22 (1 + α1 )2 )− 1 + λ2

 (−β12 α2 (1 + α1 ) + β11 α22 + β22 (1 + α1 )2 ) X 2 (t) +

(1 + α1 )

2

− α123 α2 (1 + α1 )) − (β113 α22

 1 (λ2 − α1 )(α113 α22 + α223 1 + λ2

 + β223 (1 + α1 ) − β123 α2 (1 + α1 )) X 2 (t)b0∗ + 2

 1 (λ2 − α1 )(α12 α2 (λ2 − α1 ) + α11 α22 + α22 (λ2 − α1 )2 ) − (β12 α2 (λ2 − α1 ) + β11 α22 + 1 + λ2  β22 (λ2 − α1 )2 ) U 2 (t) +

 1 (λ2 − α1 )(α223 α22 + α223 (λ2 − α1 )2 + α123 α2 (λ2 − α1 ))− 1 + λ2

 (β223 α22 + β223 (λ2 − α1 )2 + β123 α2 (λ2 − α1 )) U 2 (t)b0∗ +

 1 (λ2 − α1 )(α12 α2 (λ2 − α1 ) 1 + λ2

−α12 α2 (1 + α1 ) + 2α11 α22 − 2α22 (1 + α1 )(λ2 − α1 )) − (β12 α2 (λ2 − α1 )  −β12 α2 (1 + α1 ) + 2β11 α22 − 2β22 (1 + α1 )(λ2 − α1 )) X (t)U (t) +

 1 (λ2 − α1 )(2α113 α22 + 1 + λ2

2α223 (1 + α1 )(λ2 − α1 ) + α123 α2 (λ2 − α1 ) − α123 α2 (1 + α1 )) − (2β113 α22 + 2β223 (1 + α1 )(λ2 − α1 )+  β123 α2 (λ2 − α1 ) − β123 α2 (1 + α1 )) X (t)U (t)b0∗ +  −α23 (1 + α1 )) − (β13 α2 − β23 (1 + α1 )) X (t)b0∗ +

 1 (λ2 − α1 )(α13 α2 1 + λ2

 1 (λ2 − α1 )(α13 α2 + α23 (λ2 − α1 ))− 1 + λ2

 (β13 α2 + β23 (λ2 − α1 )) U (t)b0∗ ,

382

K. Mokni and M. Ch-Chaoui

and G 1 (X (t), U (t), b0∗ ) =

 1 (1 + α1 )(−α12 α2 (1 + α1 ) + α11 α22 + α22 (1 + α1 )2 ) + (β11 α22 − 1 + λ2

 β12 α2 (1 + α1 ) + β22 (1 + α1 )2 ) X 2 (t) +

 1 (1 + α1 )(α113 α22 + α223 (1 + α1 )2 1 + λ2

 −α123 α2 (1 + α1 )) + (β113 α22 + β223 (1 + α1 )2 − β123 α2 (1 + α1 )) X 2 (t)b0∗ +

 1 (1 + α1 )(α12 α2 (λ2 − α1 ) + α11 α22 + α22 (λ2 − α1 )2 ) + (β12 α2 (λ2 − α1 )+ 1 + λ2  β11 α22 + β22 (λ2 − α1 )2 ) U 2 (t) +

 1 (1 + α1 )(α223 α22 + α223 (λ2 − α1 )2 + 1 + λ2

 α123 α2 (λ2 − α1 )) + (β223 α22 + β223 (λ2 − α1 )2 + β123 α2 (λ2 − α1 )) U 2 (t)b0∗ +  1 (1 + α1 )(α12 α2 (λ2 − α1 ) − α12 α2 (1 + α1 ) + 2α11 α22 − 2α22 (1 + α1 )(λ2 − α1 ))+ 1 + λ2

 (β12 α2 (λ2 − α1 ) − β12 α2 (1 + α1 ) + 2β11 α22 − 2β22 (1 + α1 )(λ2 − α1 )) X (t)U (t)+  1 (1 + α1 )(2α113 α22 + 2α223 (1 + α1 )(λ2 − α1 ) + α123 α2 (λ2 − α1 ) − α123 α2 (1 + α1 ))+ 1 + λ2

 (2β113 α22 + 2β223 (1 + α1 )(λ2 − α1 ) + β123 α2 (λ2 − α1 ) − β123 α2 (1 + α1 )) X (t)U (t)b0∗ +

  1 (1 + α1 )(α13 α2 − α23 (1 + α1 )) + (β13 α2 − β23 (1 + α1 )) X (t)b0∗ + 1 + λ2   1 (1 + α1 )(α13 α2 + α23 (1 + α1 )) + (β13 α2 + β23 (1 + α1 )) U (t)b0∗ . 1 + λ2 Hereafter we determine the center manifold Wc (0, 0) of (69) about (0, 0) in a small neighborhood of b0∗ . By center manifold theorem, there exists a center manifold Wc (0, 0) that can be represented as follows: Wc (0, 0) = {(X (t), U (t)) : U (t) = h(X (t), b0∗ ) = a1 X (t)2 + a2 X (t)b0∗ + a3 b0∗2 + O((| X (t) | + | b0∗ |)2 )}.

(70)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

383

where O((| X (t) | + | b0∗ |)2 ) is a function with order at least three in their variables (X (t), b0∗ ). Moreover, the center manifold must satisfy  h − X (t) +

F1 (X (t), h(X (t), b0∗ )), b0∗ ), b0∗



−λ2 h(X (t), b0∗ ) − G 1 (X (t), h(X (t), b0∗ ), b0∗ ) = 0.

(71)

By equating coefficients of like powers to zero, we obtain a1 =

(1 + α1 )(−α12 α2 (1 + α1 ) + α11 α22 + α22 (1 + α1 )2 ) + 1 − λ22 (−β12 α2 (1 + α1 ) + β11 α22 + β22 (1 + α1 )2 ) 1 − λ22

a2 = −

(1 + α1 )(α13 α2 − α23 (1 + α1 )) + (β13 α2 − β23 (1 + α1 )) , (1 + λ2 )2 a3 = 0.

Therefore, we consider the map which is the map (69) restricted to the center manifold Wc (0, 0) f = X (t + 1) = −X (t) + ϕ1 X (t)b0∗ + ϕ2 X 2 (t) + ϕ3 X 2 (t)b0∗ + ϕ4 X 3 (t),

(72)

where   1 (λ2 − α1 )(α13 α2 − α23 (1 + α1 )) − (β13 α2 − β23 (1 + α1 )) , ϕ1 = 1 + λ2 ϕ2 =

1 1+λ2

 (λ2 − α1 )(−α12 α2 (1 + α1 ) + α11 α22 + α22 (1 + α1 )2 )−  (−β12 α2 (1 + α1 ) + β11 α22 + β22 (1 + α1 )2 )) ,

ϕ3 =

 1 (λ2 − α1 )(α113 α22 + α223 (1 + α1 )2 − α123 α2 (1 + α1 )) − (β113 α22 + β223 (1 + α1 )2 − 1 + λ2

 β123 α2 (1 + α1 )) −

 1 (λ2 − α1 )α12 α2 (λ2 − α1 ) − α12 α2 (1 + α1 ) + 2α11 α22 − (1 + λ2 )3

2α22 (1 + α1 )(λ2 − α1 ) − β12 α2 (λ2 − α1 ) − β12 α2 (1 + α1 ) + 2β11 α22 −

384

K. Mokni and M. Ch-Chaoui

  2β22 (1 + α1 )(λ2 − α1 ) (1 + α1 )α13 α2 − α23 (1 + α1 ) + (β13 α2 − β23 (1 + α1 )) +

  1 (λ2 − α1 )α13 α2 − α23 (1 + α1 ) − (β13 α2 − β23 (1 + α1 )) (1 + λ2 )(1 − λ22 )   (1 + α1 )α13 α2 + α23 (λ2 − α1 ) + (β13 α2 + β23 (λ2 − α1 )) , ϕ4 =

 1 (λ2 − α1 )(α12 α2 (λ2 − α1 ) − α12 α2 (1 + α1 ) + 2α11 α22 − (1 + λ2 )(1 − λ22 )

2α22 (1 + α1 )(λ2 − α1 )) − (β12 α2 (λ2 − α1 ) − β12 α2 (1 + α1 ) + 2β11 α22 −  2β22 (1 + α1 )(λ2 − α1 ))

(1 + α1 )(−α12 α2 (1 + α1 ) + α11 α22 + α22 (1 + α1 )2 )+

(−β12 α2 (1 + α1 ) +

β11 α22

 + β22 (1 + α1 ) ) . 2

In order for the map (72) to undergo a period-doubling bifurcation, we require that the following discriminatory quantities are non-zero [14, 27]:  π1 =

∂2 f 1 ∂ f ∂2 f + ∂ X (t)∂b0∗ 2 ∂b0∗ ∂ 2 X (t)

 π2 =

1 ∂3 f + 6 ∂ X (t)3



1 ∂2 f 2 ∂ X (t)2

 |(0,0) = 0,

2  |(0,0) = 0.

After calculating we get π1 = ϕ3 = 0, π2 = ϕ4 + ϕ22 = 0. Finally, we have the following theorem Theorem 4 If π2 = 0, and π1 = 0, then the discrete model (13)–(14) undergoes a period-doubling bifurcation about the positive fixed point (x(b0 ), u(b0 )) when b0∗ varies in a small neighborhood of b0 . Moreover, if π2 > 0 ( resp π2 < 0 ), then the period 2 points that bifurcate from (x(b0 ), u(b0 )) are stable (unstable).

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

385

5 Chaos Control In this section, we implement three different methods to control chaos influenced by NSB and PDB [12, 24, 28].

5.1 State Feedback Control This method stabilizes chaotic orbits at an unstable fixed point in the evolutionary system (13), (14). Thus, let us consider the following controlled system: x(t + 1) = b0 exp(−

u(t)2 ) exp(−x(t))x(t) + hx(t) − P(t), 2

(73)

u(t)2 −u(t) + c1 x(t) . ) exp(−x(t)) 2 2 b0 exp(− u(t) ) exp(−x(t)) + h 2 (74) Here, P(t) = α(x(t) − x(b0 )) + β(u(t) − u(b0 )) is feedback controlling force at the fixed point (x(b0 ), u(b0 )) where α, β are feedback gains. b0 represents nominal value for b0 which belongs to some chaotic regions. The Jacobian matrix J at (x(b0 ), u(b0 )) is u(t + 1) = u(t) + σ 2 b0 exp(−

J (x(b0 ), u(b0 )) = ⎛ ⎜ ⎝

u 2 (b0 ) 2 )(1 − x(b0 )) + h u 2 (b ) σ 2 b0 exp(−x(b0 )− 2 0 )c1

b0 exp(−x(b0 ) −

b0 exp(−x(b0 )−

u 2 (b0 ) 2 2 )c1 x (b0 ) − β u 2 (b0 ) b exp(−x(b0 )− 2 ) 1 − σ2 0

− α −b0 exp(−x(b0 ) −

u 2 (b0 ) 2 )+h

b0 exp(−x(b0 )−

⎞ ⎟ ⎠.

u 2 (b0 ) 2 )+h

The corresponding characteristic equation is 2   0) b0 exp(−x(b0 ) − u (b u 2 (b0 ) 2 ) ζ+ )(1 − x(b0 )) + h − α + 1 − σ 2 ζ 2 − b0 exp(−x(b0 ) − 2 2 b0 exp(−x(b0 ) − u (b0 ) ) + h

2

(75)

2    0) b0 exp(−x(b0 ) − u (b u 2 (b0 ) 2 ) )(1 − x(b0 )) + h − α 1 − σ 2 + b0 exp(−x(b0 ) − 2 2 b0 exp(−x(b0 ) − u (b0 ) ) + h

2

 u 2 (b0 ) )c1 2 b0 u 2 (b0 ) )+h 2

σ 2 b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

exp(−x(b0 ) −

u 2 (b0 ) )c1 x 2 (b0 ) + β 2

 = 0.

Let ζ1 , ζ2 are the eigenvalues of the characteristic (75) then sum and product of the roots is given by

386

K. Mokni and M. Ch-Chaoui 0) b0 exp(−x(b0 ) − u (b u 2 (b0 ) 2 ) , )(1 − x(b0 )) + h − α + 1 − σ 2 2 2 b0 exp(−x(b0 ) − u (b0 ) ) + h 2

ζ1 + ζ2 = b0 exp(−x(b0 ) −

2

(76)

2    0) b0 exp(−x(b0 ) − u (b u 2 (b0 ) 2 ) + ζ1 ζ2 = b0 exp(−x(b0 ) − )(1 − x(b0 )) + h − α 1 − σ 2 2 2 b0 exp(−x(b0 ) − u (b0 ) ) + h

2

 u 2 (b0 ) )c1 2 b0 u 2 (b0 ) )+h 2

σ 2 b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

exp(−x(b0 ) −

(77)

 u 2 (b0 ) )c1 x 2 (b0 ) + β . 2

Lemma 4 The system (73)–(74) is asymptotically stable if all the eigenvalues of the characteristic (115) lie in an open unit disc. Proof The marginal stability lines can be obtained from the conditions: ζ1 = ±1 , ζ1 ζ2 = 1. For the conditions ζ1 ζ2 = 1, the Eq. (77) gives  L1 : − 1 − σ 2

 u 2 (b0 ) 2 ) α u 2 (b0 ) 2 )+h

b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

+

u 2 (b0 ) 2 )c1 β u 2 (b0 ) 2 )+h

σ 2 b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

=

(78)

2    0) b0 exp(−x(b0 ) − u (b u 2 (b0 ) 2 ) )(1 − x(b0 )) + h 1 − σ 2 1 − b0 exp(−x(b0 ) − − 2 2 b0 exp(−x(b0 ) − u (b0 ) ) + h

2

2

u 2 (b0 ) 2 2 2 ) c1 x (b0 ) 2 , u 2 (b0 ) b0 exp(−x(b0 ) − 2 ) + h

σ 2 b0 exp(−x(b0 ) −

the (78) expresses the first condition for marginal stability. For ζ1 = 1 , the Eq. (76) yields L2 : σ 2

σ2

u 2 (b0 ) ) 2 u 2 (b0 ) )+ 2

b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

u 2 (b0 ) 2 ) u 2 (b ) b0 exp(−x(b0 )− 2 0 )+h

b0 exp(−x(b0 )−

h

α+

b0 exp(−x(b0 ) −

 b0 exp(−x(b0 ) −

2

u 2 (b0 ) )(1 2

similarly for ζ1 = −1 , one obtains

= (79)

 − x(b0 )) + h − 1 −

u 2 (b0 ) 2 2 2 ) c1 x (b0 ) 2 , 2 0) b0 exp(−x(b0 ) − u (b )+h 2

σ 2 b0 exp(−x(b0 ) −

u 2 (b0 ) )c1 2 β u 2 (b0 ) )+h 2

σ 2 b0 exp(−x(b0 ) −

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

 L3 : σ 2  σ2

u 2 (b0 ) ) 2 u 2 (b0 ) )+ 2

b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

b0 exp(−x(b0 ) −

u 2 (b0 ) 2

2  0) σ 2 b0 exp(−x(b0 ) − u (b )c1 2 −2 α+ β= u 2 (b0 ) h b0 exp(−x(b0 ) − 2 ) + h (80)





u 2 (b0 ) 2 )

b0 exp(−x(b0 ) −

)+h

387

−2

b0 exp(−x(b0 ) −

u 2 (b0 ) )(1 − x(b0 )) + h + 1 2

2

−

u 2 (b0 ) 2 2 2 ) c1 x (b0 ) 2 . u 2 (b0 ) b0 exp(−x(b0 ) − 2 ) + h

σ 2 b0 exp(−x(b0 ) −

The lines L 1 , L 2 , L 3 give the conditions for the eigenvalues to have absolute value less than 1. The triangular region bounded by these lines accommodates stable eigenvalues.

5.2 Pole Placement Method This method is a generalization of OGY method [24] for higher dimensional systems. The aim is to stabilize chaotic orbits by using small perturbations to unstable periodic orbits. The system (13)–(14) can be written in the following form x(t + 1) = b0 exp(−

u(t)2 ) exp(−c(u(t), u(t))x(t))x(t) + hx(t) = f (x(t), u(t), b0 ) 2 −u(t)+c1 x(t)

2

u(t + 1) = u(t) + σ 2 b0 exp(− u(t) 2 ) exp(−x(t))

2

b0 exp(− u(t) 2 ) exp(−x(t))+h

(81)

= g(x(t), u(t), b0 )

(82) where b0 denotes parameter for chaos control. Suppose that b0 lies in a small interval, that is,b0 ∈ (b0 − ν, b0 + ν), such that ν > 0, and b0 represents nominal value for b0 which belongs to some chaotic region. Suppose that (x(b0 ), u(b0 )) denotes an unstable fixed point for (13)–(14) in a chaotic region which is produced under the influence of Neimark–Sacker bifurcation or period doubling bifurcation. In this case (81)–(82) is approximated in the neighborhood of (x(b0 ), u(b0 )) as follows 

x(t + 1) − x(b0 ) u(t + 1) − u(b0 )





J (x(b0 ), u(b0 ), b0 )

x(t) − x(b0 ) u(t) − u(b0 )



+ B b0 − b0 , (83)



where J (x(b0 ), u(b0 ), b0 ) =  =

∂ f (x(b0 ),u(b0 ),b0 ) ∂ f (x(b0 ),u(b0 ),b0 ) ∂x ∂u ∂g(x(b0 ),u(b0 ),b0 ) ∂g(x(b0 ),u(b0 ),b0 ) ∂x ∂u

b0 γ (1 − x(b0 )) + h −b0 γ c1 x 2 (b0 ) σ 2 b0 γ c1 γ 1 − σ 2 b bγ0 +h b γ +h 0

0

 ,



388

K. Mokni and M. Ch-Chaoui

with γ = exp(−x(b0 ) − 

and B=

∂ f (x(b0 ),u(b0 ),b0 ) ∂b0 ∂g(x(b0 ),u(b0 ),b0 ) ∂b0

u 2 (b0 ) ), 2



 =

x(b0 )γ 0

 .

(84)

The system (81), (82) is controllable if the following matrix has rank 2  C = (B : J B) =

x(b0 )γ x(b0 )γ (b0 γ (1 − x(b0 )) + h) σ 2 b0 γ c1 0 x(b0 )γ b γ +h

 ,

(85)

0

this implies that rank  of C is 2. Hence  the system is controllable. Furthermore, we

x(t) − x(b0 ) ,, where K = k1 k2 , then system (83) is set b0 − b0 = −K u(t) − u(b0 ) written as follows    

x(t) − x(b0 ) x(t + 1) − x(b0 )

J − BK . (86) u(t + 1) − u(b0 ) u(t) − u(b0 ) In this case, the corresponding control system of (81)–(82) is given as follows   u(t)2 ) + h x(t), x(t + 1) = b0 − k1 (x(t) − x(b0 )) − k2 (u(t) − u(b0 )) x(t) exp(−x(t)) exp(− 2

(87)

 2  −u(t) + c1 x(t) u (t)  2  exp(−x(t)) u(t + 1) = u(t) + σ 2 b0 exp − . (88) 2 b0 exp − u 2(t) exp(−x(t)) + h

The positive fixed point point of (87)–(88) is locally stable if and only if absolute values of both eigenvalues of J − B K are less than one. Moreover, the matrix J − B K is given as follows  J − BK =

b0 γ (1 − x(b0 )) + h − γ x(b0 )k1 −b0 γ c1 x 2 (b0 ) − γ x(b0 )k2 σ 2 b0 γ c1 γ 1 − σ 2 b bγ0 +h b γ +h 0

 .

0

The characteristic equation for the matrix J − B K is given as follows  μ2 − b0 γ (1 − x(b0 )) + h − γ x(b0 )k1 + 1 − σ 2

b0 γ b0 γ + h

 μ+

(89)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

389

  b0 γ (1 − x(b0 )) + h − γ x(b0 )k1 1 − σ 2



b0 γ

b0 γ + h   2  σ b0 γ c1 + b0 γ c1 x 2 (b0 ) + γ x(b0 )k2 = 0. b0 γ + h

(90)

Assume that μ1 and μ2 represent the roots of (89), then it follows that μ1 + μ2 = b0 γ (1 − x(b0 )) + h − γ x(b0 )k1 + 1 − σ 2

b0 γ b0 γ + h

,

(91)

   γ μ1 μ2 = b0 γ (1 − x(b0 )) + h − γ x(b0 )k1 1 − σ 2 b bγ0 +h 0

   2 σ b0 γ c1 2 b0 γ c1 x (b0 ) + γ x(b0 )k2 . + b0 γ + h

(92)

Moreover, we take μ1 = ±1 and μ1 μ2 = 1. Then, the lines of marginal stability for (87)–(88) are computed as follows: 

 l1 : b0 γ (1 − x(b0 )) + h 1 − σ 2

b0 γ



 − γ x(b0 )k1 1 − σ 2

b0 γ

 +

b0 γ + h b0 γ + h (93)    2 2 σ b γ c b γ c σ 0 1 0 1 + γ x(b0 )k2 = 1, b0 γ c1 x 2 (b0 ) b0 γ + h b0 γ + h 



 l2 : b0 γ (1 − x(b0 )) + h σ 2

b0 γ

b0 γ

− γ x(b0 )k1 σ 2

b0 γ

− b0 γ + h b0 γ + h b0 γ + h (94)    2  2 σ b γ c b γ c σ 0 1 0 1 − γ x(b0 )k2 = 0, b0 γ c1 x 2 (b0 ) b0 γ + h b0 γ + h

  l3 = b0 γ (1 − x(b0 )) + h 2 − σ 2 +2 − σ 2

b0 γ b0 γ + h

 + b0 γ c1 x 2 (b0 )

− σ2

b0 γ



b0 γ + h σ 2 b0 γ c1 b0 γ + h

 − γ x(b0 )k1 2 − σ 2



 + γ x(b0 )k2

b0 γ



b0 γ + h (95)  σ 2 b0 γ c1 = 0. b0 γ + h

Then, stability region for (87), (88) is a triangular region bounded by l1 , l2 and l3 in k1 k2 - plane.

390

K. Mokni and M. Ch-Chaoui

5.3 Chaos Control Using Hybrid Method In this subsection, we apply hybrid control method to stabilize unstable periodic orbits embedded in the chaotic attractor of the system (87), (88). The controlled system is written as follows     u 2 (t) exp(−x(t))x(t) + hx(t) + (1 − ρ)x(t), x(t + 1) = ρ b0 x(t) exp − 2 (96)     u(t + 1) = ρ u(t) + σ 2 b0 exp

2 − u(t) 2

 −u(t)+c  1 x(t)

exp(−x(t))

b0 exp − u(t) 2

2

) + (1 − ρ)u(t).

exp(−x(t))+h

(97) where ρ ∈ (0, 1), illustrates the control parameters for the controlled system (96)– (97). Moreover, the Jacobian matrix for (96)–(97) at (x(b0 ), u(b0 )) for controlled system (96)–(97) as follows J (x(b0 ), u(b0 )) = ⎛ ⎜ ⎜ ⎝

⎞

u 2 (b0 ) u 2 (b0 ) 2 2 )(1 − x(b0 )) + h) + 1 − ρ −ρ(b0 exp(−x(b0 ) − 2 )c1 x (b0 ) ⎟ 2 u (b0 ) u 2 (b0 ) ⎟, 2 σ b0 exp(−x(b0 )− 2 )c1 ⎠ 2 b0 exp(−x(b0 )− 2 )

ρ(b0 exp(−x(b0 ) − ρ

b0 exp(−x(b0 )−

1 − ρσ

u 2 (b0 ) 2 )+h

b0 exp(−x(b0 )−

u 2 (b0 ) 2 )+h

(98) furthermore, the following lemma gives parametric conditions for local stability of the fixed point (x(b0 ), u(b0 )) for controlled system (96)–(97). Lemma 5 The positive fixed point (x(b0 ), u(b0 )) of the control system (96)–(97) is locally asymptotically stable if the following conditions hold • − b0 exp(−x(b0 ) −

u 2 (b0 ) 2 )(1 −

x(b0 )) + exp(−x(b0 ) −

• ρ(b0 exp(−x(b0 ) −

u 2 (b0 ) 2 )(1 −

x(b0 )) + h)(2 − ρσ 2

ρσ 2

u 2 (b0 ) 2 ) u 2 (b ) b0 exp(−x(b0 )− 2 0 )+h

b0 exp(−x(b0 )−

• (b0 exp(−x(b0 ) − σ2

u 2 (b0 ) )(1 2

(2 − ρ) −

u 2 (b0 ) 2 ) u 2 (b0 ) b0 exp(−x(b0 )− 2 )+h

b0 exp(−x(b0 )−

−

u 2 (b0 ) 2 2 2 2 2 ) c1 b0 x (b0 ) u 2 (b ) b0 exp(−x(b0 )− 2 0 )+h

σ 2 ρ 2 exp(−x(b0 )−

Proof The characteristic equation of (96)–(97) gives

,

u 2 (b0 ) 2 ) u 2 (b0 ) b0 exp(−x(b0 )− 2 )+h

b0 exp(−x(b0 )−

.

> 0,

) > 2ρ − 4+

u 2 (b0 ) 2 2 2 2 2 ) c1 b0 x (b0 ) u 2 (b ) b0 exp(−x(b0 )− 2 0 )+h

σ 2 ρ 2 exp(−x(b0 )−

− x(b0 )) + h)(1 − ρσ 2

u 2 (b0 ) 2 )(1−ρ) u 2 (b ) b0 exp(−x(b0 )− 2 0 )+h

b0 exp(−x(b0 )−

u 2 (b0 ) 2 2 2 )c1 b0 x (b0 ) + 1 − h

) < 1+

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

391

 u 2 (b0 ) )(1 − x(b0 )) + h) + 2 − ρ P(ω) = ω − ρ(b0 exp(−x(b0 ) − 2 2  0) b0 exp(−x(b0 ) − u (b ) 2 ω+ (99) −ρσ 2 2 0) b0 exp(−x(b0 ) − u (b )+h 2 2

  u 2 (b0 ) )(1 − x(b0 )) + h) + 1 − ρ ρ(b0 exp(−x(b0 ) − 2 2    2 u 2 (b0 ) 0) 2 2 2 2 b0 exp(−x(b0 ) − 2 ) b0 ρ 2 exp(−x(b0 ) − u (b 2 ) c1 σ x (b0 ) 2 1 − ρσ + = 0. 2 2 0) 0) b0 exp(−x(b0 ) − u (b b0 exp(−x(b0 ) − u (b 2 )+h 2 )+h

Now from Jury’s criterion for the stability [13] below | tr (J ) − 1 |< det (J ) < 1,

(100)

yields 2   0) 2   b0 exp(−x(b0 ) − u (b 2 )  ρ(b0 exp(−x(b0 ) − u (b0 ) )(1 − x(b0 )) + h) + 1 − ρ − ρσ 2 2  < u (b0 ) 2 b0 exp(−x(b0 ) − )+h

2

(101)

  u 2 (b0 ) ρ(b0 exp(−x(b0 ) − )(1 − x(b0 )) + h) + 1 − ρ 2  × 1 − ρσ 2 2

u 2 (b0 ) ) 2 u 2 (b0 ) )+ 2



b0 exp(−x(b0 ) − b0 exp(−x(b0 ) −

h

+

u 2 (b0 ) 2 2 2 ) c1 σ (1 − h)x 2 (b0 ) 2 2 0) b0 exp(−x(b0 ) − u (b )+h 2

b0 ρ 2 exp(−x(b0 ) −

< 1.

Simplifying (100) one gets the conditions of the lemma.

6 Numerical Simulations In order to substantiate the obtained results and explore the rich dynamics in the system (13)–(14), detailed numerical simulations are performed with respect to different parameters. Choosing the following set of parameters in consistency of Theorem 2: σ 2 = 0.8, c1 = 2, h = 0.2.

392

K. Mokni and M. Ch-Chaoui

1.6

1.7

1.5

1.6 1.5

1.4

1.4

1.3

1.3

u

u

1.2 1.1

1.1

1

1

0.9

0.9

0.8 0.7 0.2

1.2

0.8 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.7 0.3

1

0.4

0.5

0.6

x

0.7

0.8

0.9

1

0.8

0.9

1

x

(b): b0 = 2.8

(a): b0 = 2.5 1.8

1.7 1.6

1.6

1.5 1.4

1.4

1.2

u

u

1.3 1.2

1.1 1

1 0.9

0.8

0.8 0.7 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.6 0.3

1

0.4

0.5

0.6

0.7

x

x

(c): b0 = 3

(d): b0 = 3.25

Fig. 1 Phase portraits of model for the evolutionary discrete model (13)–(14) for h = 0.2 and different value of b0

For this set of parameters, we change b0 accordingly. Figure 1 shows the stable dynamics of the population and their trait, which converges to the positive fixed point (0.62, 1.24) for the initial condition (x(0), u(0)) = (1, 1). For the value b0 = 3.29. The system (13)–(14) start to lose its stability. In particular, the existence of an attracting closed invariant curve implies that the discrete-time model (13)–(14) undergoes a Neimark–Sacker bifurcation about (0.62, 1.24). To see this if b0 > 3.29 the model (13)–(14) becomes  exp

x(t + 1) = (3.29 + b0∗ )x(t) u(t + 1) = u(t) + 0.8(3.29 + b0∗ ) exp

 −

u(t)2 2

 2

− u 2(t) 1+x(t)

+ 0.2x(t),



(102)

−u(t)+2x(t)  

exp(−x(t)) (3.29+b0∗ ) exp

− u(t) 2

2

.

exp(−x(t))+0.2

(103)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

393

whose fixed point is (0.62, 1.24). Now we transform (0.62, 1.24) into (0, 0) by using the following transformation: v(t) = x(t) − 0.62,

(104a)

w(t) = u(t) − 1.24.

(104b)

where (102), (103) becomes v(t + 1) = 0.504v(t) − 0.61504w(t) − 1.60704v(t)w(t) − 0.552v 2 (t) + 0.1333248w 2 (t),

(105a) w(t + 1) = 1.28v(t) + 0.36w(t) − 0.18944v(t)w(t) − 0.256v (t) + 0.1984w 2 (t). 2

(105b) Since linear part of (105) is just as J (0.62, 1.24) about J (0.62, 1.24) with eigenvalues λ1,2 = 0.432 ± 0.8843456338 i.

(106)

Now, (105) takes the following form 

X (t + 1) U (t + 1)



 =

0.432 −0.8843456338 −0.8843456338 0.432



X (t) U (t)



 +

F(X (t), U (t)) G(X (t), U (t))

 ,

(107) where F(X (t), U (t)) = −0.279280565X 2 (t) + −0.8571034642X (t)U (t) + 0.104268953U 2 (t), G(X (t), U (t)) = −0.104198639X 2 (t) − 0.0777725189X (t)U (t) + 0.1551621324U 2 (t).

By using transformation      v(t) −0.61504 0 X (t) = , w(t) 0.432 − 0.504 −0.8843456338 U (t) Moreover, computation yields τ02 = −0.07644425 + 0.1494356732 i,

(108)

τ11 = −0.087505806 + 0.0254817467 i,

(109)

τ20 = 0.0243097733 + 0.1494356732 i,

(110)

τ21 = 0.

(111)

Using (106)–(108)–(109)–(110) and (111) in (61), one gets L = 0.20945448 > 0. Hence, the model (102)–(103) undergoes a supercritical Neimark–Sacker bifurcation if b0 > 3.29, and meanwhile, stable curve appears, which is depicted in Fig. 2. For

K. Mokni and M. Ch-Chaoui

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

u

u

394

1

1

0.8

0.8

0.6 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.6 0.3

1

0.4

0.5

0.6

0.7

0.8

x

x

(a): b0 = 3.29

(b): b0 = 3.34

0.9

1

2

1.8

1.8

1.6

1.6

1.4

u

u

1.4 1.2

1.2 1

1

0.8

0.6 0.3

0.8

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.3

0.4

0.5

0.6

0.7

0.8

x

x

(c): b0 = 3.4

(d): b0 = 3.6

0.9

1

1.1

Fig. 2 Supercritical Neimark–Sacker bifurcation for the system (13)–(14)

exploring the complexity of system (13)–(14), four bifurcation diagrams are plotted in Fig. 3 with respect to b0 for different value of h. The evolutionary system (13)–(14) exhibits a range of period doubling route to chaos phenomenon. Now, some numerical computations are performed to stabilize the chaos in the system (13)–(14) influenced by Neimark–Sacker and period doubling bifurcations respectively. For the first method, state feedback control, It is exhibited for b¯0 = 3.4. As reported in Fig. 2c the system shows closed curve . In order to implement this method Lemma 4 gives the following lines of marginal stability of the system (73)–(74) L 1 : −0.352690278α + 1.2946194445β = −0.294619445, L 2 : −0.647309722α − 1.2946194445β = 1.153180741, L 3 : −1.352690278α + 1.2946194445β = −2.90294468.

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

(a): h = 0.2

(b): h = 0.4

(c): h = 0.5

(d): h = 0.7

395

Fig. 3 Bifurcation diagrams for system (13)–(14) with respect to b0

The system (73)–(74) is stable for the triangular region bounded by the marginal lines L 1 , L 2 and L 3 . Now, in order to make the fixed point (0.62, 1.24) locally asymptotically stable, consider the feedback controlling force U (t) = α(x(t) − x ∗ ) + β(u(t) − u ∗ ) with feedback gains are α = 2, β = 0.3, chosen from the triangular region from Fig. 4 (d). For these values, a time series are drawn in Fig. 4 which shows that the system (73)–(74) achieves stability and converges to the positive fixed point(0.62, 1.24). Finally, we apply using the hybrid control strategy to system (96)–(97) to control chaos introduced by Neimark–Sacker bifurcation for (b0 , h, c1 , σ 2 ) = (3.4, 0.2, 2, 0.8), the controlled system associated to (13)–(14) is   u 2 (t)   exp − 2 + 0.2x(t) + (1 − ρ)x(t), x(t + 1) = ρ 3.4x(t) 1 + x(t)

(112)

396

K. Mokni and M. Ch-Chaoui

2

2 x u

1.8 1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

50

100

150

200

250

300

350

x u

1.8

0

400

0

50

100

time (t)

200

250

300

350

400

time (t)

(a)

(b)

2

4 x u

1.8

L1 L2 L3

3

1.6

2

1.4

1

1.2

β

1 0.8

0 -1

0.6

-2

0.4

-3

0.2 0

150

0

50

100

150

200

250

300

350

400

-4 -3

-2

-1

0

1

2

3

4

α

time (t)

(d)

(c)

Fig. 4 Time series of x and for u for the model (73)–(74) for b0 = 3.4. (a) Chaotic behavior. In (b) the chaos is controlled after time t = 40, and in (c) the chaos is controlled after time t = 300. The panel (d) stability region of the controlled system (73)–(74)

   u(t)2 u(t + 1) = ρ u(t) + 0.8 ∗ 3.4 exp − 2  −u(t) + 2x(t)   + (1 − ρ)u(t). (113) × exp(−x(t)) u(t)2 exp(−x(t)) + 0.2 3.4 exp − 2 The jacobian matrix of (112)–(113) is  J=

−0.477808308ρ + 1 −0.6518446647ρ 1.2946194445ρ 1 − 0.6473097223ρ

 .

(114)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

397

The characteristic polynomial equation of (114) is P(ω) = ω2 − (2 − 1.12511803ρ)ω + 1 − 1.12511803ρ + 1.1531807409ρ 2 = 0, (115) where (116) tr J (x ∗ , u ∗ )ρ = 2 − 1.12511803ρ, det J (x ∗ , u ∗ )ρ = 1 − 1.12511803ρ + 1.1531807409ρ 2 .

(117)

Replacing the control parameter ρ = 0.98 in (116) and (117), and using the jury’s criteria (100), one gets 1 − tr J (x ∗ , u ∗ )ρ=0.98 + det J (x ∗ , u ∗ )ρ=0.98 = 1.1075147836 > 0,

(118)

1 + tr J (x ∗ , u ∗ )ρ=0.98 + det J (x ∗ , u ∗ )ρ=0.98 = 2.9022834448 > 0,

(119)

det J (x ∗ , u ∗ )ρ=0.98 = 1.0048991142 > 1.

(120)

For the control parameter ρ = 0.98, the stability conditions of Lemma 5 for hybrid control method are not all verified because det J (x ∗ , u ∗ )ρ=0.98 = 1.0048991142 > 1. For this value of ρ = 0.98, a time series is plotted in Fig. 5 which shows the instability for the system. Now we choose ρ = 0.94, one gets 1 − tr J (x ∗ , u ∗ )ρ=0.94 + det J (x ∗ , u ∗ )ρ=0.94 = 1.0189505027 > 0,

(121)

1 + tr J (x ∗ , u ∗ )ρ=0.94 + det J (x ∗ , u ∗ )ρ=0.94 = 2.9037286063 > 0,

(122)

det J (x ∗ , u ∗ )ρ=0.94 = 0.9613395545 < 1 1.8

(123)

3 x u

1.7 2.5

1.6 1.5

2

u

1.4 1.3

1.5

1.2 1

1.1 1

0.5

0.9 0.8 0.3

0.4

0.5

0.6

0.7

x

0.8

0.9

1

0

0

50

100

150

200

250

300

350

400

time t

Fig. 5 Phase portrait and time series of the evolutionary discrete model (112)–(113) respectively for b0 = 3.4 for ρ = 0.98

398

K. Mokni and M. Ch-Chaoui

1.8

3 x u

1.7 2.5

1.6 1.5

2

u

1.4 1.3

1.5

1.2 1

1.1 1

0.5

0.9 0.8 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

x

0

50

100

150

200

250

300

350

400

time t

Fig. 6 Phase portraits and time series of the controlled system (112)–(113) respectively, for h = 0.2 and b0 = 3.4 and ρ = 0.94

For the control parameter ρ = 0.94, the eigne values of above matrix lie in an open unit disk and stability conditions of Lemma 5 for hybrid control method are verified. For this value of ρ, a time series is plotted in Fig. 6 which shows stability for the system. To implement the pole-placement method, The system (13)–(14) can be written as   u(t)2 exp(−x(t))x(t) + 0.2x(t), (124) x(t + 1) = b0 exp − 2  u(t + 1) = u(t) + 0.8 ∗ b0 exp

−

u(t)2 2

 

exp(−x(t)) b0 exp

−u(t) + 2x(t)  , 2 − u(t) exp(−x(t)) + 0.2 2

(125) where b¯0 is the nominal value of b0 taken from a chaotic region produced under the influence of Period doubling bifurcation see Fig. 3a. As for state feedback method, let b0 = 5. Then the linearized system of (124)–(125) approximated at the positive fixed point (x(b0 ), u(b0 )) = (0.62, 1.24) is 

x(t + 1) − 0.62 u(t + 1) − 1.24





J (0.62, 1.24, 5) 

where J (0.62, 1.24, 5) =





+ B b0 − 5 ,

(126)

 1.8373087474 −3.3125341184 . 1.5290262189 0.2354868906

 B=

x(t) − 0.62 u(t) − 1.24

 0.5342795097 . 0

(127)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

399

Then, it follows that  C = (B : J B) =

 0.5342795097 0.9816364167 , 0 0.8169273786

(128)

is of rank 2. Hence the system is controllable. Taking b0 = (b0 − k1 (x(t) − x(b0 )) − k2 (u(t) − u(b0 ))). The controlled system corresponding to (87)–(88) for b0 = 5 becomes  exp x(t + 1) = (5 − k1 (x(t) − 0.62) − k2 (u(t) − 1, 24))x(t)  u(t + 1) = u(t) + 0.8 ∗ 5 exp

−

u(t)2 2

 

exp(−x(t)) 5 exp

2 − u(t) 2

1 + x(t)

 + 0.2x(t),

(129)

−u(t) + 2x(t)  . 2 − u(t) exp(−x(t)) + 0.2 2

(130) The Jacobian matrix of the system (129)–(130) becomes  J (0.62, 1.24) =

1.8373087474 − 0.5342795097k1 −3.3125341184 − 0.5342795097k2 1.5290262189 0.2354868906

 ,

whose corresponding characteristic equation is μ2 − (2.072795638 − 0.5342795097k1 )μ + 5.497613642 −0.1258158205k1 + 0.8169273786k2 = 0.

(131)

Assume that μ1 and μ2 represent the roots of (131), then it follows that μ1 + μ2 = 2.072795638 − 0.5342795097k1 , μ1 μ2 = 5.497613642 − 0.1258158205k1 + 0.8169273786k2 . The lines of marginal stability for (129)–(130) are drawn in Fig. 7. Setting the value k2 = −6 and k1 ∈ (5.6, 6.5) such that the set (k1 , k2 ) satisfies the inequality | μi |< 1 for i = 1, 2. The dense chaotic region is reduced to periodic and quasi-periodic windows reproduced in Fig. 8 respective to Fig. 3. Additional simulations are carried out. The fixed point of corresponding controlled system is locally asymptotically stable for k1 = 0.17 and k2 ∈ (−7, 4.5). In Fig. 9, the system exhibits a control phenomenon where chaos converges on a stable window.

400

K. Mokni and M. Ch-Chaoui 15

L1 L2 L3

10

k1

5

0

-5

-10 -10

-9

-8

-7

-6

-5

-4

-3

-2

k2

Fig. 7 Stability region for the controlled system (129)–(130)

Fig. 8 Bifurcation diagrams for the controlled system (87)–(88) with respect to k1 at k2 = −6 and k1 ∈ (5.6, 6.5)

Asymptotic Stability, Bifurcation Analysis and Chaos Control …

401

Fig. 9 Bifurcation diagram for the controlled system (87)–(88) with respect to k2 at k1 = 0.17 and k2 ∈ (−7, 4.5)

7 Concluding Remakes In this paper, we have derived a coupled evolutionary Ricker model incorporating the effect of immigration, by using Evolutionary Game Theory methodology, which is capable of modeling the interaction of population dynamics and evolutionary dynamics. The qualitative behavior of the discrete derived model was explored, and asymptotic stability results for the positive fixed point were obtained. We established that the system (13)–(14) goes through Neimark–Sacker bifurcation and period doubling bifurcation over the vast range of the bifurcation’s parameter b0 to support the complexity of the resulting evolutionary model (13)–(14). The aforementioned bifurcation analysis is developed by using both bifurcation theory and center manifold theory. We employed different techniques for controlling chaos that were influenced by Neimark–Sacker bifurcation and period doubling bifurcation. These findings show that the intrinsic growth rate is crucial to the stability of the discrete evolutionary Ricker model. This means that the nonlinear dynamical behaviors of this ecological model are not only dependent on bifurcation parameters, but also sensitive to parameter changes. Numerical simulations give evidence of the successful implementation of these three methods. The current evolutionary model can readily incorporate the strong Allee effect. We are working on two populations with different traits. We want to do an asymptotic study of the global stability of the present discrete single species model. As a result, new methods for determining the global stability of fixed points, such as the method of Lyapunov functions, are required [16].

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Acknowledgements The authors would like to thank Professor Saber Elaydi for his comments on an early draft of this manuscript. K.M. thanks her main supervisor, Professor Saber Elaydi, for the excellent advice and support. The authors also thank the anonymous referees for their valuable comments and suggestions.

References 1. Auerbach, D., Grebogi, C., Ott, E., Yorke, J.A.: Controlling chaos in high dimensional systems. Phys. Rev. Lett. 69(24), 3479 (1992) 2. Assas, L., Elaydi, S., Kwessi, E., Livadiotis, G., Ribble, D.: Hierarchical competition models with Allee effects. J. Biol. Dyn. 9(sup1), 32–44 (2015). https://doi.org/10.1080/17513758. 2014.923118 3. Assas, L., Dennis, B., Elaydi, S., Kwessi, E., Livadiotis, G.: Hierarchical competition models with the Allee effect II: the case of immigration. J. Biol. Dyn. 9(1), 288–316 (2015) 4. Cushing, J.M.: A darwinian ricker equation. In: Baigent, S., Bohner, M., Elaydi, S. (eds.) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol. 341. Springer, Cham (2020). https://doi.org/10. 1007/978-3-030-60107-2_10 5. Cushing, J.M.: A strong ergodic theorem for some nonlinear matrix models. Nat. Resour. Model. 3(3), 331–357 (1989) 6. Cushing, J.M.: An evolutionary Beverton-Holt model. In: AlSharawi, Z., Cushing, J.M., Elaydi, S. (eds.) Theory and Applications of Difference equations and Discrete Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol. 102, pp. 127–141 (2014) 7. Cushing, J.M., Levarge, S., Chitnis, N., Henson, S.M.: Some discrete competition models and the competitive exclusion principle. J. Diff. Equ. Appl. 10(13–15), 1139–1151 (2004) 8. Cushing, J., Stefanco, K.: A darwinian dynamics model for the evolution of post-reproduction survival. J. Biol. Syst. 29(02), 433–450 (2021) 9. Cushing, J., et al.: The evolutionary dynamics of a population model with strong Allee effect. Math. Biosci. Eng. 12, 4 (2015) 10. Choua, Y.H., Chowb, Y., Huc, X., Jang, S.R.J.: A Ricker type predator prey system with hunting cooperation in discrete time. Math. Comput. Simul. 190, 570–586 (2021) 11. Darwin, C.: On the Origin of Species. Harvard University Press, Cambridge (2001) 12. Din, Q.: Complexity and chaos control in a discrete time prey-predator model. Commun. Nonlinear Sci. Numer. Simul. 49, 113–34 (2017) 13. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005) 14. Elaydi, S.: Discrete Chaos, Applications in Science and Engineering, 2nd edn. Chapman and Hall/CRC, London (2008) 15. Elaydi, S., Kwessi, E., Livadiotis, G.: Hierarchical competition models with the Allee effect III: multi-species. J. Biol. Dyn. 12(1), 271–287 (2018) 16. Elaydi, S.: Global dynamics of discrete dynamical systems and difference equations. In: Elaydi S., Potzsche C., Sasu A. (eds.) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics & Statistics, vol. 287. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20016-9_3 17. Feia, L., Chena, X., Hanb, B.: Bifurcation analysis and hybrid control of a discrete-time predator-prey model. J. Diff. Equ. Appl. 27(1), 102–117 (2021) 18. Liu, X.L., Xiao, D.M.: Complex dynamic behaviors of a discrete-time predator-prey system. Chaos, Solitons Fractals 32, 80–94 (2007) 19. Li, B., He, Z.: Bifurcations and chaos in a two-dimensional discrete Hindmarsh Rose model. Nonlinear Dyn. 76(697–715), 20 (2014) 20. Livadiotis, G., Elaydi, S.: General Allee effect in two-species population biology. J. Biol. Dyn. 6, 959–973 (2012)

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21. Livadiotis, G., Assas, L., Elaydi, S., Kwessid, E., Ribblef, D.: Competition models with Allee effects. J. Diff. Eq. 20(8), 1127–1151 (2014) 22. Mokni, K., Elaydi, S., Ch-Chaoui, M., Eladdadi, A.: Discrete evolutionary population models: a new approach. J. Biol. Dyn. 14(1), 454–478 (2020) 23. Murry, J.D.: Mathematical Biology. Springer, New York (1989) 24. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990) 25. Khan, A.Q., Ma, J., Dongmei, X.: Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect. J. Biol. Dyn. 11(1), 121–146 (2017) 26. Khan, A.Q., Abdullah, E., Ibrahim, T.F.: Supercritical Neimark Sacker bifurcation and hybrid control in a discrete time glycolytic oscillator model. Math. Problems Eng. 2020, Article ID 7834076, 15 27. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004) 28. Romeiras, F.J., Grebogi, C., Ott, E., Dayawansa, W.: Controlling chaotic dynamical systems. Physica D 58(1–4), 165–192 (1992) 29. Tan, W., Gao, J., Fan, W.: Bifurcation analysis and chaos control in a discrete epidemic system. Discr. Dyn. Nat. Soc. 2015, Article ID 974868 (2015) 30. Vincent, T.L., Brown, J.S.: Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, Cambridge, UK (2005) 31. Zhang, L., Zou, L.: Bifurcations and control in a discrete predator prey model with strong Allee effect. Int. J. Bifur. Chaos 28(5) (2018) 32. Zhu, J., Wu, R., Chen, M.: Bifurcation analysis in a predator prey model with strong Allee effect. Zeitschrift fur Naturforschung A 76(12), 000010151520210178 (2021). https://doi.org/ 10.1515/zna-2021-0178

Weighted Norms In Advanced Volterra Difference Equations Youssef N. Raffoul

Abstract In this research we explore the existence of bounded solutions, and periodic solutions of Advanced type Volterra difference equations of the form x(n) = f (n, x(n), x(n − h)) −

∞ 

Q(s, x(s), x(s − h))C(n − s) + p(n)

s=n

using weighted norms and the contraction mapping principle in a suitable space. The suitable space will be constructed based on the magnitude and properties of p and the convergence properties of the term C. Keywords Advanced · Volterra · Weighted norm · Periodicity · Stability

1 Introduction We assume the reader is familiar with the calculus of difference equations and for references we mention [3, 5]. Few studies exist on advanced type difference equations. In [2] the authors considered an advanced type difference equation of kth order and established an analogue of a result by Levin and May which is concerned with the asymptotic stability of the zero solution. On the other hand, the authors in [15] considered a difference equations with advanced argument and studied the oscillation properties of its solutions. For more on asymptotic behavior of solutions of difference equations we refer to [8–10]. In the past hundred and fifty years, Lyapunov functions/functionals have been exclusively and successfully used in the study of stability and existence of periodic and bounded solutions. This research is devoted to the study of rate of convergence of solutions using weighted norm and fixed point theory. Very few papers exist on advanced Volterra difference equations and this paper should be considered as a foundation for future studies. Most of real life applications are Y. N. Raffoul (B) Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_18

405

406

Y. N. Raffoul

modeled by nonlinear systems for which implicit solutions can not be explicitly stated. This necessitates the qualitative analysis of such systems and in particular the study of how solutions behave with time. Biologists are interested in solutions remaining bounded and the exhibition of periodic behavior of solutions. For example, in [6] it is shown that there is a direct connection between boundedness of solutions and for solutions to exhibit a periodic behavior. In the paper [4] the authors considered a dynamical system and proved ultimate boundedness implied periodicity provided given functions are periodic. In addition, in the papers [11–13], the author used the notions of Lyapunov functionals and fixed point theory and obtained necessary conditions for the boundedness and ultimate boundedness and the existence of periodic solutions of functional difference equations. Advanced differential equations have been studied in [1] and in some of the references therein. Let Z+ , N, and R denote the set of non-negative integers the set of natural numbers and the set of all real numbers, respectively. Consider the neutral delay functional difference equation x(n + 1) = αx(n + 1 − h) + ax(n) − q(n, x(n), x(n − h)), n ∈ Z+ , h ∈ N (1) where the function q : Z+ × R × R → R is continuous and α and a are constants. Let a be a constant such that a > 1, and notice that expression (1) is equivalent to      (x(n) − αx(n − h)a 1−n ) = aαx(n − h) − q(n, x(n), x(n − h)) a −n .

(2)

We search for a solution of (1) having the property (x(n) − αx(n − h))a −n → 0, as n → ∞. Hence, by summing (2) from 0 to ∞ we get the following advanced type Volterra difference equation, x(n) = αx(n − h) −

∞    aαx(s − h) − q(s, x(s), x(s − h)) a n−s

(3)

s=n

which is an indication to study the general advanced type Volterra summation equation x(n) = f (x(n − h)) +

∞ 

Q(s, x(s), x(s − h))C(n − s) + p(n), n ∈ Z+ , h ∈ N

s=n

where the functions f : R → R, Q : Z+ × R × R → R are continuous.

(4)

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407

2 Boundedness of Solutions Under suitable conditions, we use fixed point theory to explore the boundedness of solutions of (4) by constructing suitable Banach spaces. In preparation for the next theorem we make the following assumptions. There exists an α ∈ [0, 1) such that for all x, y ∈ R, | f (x) − f (y)| ≤ α|x − y|.

(5)

There exists k ∈ [0, 1) such that for all x, y, w, z ∈ R, and n ∈ Z+ |Q(n, x, y) − Q(n, w, z)| ≤ k|x − w| + (1 − k)|y − z|.

(6)

We also assume that |Q(n, 0, 0)| ≤ 1, and

∞ 

|C(−u)| < 1 − α =: C0 .

(7)

(8)

u=0

Theorem 1 Let p(n) be bounded uniformly for all n ∈ Z+ . Assume conditions (5)– (8) hold. If in addition we require that ζ := α + C0 < 1, then (4) has a unique bounded solution. Proof Let (B, || · ||) be the Banach space of bounded sequences ϕ : R → R with the supremum norm. For ϕ ∈ B, define P:B→B by ∞    Q(s, ϕ(s), ϕ(s − h))C(n − s) + p(n), n ∈ Z+ . Pϕ (n) = f (ϕ(n − h)) + s=n

(9) Since the functions f and Q are continuous, we have P is continuous. Next we show the map P given by (9) is bounded. For ϕ ∈ B, which is bounded, we have | f (x)| = | f (x) − f (0) + f (0)| ≤ α|x| + | f (0)|. Similarly, using conditions (6) and (7), we have for x, y ∈ B that

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Y. N. Raffoul

|Q(n, x, y)| = |Q(n, x, y) − Q(n, 0, 0) + Q(n, 0, 0)| ≤ |Q(n, x, y) − Q(n, 0, 0)| + |Q(n, 0, 0)| ≤ k|x − 0| + (1 − k)|y − 0| + 1 = 1 + k|x| + (1 − k)|y|. This shows f and Q are bounded; say by positive constants M1 and M2 . Thus, ∞        Pϕ (n) =  f (ϕ(n − h)) + Q(s, ϕ(s), ϕ(s − h))C(n − s) + p(n)

≤ | f (ϕ(n − h))| +

s=n ∞ 

|Q(s, ϕ(s), ϕ(s − h))||C(n − s)| + | p(n)|

s=n

≤ M1 + M2

∞ 

|C(−u)| + | p(n)|.

u=0

Utilizing condition (8) we conclude the map P is bounded. Next we show the map P is a contraction. For ϕ, ψ ∈ B, we have that        Pϕ (n) − Pψ (n) ≤  f (ϕ(n − h)) − f (ψ(n − h))| +

∞      Q(s, ϕ(s), ϕ(s − h)) − Q(s, ψ(s), ψ(s − h))C(n − s) s=n

≤ α|ϕ − ψ| +

∞ 

[k|ϕ(s) − ψ| + (1 − k)|ϕ(s − h) − ψ(s − h)|]|C(n − s)|

s=n

≤ α||ϕ − ψ|| + ||ϕ − ψ||

∞ 

|C(−u)|

u=0

= ζ ||ϕ − ψ||.

Thus P is a contraction with unique fixed point. This completes the proof. The next result is concerned with the existence of a unique periodic solution and its proof is easily concluded from Theorem 1. Corollary 1 Assume the hypothesis of Theorem 1 with the addition of the periodicity condition Q(n + T, x, y) = Q(n, x, y), p(n + T ) = p(n), (10) for a least positive integer T , then (4) has a unique periodic solution. Proof Let P be defined as in Theorem 1. We denote the space of T -periodic sequences with the maximum norm ||x|| = max {|x(n)|} 1≤n≤T

Weighted Norms In Advanced Volterra Difference Equations

409

by (PT , || · ||). Then, (PT , || · ||) is a Banach space. It follows from Theorem 1, that P has a unique fixed point. A change of variable shows that for ϕ ∈ PT implies that Pϕ ∈ PT . Thus, the uniques fixed point is in PT . This completes the proof. For more on the connection between boundedness and the existence of periodic solutions we refer to [4]. In addition, for references on periodicity we refer to [15, 16].

3 Weighted Norm Depending on the function p(n) we choose a positive constant K and a function W : [−h, ∞) → [1, ∞) with

 p(n)  W (n − h)   ≤ K. max   < ∞ and n≥−h W (n) W (n)

(11)

Here the function W is the weight for a norm on Banach space. If we assume W is increasing, then our calculations will be much simplified, but we don’t make such assumption. In addition, W could be taken to be small and such choice depends on the function p.   Define B, | · |W as the Banach space of sequences ϕ : [−h, ∞) → R with norm

 ϕ(n)    |ϕ|W := max  . n≥−h W (n)

We want W to be compatible with C(n − s) and so we ask that max n≥0

∞   W (s)    ] < ∞. C(n − s)[ W (n) s=n

(12)

Finally, in order to have a contraction mapping we must ask for the existence a positive constant ζ such that ∞   W (s)      ζ = α K + k + K (1 − k) max ] < 1. C(n − s)[ n≥0 W (n) s=n

(13)

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Y. N. Raffoul

Theorem 2 Assume conditions (5)–(7) and (11)–(13) hold. Let ψ : [−h, 0] → R be a given sequence. Then there exists a sequence x ∈ B satisfying (4) for n ≥ 0 and x(n) = ψ(n) on [−h, 0]. Proof Let B ∗ = {ϕ ∈ B : ϕ(n) = ψ(n) on [−h, 0]}. Let ϕ ∈ B ∗ , and define P : B∗ → B∗ by (Pϕ)(n) = ψ(n) for n ∈ [−h, 0], and ∞    Q(s, ϕ(s), ϕ(s − h))C(n − s) + p(n), n ∈ Z+ . Pϕ (n) = f (ϕ(n − h)) + s=n

By using (11)-(12) we have that   ∞  P ϕ (n)   f (ϕ(n − h)) W (n − h)    Q(s, ϕ(s), ϕ(s − h))C(n − s)   p(n)    ≤ + +  W (n) W (n) W (n − h) W (n) W (n) s=n ∞    f (0  1 |ϕ(s)|  + α K |ϕ(n − h)| + ≤ +k W (n) W (n − h) s=n W (s) W (s)   ϕ(s − h) W (s)  p(n)  |C(n − s)| + < ∞, + (1 − k)K W (s − h) W (n W (n)

where we have used WW(n−h) ≤ K . Taking the maximum for n ≥ 0 we conclude that (n)   Pϕ  exists. Left to show that P is a contraction over the set B ∗ . Let ϕ1 , ϕ2 ∈ B ∗ . W Using (5) and (6) and by a similar reasoning as in the previous work we arrive at      P ϕ1 (n) − P ϕ2 (n)      ≤ α K ϕ1 − ϕ2  W W (n)

∞       W (s) + k ϕ1 − ϕ2 W + (1 − k)K ϕ1 − ϕ2 W |C(n − s)| W (n s=n     < ζ ϕ1 − ϕ2 W .

This shows P is a contraction over the set B ∗ . We conclude that the mapping P has a unique fixed point in B ∗ . This completes the proof.

Weighted Norms In Advanced Volterra Difference Equations

411

4 Applications In this section we provide three examples as applications to the results of Sect. 3. For the purpose of illustration and for a constant a > 1, we let x(n) = α1 x(n − h) + α2

∞ 

x(s)a b(n−s) + p(n), n ∈ Z+ , h ∈ N

(14)

s=n

where b is a positive constant. Example 1 If p(n) is bounded for all n ∈ Z+ , and |α1 | + α2

ab < 1, ab − 1

(15)

then (14) has a unique solution. Proof Since p(n) is bounded, we take W (n) = 1. Then condition (13) reduces to ζ = |α1 | + |α2 | max n≥0

∞ 

a b(n−s) < 1.

s=n

To be more precise, let u = s − n and γ = a1 . Then ∞ 

a

b(n−s)

=

s=n

∞ 

a b(n−s)

u=0

=

∞ 

a −bu =

u=0

∞ 

γ bu

u=0

γ bu ∞ 1 |u=0 = − b = b γ −1 γ −1 ab . = b a −1 Thus, if |α1 | + |α2 |

ab < 1, ab − 1

then (14) has a unique solution. In the next example we assume p(n) is unbounded and find an estimate on the rate of growth of the solutions.

412

Y. N. Raffoul

Example 2 Suppose p(n) = n for n ∈ Z+ , and ξ = |α1 |K + α2 max n≥0

a b (a b − 1)(n + h) + a b < 1, (a b − 1)2 (n + h + 1)

(16)

then (14) has a unique solution x(n, 0, ψ) satisfying the growth condition max | n≥0

x(n) | < ∞. (n + h + 1)

where ψ is an initial function satisfying x(n) = ψ(n) on [−h, 0], and K is a positive constant to be determined in the proof. Proof Set W (n) = n + h + 1. Then

p(n) W (n − h) n+1 < ∞ and = < 1 =: K . W (n) W (n) n+h+1

Next we compute condition (13). Since f (x(n − h)) = α1 x(n − h) and Q(n, x(n), x(n − h)) = α2 x(n), condition (13) reduces to ξ = α1 K + α2 max n≥0

∞   W (s)    ]. C(n − s)[ W (n) s=n

Thus, ∞  ∞  W (s)   s+h+1   ]= a b(n−s) C(n − s)[ W (n) n+h+1 s=n s=n ∞

 1 = (s + h + 1)a b(n−s) . n + h + 1 s=n To further reduce the above expression, we use the summation by parts formula that says for any two sequences y and z we have that 

zy = yz −



E yz,

where E y(n) = y(n + 1). Hence, we let z = s + h + 1 and y = a b(n−s) .

Weighted Norms In Advanced Volterra Difference Equations

Then y= which implies that E y = ∞ 

a b(n−s) and z = 1, a −b − 1

a b(n−s−1) . a −b −1

As a consequence, we have that

(s + h + 1)a b(n−s) =

s=n

413

∞  a b(n−s−1) s + h + 1 b(n−s) ∞ a −  s=n a −b − 1 a −b − 1 s=n

n+h+1 a −b  a b  − a −b − 1 a −b − 1 a b − 1 n+h+1 ab = + b −b 1−a (a − 1)2 ab a b (n + h + 1) + b = . b a −1 (a − 1)2 =−

Thus,  a b (n + h + 1)  1 ab 1 + (s + h + 1)a b(n−s) = n + h + 1 s=n n+h+1 ab − 1 (a b − 1)2 ∞

=

a b (a b − 1)(n + h) + a b . (a b − 1)2 (n + h + 1)

Thus, if ξ < 1, where ξ is given by (16) then (14) has a unique solution x(n) = x(n, 0, ψ) satisfying the growth condition x(n) | < ∞. max | n≥0 (n + h + 1) Example 3 Let c be a positive number satisfying 0 < c < b. Suppose p(n) = a nc for n ∈ Z+ . Let W (n) = a nc a hc , If ξ = |α1 |K + α2

1 < 1, 1 − a (c−b)

(17)

414

Y. N. Raffoul

then (14) has a unique solution x(n, 0, ψ) satisfying the growth condition max | n≥0

x(n) | < ∞. a nc

where ψ is an initial function satisfying x(n) = ψ(n) on [−h, 0], and K is a positive constant to be determined in the proof. Proof We easily have p(n) W (n − h) = a −hc < ∞ and = a −c =: K . W (n) W (n) Next we compute condition (13). Again, since f (x(n − h)) = α1 x(n − h) and Q(n, x(n), x(n − h)) = α2 x(n), condition (13) reduces to ξ = α1 K + α2 max n≥0

∞   W (s)    ]. C(n − s)[ W (n) s=n

Thus, ∞  ∞  W (s)   a sc a hc   a b(n−s) nc hc ]= C(n − s)[ W (n) a a s=n s=n

= a (b−c)n = a (b−c)n

∞ 

a (c−b)s

s=n (c−b)s

a

a (c−b) − 1

∞  

s=n

1 = , 1 − a (c−b) due to the assumption 0 < c < b. Thus, if ξ = |α1 |K + α2

1 < 1, 1 − a (c−b)

then (14) has a unique solution x(n) = x(n, 0, ψ) satisfying the growth condition max | n≥0

x(n) | < ∞. a nc

Weighted Norms In Advanced Volterra Difference Equations

415

5 Concluding Remarks Again, let’s consider the neutral delay functional difference equation x(n + 1) = αx(n + 1 − h) + ax(n) − q(n, x(n), x(n − h)), n ∈ Z+ , h ∈ N (18) where the function q : Z+ × R × R → R is continuous and α and a are constants with a > 0. We assume the periodicity condition q(n + T, x, y) = q(n, x, y)

(19)

for a least positive integer T. We denote the space of T -periodic sequences with the maximum norm ||x|| = max {|x(n)|} 1≤n≤T

by (PT , || · ||). Then, (PT , || · ||) is a Banach space. More results on such equations can be found in [7, 14]. Let x(n) ∈ PT and sum (2) from n − T to n − 1. Using the fact that x(n − T ) = x(n) gives the Volterra summation equation n−1    1 aαx(s − h) − q(s, x(s), x(s − h)) a n−s . T 1 − a s=n−T (20) The work we did for (4) is of more general nature and is not applicable to (20). We have the following theorem regarding the existence of a unique periodic solution of (20). A more general type of equations were studied in Chap. 4 of [14].

x(n) = αx(n − h) +

Theorem 3 Assume 1 − a T = 0 for all n ∈ [1, T − 1]. Suppose (19) holds and there exists k ∈ [0, 1) such that for all x, y, w, z ∈ R, and n ∈ Z+ |q(n, x, y) − q(n, w, z)| ≤ k|x − w| + (1 − k)|y − z|. (21) If ξ = |α| + (a|α| + 1)

a < 1, (1 − a)

then (18) has a unique periodic solution. Proof For ϕ ∈ PT define

P : PT → PT

(22)

416

Y. N. Raffoul

by n−1    1 aαϕ(s − h) − q(s, ϕ(s), ϕ(s − h)) a n−s . 1 − a T s=n−T (23) Then P is continuous in ϕ. Left to show the mapping P is a contraction. For ϕ, ψ ∈ PT , we have that

  Pϕ (n) = αϕ(n − h) +

n−1        Pϕ (n) − Pψ (n) ≤ |α|||ϕ − ψ|| + (|a||α| + 1) ||ϕ − ψ|| a n−s 1 − aT s=n−T

(a|α| + 1) a(1 − a T ) = |α|||ϕ − ψ|| + ||ϕ − ψ|| T 1−a (1 − a)  a ||ϕ − ψ|| = |α| + (a|α| + 1) (1 − a) = ξ ||ϕ − ψ||. Thus P is a contraction with unique fixed point in PT . a Note that in condition (22) the term 1−a is exposed and it has to satisfy the condition a < 1, which is true for a < 1/2. Thus, in order for (22) to be valid, we must require 1−a 0 < a < 1/2. At the expense of losing uniqueness, the restriction 0 < a < 1/2 can be relaxed a bit by using Krasnoseslkii’s fixed point theorem and for such study we refer to [14].

References 1. Burton, Theodore: Integral equations, implicit functions, and fixed points. Proc. Am. Math. Soc. 124, 2383–2390 (1996) 2. Dannan, Fozi, Elaydi, Saber: Asymptotic stability of linear difference equations of advanced type. J. Comput. Anal. Appl. 6(2), 173–187 (2004) 3. Elaydi, Saber: An Introduction to Difference Equations. Springer, New York (1999) 4. Kaufmann, E., Kosmatov, N., Raffoul, Y.: The connection between boundedness and periodicity in nonlinear functional neutral dynamic equations on a time scales. Nonlinear Dyn. Syst. Theory 9(1), 89–98 (2009) 5. Kelley, W., Peterson, A.: Difference Equations an Introduction with Applications. Academic (2001) 6. Li, Wan-Tong., Huo, Hai-Feng.: Positive periodic solutions of delay difference equations and applications in population dynamics. J. Comput. Appl. Math. 176(2), 357–369 (2005) 7. Maroun, Mariette, Raffoul, Youssef: Periodic solutions in nonlinear neutral difference equations with functional delay. J. Korean Math. Soc. 42(2), 255–268 (2005) 8. Migda, Janusz: Asymptotic behavior of solutions of nonlinear difference equations. Math. Bohem. 129(4), 349–359 (2004) 9. Islam, M., Yankson, E.: Boundedness and stability for nonlinear delay difference equations employing fixed point theory. Electron. J. Qual. Theory Wiffer. Equ. No. 26, 18 pp (2005)

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10. Qian, Chuanxi, Sun, Yijun: On global attractivity of nonlinear delay difference equations with a forcing term. J. Differ. Equ. Appl. 11(3), 227–243 (2005) 11. Raffoul, Youssef: Periodicity in general delay non-linear difference equations using fixed point theory. J. Differ. Equ. Appl. 10(13–15), 1229–1242 (2004) 12. Raffoul, Youssef: Periodicity in general delay non-linear difference equations using fixed point theory. J. Differ. Equ. Appl. 10(13–15), 1229–1242 (2004) 13. Raffoul, Youssef: General theorems for stability and boundedness for nonlinear functional discrete systems. J. Math. Anal. Appl. 279, 639–650 (2003) 14. Raffoul, Y.: Qualitative Theory of Volterra Difference Equations. Springer Nature Switzerland (2018) 15. Chatzarakis, G., Stavroulakis, I.P.: Oscillations of difference equations with general advanced argument. Cent. Eur. J. Math. ? 10(2), 807–823 (2012) 16. Zhu, Huiyan; Huang, Lihong, Asymptotic behavior of solutions for a class of delay difference equation., Ann. Differential Equations 21 (2005), no. 1, 99–105

Comparison of Tests for Oscillations in Delay/Advanced Difference Equations with Continuous Time Andrea Rožnjik, Hajnalka Péics, and George E. Chatzarakis

Abstract In this paper we compare sufficient conditions for the oscillation of all solutions of the delay (advanced) difference equation with continuous time inspired by our results published in [Filomat 34(8), 2693–2704 (2020)] to relevant results in the literature. The oscillatory conditions appear for various functional equations, but their comparison is possible for some special forms. We provide various examples with constant delays (advances) and with variable or constant coefficients, on which we have shown the independency of our conditions. Keywords Functional equations · Difference equations with continuous time · Oscillatory solutions

1 Introduction The oscillatory behavior of the solutions of difference equations with continuous time has been a prominent research subject. There have been several results that have broken new ground and very significantly advanced our understanding on this topic. Thus, notable work in this field includes the studies by Golda and Werbowski [2] and Shen and Stavroulakis [9, 10] who have considered second-order linear functional equations. Nowakowska and Werbowski in [4–8] and Zhang and Choi in [11] investigated higher-order linear functional equations. Scalar delay difference equations with continuous arguments have been studied by Ladas, Pakula and Wang [3], Zhang and Yan [12], Zhang, Yan and Zhao [14], Zhang, Yan and Choi [13].

A. Rožnjik (B) · H. Péics Faculty of Civil Engineering, University of Novi Sad, Kozaraˇcka 2/A, Subotica, Serbia e-mail: [email protected] G. E. Chatzarakis Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), Marousi, 15122 Athens, Greece © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_19

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In paper [1] we proposed sufficient conditions for the oscillation of solutions of difference equations with continuous time caused by several deviating arguments. The observed delay difference equation was Δx(t) +

m 

p¯ i (t)x(t − k¯i (t)) = 0, t ≥ t0 ,

(1)

i=1

and its dual, the advanced difference equation, ∇x(t) −

m 

p¯ i (t)x(t + k¯i (t)) = 0, t ≥ t0 ,

(2)

i=1

where Δx(t) = x(t + 1) − x(t), ∇x(t) = x(t) − x(t − 1), m ∈ {1, 2, . . .}, functions p¯ i : [t0 , ∞) → (0, ∞), i = 1, 2, . . . , m, are continuous, functions k¯i : [t0 , ∞) → {1, 2, . . .}, i = 1, 2, . . . , m are piecewise constant and t0 ∈ IR. In this paper, we aim to compare the proposed conditions to relevant results in the literature for which the comparison had been the most interesting or challenging. Setting delays and advances to be constant, our results became comparable with results in the abovementioned literature. Therefore, in this paper, we consider the delay difference equation with constant delays x(t) − x(t − 1) +

m 

Pi (t)x(t − K i ) = 0, t ≥ t0 + 1

(3)

i=1

and the advanced difference equation with constant advances x(t + 1) − x(t) −

m 

Pi (t)x(t + K i ) = 0, t ≥ t0 − 1,

(4)

i=1

where K i are integers greater than 2, i = 1, 2, . . . , m. Equation (3) is obtained replacing t + 1 by t in Eq. (1) and taking k¯i (t − 1) + 1 ≡ K i and Pi (t) = p¯ i (t − 1) for i = 1, 2, . . . , m. Similarly, replacing t − 1 by t in Eq. (2) and taking k¯i (t) + 1 ≡ K i and Pi (t) = p¯ i (t + 1) for i = 1, 2, . . . , m, the Eq. (4) is obtained.

2 The Main Test Let us start with the main tests used in comparison. First we state the test for oscillatory solutions of delay differential equation (3) proposed in [1]. Theorem 1 ([1, Corollary 4.1]) If, for constant delays K i ∈ {2, 3, . . .}, and positive and continuous functions Pi , i = 1, 2, . . . , m, the condition

Comparison of Tests for Oscillations in Delay/Advanced Difference …

lim inf t→∞

t−1    K −1 τ =t− i2

421

Pi (τ ) > 0 for K i ≥ 3, i = 1, 2, . . . , m

(5)

in conjunction with the condition ⎛ ⎛ m m   ⎝ ⎝ lim inf or

i=1

j=1

⎛

⎛

t→∞

t−1 

⎞⎞ m1 Pi (τ )⎠⎠ >

τ =t−K j +1

1 e

⎞ 21 ⎞2 1 ⎜ 1 ⎟ ⎝lim inf Pi (τ )⎠ ⎠ > ⎝ t→∞ m i=1 e τ =t−K +1 m 

t−1 

(6)

(7)

i

is fulfilled, then all solutions of (3) are oscillatory. The formulation of the test for oscillatory solutions of advanced differential equation (4) is the following. Theorem 2 ([1, from Theorem 3.2]) If, for constant advances K i ∈ {2, 3, . . .}, and positive and continuous functions Pi , i = 1, 2, . . . , m, the condition   K −1 t+ i2

lim inf t→∞



Pi (τ ) > 0 for K i ≥ 3, i = 1, 2, . . . , m

(8)

τ =t+1

and one of the conditions ⎞⎞ m1 ⎛ t+K j −1 m m    1 ⎝ ⎝ lim inf Pi (τ )⎠⎠ > t→∞ e τ =t+1 i=1 j=1 ⎛

or

⎛

21 ⎞2 t+K m i −1  1 ⎝ 1 Pi (τ ) ⎠ > , lim inf m i=1 t→∞ τ =t+1 e

(9)

(10)

are satisfied, then every solution of (4) oscillates. Since comparison with some of results will be presented for delay difference equations with constant coefficients, i.e., for Pi (t) ≡ pi ∈ (0, ∞) for i = 1, 2, . . . , m, we also state the oscillatory conditions for the delay difference equation

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x(t) − x(t − 1) +

m 

pi x(t − K i ) = 0, t ≥ t0 + 1.

(11)

i=1

Theorem 3 ([1, Corollary 4.2]) For constants K i ∈ {2, 3, . . .} and pi ∈ (0, ∞), i = 1, 2, . . . , m, the condition m  i=1

or 1 m

⎞

m1 ⎛ m  ⎝ (K j − 1)⎠ > 1 pi e j=1

m 

2 pi (K i − 1)

>

i=1

1 e

(12)

(13)

implies that Eq. (11) possesses only oscillatory solutions. In [1], we already illustrated by examples that our stated conditions are independent of each other, but this feature will also appear in some of the following examples.

3 The Comparison to the Higher-Order Functional Equations The first group of oscillatory conditions which are comparable with our oscillatory conditions forms conditions for the higher-order functional equations. In these comparisons, we can compare results for delay differential equations and for advanced differential equations.

3.1 Nowakowska-Werbowski Conditions Equations (3) and (4) are special cases of the higher-order functional equations. Nowakowska and Werbowski in [5–7] observed higher-order functional equation of the form s−1 m+1   x(g s (t)) = Q i (t)x(g i (t)) + Q i (t)x(g i (t)), (14) i=0

i=s+1

with m ∈ {1, 2, . . .}, s ∈ {1, 2, . . . , m}, nonnegative real-valued functions Q i , i = 0, 1, . . . , s − 1, s + 1, s + 2, . . . , m + 1, and the iterative function g defined by g 0 (t) = t, g i+1 (t) = g(g i (t)), i = 0, 1, . . . .

(15)

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The special case of Eq. (14) for s = 1 appeared in Nowakowska and Werbowski [4]. Equation (14) yields (3) setting the parameters and coefficients as follows: s = 1,Q 0 (t) ≡ 1 and Q i (t) = Pi−1 (t), i = 2, 3, . . . , m + 1 and g(t) = t − 1. Taking same parameters and coefficients, but g(t) = t + 1, Eq. (14) yields (4). The higher-order functional equation x(g(t)) =

m 

Pi (t)x(g

i+1

(t)) +

i=1

l 

A j (t)x(g − j (t)),

(16)

j=0

where m ∈ {1, 2, . . .}, l ∈ {0, 1, 2, . . .}, Pi for i = 1, 2, . . . , m and A j for j = 1, 2, . . . , l are nonnegative real-valued functions, A0 is positive real-valued function, and function g is defined by (15) and g −1 is its inverse function, is studied by Nowakowska and Werbowski in [8]. Equation (16) becomes (3) taking l = 0, A0 (t) ≡ 1, and g(t) = t − 1, but with g(t) = t + 1 it becomes (4). In paper [1] we already showed that tests for oscillation stated in Theorem 1 are independent of the conditions in papers Nowakowska and Werbowski [4–8], therefore we omit stating those conditions and comparisons in this paper. To illustrate the duality in the oscillatory conditions we use Example 4.4 from [1] and comparison of our results to Nowakowska and Werbowski’s results from [6]. But, first we state the sufficient oscillatory conditions from [6] formulated to Eqs. (3) and (4). Theorem 4 ([6, Theorems 2, 4]) Let B(t) =

m 

Pi (t)

i=1

and B1 (t) = B(t) + B(g(t)) + B(g(t))B(g 2 (t)). If 1 4

(17)

lim sup B1 (t) > 1

(18)

lim inf B(t) > t→∞

or t→∞

or B(t) ≥ δ > 0, δ
1 − δ 2 , 4 t→∞

(19)

then, for g(t) = t − 1, all solutions of (3) are oscillatory and, for g(t) = t + 1, all solutions of (4) are oscillatory.

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Notice that condition (17) does not depend on the choice of function g, so it is same for the observed delay and advance equations. Example 1 Consider the delay difference equation sin 2t + 8 13 x(t − 2) + x(t − 3) = 0, t ≥ 1 100 60

(20)

sin 2t + 8 13 x(t + 2) − x(t + 3) = 0, t ≥ −1. 100 60

(21)

x(t) − x(t − 1) + and the dual equation x(t + 1) − x(t) −

13 2t+8 Here m = 2, P1 (t) ≡ 100 , P2 (t) = sin 60 , K 1 = 2 and K 2 = 3. Thus our conditions are fulfilled, so every solution of (20) and (21) oscillates. Namely, since 1 ≈ 0.367879, e

Delay equation

Advance equation

condition (5) :

condition (8) :

1 7 1 7 2 2 − = > 0 lim inf P2 (t + 1) = − = >0 lim inf P2 (t − 1) = t→∞ t→∞ 15 60 60 15 60 60 condition (6) : 2   lim inf Pi (t − 1)+

condition (9) : 2   lim inf Pi (t + 1)+

t→∞

i=1

+ lim inf (Pi (t − 2) + Pi (t − 1)) t→∞

≈ 0.37746 >

2

t→∞

i=1

 1

 1

≈ + lim inf (Pi (t + 1) + Pi (t + 2)) t→∞

≈ 0.37746 >

1 e

2

≈

1 e

condition (7) :  1  lim inf P1 (t − 1)+ t→∞ 2 2  + lim inf (P2 (t − 2) + P2 (t − 1)) ≈

condition (10) :  1  lim inf P1 (t + 1)+ t→∞ 2 2  + lim inf (P2 (t + 1) + P2 (t + 2)) ≈

≈ 0.369121 >

≈ 0.369121 >

t→∞

1 e

t→∞

1 e

Considering Nowakowska and Werbowski’s conditions we have B(t) = P1 (t) + P2 (t) =

1 79 + sin 2t, 300 60

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so lim inf B(t) = t→∞

1 37 ≈ 0.246667 < . 150 4

Consequently condition (17) is not fulfilled. For function B1 holds the following. Delay equation (g(t) = t − 1) Advance equation (g(t) = t + 1) B1 (t) = B(t) + B(t − 1)+

B1 (t) = B(t) + B(t + 1)+

+B(t − 1)B(t − 2)

+B(t + 1)B(t + 2)

lim inf B1 (t) ≈ 0.612428 t→∞

lim inf B1 (t) ≈ 0.612428 t→∞

Hence, condition (18) is not satisfied for both considered equations. Due to for δ such that 1 B(t) ≥ δ > 0 and δ < 4 it holds  2 1 15 1−δ >1− ≈ 0.9375 = 4 16 2

and therefore lim inf B1 (t) < 1 − δ 2 . t→∞

Thus, condition (19) is not fulfilled for Eqs. (20) and (21), as well. As the example indicates, our oscillatory conditions are independent from Nowakowska and Werbowski’s conditions (17)–(19) for delay difference equation (3) and advanced difference equation (4).

3.2 Zhang-Choi Conditions Equations (3) and (4) are also special cases of the higher-order functional equation x(g(t)) = p(t)x(t) +

m 

Pi (t)x(g L+i (t))

(22)

i=1

studied by Zhang and Choi [11]. Parameters m, L ∈ {1, 2, . . .}, function g is defined by (15), and functions p and Pi , i = 1, 2, . . . , m, are positive real-valued. Hence, for

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p(t) ≡ 1 and g(t) = t − 1 or g(t) = t + 1 we obtain Eq. (3) or (4) with K i = L + i, i = 1, 2, . . . , m. In the following theorem, we state the sufficient oscillatory conditions proposed in [11] formulated for delay differential equation with constant coefficients (11). Theorem 5 ([11, Theorem 4.1]) Let B2 =

m 

pj.

j=1

If B2 > or

(K 1 − 1) K 1 −1

(23)

K 1K 1

  K −1 m  i 1 Ki − 1 r,

(25)

1

or  where r is the largest positive root of the equation r = 1 − then all solutions of (11) are oscillatory.

 B2 K 1 −1 , r

Let us consider what happens when m = 1. In that case, Zhang and Choi’s first condition, i.e., inequality (23), becomes p1 >

(K 1 − 1) K 1 −1 K 1K 1

(26)

and our both conditions, i.e., conditions (12) and (13), become p1 (K 1 − 1) >

1 . e

(27)

1 , e

(28)

Due to for every positive integer K 1 it holds (K 1 − 1) K 1 K 1K 1


1 our second condition is independent of Zhang and Choi’s conditions (23)–(25).

Comparison of Tests for Oscillations in Delay/Advanced Difference …

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Example 2 For the delay difference equation x(t) − x(t − 1) + m = 2, p1 =

3 , 50

p2 =

19 , 100

3 19 x(t − 2) + x(t − 3) = 0, t ≥ 0, 50 100

(29)

K 1 = 2, K 2 = 3, and condition (13) gives

2  √  1  1 1  11 + 57 ≈ 0.370997 > . p1 (K 1 − 1) + p2 (K 2 − 1) = 2 500 e This yields that Eq. (29) has only oscillatory solutions. Since 1 B2 = p1 + p2 = 4 and

(K 1 − 1) K 1 −1 K 1K 1

=

1 , 4

condition (23) does not hold. Moreover, condition (24) does not hold, as well, since K1 − 1 2K 1



1 2 p1 K 1

 K 1−1 1

K2 − 1 + 2K 2

The equation



1 2 p2 K 2

 K 1−1 2

√ 25 5 114 + ≈ 1.35386 > 1. = 24 171

  B2 K 1 −1 r = 1− r

is indeed r =1−

1 , i.e., 4r 2 − 4r + 1 = 0 4r

and its solutions are r = 0.5, Therefore, condition (25) takes the form 1 > 0.5, 4 which is not fulfilled. Notice that for previous example condition (12), i.e. our first condition, does not holds because √ 1 3 114 √ ≈ 0.32031 < . p1 p2 (K 1 − 1 + K 2 − 1) = 100 e As our second condition is satisfied and the first one is not satisfied, it is illustrated that our second condition is independent of the first one.

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3.3 The Comparison to the Second-Order Functional Equations For m = 1 and K 1 = 2 Eqs. (3) and (4) are comparable with the second-order functional equations (Eq. (14) with s = 1 and M = 1) studied in Golda and Werbowski in [2] and Shen and Stavroulakis in [9] and [10]. The conclusion from those papers is that the condition 1 (30) lim inf P1 (t) > t→∞ 4 ensures that every solution of (3) and (4) with m = 1 and K 1 = 2 oscillates. The condition (30) is a so called “sharp” condition since it is a necessary and sufficient condition when coefficients are constant. Conditions of Theorems 1 and 2 with m = 1 and K 1 = 2 reduce to lim inf P1 (t − 1) >

1 e

(31)

lim inf P1 (t + 1) >

1 , e

(32)

t→∞

and t→∞

respectively. Since 1 1 > , e 4 if (31) or (32) holds, then (30) is also satisfied. Therefore, the fulfillment of our conditions for m = 1 and K 1 = 2 implies the fulfillment of “sharp” condition.

4 The Comparison to the Delay Difference Equations In this section, we provide comparisons with the oscillatory results of the literature dealing with delay differential equations.

4.1 Ladas-Pakula-Wang Conditions Ladas, Pakula and Wang in [3] investigated the oscillatory behavior of delay differential equations with constant coefficients. The most general form of the differential equation considered by them has the form x(t) − px(t − T ) +

m  i=1

pi x(t − K i ) = 0.

Comparison of Tests for Oscillations in Delay/Advanced Difference …

429

The above equation, for p = 1 and T = 1, reduces (11) with sufficient oscillatory condition equivalent to m  K iK i pi >1 (33) (K i − 1) K i −1 i=1 (see [3], Theorem 3). For m = 1 condition (33) becomes the inequality p1

K 1K 1 > 1, (K 1 − 1) K 1 −1

which we saw in comparison to Zhang-Choi condition, i.e., condition (26). Our conditions for this case become inequality (27). Therefore, the fulfillment of our conditions for m = 1 and constant coefficients implies the fulfillment of LadasPakula-Wang condition. But, the independence of our first condition, condition (12), of Ladas, Pakula and Wang’s condition (33) is shown in next example, where m > 1. Example 3 For the delay difference equation x(t) − x(t − 1) +

1 1 x(t − 2) + x(t − 13) = 0, t ≥ 0, 10 100

(34)

1 1 m = 2, p1 = 10 , p2 = 100 , K 1 = 2, and K 2 = 13. Our condition (12), gives

√

√ 1 13 ·10 ≈ 0.411096 > . p1 p2 (K 1 − 1 + K 2 − 1) = 100 e

Thus, all solutions of (34) are oscillatory. On the other hand, condition (33) takes the form p1

K 1K 1 K 2K 2 + p ≈ 0.739695 < 1. 2 (K 1 − 1) K 1 −1 (K 2 − 1) K 2 −1

Therefore, the condition (33) is not fulfilled. As 2  √  1  1  1 11 + 2 30 ≈ 0.219545 < , p1 (K 1 − 1) + p2 (K 2 − 1) = 2 100 e our second condition (13) is not satisfied for Eq. (34). Therefore, the delay difference equation (34) confirms that our first condition is independent from the second one.

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4.2 Zhang-Yan-Zhao Conditions Zhang, Yan and Zhao in [14] considered the difference equation with constant delay and variable coefficients x(t) − x(t − T ) + P1 (t)x(t − K 1 ) = 0,

(35)

where T and K 1 are positive constants such that T < K 1 and P1 is positive, continuous, real-valued function. Since the special case of Eq. (3) with m = 1 is equivalent to the special case of (35) for T = 1, our oscillatory conditions are comparable to the oscillatory criteria established in [14], through the equation x(t) − x(t − 1) + P1 (t)x(t − K 1 ) = 0.

(36)

Zhang, Yan and Zhao sufficient conditions from [14] stated for Eq. (36) have the following forms. Theorem 6 ([14, Theorem 2]) The condition lim inf P1 (t) >

(K 1 − 1) K 1 −1

t→∞

K 1K 1

(37)

ensures that every solution of (36) oscillates. Taking coefficient to be constant, i.e., for P1 (t) ≡ p1 ∈ (0, ∞), condition (37) obtains the form of inequality (26). In view of our oscillatory conditions for Eq. (36) with P1 (t) ≡ p1 obtain the form of inequality (27), we have that the fulfillment of our oscillatory conditions for Eq. (36) with a constant coefficient imply the fulfillment of Zhang, Yan and Zhao’s conditions for a constant coefficient. Actually, this was expected since in [10] it is emphasized that condition (37) is a “sharp” condition. Furthermore, if K 1 = 2, then (K 1 − 1) K 1 −1 K 1K 1

=

1 4

and condition (37) becomes the “sharp” condition (30). Therefore, in this case, the fulfillment of our oscillatory conditions for Eq. (36), which are reduced to (31), imply the fulfillment of compared condition (37). But, when we compare our conditions (conditions (6) and (7)) for scalar delay difference equations with variable coefficients and delay K 1 > 2 with Zhang, Yan

Comparison of Tests for Oscillations in Delay/Advanced Difference …

431

and Zhao’s condition (37), we obtained independency, as it is shown in following example. Example 4 For the delay difference equation  x(t) − x(t − 1) + P1 (t) = 18 sin 2t + becomes

13 50

 1 13 x(t − 3) = 0, t ≥ 0, sin 2t + 8 50

(38)

and K 1 = 3. The common form of conditions (6) and (7)

lim inf (P1 (t − 2) + P1 (t − 1)) > t→∞

1 . e

Then condition (5) takes the form lim inf P1 (t − 1) > 0. t→∞

Clearly,  lim inf P1 (t − 1) = lim inf t→∞

t→∞

1 13 sin 2(t − 1) + 8 50

 =

13 1 27 − = >0 50 8 200

and   1 1 13 sin 2(t − 2) + sin 2(t − 1) + lim inf (P1 (t − 2) + P1 (t − 1)) = lim inf t→∞ t→∞ 8 8 25   1 2(t − 2) + 2(t − 1) 2(t − 2) − 2(t − 1) 13 · 2 sin cos + = lim inf t→∞ 8 2 2 25   1 13 sin (2t − 3) cos (−1) + = lim inf t→∞ 4 25 1 13 1 − cos 1 ≈ 0.384924 > . = 25 4 e Thus, condition (6), i.e., condition (7) is fulfilled and implies that every solution of (38) oscillates. Observe, however, that   1 13 27 lim inf P1 (t) = lim inf sin 2t + = = 0.135, t→∞ t→∞ 8 50 200 (K 1 − 1) K 1 −1 4 ≈ 0.148148, = K1 27 K1 that is Zhang, Yan and Zhao’s condition (37) is not satisfied.

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4.3 Zhang-Yan-Choi Conditions Difference equation with constant delay and variable coefficients (35) is also studied by Zhang, Yan and Choi in [13]. Therefore our oscillatory conditions are comparable to the oscillatory criteria presented in [13] through Eq. (36). In the following theorem, sufficient conditions from [13] are stated for Eq. (36). Theorem 7 ([13, Theorem 2.1]) Let q(t) = min P1 (s) and E = {λ > 0 | 1 − λq(t) > 0 eventually} . t−1≤s≤t

(39)

If lim sup q(t) > 0

(40)

t→∞

and there exists τ > 0 such that sup λ λ∈E t≥τ

K 1 −1

(1 − λq(t − i)) < 1,

(41)

i=1

then every solution of (36) oscillates. For a constant coefficient, P1 (t) ≡ p1 ∈ (0, ∞), function q and set E from the last theorem become q(t) ≡ p1

    1  and E = λ > 0  >λ . p1

Hence, condition (40) is satisfied due to p1 is a positive constant and condition (41) obtains the form (42) sup λ (1 − λp1 ) K 1 −1 < 1. 0 1 are also independent from Zhang, Yan and Choi’s sufficient conditions (41). Example 5 We already showed that our conditions (6) and (7) are satisfied for difference equation (38). Considering conditions (40) and (41) we have  q(t) = min P1 (s) = min t−1≤s≤t

t−1≤s≤t

13 1 sin 2s + 8 50

 ≥

13 1 27 − = > 0. 50 8 200

Therefore lim sup q(t) > 0. t→∞

To show that condition (41) is not fulfilled, we are looking for variables λ and t such that the expression λ (1 − λq(t − 1)) (1 − λq(t − 2)) is greater than 1. Due to the 27 definition of set E in (39) and the fact that the minimal value of function q is 200 , we get 200 1 ≤ . 0 0, we have that sup λ (1 − λq(t − 1)) (1 − λq(t − 2)) ≥ λ∈E t≥τ

≥

   200 27 200 27 800 200 1− · 1− · = > 1. 81 81 200 81 200 729

Consequently, Zhang, Yan and Choi’s condition (41) is not fulfilled.

4.4 Zhang-Yan Conditions The oscillatory behavior of constant delay differential equations with variable coefficients have been also investigated by Zhang and Yan [12] in the form x(t) − x(t − T ) +

m 

Pi (t)x(t − K i ) = 0

(43)

i=1

with positive continuous real-valued functions Pi , i = 1, 2, . . . , m and positive constants T and K i , i = 1, 2, . . . , m, such that T ≤ K 1 < K 2 < . . . < K m . Taking T = 1 Eq. (43) is reduced to Eq. (3) with oscillation criteria in the following form. Theorem 8 ([12, Theorems 1–2]) If lim inf Pi (t) > 0, i = 1, 2, . . . , m t→∞

and one of the conditions

Comparison of Tests for Oscillations in Delay/Advanced Difference … m  i=1

or m

m  i=1

K iK i lim inf Pi (t) > 1 (K i − 1) K i −1 t→∞

m1

m KK 1  > 1, where K = Ki , (K − 1) K −1 m i=1

lim inf Pi (t) t→∞

435

(44)

(45)

or K i ≥ 2, i = 1, 2, . . . , m and there exists an nonnegative increasing sequence {tn }∞ n=1 such that lim tn = ∞

n→∞

and lim sup t→∞

m  i=1

min

τ ∈[tn ,tn +1]

Pi (τ ) > 1 −

m  i=1

lim inf Pi (t) t→∞

(46)

is fulfilled, then Eq. (3) possesses only oscillatory solutions. When m = 1, then condition (44) becomes Zhang-Yan-Zhao condition (37), i.e., a “sharp” condition, for which we have already showed that the fulfillment of our oscillatory conditions imply it for a constant coefficient or K 1 = 2. Even for m = 2 and constant coefficients we cannot obtain that our conditions are independent from Zhang-Yan conditions. Namely, when m = 2, P1 (t) ≡ p1 and P2 (t) ≡ p2 , condition (45) is √ 2 p1 p2

KK K1 + K2 > 1, with K = (K − 1) K −1 2

(47)

and our first condition becomes √

p1 p2 (K 1 + K 2 − 2) >

1 . e

Since K 1 + K 2 = 2K , (48) is equivalent to 1 √ 2 p1 p2 (K − 1) > . e Further, using the fact that

1 e

>

(K −1) K KK

, we obtain

(K − 1) K √ , 2 p1 p2 (K − 1) > KK which is equivalent to (47). Therefore, condition (48) implies condition (47).

(48)

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But, with m = 2 and variable coefficients we can show that our oscillatory conditions (6) and (7) are independent from Zhang and Yan oscillatory conditions (44)– (46), as it is illustrated by the following example. Example 6 Consider the delay difference equation     31 16 x(t − 3) + sin 7t + x(t − 4) = 0, t ≥ 0. (49) x(t) − x(t − 1) + sin 7t + 15 30

Now m = 2, P1 (t) = sin 7t + condition (5) becomes lim inf P1 (t − 1) = t→∞

16 , 15

P2 (t) = sin 7t +

K 1 = 3 and K 2 = 4, thus

31 , 30

1 1 > 0, lim inf P2 (t − 1) = > 0, t→∞ 15 30

and it is fulfilled. Due to t−1 

lim inf t→∞

lim inf t→∞

P1 (τ ) ≈ 0.26042,

τ =t−K 1 +1 t−1 

P2 (τ ) ≈ 0.193753,

τ =t−K 1 +1

lim inf t→∞

t−1 

P1 (τ ) ≈ 0.692195,

τ =t−K 2 +1

lim inf t→∞

t−1 

P2 (τ ) ≈ 0.592195,

τ =t−K 2 +1

conditions (6) and (7) take the respective forms 2  lim inf i=1

t→∞

t−1 

Pi (τ ) + lim inf

τ =t−K 1 +1

t→∞

21

t−1 

Pi (τ )

≈ 0.865279 >

τ =t−K 2 +1

1 , e

 ⎞2 ⎛   t−1 t−1     1 ⎝ 1 lim inf P1 (τ ) + lim inf P2 (τ )⎠ ≈ 0.819016 > t→∞ t→∞ 2 e τ =t−K +1 τ =t−K +1 1

2

and are fulfilled. Consequently, all solutions of (49) are oscillatory. At the same time, lim inf P1 (t) = t→∞

1 1 > 0, lim inf P2 (t) = > 0, t→∞ 15 30

but no one of the conditions (44) and (45) are satisfied. Namely,

Comparison of Tests for Oscillations in Delay/Advanced Difference … 2  i=1

437

K iK i lim inf Pi (t) ≈ 0.766049 < 1, (K i − 1) K i −1 t→∞



2 lim inf P1 (t) · lim inf P2 (t) t→∞

t→∞

KK ≈ 0.765265 < 1, (K − 1) K −1 K1 + K2 7 with K = = . 2 2

Further, since period of functions P1 and P2 is less than 1, for every nonnegative increasing sequence {tn }∞ n=1 such that lim n→∞ tn = ∞, it holds min

τ ∈[tn ,tn +1]

Pi (τ ) = lim inf Pi (t). t→∞

Hence, condition (46) becomes lim sup t→∞

2  i=1

lim inf Pi (t) > 1 − t→∞

2 

lim inf Pi (t),

i=1

t→∞

i.e., 2

2  i=1

lim inf Pi (t) > 1. t→∞

As 2

2  i=1

lim inf Pi (t) = t→∞

1 , 5

condition (46) is not satisfied, as well.

5 Summary In this paper, we compared our conditions for oscillatory solutions of the delay/ advanced difference equations with continuous time and several deviating arguments from [1] to conditions that appeared in the literature. Results have been comparable only for special forms of considered equations, although various general forms can be found. For difference equations with two deviating arguments (m = 1) and constant coefficients there are no better conditions than “sharp” conditions, but considering variable coefficients we showed the independence of our conditions of the compared ones. In some comparisons, we reached the independency with constant coefficients,

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but with more than two deviating arguments (m > 1). There were comparisons where delays greater than 2 ensured that our conditions are independent of the compared ones. At final, we would like to emphasize that our oscillatory conditions and the compared conditions are, in original form, valid for more general forms of functional equations than Eqs. (3) and (4). Acknowledgements The research of H. Péics is supported by the Serbian Ministry of Science, Technology and Development for Scientific Research Grant No. III44006.

References 1. Chatzarakis, G.E., Péics, H., Rožnjik, A.: Oscillations in difference equations with continuous time caused by several deviating arguments. Filomat 34(8), 2693–2704 (2020). https://doi.org/ 10.2298/FIL2008693C 2. Golda, W., Werbowski, J.: Oscillation of linear functional equations of the second order. Funkcial. Ekvac. 37, 221–227 (1994) 3. Ladas, G., Pakula, L., Wang, Z.: Necessary and sufficient conditions for the oscillation of difference equations. Panamer. Math. J. 2(1), 17–26 (1992) 4. Nowakowska, W., Werbowski, J.: Oscillation of linear functional equations of higher order. Arch. Math. (Brno) 31(4), 251–258 (1995) 5. Nowakowska, W., Werbowski, J.: Oscillatory behavior of solutions of functional equations. Nonlinear Anal. 44(6) 767–775 (2001). Ser. A: Theory Methods. https://doi.org/10.1016/ S0362-546X(99)00306-5 6. Nowakowska, W., Werbowski, J.: Conditions for the oscillation of solutions of iterative equations. Abstr. Appl. Anal. 7, 543–550 (2004). https://doi.org/10.1155/S1085337504306305 7. Nowakowska, W., Werbowski, J.: Oscillatory solutions of linear iterative functional equations. Indian J. Pure Appl. Math. 35(4), 429–439 (2004) 8. Nowakowska, W., Werbowski, J.: Oscillation of all solutions of iterative equations. Nonlinear Oscil. 10(3), 351–366 (2007). https://doi.org/10.1007/s11072-007-0029-6 9. Shen, J., Stavroulakis, I. P.: An oscillation criteria for second order functional equations. Acta Math. Sci . Ser. B Engl. Ed. 22(1), 56–62 (2002). https://doi.org/10.1016/S02529602(17)30455-1 10. Shen, J., Stavroulakis, I.P.: Sharp conditions for nonoscillation of functional equations. Indian J. Pure Appl. Math. 33(4), 543–554 (2002) 11. Zhang, B.G., Choi, S.K.: Oscillation and nonoscillation of a class of functional equations. Math. Nachr. 227, 159–169 (2001). https://doi.org/10.1002/1522-2616(200107)227:13.0.CO;2-D 12. Zhang, Y.Z., Yan, J.R.: Oscillation criteria for difference equations with continuous arguments. Acta Math. Sinica 38(3), 406–411 (1995). (in Chinese) 13. Zhang, B.G., Yan, J., Choi, S.K.: Oscillation for difference equations with continuous variable. Comput. Math. Appl. 36(9), 11–18 (1998). https://doi.org/10.1016/S0898-1221(98)00189-8 14. Zhang, Y., Yan, J., Zhao, A.: Oscillation criteria for a difference equation. Indian J. Pure Appl. Math. 28(9), 1241–249 (1997)

Krause Mean Processes Generated by Cubic Stochastic Matrices IV: Off-Diagonally Uniformly Positive Nonautonomous Cubic Stochastic Matrices Mansoor Saburov and Khikmat Saburov Abstract A multi-agent system is a system composed of multiple interacting socalled intelligent agents who possibly have different information and/or diverging interests. The agents could be robots, humans or human teams. Opinion dynamics is a process of individual opinions in which a group of interacting agents continuously fuse their opinions on the same issue based on established rules to reach a consensus in the final stage. Historically, the idea of reaching consensus through repeated averaging was introduced by DeGroot for a structured time-invariant and synchronous environment. Since that time, the consensus which is the most ubiquitous phenomenon of multi-agent systems becomes popular in the various scientific fields such as biology, physics, control engineering and social science. To some extent, the Krause mean process is a general model of opinion sharing dynamics in which the opinions are represented by vectors. In this paper, we present an opinion sharing dynamics by means of Krause mean processes that are generated by offdiagonally uniformly positive nonautonomous cubic doubly stochastic matrices and we then establish the consensus in the multi-agent system. Keywords Multi-agent system · Consensus · Krause mean process · Cubic stochastic matrix · Quadratic operator

M. Saburov (B) College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait e-mail: [email protected] K. Saburov Tashkent State University of Economics, Tashkent, Uzbekistan e-mail: [email protected] YEOJU Technical Institute in Tashkent, Tashkent, Uzbekistan National University of Uzbekistan, Tashkent, Uzbekistan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_20

439

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1 Introduction A multi-agent system is a system composed of multiple interacting so-called intelligent agents who possibly have different information and/or diverging interests. The agents could be robots, humans or human teams. The humans are complex individuals whose behaviors are governed by many aspects, related to social context, culture, law, and other factors. In spite of these many factors, human societies are characterized by stunning global regularities in which we can see transitions from disorder to order. These macroscopic phenomena naturally call for a mathematical model to understand social behavior, i.e., a model to understand regularities at large scale as collective effects of the interaction among single individuals. Opinions are at the basis of human behavior and can be seen as the internal state of individuals that drives a certain action. Opinion dynamics is a fusion process of individual opinions in which a group of interacting agents continuously fuse their opinions on the same issue based on established fusion rules to reach a consensus, polarization, or fragmentation in the final stage Different mathematical models have been constructed to study the evolution of the opinions of a group of interacting individuals. The majority of the concerned models are linear. Typically researchers are more focused on the consensus problem and try to find out how to reach it. Historically, an idea of reaching consensus for a structured time-invariant and synchronous environment was introduced by DeGroot [5]. Later, Chatterjee and Seneta [4] generalized DeGroot’s model for a structured time-varying and synchronous environment. In these models, an opinion sharing dynamics of a structured time-varying synchronous multi-agent system is presented by the backward product of square stochastic matrices. Meanwhile, a non-homogeneous Markov chain is presented by the forward product of square stochastic matrices. Therefore, the consensus in the multi-agent system and the ergodicity of the Markov chain are dual problems to each other. Since that time, the consensus which is the most ubiquitous phenomenon of multi-agent systems becomes popular in various scientific communities such as biology, physics, control engineering and social science (see the references [12, 23, 44, 45]). Recently, some nonlinear models have been constructed to characterize the opinion dynamics in social communities (see the references [9–11, 16–19, 22]). A more general model of the opinion sharing dynamics is the Krause mean process in which the opinions are represented by vectors. The reader may refer to the monograph [20] for a complete exposition of the Krause mean process. In the series of the papers [33–38], the correlation between the Krause mean processes and quadratic stochastic processes was established. A quadratic stochastic process [6, 42] is the simplest nonlinear Markov chain. The analytic theory of the quadratic stochastic process generated by cubic stochastic matrices was established in the papers [6, 42]. Historically, a quadratic stochastic operator (in short QSO) was first introduced by Bernstein [2]. The quadratic stochastic operator was considered an important source of analysis for the study of dynamical properties and modeling in various fields such as biology [13, 21], physics [46], control system [33–38]. The fixed point sets and omega limiting sets of quadratic

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stochastic operators defined on the finite-dimensional simplex were studied in the references [39–41]. Ergodicity and chaotic dynamics of quadratic stochastic operators on the finite dimensional simplex were studied in the papers [26–32]. A long self-contained exposition of recent achievements and open problems in the theory of quadratic stochastic operators and processes was presented in the reference [8]. In this paper, we present an opinion sharing dynamics by means of Krause mean processes that are generated by off-diagonally uniformly positive nonautonomous cubic doubly stochastic matrices and we then establish the consensus in the multiagent system.

2 Krause Mean Processes We first review a general model of opinion sharing dynamics of the multi-agent system presented in the paper [10] which encompasses all classical models of opinion sharing dynamics [1, 4, 5]. Consider a group of m individuals Im := {1, · · · , m} acting together as a team or committee, each of whom can specify his/her own subjective distribution for some given task. It is assumed that if the individual i is informed of the distributions of each of the other members of the group then he/she might wish to revise his/her subjective distribution to accommodate the information. Let x(t) = (x1 (t), · · · , xm (t))T be the subjective distributions of the multi-agent system at the time t where xi (t) ≥ 0 for all i ∈ Im . Let pi j (t, x(t)) denote the weight that the individual i assigns to x j (t) when he/she makes the revision at the time t + 1. It is assumed that m 

pi j (t, x(t)) = 1 and pi j (t, x(t)) ≥ 0, ∀ i, j ∈ Im .

j=1

After being informed of the subjective distributions of the other members of the group, the individual i revises his/her own subjective distribution from xi (t) to xi (t + 1) as follows m  pi j (t, x(t))x j (t), ∀ i ∈ Im xi (t + 1) = j=1

Let P (t, x(t)) denote an m × m row-stochastic matrix whose (i j) element is pi j (t, x(t)). A general model of the structured time-varying synchronous system is defined as follows x(t + 1) = P (t, x(t)) x(t), ∀ t ∈ N.

(1)

We may then obtain all classical models [1, 4, 5, 10, 11] by choosing suitable matrices P (t, x(t)).

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We say that a consensus is reached in the structured time-varying synchronous multi-agent system (1) if x(t) converges to c = (c, · · · , c)T as t → ∞. It is worth mentioning that the consensus c = c(x(0)) might depend on an initial opinion x(0). A more general model of the opinion sharing dynamics is the Krause mean process in which the opinions are represented by vectors. The reader may refer to an excellent monograph [20] by Krause for a detailed exposition of the mean processes. Let S be a non-empty convex subset of Rd and Sm be the m−fold Cartesian product of S. m Definition 1 (The Krause mean process [20]) A given sequence {x(t)}∞ t=0 ⊂ S T m where x(t) = (x1 (t), · · · , xm (t)) is called a Krause mean process on S if one has that xi (t + 1) ∈ conv{x1 (t), · · · , xm (t)}, ∀ i ∈ Im , t ∈ N,

where conv{·} is a convex hull of a set. m In other words, a given sequence {x(t)}∞ t=0 ⊂ S is the Krause mean process if one has that

conv{x1 (t + 1), · · · , xm (t + 1)} ⊂ conv{x1 (t), · · · , xm (t)}, ∀ t ∈ N Definition 2 (The Krause mean operator [20]) A mapping T : Sm → Sm is called a t Krause mean operator if its trajectory {x(t)}∞ t=0 , x(t) = T (x(0)) starting from any m initial point x(0) ∈ S generates a Krause mean process on Sm . It is worth mentioning that the nonlinear model of opinion sharing dynamics given by (1) is a Krause mean process due to the fact that the action of a stochastic matrix P = ( pi j )i,m j=1 on a vector x = (x1 , · · · , xm )T can be viewed as formation of arithmetic means m  pi j x j , ∀ i ∈ Im (Px)i = j=1

with weights pi j . The various kinds of nonlinear models of mean processes have been studied in the series of papers [10, 11, 16–19].

3 Quadratic Stochastic Processes m m Let {ek }m k=1 be the standard basis of the space R . Suppose that R is equipped with the l1 −norm m  x1 := |xk | k=1

where x = (x1 , · · · , xm )T ∈ Rm . We say that x ≥ 0 (respectively, x > 0) if xk ≥ 0 (respectively, xk > 0) for all k ∈ Im . Let

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443

  Sm−1 = x ∈ Rm : x ≥ 0, x1 = 1 be the (m − 1)−dimensional standard simplex. An element of the simplex Sm−1 is called a stochastic vector. Let c = ( m1 , · · · , m1 )T be the center of the simplex Sm−1 and intSm−1 = {x ∈ Sm−1 : x > 0} and ∂Sm−1 = Sm−1 \ intSm−1 be, respectively, an interior and boundary of the simplex Sm−1 . We now provide some necessary definitions of non-homogeneous Markov chains and quadratic stochastic processes by following the papers [6, 7, 24, 42, 43]. Let P = ( pi j )i,m j=1 be a matrix. We define the following vectors (pi∗ )T := ( pi1 , · · · , pim ) and p∗ j := ( p1 j , · · · , pm j )T , ∀ i, j ∈ Im . A square matrix P = ( pi j )i,m j=1 is called row-stochastic (respectively, columnstochastic) if pi∗ (respectively, p∗ j ) is a stochastic vector for all i ∈ Im (respectively, for all j ∈ Im ). A square matrix P = ( pi j )i,m j=1 is called doubly stochastic if it is simultaneously row-stochastic and column-stochastic. We say that P ≥ 0 (respectively, P > 0) if pi∗ ≥ 0 (respectively, pi∗ > 0) for all i ∈ Im . During the last few decades, the huge efforts have been made to construct various necessary and/or sufficient conditions for the ergodicity of non-homogeneous Markov chains [24, 43]. One of the major areas of study in non-homogeneous Markov chains is that of finding conditions under which a chain is weakly/strongly ergodic. A basic technique for doing this is to establish that all finite products are regular and then require some condition on the size of the positive entries in the transition matrices [43]. In looking for sets of square stochastic matrices which can be used in forming weakly/strongly ergodic non-homogeneous Markov chains, one needs to find subsets of regular square stochastic matrices which form semi-groups. A set of scrambling square stochastic matrices is one of these sets [43]. A stochastic matrix P = ( pi j )i,m j=1 is called scrambling if for any i, j ∈ Im there exists k ∈ Im such that pik p jk > 0, i.e., any two rows of the square stochastic matrix are not orthogonal. A square stochastic matrix with identical rows is called a stable (constant) matrix. It is obvious that any positive square stochastic matrix is scrambling. Moreover, any stable (constant) square stochastic matrix is also a scrambling matrix. One of the classical results in the theory of linear Markov chains that a stochastic matrix is strongly ergodic, i.e., its powers converge to some stable/constant stochastic matrix if and only if its some power is a scrambling matrix. A family of square row-stochastic matrices  m  [r,t] P[r,t] = pik

i,k=1

 : r, t ∈ N, t − r ≥ 1

is called a discrete time non-homogeneous Markov chain if for any natural numbers r, s, t with r < s < t the following condition, known as the Chapman–Kolmogorov equation, is satisfied

444

M. Saburov and K. Saburov [r,t] pik =

m 

[s,t] pi[r,s] j p jk , ∀ i, k ∈ Im .

j=1

A linear operator L[r,t] : Sm−1 → Sm−1 associated with the square row-stochastic 

[r,t] matrix P[r,t] = pik

m

i,k=1

where

m  [r,t]

[r,t] L (x) k = xi pik , ∀ k ∈ Im , i=1

is called a linear stochastic (Markov) operator [24, 43]. Notice that the Chapman–Kolmogorov equation can be written in the following form L[r,t] = L[s,t] ◦ L[r,s] ,

r < s < t.

Remark 1 It is worth mentioning that a discrete time non-homogeneous Markov chain generates a nonautonomous dynamical system (see [14]). Indeed, if we define a continuous mapping ϕ(t0 , t0 , x) := x, ϕ(t, t0 , x) := L[t0 ,t] (x) = xP[t0 ,t] ∀ t > t0 , ∀ x ∈ Sm−1 , then due to the Chapman–Kolmogorov equation L[r,t] = L[s,t] ◦ L[r,s] ,

r < s < t,

a family of continuous mappings (t, t0 , x) → ϕ(t0 , t0 , x) ∈ Sm−1 for t, t0 ∈ N and x ∈ Sm−1 with t ≥ t0 generates a nonautonomous dynamical system (or two-parameter semi-group process): (i) ϕ(t0 , t0 , x) = x for all t0 ∈ N and x ∈ Sm−1 ; (ii) ϕ(t2 , t0 , x) = ϕ(t2 , t1 , ϕ(t1 , t0 , x)) for all t0 ≤ t1 ≤ t2 and x ∈ Sm−1 . Let P = ( pi jk )i,m j,k=1 be a cubic matrix (see the Refs. [6, 7, 42]). We define the following vectors pi j∗ := ( pi j1 , · · · , pi jm )T , ∀ i, j ∈ Im . A cubic matrix P = ( pi jk )i,m j,k=1 is called stochastic if pi j∗ is a stochastic vector for all i, j ∈ Im . A family of cubic stochastic matrices  m  P [r,t] = pi[r,t] jk

i, j,k=1

 [r,t] : pi[r,t] = p , r, t ∈ N, t − r ≥ 1 jk jik

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with an initial distribution x(0) ∈ Sm−1 is called a discrete time quadratic stochastic process if for any natural numbers r, s, t with r < s < t one of the following conditions, so-called nonlinear Chapman–Kolmogorov equations, is satisfied (A) (B)

pi[r,t] jk = pi[r,t] jk =

where xk(ν) =

m  α,β=1 m 

(s) [s,t] pi[r,s] jα x β pαβk ,

α,β,γ ,δ=1 m 

i, j=1

i, j, k ∈ Im ,

[r,s] (r ) [r,s] [s,t] xα(r ) piαβ xγ p jγ δ pβδk ,

i, j, k ∈ Im ,

[0,ν] xi(0) x (0) j pi jk .

We remark that the conditions (A) and (B) are not equivalent to each other. The reader may refer to the papers [6, 42] for the exposition of quadratic stochastic processes. [r,t] A continuousmapping : Sm−1 → Sm−1 associated with the cubic stochastic  Q

matrix P [r,t] = pi[r,t] jk

m

i, j,k=1

where

m  [r,t]

xi x j pi[r,t] Q (x) k = jk , k ∈ Im . i, j=1

is called a quadratic stochastic operator (a nonlinear Markov operator). Obviously, we have that x(r ) = Q[0,r ] (x(0) ). Notice that the nonlinear Chapman–Kolmogorov equation can be written in the following form Q[r,t] (x(r ) ) = Q[s,t] (Q[r,s] (x(r ) )), ∀ r < s < t. Remark 2 It is worth mentioning that a discrete time quadratic stochastic process generates a nonautonomous dynamical system (see [14]). Indeed, if we define a continuous mapping ϕ(t0 , t0 , x) := x, ϕ(t, t0 , x) := Q[t0 ,t] (x) =

m 

0 ,t] xi x j pi[tj∗ ∀ t > t0 , x ∈ Sm−1 ,

i, j=1

then due to the nonlinear Chapman–Kolmogorov equation Q[r,t] (x(r ) ) = Q[s,t] (Q[r,s] (x(r ) )), ∀ r < s < t, a family of continuous mappings (t, t0 , x) → ϕ(t0 , t0 , x) ∈ Sm−1 for t, t0 ∈ N and x ∈ Sm−1 with t ≥ t0 generates a nonautonomous dynamical system (or two-parameter semi-group process):

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(i) ϕ(t0 , t0 , x) = x for all t0 ∈ N and x ∈ Sm−1 ; (ii) ϕ(t2 , t0 , x) = ϕ(t2 , t1 , ϕ(t1 , t0 , x)) for all t0 ≤ t1 ≤ t2 and x ∈ Sm−1 . We define the following stochastic vectors and square row-stochastic matrices associated with the cubic stochastic matrix P = ( pi jk )i,m j,k=1 as follows pi j∗ : = ( pi j1 , pi j2 , · · · , pi jm )T , i, j ∈ Im , m Pi∗∗ : = ( pi jk ) j,k=1 , i ∈ Im , Px : =

m 

xi Pi ,

x ∈ Sm−1 .

i=1

It is easy to check that the quadratic stochastic operator has the following vector and matrix forms Q(x) =

m 

(Vector form)

xi x j pi j∗

(2)

i, j=1 m 

T T

Q(x) = xT Px = PxT x = xi Pi∗∗ x

(Matrix form)

(3)

i=1

Definition 3 (The nonlinear Markov operator [15]) A continuous mapping M : Sm−1 → Sm−1 is called a nonlinear Markov operator if one has that M(x) = Mx x, ∀ x ∈ Sm−1

m where Mx = pi j (x) i, j=1 is a column-stochastic matrix depends on x ∈ Sm−1 (which introduces non-linearity). The quadratic stochastic operator Q : Sm−1 → Sm−1 given by (2) is indeed a nonlinear Markov operator since it can be written in the matrix form defined by (3). It is worth mentioning that there are some nonlinear Markov operators which are not quadratic stochastic operators [15]. Therefore, the set of all quadratic stochastic operators cannot cover the set of all nonlinear Markov operators.

4 The Krause Mean Process Generated by the Quadratic Stochastic Operator In this section, we establish some correlation between the Krause mean processes and quadratic stochastic operators. We first introduce some notions and notations. Definition 4 A cubic matrix P = ( pi jk )i,m j,k=1 is called doubly stochastic if one has that

Krause Mean Processes Generated by Cubic Stochastic Matrices IV … m 

pi jk =

j=1

m 

447

pi jk = 1, pi jk ≥ 0, ∀ i, j, k ∈ Im .

k=1

Remark 3 In this paper, we do not require the condition pi jk = p jik for any i, j, k ∈ Im . Let P = ( pi jk )i,m j,k=1 be a cubic doubly stochastic matrix and P∗∗k = ( pi jk )i,m j=1 be a square matrix for fixed k ∈ Im . It is clear that P∗∗k = ( pi jk )i,m j=1 is also a square stochastic matrix. In the sequel, we write P = (P∗∗1 | · · · |P∗∗m ) for the cubic doubly stochastic matrix. We define a quadratic stochastic operator Q : Sm−1 → Sm−1 associated with the cubic doubly stochastic matrix P = (P∗∗1 | · · · |P∗∗m ) as follows (Q(x))k =

m 

pi jk xi x j , ∀ k ∈ Im .

(4)

i, j=1

We also define a linear stochastic operator Lk : Sm−1 → Sm−1 associated with the square stochastic matrix P∗∗k = ( pi jk )i,m j=1 as follows m 

pi jk xi , ∀ j ∈ Im . (Lk (x)) j = xT P∗∗k j =

(5)

i=1

It follows from (4) and (5) that (Q(x))k =

m m   j=1

pi jk xi

i=1

xj =

m 



(Lk (x)) j x j = Lk (x), x , ∀ k ∈ Im

j=1

where (·, ·) stands for the standard inner product of two vectors. Therefore, the quadratic stochastic operator Q : Sm−1 → Sm−1 given by (4) can be written as follows Q(x) =





 T L1 (x), x , · · · , Lm (x), x

where Lk : Sm−1 → Sm−1 is defined by (5) for all k ∈ Im . We now define an m × m matrix as follows

(6)

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⎛ L1 (x) 1 ⎜ L2 (x) 1 ⎜ P(x) = ⎜ .. ⎝ .

Lm (x) 1



L1 (x) 2 L2 (x) 2 .. .

Lm (x) 2



⎞ · · · L1 (x) m · · · L2 (x) m ⎟ ⎟ ⎟. .. .. ⎠ . .

· · · Lm (x) m

(7)

We show that P(x) is doubly stochastic matrix for every x ∈ Sm−1 . In fact we

m know that P(x) = pk j (x) k, j=1 where m 

pk j (x) = Lk (x) j = pi jk xi .

(8)

i=1

Therefore, we have that m 

pk j (x) =

k=1 m  j=1

pk j (x) =

m m   k=1

i=1

j=1

i=1

m m  

=

pi jk xi

pi jk xi

m m   i=1

k=1

i=1

j=1

pi jk xi =

m 

xi = 1,

i=1

⎛ ⎞ m m m    ⎝ pi jk ⎠ xi = xi = 1. = i=1

Hence, it follows from (6) and (7) that Q(x) = P(x)x

(9)

and we call it a matrix form of the quadratic stochastic operator (4) associated with the cubic doubly stochastic matrix P = (P∗∗1 | · · · |P∗∗m ). We now present the nonlinear opinion sharing dynamics of the multi-agent system. (n) P ROTOCOL –DSM: Let Pn = (P(n) ∗∗1 | · · · |P∗∗m ) be a cubic doubly stochastic m−1 m−1 →S be a quadratic stochastic operator assomatrix for all n ∈ N and Qn : S (n) ciated with a cubic doubly stochastic matrix Pn = (P(n) ∗∗1 | · · · |P∗∗m ) for all n ∈ N. Suppose that an opinion sharing dynamics of the multi-agent system is generated by a quadratic stochastic operator Qn : Sm−1 → Sm−1 as follows



x(n+1) := Qn x(n) = Pn x(n) x(n) ,

x(0) ∈ Sm−1

T  where x(n) = x1(n) , · · · , xm(n) is the subjective distribution after n revisions. We propose the multi-agent system interpretation of P ROTOCOL –DSM. We assume that each agent may revise his/her own opinion on some issue after the influences of all possible groups of 2-agents which obviously create non-linearity in the proposed model. In order to keep the model homogeneous, we interpret the influence of a single agent as the influence of a group of two identical agents. Precisely, the following assumptions are made:

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• There is a group of m agents Im := {1, · · · , m} acting together as a team or committee; • Each agent can specify his/her own opinion on some issue. An opinion is a generic concept that represents an agent’s belief or behavior or attitude; • An opinion profile at time n is a stochastic vector x(n) = (x1(n) , · · · , xm(n) )T ; • Each agent, say k, is influenced by a group of 2-agents, say {i, { j}}, in which an agent j is a spokesman of the group; • At time n, the influence of a group of 2-agents, say {i, { j}} (resp. { j, {i}}) with (n) the spokesman j (resp. i), on an agent k is pi(n) j,k (resp. p ji,k ). In general, it is (n) possible that pi(n) j,k = p ji,k ; • At time n, the influence profile of the group {i, { j}} with the spokesman j is a (n) (n) (n) T stochastic vector pi(n) j∗ := ( pi j,1 , pi j,2 , · · · , pi j,m ) ; • At time n, the influences of all possible groups of 2-agents on an agent k is a (n) m square stochastic matrix P(n) ∗∗k = ( pi j,k )i, j=1 ; • An agent believes that a group of 2-agents is trusted/influential if its influence on an agent is high; (n) • At time n, the trust pk(n) j (x ) of an agent k in an agent j with the opinion (n) profile x is the average influence of all possible groups of 2-agents having j (n) as a spokesman on an agent k over the opinion profile x(n) , i.e., pk(n) j (x ) := m  xi(n) pi(n) j,k ; i=1

• At time n, the trust matrix with the opinion profile x(n) is a square stochastic (n) (n) m (n) matrix Pn (x ) = pk j (x ) k, j=1 ; • The opinion profile at time n + 1 is then revised as follows x(n+1) :=

Pn x(n) x(n) . It follows from (9) that the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSM can be written as

x(n+1) = Pn x(n) x(n) , x(0) ∈ Sm−1 T  where x(n) = x1(n) , · · · , xm(n) is the subjective distribution after n revisions. This means that, due to the matrix form (1), the opinion sharing dynamics of the multiagent system given by P ROTOCOL –DSM generates a Krause mean process. Consequently, we have the following result (see [37]). (n) Proposition 1 Let Pn = (P(n) ∗∗1 | · · · |P∗∗m ) be a cubic doubly stochastic matrix for all m−1 m−1 →S be a quadratic stochastic operator associated with a n ∈ N and Qn : S (n) cubic doubly stochastic matrix Pn = (P(n) ∗∗1 | · · · |P∗∗m ) for all n ∈ N. Then the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSM generates the Krause mean process.

Remark 4 It is worth mentioning that the proposed P ROTOCOL –DSM also generates a nonautonomous dynamical system (see [14]). Indeed, if we define a continuous mapping

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ϕ n 0 , n 0 , x(n 0 ) := x(n 0 ) ,





ϕ n, n 0 , x(n 0 ) := Qn−1 · · · Qn 0 x(n 0 ) · · · , ∀ n > n 0 then a family of continuous mappings (n, n 0 , x(n 0 ) ) → ϕ(n, n 0 , x(n 0 ) ) ∈ Sm−1 for n, n 0 ∈ N and x(n 0 ) ∈ Sm−1 with n ≥ n 0 generates a nonautonomous dynamical system (or two-parameter semi-group process): (i) ϕ(n 0 , n 0 , x(n 0 ) ) = x(n 0 ) for all t0 ∈ N and x ∈ Sm−1 ; (ii) ϕ(n 2 , n 0 , x(n 0 ) ) = ϕ(n 2 , n 1 , ϕ(n 1 , n 0 , x(n 0 ) )) for all n 0 ≤ n 1 ≤ n 2 .

  x It is obvious that for any nonzero vector x ≥ 0 we have Q(x) = x21 Q x 1   x x m−1 m−1 (0) in which x ∈ S ∈ S and Q . Consequently, if x  > 1 then 1 x1 1 lim x(n) 1 = lim x(0) 21 = ∞ n

n→∞

n→∞

and

if

x(0) 1 < 1

lim x(0) 21 = 0. n

then

lim x(n) 1 =

n→∞

n→∞

Hence, unlike the linear case, the simplex Sm−1 = {x ∈ Rm : x ≥ 0, x1 = 1} is only an

invariant set under the action of the quadratic stochastic operator, i.e., Q Sm−1 ⊂ Sm−1 . In other words, for any initial opinion x(0) ∈ Sm−1 the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSM gen(n) ∈ Sm−1 for all n ∈ N. Similar erates a vector sequence {x(n) }∞ n=0 for which x to the linear case, we say that a consensus is reached in the structured timevarying synchronous multi-agent system if a vector sequence {x(n) }∞ n=0 converges to c = (c, · · · , c)T ∈ Sm−1 as n → ∞. However, it is obvious that the only stochastic vector of the simplex with equal distributions is the center of the simplex, i.e., c = ( m1 , · · · , m1 )T . Consequently, unlike the linear case, the consensus in the opinion sharing dynamics given by P ROTOCOL –DSM does not depend on the initial opinion x(0) ∈ Sm−1 and it can be stated as follows. Definition 5 (Consensus) We say that the multi-agent system eventually reaches a consensus if an opinion sharing dynamics converges to the center c = ( m1 , · · · , m1 )T of the simplex Sm−1 .

5 The Main Result Let us first introduce a notion of off-diagonally positive cubic stochastic matrices. Definition 6 (Off-diagonally positive cubic stochastic matrix) A family {Pn }∞ n=0 m of cubic stochastic matrices Pn = ( pi(n) ) is called off-diagonally uniformly jk i, j,k=1 positive if there exists δ > 0 such that pi(n) j∗ > δ for any i, j ∈ Im with i = j and for any n ∈ N. We are now ready to state the main result of this paper.

Krause Mean Processes Generated by Cubic Stochastic Matrices IV …

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Theorem 1 (Consensus of cubic doubly stochastic matrix) Let Pn = (n) m−1 → (P(n) ∗∗1 | · · · |P∗∗m ) be a cubic doubly stochastic matrix for all n ∈ N and Qn : S m−1 be a quadratic stochastic operator associated with a cubic doubly stochasS (n) T tic matrix Pn = (P(n) ∗∗1 | · · · |P∗∗m ) for all n ∈ N. Let e1 = (1, 0, 0, · · · , 0) , e2 = T T m−1 and (0, 1, 0, · · · ,  0) , and em = (0, 0, 0, · · · , 1) be vertices of the simplex S (n+1) (n) (0) ∞ ek := Qn ek where ek := ek for all k ∈ Im and n ∈ N. A family {Pn }n=0 of m cubic stochastic matrices Pn = ( pi(n) jk )i, j,k=1 is off-diagonally uniformly positive and moreover if for each k ∈ Im one has ek(n k ) ∈ intSm−1 for some n k ∈ N then the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSM eventually reaches a consensus for any initial opinion.

Multi- Agent System Interpretation: Suppose that an opinion sharing dynamics of the multi-agent system is given by P ROTOCOL –DSM. If each and every agent has an eventually positive opinion and moreover if the influence of each and every group of two different agents on any other agent is positive then the multi-agent system reaches a consensus. Proof Let e1 = (1, 0, 0, · · · , 0)T , e2 = (0, 1, 0, ·· · , 0)T , and em = (0, 0, ∞ 0, · · · , 1)T be vertices of the simplex Sm−1 and let x(n) n=0 where x(n+1) = Qn (x(n) ) m−1 be a trajectory of the quadratic stochastic operators → Sm−1 starting from   Qn : S an initial point x(0) ∈ Sm−1 . Particularly, let ek(n)

∞

n=0

be a trajectory of the quadratic

→S starting from a vertex ek of the simplex Sm−1 stochastic operator Qn : S for all k ∈ Im . According to the definition, the multi-agent system eventually reaches 1 1 T a consensus if {x(n) }∞ n=0 converges to the center c = ( m , · · · , m ) of the simplex Sm−1 for any initial point x(0) ∈ Sm−1 . We accomplish it under two hypotheses: m−1

m−1

(i) For each k ∈ Im one has ek(n k ) ∈ intSm−1 for some n k ∈ N; (n) m (ii) A family {Pn }∞ n=0 of cubic stochastic matrices Pn = ( pi jk )i, j,k=1 is off-diagonally uniformly positive, i.e., there exists δ > 0 such that pi(n) j∗ > δ for any i, j ∈ Im with i = j and for any n ∈ N. Step-1 We first show that Qn (intSm−1 ) ⊂ intSm−1 for any n ∈ N. Indeed,

m let x ∈ intSm−1 . This means that xi > 0 for all i ∈ Im . Since Pn (x) = pk(n) j (x) k, j=1 is a square doubly stochastic matrix and Qn (x) = Pn (x)x, we derive that 0 < min x j ≤ j∈Im

m 

pk(n) j (x)x j = (Qn (x))k

∀ k ∈ Im .

j=1

This means that Qn (x) ∈ intSm−1 . Step-2 We now show that there exists n 0 ∈ N such that for any initial point x(0) ∈ Sm−1 one has x(n 0 ) ∈ intSm−1 . It is worth mentioning that n 0 does not depend on an initial point x(0) ∈ Sm−1 . Since for each k ∈ Im one has ek(n k ) ∈ intSm−1 for some n k ∈ N, it then follows from the previous step that for each k ∈ Im one has ek(n) ∈ intSm−1 for any n > n k .

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Let n 0 := max n k . Then ek(n 0 ) ∈ intSm−1 for all k ∈ Im . We now show that x(n 0 +1) = k∈Im

Q n 0 (· · · (Q0 (x(0) )) · · · ) ∈ intSm−1 for any initial point x(0) ∈ Sm−1 . Since Pn (λx + (1 − λ)y) = λPn (x) + (1 − λ)Pn (y) for any x, y ∈ Sm−1 and 0 < λ < 1, we have that x(1) = Q0 (x(0) ) = P0 (x(0) )x(0) = x1(0) P0 (x(0) )e1 + · · · + xm(0) P0 (x(0) )em m  2   (0) (0) xi(0) P0 (ei )ei + 2 = xi x j P0 (e j )ei i=1

i< j

Let ei(11) := P0 (e j )ei for any i = j. We then obtain that j x(1) =

m  

xi(0)

2

ei(1) + 2

i=1



(11) xi(0) x (0) j ei j

(10)

i< j

Similarly, we may get the following x(2) = P1 (x(1) )x(1) =

m  

xi(0)

2

P1 (x(1) )ei(1) + 2

i=1

=

m  m   i=1 k=1 m 

+2

xi(0)

2 

xk(0)

2



(1) (11) xi(0) x (0) j P1 (x )ei j

i< j

P1 (ek(1) )ei(1) + 2

m  

xi(0)

2

(11) (1) xk(0) xl(0) P1 (ekl )ei

i=1 k 0

(13)

for any x(0) ∈ Sm−1 . This shows that x(n 0 ) ∈ intSm−1 .  

Step-3 We now show that for any x(0) ∈ Sm−1 an omega limit set ω x(n) (n) ∞ m−1 of the simplex of the sequence  }n=0 ism−1a subset of the interior intS  (n){x m−1  intS . Indeed, it follows from the previous step that i.e., ω x S Qn 0 (· · · (Q0 (Sm−1 )) · · · )  intSm−1 . Since it is a compact set, there exists α > 0 such that Qn 0 (· · · (Q0 (x)) ≥ αe := (α, α, · · · , α)T

∀ x ∈ Sm−1 .

On the other hand, the interior intSm−1 of the simplex Sm−1 is an invariant set (see Step-1) and Qn (· · · (Q0 (Sm−1 )) · · · ) ⊂ Qn 0 (· · · (Q0 (Sm−1 )) · · · ) for any n > n 0 , (n) ≥ αe for any n > n 0 where we have that {x(n) }∞ n=n 0 ⊂ Sα , i.e., x Sα := {x ∈ Sm−1 : x ≥ αe}.  (n) 

{x(n) }∞ Consequently, an omega n=0 is a subset  set ω x m−1of the sequence  (n)limit (0) m−1 ⊂ Sα ⊂ intS any x ∈ S . of the set Sα , i.e., ω x Step-4 As we showed in the previous step that Qn (· · · (Q0 (Sm−1 )) · · · ) ⊂ Sα for any n > n 0 , it is therefore enough to study the dynamics of the quadratic stochastic operator over the set Sα which is an invariant set. Let x(0) ∈ Sα . Then x(n) ∈ Sα , i.e., x(n) ≥ αe for any n ∈ N. It follows from the matrix form (9) of the quadratic stochastic operator that





x(n+1) = Qn (x(n) ) = Pn x(n) x(n) = Pn x(n) · · · P1 x(1) P0 x(0) x(0) where Pn (x) is the square doubly stochastic matrix defined by (7). Let us set for any two integer numbers n > r







(n) (r ) P[x ,x ] := Pn x(n) Pn−1 x(n−1) · · · Pr +1 x(r +1) Pr x(r ) . We then obtain for any n ≥ r ≥ 0 that (n) (0) (n) (r ) x(n+1) = P[x ,x ] x(0) = P[x ,x ] x(r ) .

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We set ρ := inf min pi(n) jk > 0. n∈N i, j,k∈Im i = j



(n) m Then, for a stochastic matrix Pn (x(n) ) = pk(n) j (x ) k, j=1 it follows from (8) (n) pk(n) j (x )

=

m 

(n) pi(n) jk x i

≥

i=1

m 

(n) pi(n) ≥ (m − 1)ρα > 0 jk x i

i=1 i = j



m for any j, k ∈ Im and n ∈ N. Consequently, Pn (x(n) ) = pk j (x(n) ) k, j=1 is a positive square stochastic matrix for any n ∈ N. m  | pi1 j − pi2 j | be Dobrushin’s ergodicity coefficient Step-5 Let δ(P) = 21 max i 1 ,i 2 j=1

m [43] of a square stochastic matrix P = ( pi j )i,m j=1 . Since Pn (x(n) ) = pk j (x(n) ) k, j=1 is positive and its entries are uniformly bounded away from zero for any n ∈ N, we then obtain that

δ Pn (x(n) ) ≤ λ < 1, ∀ n ∈ N. Hence, we have that  (n) (0)  δ P[x ,x ] ≤ λn+1 ,

(14)

 (n) (0)  lim δ P[x ,x ] = 0.

n→∞

Therefore, the backward product of doubly stochastic matrices {Px(n) }∞ n=0 is weakly ergodic (see [43]) which is also strongly ergodic (see [43]), i.e., (n) (0) lim P[x ,x ] = mc T c and

n→∞

(n) (0) lim x(n+1) = lim P[x ,x ] x(0) = c, ∀ x(0) ∈ Sα ,

n→∞

n→∞

where c = ( m1 , · · · , m1 )T . This completes the proof.

6 Discussions We now discuss the main result of this paper and compare it with the previous results [3, 25, 33–38] on the consensus problem. Definition 7 A cubic matrix P = ( pi jk )i,m j,k=1 is called triply stochastic if one has that m m m    pi jk = pi jk = pi jk = 1, pi jk ≥ 0, ∀ i, j, k ∈ Im . i=1

j=1

k=1

Krause Mean Processes Generated by Cubic Stochastic Matrices IV …

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It is obvious that if P = ( pi jk )i,m j,k=1 is a cubic triply stochastic matrix then it is also a cubic doubly stochastic matrix. Consequently, a quadratic stochastic operator associated with a cubic triply stochastic matrix also generates a Krause mean process. P ROTOCOL –TSM: Let P = (P∗∗1 | · · · |P∗∗m ) be a cubic triply stochastic matrix and let Q : Sm−1 → Sm−1 be a quadratic stochastic operator associated with a cubic triply stochastic matrix P = (P∗∗1 | · · · |P∗∗m ). Suppose that an opinion sharing dynamics of the multi-agent system is generated by a quadratic stochastic operator Q : Sm−1 → Sm−1 as follows

x(n+1) = Q x(n) , x(0) ∈ Sm−1 T  where x(n) = x1(n) , · · · , xm(n) is the subjective distribution after n revisions. The following result was proved in the papers [33–38]. Theorem 2 (Consensus of cubic triply stochastic matrix [33–38]) Let P = (P∗∗1 | · · · |P∗∗m ) be a cubic triply stochastic matrix and let Q : Sm−1 → Sm−1 be the associated quadratic stochastic operator. If P > 0 then the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –TSM eventually reaches a consensus. We would like to emphasize that since we do not require the condition pi jk = p jik for any i, j, k ∈ Im , in general, the double stochasticity does not imply the triple stochasticity of cubic matrices. Let us now provide some supporting examples. Example 1 We consider the cubic doubly stochastic matrix P = (P1∗∗ |P2∗∗ |P3∗∗ ) where P1∗∗ , P2∗∗ , and P3∗∗ are square doubly stochastic matrices given as follows ⎛

P1∗∗

⎛ ⎛ ⎞ ⎞ ⎞ p111 p112 p113 p211 p212 p213 p311 p312 p313 = ⎝ p121 p122 p123 ⎠ P2∗∗ = ⎝ p221 p222 p223 ⎠ P3∗∗ = ⎝ p321 p322 p323 ⎠ p131 p132 p133 p231 p232 p233 p331 p332 p333

The following quadratic stochastic operator Q : S2 → S2 presents Protocol–DSM T

T

T

x + x2 P2∗∗ x + x3 P3∗∗ x = P (x) x Q(x) = x1 P1∗∗

(15)

T T T + x2 P2∗∗ + x3 P3∗∗ is a square doubly stochastic matrix. where P (x) = x1 P1∗∗ It was shown in the papers [33–38] that if the square doubly stochastic matrices P1∗∗ > 0, P2∗∗ > 0, and P3∗∗ > 0 are positive and

⎛

P1∗∗ + P2∗∗ + P3∗∗

⎞ 111 = ⎝1 1 1⎠ 111

(16)

then the consensus is established in the system described by Protocol–TSM. Later, this result was generalized in the paper [3]. Namely, the consensus was established for

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the positive square doubly stochastic matrices P1∗∗ > 0, P2∗∗ > 0, P3∗∗ > 0 without the constraint (16). However, these results were further improved in the paper [25]. Namely, without positivity of the square doubly stochastic matrices P1∗∗ , P2∗∗ , P3∗∗ and without the constraint (16), the consensus is still established in the system described by Protocol–DSM if the cubic doubly stochastic matrix P is (only) diagonally primitive, i.e., for some s ∈ N we have ⎛ ⎞s p111 p112 p113  s diag(P) = ⎝ p221 p222 p223 ⎠ > 0. p331 p332 p333 It is worth mentioning that there is still a room to further improve these results. Indeed, Theorem 1 is further extension and generalization of the results published in the papers [3, 25, 33–38]. To illustrate it, let us consider the following example. Example 2 We consider the following square doubly stochastic matrices ⎛1 ⎜ P1∗∗ = ⎝

2 1 4 1 4

1 2 1 4 1 4

0 1 2 1 2

⎞

⎛1

⎟ ⎜ ⎠ P2∗∗ = ⎝

4 1 2 1 4

1 4 1 2 1 4

1 2

⎞

⎛1

⎜ 0⎟ ⎠ P3∗∗ = ⎝ 1 2

4 1 4 1 2

1 4 1 4 1 2

1 2 1 2

⎞ ⎟ ⎠

0

We then obtain the following quadratic stochastic operator Q : S2 → S2 Q(x) = (x12 + x22 + x32 )

e1 + e2 e1 + e2 + 2e3 + (x1 x2 + x1 x3 + x2 x3 ) 2 2

where e1 = (1, 0, 0)T , e2 = (0, 1, 0)T , and e3 = (0, 0, 1)T are vertices of the simplex S2 . On the one hand, since for any s ∈ N ⎛1  s ⎜ diag(P) = diag(P) = ⎝

2 1 2 1 2

1 2 1 2 1 2

0

⎞

0⎟ ⎠, 0

the cubic doubly stochastic matrix P = (P1∗∗ |P2∗∗ |P3∗∗ ) is not diagonally primitive. On the other hand, since pi j∗ = ( 41 , 41 , 21 ) > 0 for all i, j ∈ I3 with i = j, i.e., P = (P1∗∗ |P2∗∗ |P3∗∗ ) is off-diagonally positive, due to Theorem 1, the opinion sharing dynamics of the multi-agent system given by P ROTOCOL –DSM eventually reaches a consensus for any initial opinion x ∈ S2 . In this sense, Theorem 1 extends and generalizes all results of the papers [3, 25, 33–38].

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7 Conclusion Historically, an idea of reaching consensus for a structured time-invariant and synchronous environment was introduced by DeGroot [5]. Later, Chatterjee and Seneta [4] generalized DeGroot’s model for a structured time-varying and synchronous environment. Since that time, the consensus which is the most ubiquitous phenomenon of multi-agent systems becomes popular in various scientific communities such as biology, physics, control engineering and social science. To some extent, the Krause mean process is a general model of opinion sharing dynamics in which the opinions are represented by vectors. In this paper, we present an opinion sharing dynamics by means of Krause mean processes which are generated by cubic doubly stochastic matrices. Arguably, the proposed model P ROTOCOL –DSM is a feasible generalization of the classical DeGroot model from square stochastic matrices to cubic stochastic matrices. Namely, unlike the classical case, we assumed that each agent may revise his/her own opinion on some issue after the influences of all possible groups of 2-agents. In order to keep the model homogeneous, we interpreted the influence of a single agent as the influence of a group of two identical agents. We then showed that if each and every agent has an eventually positive opinion and moreover if the influence of each and every group of two different agents on any other agent is positive then the multi-agent system reaches a consensus. The main result (Theorem 1) of this paper extends and generalizes all results of the papers [3, 25, 33–38]. Acknowledgements The authors are greatly indebted to anonymous reviewer for several useful suggestions and comments which improved the presentation of the paper.

References 1. Berger, R.L.: A necessary and sufficient condition for reaching a consensus using DeGroot’s method. J. Am. Stat. Assoc. 76, 415–418 (1981) 2. Bernstein, S.: Solution of a mathematical problem connected with the theory of heredity. Ann. Math. Stat. 13, 53–61 (1942) 3. Candan, T., Saburov, M., Ufuktepe, U.: Reaching a consensus via Krause Mean processes in multi-agent systems: quadratic stochastic operators in the book. Progress on Difference Equations and Discrete Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol. 341 , pp. 397–409 (2020) 4. Chatterjee, S., Seneta, E.: Towards consensus: some convergence theorems on repeated averaging. J. Appl. Prob. 14, 89–97 (1977) 5. De Groot, M.H.: Reaching a consensus. J. Am. Stat. Assoc. 69, 118–121 (1974) 6. Ganihodzhaev, N.: On stochastic processes generated by quadratic operators. J. Theor. Prob. 4, 639–653 (1991) 7. Ganikhodjaev, N., Akin, H., Mukhamedov, F.: On the ergodic principle for Markov and quadratic stochastic processes and its relations. Linear Algebra App 416, 730–741 (2006) 8. Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and Open Problems. Inf. Dim. Anal. Quan. Prob. Rel. Top. 14(2), 279–335 (2011)

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9. Girejko, E., Machado, L., Malinowska, A.B., Martins, N.: Krause’s model of opinion dynamics on isolated time scales. Math. Meth. Appl. Sci. 39(18), 5302–5314 (2016) 10. Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence: models, analysis and simulation. J. Art. Soc. Soc. Sim. 5(3), 1–33 (2002) 11. Hegselmann, R., Krause, U.: Opinion dynamics driven by various ways of averaging. Comp. Econ. 25, 381–405 (2005) 12. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 985–1001 (2003) 13. Kesten, H.: Quadratic transformations: a model for population growth I. Adv. App. Prob. 2, 1–82 (1970) 14. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, vol. 176 (2011) 15. Kolokoltsov, V.: Nonlinear Markov Processes and Kinetic Equations. Cambridge University Press (2010) 16. Krause, U.: A discrete nonlinear and non-autonomous model of consensus formation. In: Elaydi, S., et al. (eds.) Communications in Difference Equations, pp. 227–236. Gordon and Breach, Amsterdam (2000) 17. Krause, U.: Compromise, consensus, and the iteration of means. Elem. Math. 64, 1–8 (2009) 18. Krause, U.: Markov chains, Gauss soups, and compromise dynamics. J. Cont. Math. Anal. 44(2), 111–116 (2009) 19. Krause, U.: Opinion dynamics – local and global. In: Liz, E., Manosa, V. (eds.) Proceedings of the Workshop Future Directions in Difference Equations, pp. 113–119. Universidade de Vigo, Vigo (2011) 20. Krause, U.: Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications. Walter de Gruyter (2015) 21. Lyubich, Y.I.: Mathematical Structures in Population Genetics. Springer (1992) 22. Malinowska, A.B., Schmeidel, E., Zdanowicz, M.: Discrete leader-following consensus. Math. Meth Appl. Sci. 40(18), 7307–7315 (2017) 23. Moreau, L.: Stability of multiagent systems with time-dependent communication links. IEEE Trans. Autom. Control 50(2), 169–182 (2005) 24. Pulka, M.: On the mixing property and the ergodic principle for non-homogeneous Markov chains. Linear Algebra App. 434, 1475–1488 (2011) 25. Saburov, Kh.: Krause Mean Processes Generated by Cubic Stochastic Matrices I: Diagonally primitive cubic stochastic matrices (Submitted) 26. Saburov, M.: Ergodicity of nonlinear Markov operators on the finite dimensional space. Non. Anal. Theo. Met. Appl. 143, 105–119 (2016) 27. Saburov, M.: Quadratic stochastic Sarymsakov operators. J. Phys: Conf. Ser. 697, 012015 (2016) 28. Saburov, M.: On regularity of diagonally positive quadratic doubly stochastic operators. Results Math. 72, 1907–1918 (2017) 29. Saburov, M.: On regularity of positive quadratic doubly stochastic operators. Math. Notes 103(2), 328–333 (2018) 30. Saburov, M.: Ergodicity of p−majorizing quadratic stochastic operators. Markov Processes Relat. Fields 24(1), 131–150 (2018) 31. Saburov, M.: Ergodicity of p−majorizing nonlinear Markov operators on the finite dimensional space. Linear Algebra Appl. 578, 53–74 (2019) 32. Saburov, M.: The discrete-time Kolmogorov systems with historic behavior. Math. Meth Appl. Sci. 44(1), 813–819 (2021) 33. Saburov, M., Saburov, Kh.: Reaching a consensus in multi-agent systems: a time invariant nonlinear rule. J. Educ. Vocation. Res. 4(5), 130–133 (2013) 34. Saburov, M., Saburov, Kh.: Mathematical models of nonlinear uniform consensus. ScienceAsia 40(4), 306–312 (2014) 35. Saburov, M., Saburov, Kh.: Reaching a nonlinear consensus: polynomial stochastic operators. Int. J. Cont. Auto. Sys. 12(6), 1276–1282 (2014)

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Linearization for Difference Equations with Infinite Delay Lokesh Singh

Abstract In this article, we construct a conjugacy map for a linear difference equation with infinite delay and corresponding nonlinear perturbation. We also prove that the conjugacy map is one-to-one with some additional conditions. As an application of our result, we show that the cases of (uniform) exponential dichotomy follow from our result. Keywords Delay difference equations · Hartman-grobman theorem · Linearization

1 Introduction The classical Hartman and Grobman theorem [5, 6] is a fundamental result in the local theory of differential equations and dynamical systems. This celebrated theorem provides a topological conjugacy (near a hyperbolic equilibrium point x ∗ = 0) between the dynamics of a nonlinear differential equation x  = Ax + f (x) (on a finite-dimensional space) and the dynamics corresponding to the linear equation x  = Ax. Later, this result was generalized by Pugh [3] and Palis [4] to the Banach space setting. Some other improvements of the theorem are due to Reinfelds [7], and Bates and Lu [8]. It is well known that in general, the conjugacy in the GrobmanHartman theorem is only locally Hölder continuous. On the other hand, there has been a significant amount of work concerned with formulating sufficient conditions under which the conjugacy exhibits higher regularity properties. Some of the early work in this direction is by Sternberg [11] and Belitskii [9, 10], while Rodrigues et al. [13], ElBialy [12] and Zhang and Zhang along with Lu [29–31] contributed recently. Palmer [14] proved the first version of the Grobman-Hartman theorem for nonautonomous differential equations by assuming that the linear part admits an exponenL. Singh (B) University of Rijeka, Rijeka, Croatia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_21

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tial dichotomy (see Sect. (3.1) for definition). Later Shi and Xiong [17] obtained an improvement on Palmer’s result and established Hölder continuity of conjugacies. The version of Palmer’s theorem in the case of discrete time was first established by Aulbach and Wanner [15]. Castañeda and Robledo [28] discussed the regularity of conjugacy map for nonautonomous differential equations. More recently, Barreira and Valls [16] dealth with the case when the linear part admits a nonuniform exponential dichotomy. More recently, Dragiˇcevi´c, Zhang and Zhang [18, 19] discussed the higher regularity of conjugacies in the nonautonomous setting. Some other important contributions to nonautonomous linearization are given in [20, 21, 27]. For delay differential equations, due to several complexities, there has not been much progress. As Sternberg mentioned in [26], in the case of delay differential equations, the solutions form semiflows (instead of flows) and sometimes solutions may not exist in backward time. Later, he obtained Hartman-Grobman theorem [25] for finite delay differential equations in the finite-dimensional setting under some restrictive conditions. Namely, he assumed the existence of a compact global attractor on which the semiflow is one-to-one. Benkhalti and Ezzinbi [24] obtained some improvement over Sternberg’s result. Farkas [22], proved the Hartman-Grobman theorem for autonomous delay differential equations admitting uniform exponential dichotomy in finite-dimensional settings. In recent work, Barreira and Valls [23] extended the result in continuous case of nonautonomous differential equations with finite delay in Banach space setting. They also assumed that the linear part admits a uniform exponential dichotomy. In this work, our objective is to obtain a conjugacy map for nonautonomous discrete equations with infinite delay and corresponding nonlinear delay equations given in (1) and (6) respectively. To obtain the conjugacy map, we assume a sufficient condition in terms of a Green type function (given in (4)) associated with linear delay equation. With some additional assumptions, we also proved that the obtained conjugacy map is one-to-one. As an application of our result, we showed that if the linear delay equation admits (uniform) exponential dichotomy, then the corresponding Green type function satisfies the assumptions of our Theorem 2 and the result follows. With respect to existing work, our work has novelty in two senses, we considered infinite delay difference equation and we proved the result with more general conditions. This article is organized as follows. We describe our setup in Sect. 2. In Sect. 3, we prove our main result and give an application of Theorem 2.

2 Preliminaries Let Z, Z+ and Z− denote the set of all integers, set of nonnegative integers and set of nonpositive integers, respectively. Let (X, | · |) be an arbitrary Banach space. Given a sequence x : Z → X and m ∈ Z, we define xm : Z− → X by xm ( j) = x(m + j) for all j ∈ Z− .

Linearization for Difference Equations with Infinite Delay

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Next we consider a Banach space (B,  · B ) of all sequences φ : Z− → X satisfying the following assumption: (A) There exists J > 0, and K , M : Z+ → [0, ∞), such that if x : Z → X with x0 ∈ B, then for all n ∈ Z+ , xn ∈ B and J |x(n)| ≤ xn B ≤ K (n) sup |x( j)| + M(n)x0 B . 0≤ j≤n

As noted in [1], one such space satisfying the assumption (A) is Banach space (B β ,  · Bβ ), defined by     B β = φ : Z− → X φBβ < ∞ , φBβ := sup [ |φ( j)|eβ j ], j∈Z−

where β is a real constant. Observe that the Banach Space (B β ,  · Bβ ) satisfies (A) with J = 1, M(n) = e−βn and K (n) = 1 if β ≥ 0 and K (n) = e−βn if β < 0. Given a sequence of linear operators Am : B → X, m ∈ Z+ , we consider a linear difference equation given by, x(m + 1) = Am xm ,

for all m ∈ Z+ .

(1)

Given (n, φ) ∈ Z+ × B, there exists a unique sequence x : Z → X such that xn = φ and the sequence x(m) satisfies Eq. (1) for all m ≥ n ≥ 0. The sequence x is called a solution of Eq. (1) through (n, φ) and is denoted by x = x(·; n, φ). Now we define a two parameter solution operator A(m, n) : B → B of the linear equation (1) by, A(m, n)φ = xm (·; n, φ)

for all m ≥ n ≥ 0, and φ ∈ B.

(2)

Clearly, (A(m, n))m≥n≥0 is a discrete evolution family corresponding to linear equation (1) and it satisfies A(n, n) = I dB A(m, k)A(k, n) = A(m, n)

for all n ≥ 0, for all m ≥ k ≥ n ≥ 0.

Here I dB denotes the identity operator on B. Next we consider a sequence of projection maps (Pn )n∈Z+ on B such that Pm A(m, n) = A(m, n)Pn ,

for all m ≥ n ≥ 0,

and  A(m, n)Ker Pn : Ker Pn → Ker Pm

is invertible for m ≥ n ≥ 0.

(3)

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 −1  Also, for n ≥ m ≥ 0, we denote A(m, n) := A(n, m)Ker Pm : Ker Pn → Ker Pm + + and Q m := I dB − Pm for each m ∈ Z . Now, for m, n ∈ Z , we define another operator  A(m, n)Pn for m ≥ n; G(m, n) := (4) −A(m, n)Q n for m < n. This operator is usually called Green operator. With respect to projection maps (Pn )n∈Z+ , we have B = E n ⊕ Fn for each n ∈ Z+ . Here E n and Fn are ranges of projections Pn and Q n respectively. Now, let us consider a space M consisting of continuous functions, η : {(n, φ) : n ∈ Z+ , φ ∈ Fn } → B such that η∞ := sup{η(n, φ)B : n ∈ Z+ and φ ∈ Fn } < ∞. Clearly, (M,  · ∞ ) is a Banach space. We also write ηn = η(n, ·) and

h n = I d Fn + ηn .

Here I d Fn is identity map on subspace Fn . Furthermore, for each m ∈ Z+ , let f m : B → X , be a sequence of maps such that f m (0) = 0 for all m ∈ Z+ and there exist numbers cm > 0 satisfying | f m (φ) − f m (ψ)| ≤ cm min (1, φ − ψB ),

(5)

for every m ∈ Z+ and φ, ψ ∈ B. Finally, we consider a semilinear difference equation given by x(m + 1) = Am xm + f m (xm )

for all m ∈ Z+ .

(6)

Given (n, φ) ∈ Z+ × B, there exists a unique sequence x : Z → X such that xn = φ and x satisfies the semilinear difference equation (6). We write the solution of Eq. (6) in terms of operator R(m, n) given by, xm = R(m, n)xn

for all m ≥ n ≥ 0.

(7)

Now we recall [1] and [2] to give the variation of constants formula for a difference equation given by x(m + 1) = Am xm + pm , where ( pm )m∈Z+ is a sequence in X . Define Γ : Z− → L(X ) by

(8)

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465



I d X for j = 0; 0 X for j < 0.

Γ ( j) =

The symbol L(X ) denotes the space of bounded linear operators in X. For v ∈ X , define Γ v : Z− → X by  (Γ v)( j) := Γ ( j)v =

v for j = 0; 0 for j < 0.

If x : Z → X is defined by x( j) = 0 for j ≤ 0 and x( j) = v for j > 0, then x0 = 0 and x1 = Γ v. Since x0 = 0 ∈ B, by assumption (A), we have that Γ v = x1 ∈ B and Γ vB ≤ K (1)|v|.

(9)

Theorem 1 ([1]) Let φ ∈ B. A sequence x : Z → X is a solution of (8) with initial value x0 = φ if and only if for m ≥ 0, the segment xm satisfies the following relation in B, xm = A(m, 0)φ +

m−1 

A(m, k + 1)(Γ pk ), m ≥ 0.

(10)

k=0

As a corollary of above result, we have that x : Z → X is a solution of (6) if and only if φ = x0 ∈ B and xm = A(m, 0)φ +

m−1 

A(m, k + 1)(Γ f k (xk )), m ≥ 0.

k=0

Let x = x(·; 0, φ) be a solution of Eq. (6) for φ ∈ B, then using constants of variation formula from Theorem 1, for m ≥ n ≥ 0, we have xm = A(m, 0)φ +

m−1 

A(m, k + 1)(Γ f k (xk ))

k=0

= A(m, 0)φ +

n−1 

A(m, k + 1)(Γ f k (xk )) +

k=0

m−1 

A(m, k + 1)(Γ f k (xk ))

k=n

n−1 

A(n, k + 1)(Γ f k (xk )) = A(m, n) A(n, 0)φ + k=0

+

m−1  k=n

= A(m, n)xn +

m−1  k=n

A(m, k + 1)(Γ f k (xk )),

A(m, k + 1)(Γ f k (xk ))

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for all m ≥ n ≥ 0. Therefore, we obtain that x : Z → X is a solution of (6) if and only if x0 ∈ B and for m ≥ n ≥ 0 xm = A(m, n)xn +

m−1 

A(m, k + 1)(Γ f k (xk )).

(11)

k=n

3 Main Result The following theorem is our main result. Theorem 2 Assume that the semilinear Eq. (6) admits q := sup

m∈Z+



cn G(m, n + 1) ,

with K (1)q < 1.

(12)

n∈Z+

Then there exists a function η ∈ M, such that for every m ≥ n ≥ 0 h m ◦ A(m, n) = R(m, n) ◦ h n ,

on Fn .

(13)

Moreover, each map h n is one-to-one provided that the condition (12) holds with a constant sequence of Lipschitz constants (cn )n∈Z+ . First we prove a lemma which will be used to establish that h n is one-to-one for n ∈ Z+ . Set am,n := G(m, n).

(14)

Lemma 1 Assume that the sequence of Lipschitz constants (cn )n∈Z+ in (5) is constant and that (12) holds. Then, for each fixed m ∈ Z+ , we have that lim am,n = 0.

n→∞

Proof et C > 0 be such that cn = C for all n ∈ Z+ . By (12), it follows that sup

m∈Z+





  cn G(m, n + 1) = C sup am,n+1 < ∞. m∈Z+

n∈Z+

Therefore, for each fixed m ∈ Z+ ,  n∈Z+

am,n+1 < ∞.

n∈Z+

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Hence, limn→∞ am,n+1 = 0 for each fixed m ∈ Z+ . Equivalently, lim am,n = 0

n→∞

for each fixed m ∈ Z+ .

(15)

Now we give our proof of Theorem 2. Proof Let us consider a map F : M → M given by 

F(η)(n, φ) =

  G(n, m + 1)Γ f m A(m, n)φ + ηm (A(m, n)φ) ,

(16)

m∈Z+

where η ∈ M and (n, φ) ∈ Z+ × Fn . Furthermore, since f m (0) = 0 for each m ∈ Z+ and using (5), we have, | f m (φ)| ≤ cm min (1, φB ) ≤ cm ,

for all φ ∈ B.

Therefore, using above estimate,

  cm G(n, m + 1) = K (1)q < ∞. F(η)∞ ≤ sup K (1) n∈Z+

m∈Z+

Therefore, the operator F is well defined. We now claim that F is a contraction map. For each η, ξ ∈ M we have |F(η)(n, φ) − F(ξ )(n, φ)|    G(n, m + 1)| f m A(m, n)φ + ηm (A(m, n)φ) − ≤ K (1) m∈Z+

≤ K (1)



  f m A(m, n)φ + ξ m (A(m, n)φ) | G(n, m + 1)cm |ηm (A(m, n)φ) − ξ m (A(m, n)φ)|

m∈Z+

≤ K (1)



cm G(n, m + 1)η − ξ ∞ .

m∈Z+

Therefore, F(η) − F(ξ )∞ ≤ K (1)q η − ξ ∞ . As K (1)q < 1, the operator F is a contraction map. Therefore it has a unique fixed point function, say η i.e. F(η) = η. Now, for φ ∈ Fn , we have

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ηn (φ) =

 m∈Z+

=



  G(n, m + 1)Γ f m A(m, n)φ + ηm (A(m, n)φ)   G(n, m + 1)Γ f m h m (A(m, n)φ) .

(17)

m∈Z+

Note that, Q n ηn (φ) = −



  A(n, m + 1)Q m+1 Γ f m h m (A(m, n)φ) .

(18)

m≥n

Take 0 ≤ p ≤ n. By (18), for F p ψ = A( p, n)φ we have that Q p η p (ψ) = −



  A( p, m + 1)Q m+1 Γ f m h m (A(m, p)ψ) .

(19)

m≥ p

Also from (18), A( p, n)Q n ηn (φ) = −



  A( p, m + 1)Q m+1 Γ f m h m (A(m, n)φ) .

m≥n

Using (19) and the above relation, we obtain that A( p, n)Q n ηn (φ) = Q p η p (A( p, n)φ) +

n−1 

  A( p, m + 1)Q m+1 Γ f m h m (A(m, n)φ) .

m= p

Equivalently, Q n ηn (φ) = A(n, p)Q p η p (A( p, n)φ) +

n−1 

  A(n, m + 1)Q m+1 Γ f m h m (A(m, n)φ) . (20)

m= p

Similarly, from (17), it follows that Pn ηn (φ) =



  A(n, m + 1)Pm+1 Γ f m h m (A(m, n)φ) .

(21)

m p ≥ 0.

Now let n > p ≥ 0 and φ1 , φ2 ∈ F p such that h p (φ1 ) = h p (φ2 ). Using above relation and (25), we have, h n (A(n, p)φ1 ) = h n (A(n, p)φ2 ).

(25)

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Thus,   A(n, p)(φ1 − φ2 ) = − ηn (A(n, p)φ1 ) − ηn (A(n, p)φ2 ) .

(26)

Note that, as φ1 , φ2 ∈ F p ,   φ1 − φ2 = A( p, n)A(n, p)Q p φ1 − φ2 = A( p, n)Q n A(n, p)(φ1 − φ2 ).  Also, A( p, n)Q n is a nonzero map in B, as A( p, n) Fn is invertible. Therefore, we have, φ1 − φ2 B A( p, n)Q n  φ1 − φ2 B . = a p,n

A(n, p)(φ1 − φ2 )B ≥

If φ1 = φ2 , then it follows from (15) that the function n → A(n, p)(φ1 − φ2 ) is unbounded. However, the right hand side of (26), is bounded by 2η∞ , which leads to a contradiction. Therefore φ1 = φ2 and hence h p is one-to-one for all p ∈ Z+ . This completes the proof of our Theorem 2.

3.1 Uniform Exponential Dichotomy Case Let us assume that (1) admits uniform exponential dichotomy, i.e. for each n ∈ Z+ , there exists projection Pn such that: 1. A(m, n)Pn = Pm A(m,  n) for all m ≥ n ≥ 0; 2. for m ≥ n, A(m, n)Ker Pn : Ker Pn → Ker Pm is an invertible map and we denote   −1 the inverse map A(m, n)Ker Pn by A(n, m); 3. there exists D, λ > 0, such that A(m, n)Pn  ≤ De−λ(m−n)

for all m ≥ n;

and   A(m, n) I dB − Pn  ≤ De−λ(n−m)

for all m < n.

In particular, we have that, G(m, n) ≤ De−λ|m−n|

for all m, n ∈ Z+ .

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By comparing the above expression with (14), we have that am,n ≤ De−λ|m−n| . Finally, let C > 0 be a constant such that cm = C for all m ∈ Z+ . Using above estimates, we have

 

  am,n+1 ≤ C sup De−λ|m−n−1| C sup m∈Z+

n∈Z+

m∈Z+

= C D sup

n∈Z+

−λ

m∈Z+

e (1 − e−λ(m−1) ) 1  < ∞. + 1 − e−λ 1 − e−λ

Using the constant C, we can make sure that Eq. (12) is satisfied. Therefore, if the linear delay difference equation (1) satisfies uniform exponential dichotomy then our Theorem 2 is applicable. Acknowledgements Author would like to thanks Prof. D. Dragiˇcevi´c for his valuable suggestions throughout the process of solving and writing this article. The Author is supported by Croatian Science Foundation under the project IP-2019-04-1239.

References 1. Murakami, S.: Representation of solutions of linear functional difference equations in phase space. Nonlinear Anal. 30, 1153–1164 (1997) 2. Dragiˇcevi´c, D., Pituk, M.: Shadowing for nonautonomous difference equations with infinite delay. Appl. Math. Lett. 120, 107284 (2021) 3. Pugh, C.: On a Theorem of P. Hartman. Am. J. Math. 91(2), 363–367 (1969) 4. Palis, J.: On the local structure of hyperbolic points in Banach spaces. An. Acad. Brasil. Cienc. 40, 263–266 (1968) 5. Hartman, P.: On the local linearization of differential equations. Proc. Am. Math. Soc. 14, 568–573 (1963) 6. Grobman, D.: Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128, 880–881 (1959) 7. Reinfelds, A.: A generalized theorem of Grobman and Hartman. Latv. Mat. Ezhegodnik 29, 84–88 (1985) 8. Bates, W., Lu, K.: A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations. J. Dynam. Differ. Equ. 6(1), 101–145 (1994) 9. Belitskii, G.R.: Functional equations and the conjugacy of difeomorphism of finite smoothness class. Funct. Anal. Appl. 7, 268–277 (1973) 10. Belitskii, G.R.: Equivalence and normal forms of germs of smooth mappings. Russian Math. Surv. 33, 107–177 (1978) 11. Sternbger, S.: Local contractions and a theorem of Poincare. Am. J. Math. 79, 809–824 (1957) 12. ElBialy, M.S.: Smooth conjugacy and linearization near resonant fixed points in Hilbert spaces. Houston J. Math. 40, 467–509 (2014) 13. Rodrigues, H.M., Solà-Morales, J.: Invertible contractions and asymptotically stable ODEs that are not C1-linearizable. J. Dyn. Difer. Equ. 18, 961–974 (2006) 14. Palmer, K.: A generalization of Hartmans linearization theorem. J. Math. Anal. Appl. 41, 753–758 (1973) 15. Aulbach, B., Wanner, T.: Topological simplifcation of nonautonomous diference equations. J. Differ. Equ. Appl. 12, 283–296 (2006)

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16. Barreira, L., Valls, C.: A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics. J. Differ. Equ. 228, 285–310 (2006) 17. Shi, J.L., Xiong, K.Q.: On Hartmans linearization theorem and Palmers linearization theorem. J. Math. Anal. Appl. 192, 813–832 (1995) 18. Dragiˇcevi´c, D., Zhang, W., Zhang, W.: Smooth linearization of nonautonomous diference equations with a nonuniform dichotomy. Math. Z. 292, 1175–1193 (2019) 19. Dragiˇcevi´c, D., Zhang, W., Zhang, W.: Smooth linearization of nonautonomous diferential equations with a nonuniform dichotomy. Proc. Lond. Math. Soc. 121, 32–50 (2020) 20. Backes, L., Dragiˇcevi´c, D.: A generalized Grobman-Hartman theorem for nonautonomous dynamics. Collect, Math (2021) 21. Barreira, L., Dragiˇcevi´c, D., Valls, C.: Existence of conjugacies and stable manifold theorem via suspensions. Elec. J. Differ. Equ. 2017(172), 1–11 (2017) 22. Farkas, G.: A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria. Z. Angew. Math. Phys. 52, 421–432 (2001) 23. Barreira, L., Valls, C.: Perturbations of delay equations. J. Differ. Equ. 269, 7015–7041 (2020) 24. Benkhalti, R., Ezzinbi, K.: A Hartman and Grobman theorem for some partial functional differential equations. Int. J. Bif. and Cha. 10(5), 1165–1169 (2000) 25. Sternberg, N.: A Hartman-Grobman theorem for a class of retarded functional differential equations. J. Math. Anal. Appl. 176, 156–165 (1993) 26. Sternberg, N.: A Hartman-Grobman theorem for maps. In: Ordinary and Delay Differential Equations, Edinburg, TX, 1991, in: Pitman Res. Notes Math. Ser., vol. 272, Longman Sci. Tech., Harlow, pp. 223–227 (1992) 27. Dragiˇcevi´c, D.: Global smooth linearization of nonautonomous contractions on Banach spaces. Electron. J. Qual. Theory Differ. Equ., Paper No. 12, pp. 1–19 (2022) 28. Castañeda, A., Robledo, G.: Differentiability of Palmer’s linearization theorem and converse result for density function. J. Differ. Equ. 259, 4634–4650 (2015) 29. Zhang, W.M., Zhang, W.N.: C1 linearization for planar contractions. J. Funct. Anal. 260(7), 2043–2063 (2011) 30. Zhang, W.M., Zhang, W.N.: Sharpness for C1 linearization of planar hyperbolic diffeomorphisms. J. Differ. Equ. 257, 4470–4502 (2014) 31. Zhang, W.M., Lu, K., Zhang, W.N.: Differentiability of the conjugacy in the Hartman-Grobman Theorem. Trans. Am. Math. Soc. 369, 4995–5030 (2017)

A Method to Derive Discrete Population Models Sabrina H. Streipert and Gail S. K. Wolkowicz

Abstract We propose a derivation method to obtain discrete multi-species population models based on the assumption that the population sizes at time t + 1 can be expressed as a multiple of the population sizes at time t. The multiplicative term is determined by placing the growth processes in the numerator and the decline processes in the denominator of the fitness of each population. Each resulting discrete model can be related to a continuous population model based on the same model assumptions using the relationship between discrete and continuous compounding in finance. We exploit this relationship to compare the stability for the continuous and the discrete models and argue that the corresponding continuous model analogues are more stable. We illustrate the derivation technique by providing several examples of discrete models of single and multi-species derived using this method and compare their stability properties with their continuous analogues. Keywords Population modeling · Difference equations · Local stability · Fitness · Per-capita growth rate

1 Introduction Differential equations are a well-established method to describe population dynamics. Among the most well-known models are the single species logistic growth model [51] and the multi-species Gaus e-Lotka-Volterra model [36]. The differential equations can often be constructed by making assumptions on the per-capita growth rate of each species. For example, the logistic single species model assumes that the per-capita growth rate is a linear decreasing function of the population size of the S. H. Streipert University of Pittsburgh, 4200 Fifth Ave, 15260, Pittsburgh, PA, USA S. H. Streipert · G. S. K. Wolkowicz (B) McMaster University, L8S3L8, Hamilton, ON, Canada e-mail: [email protected] URL: https://ms.mcmaster.ca/wolkowic © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_22

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species. This is consistent with the assumption in the Gause-Lotka-Volterra model in which the per-capita growth rate of each species is a linear decreasing function with respect to all of the species representing the decline in each population due to intra- and inter-specific competition. Refinements of the logistic model that consider spatial movement and interactions with other species have been discussed (e.g., see [3, 11, 21, 23, 33, 38, 39, 45, 46, 48, 50]). Other refinements include the consideration of stochastic effects (e.g., see [22, 37, 40, 49, 53]) and more recently, fractional versions of the above models (e.g., see [1, 4, 19, 30]). Other population models may include a nonlinear per-capita growth rate, for example, to account for an Allee effect. Then, the per-capita growth rate of the species may be a quadratic polynomial of the population size [3]. In this work, we provide a method to derive discrete population models based on the underlying assumption that the population at time t + 1 can be expressed as a multiple (fitness function) of the population size at time t. This can be interpreted as the discrete analogue of the assumption used to derive some continuous models, that the (instantaneous) rate of change of the population is the product of the population at time t and the per-capita growth rate. The study of discrete population models may be motivated, for example, by the interest in populations with non-overlapping generations, population levels known at certain discrete time points, or simply a computational preference. It is well-known that a discrete model can exhibit significantly different behavior as compared to a continuous model that was formulated for the same set of model assumptions. A standard example is the logistic differential equation x  = r x(1 − x), where solutions converge monotonically to the carrying capacity for positive initial conditions. A popularized discretization is the logistic map X t+1 = r X t (1 − X t ) that, in contrast, can exhibit chaotic dynamics. The logistic map and the related logistic difference equation X t+1 − X t = r X t (1 − X t ) are obtained by assuming that the per-capita growth rate of the continuous model expresses the (discrete) rate of change of the population. The forward Euler scheme replaces a derivative operator in a continuous system by the forward Euler (derivative) operator. Although this method is used to numerically approximate solutions to continuous models, it often lacks biological relevance as solutions can become negative. Despite these obvious disadvantages, the application of the forward Euler scheme remains a common practice in formulating single and multi-species discrete models (e.g., see [2, 28, 31, 32]). The Ricker map, X t+1 = X t e−r (1−X t ) , is yet another accepted discretization of the continuous logistic growth model. Originally derived to fit stock recruitment data, Ricker recognized a lognormal relation in the data that can be written as the above exponential relation. In contrast to the logistic difference equation, the Ricker model overcomes the negativity of solutions for positive initial conditions and thus, satisfies a necessary condition for biologically meaningful descriptions. However, since the Ricker map exhibits chaotic behavior, it is arguable to what extent it should be considered as “the discrete analogue of the logistic growth model”. Different derivations of the Ricker model to justify the specific form from a data independent point of view have since been proposed (e.g., see [9, 10, 44]). These proposed derivations were however limited to single species populations. Nevertheless, several different formulations of Ricker type models to describe species interactions have

A Method to Derive Discrete Population Models

475

been analysed (e.g., see [6, 13, 17, 25, 35]). Unfortunately, none of these papers include a derivation method to justify their specific formulation of the Ricker type multi-species model. ν Xt (ν > 1), is yet another discrete The Beverton–Holt recurrence, X t+1 = 1+(ν−1)X t model that has been accepted as discretization of the logistic growth model [10]. Originally, the recurrence was derived based on an underlying continuous growth process that follows the logistic model [7]. It is therefore not surprising that the dynamics are consistent with the behavior of solutions for the continuous logistic model. Solutions converge monotonically to the unique positive equilibrium for positive initial population levels. The generalization of this derivation procedure to describe interacting species is however generally not possible because it relies on the analytic solution of an underlying continuous model. Nevertheless, other multi-species models that resemble the Beverton–Holt structure have been proposed (e.g., see [12, 14, 16]). Unfortunately, most of these models are not derived nor biologically justified. In this paper, we propose a method for deriving discrete multi-species population models based on the underlying biological assumptions. The axiom of parenthood is satisfied and populations levels cannot become negative for non-negative initial conditions in the models derived using this method. Thus, our proposed models overcome one of the disadvantages of the commonly applied forward Euler scheme method. We provide examples of single and multi-species models derived using this method. We then compare the equilibria and their stability of discrete and continuous models derived based on the same underlying assumptions.

2 Discrete Model Derivation We consider k different populations and let X t(i) denote population i at time t, where i = 1, 2, . . . , k and t ∈ N0 , where N0 denotes the nonnegative integers. We assume that the population size at time t + 1 is equal to a multiple, f i (t, Xt ), of the pop(i) = f i (t, Xt )X t(i) , where Xt = (X t(1) , X t(2) , . . . , X t(k) ). ulation at time t, that is, X t+1 The multiplicative term, f i , is considered to be the fitness function of species X (i) and is assumed to capture growth contributions in pi (t, Xt ) ≥ 0, which we place in the numerator and decline contributions in qi (t, Xt ) ≥ 0, which we place in the denominator, so that the population model is given by: (i) = F(t, Xt ) = f i (t, Xt )X t(i) = X t+1

1 + pi (t, Xt ) (i) Xt , 1 + qi (t, Xt )

t ∈ N0 ,

(1)

and i ∈ {1, 2, . . . , k}. If there are no processes that contribute to the growth nor to the decline of the population, then both pi (t, Xt ) = 0 and qi (t, Xt ) = 0, respectively. (i) = X t(i) for all t. In this case, f i (t, Xt ) = 1 and X t+1 In order to obtain the fitness functions, f i (t, Xt ), the growth contributions and the decline contributions are determined based on the assumptions in the scenario modelled.

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A corresponding continuous model, based on the same growth and decline assumptions, with k species at time t, x(t) = (x1 (t), x2 (t), . . . , xk (t)), would be given by, ˜ x(t)) = f˜i (t, x(t))xi (t) = ( pi (t, x(t)) − qi (t, x(t)))xi (t), xi (t) = F(t,

(2)

t ∈ R, and i ∈ {1, 2, . . . , k}. Note that here, the per-capita growth rate of the population i, f˜i (t, x(t)), plays the role of the fitness function in the discrete model. As a simple example, consider a model involving only a single population with birth rate b > 0 and death rate d > 0. Thus, the birth rate of the population contributes to the growth process, so that p(t, X t ) = b and the death rate determines the decline 1+b X t , which process, q(t, X t ) = d. The recurrence (1) is then given by X t+1 = 1+d would correspond to the continuous (ordinary differential equations) model, x  (t) = (b − d)x(t). The claim that (2) is a natural continuous analogue of (1) is based on the observation that in the case of a single population, the solution of (2) for p, q ∈ R satisfies x(t + 1) = Since

e

 t+1

e

t

 t+1 t

p ds q ds

x(t) =

ep x(t). eq

(3)

 p n e p = lim 1 + , n→∞ n

e p represents the continuous occurrence of (reproductive-)events during the time n interval (t, t + 1), where as 1 + np represents n such events distributed equally over the same time interval. If only one such event occurs, that is, n = 1, then 1 + p should be applied instead of e p . This is related to discrete and continuous compounding in finance. An investment I is worth e p I at the end of a year, where e p represents the continuous compounding of an investment with an annual interest rate p. If instead the investment is compounded discretely, once a year, the value of the investment is (1 + p)I . This relation suggests that the continuous analogue of p 1+ p is eeq . 1+q

3 Analysis of the Discrete Model Note that the trivial equilibrium, E 0 = (0, 0, . . . , 0), is always an equilibrium of (1). Another observation is that (1) satisfies the axiom of parenthood, that is, if X 0(i) = 0, then X t(i) = 0 for all t ≥ 0. As well, if X 0(i) > 0, then X t(i) > 0 for all t ≥ 0. Therefore, the proposed discrete models in (1) do not suffer from the possibility that population levels could become negative, thus overcoming a common disadvantage of discrete models that are based on the Euler discretization scheme. In this section, we consider the autonomous version of (1),

A Method to Derive Discrete Population Models (i) X t+1 = f i (Xt )X t(i) =

477

1 + pi (Xt ) (i) Xt , 1 + qi (Xt )

i = 1, 2, . . . , k,

(4)

where pi (Xt ), qi (Xt ) ≥ 0 for all Xt ∈ [0, ∞)k , pi , qi ∈ C 1 , and t ∈ N0 . The Jacobian of (4) at a point Z = (Z (1) , Z (2) , . . . , Z (k) ) ∈ Rk is (J (Z))i j =

⎧ ⎨ f i (Z) + Z (i) ∂ fi (Z) , i = j, ∂ Z (i) ⎩ Z (i) ∂ fi (Z) ,

(5)

i = j.

∂ Z ( j)

k Let en , n = 1, 2, . . . , k denote the standard

k unit-basis vectors in R and let E denote an equilibrium of (4), where E = i=1 bi ei with bi ≥ 0. We subdivide the k species into two groups by distinguishing between zero and nonzero equilibrium components. Let I 0 = {i ∈ {1, 2, . . . , k} : bi = 0} and I + = {i ∈ {1, 2, . . . , k} : bi > 0}. Note that I + includes only bi > 0, because bi < 0 is not biologically feasible. We say that population X (i) belongs to group G 0 if i ∈ I 0 . Otherwise, if i ∈ I + , then X (i) belongs to group G + . Without loss of generality, assume I 0 = {1, 2, . . . , m} and I + = {m + 1, m + 2, . . . , k} and assume that I + = ∅. By (5), we have

 (J (E))i j =

where κi j =

⎧ f i (E), ⎪ ⎪ ⎪ ⎨ (i) f i (E) + E κii , j = i, 1 + E (i) κii , = (i) j = i, ⎪ E κi j , 0, ⎪ ⎪ ⎩ (i) E κi j ,

∂ f i (E) . ∂ X ( j)

j j j j

= i ∈ I 0, = i ∈ I +, = i, i ∈ I 0 , = i, i ∈ I + ,

Thus, the Jacobian is of block form,

⎡

f 1 (E) .. ⎢ ⎢ . ⎢ ⎢ 0 ⎢ J (E) = ⎢ ⎢ (m+1) ⎢E κm+1,1 ⎢ . ⎢ . ⎣ . E (k) κk,1

... .. . ...

0 .. . f m (E)

0 .. . 0

. . . E (m+1) κm+1,m 1 + E (m+1) κm+1,m+1 . . . . . . . . . (k) (k) ... E κk,m E κk,m+1

... .. . ... ... . . . ...

0 .. . 0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ (m+1) E κm+1,k ⎥ ⎥ . ⎥ . ⎦ . 1 + E (k) κk,k

Therefore, for the analysis of the stability of E, we can split the group of populations into two groups, G 0 and G + and consider their stability separately. We will see that for E to be stable, the equilibrium needs to be stable for both subgroups. Since G 0 = {X (1) , . . . , X (m) } and G + = {X (m+1) , . . . , X (k) }, we can consider two separate subsystems 1 + Pi (Ut ) (i) (i) Ut , = 1 ≤ i ≤ m, (6) Ut+1 1 + Q i (Ut ) and

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S. H. Streipert and G. S. K. Wolkowicz (i) Vt+1 =

1 + P i (Vt ) 1 + Q i (Vt )

Vt(i) ,

1 ≤ i ≤ k − m,

(7)

where Pi (Ut )

= pi (U (1) , . . . , U (m) , E (m+1) , . . . , E (k) ),

P i (Vt ) = pi (0, . . . , 0, V (1) , . . . , V (k−m) ),

Q i (Ut )

= qi (U (1) , . . . , U (m) , E (m+1) , . . . , E (k) ),

Q i (Vt ) = qi (0, . . . , 0, V (1) , . . . , V (k−m) ).

The Jacobian of (4) can now be expressed as  JU (0) 0 J (E) = C JV (E) 

for an appropriate matrix C, where JU (0) is the m × m Jacobian of (6) evaluated at U0 = (0, . . . , 0) ∈ Rm , and JV (E) is the (k − m) × (k − m) Jacobian evaluated at E = (E (m+1) , . . . , E (k) ). Since det(J (E) − λIk ) = det(JU (0) − λIk )det(JV (E) − λIk )  m   K − λIm−k ), ( f i (E) − λ) det(Im−k + D =

(8)

i=1

where

 = diag(E(m+1) , E(m+2) , . . . , E(k) ), D

Kij = ˇm+i,m+j ,

(9)

the eigenvalues of the k × k Jacobian of (4) are given by the eigenvalues of the m × m Jacobian corresponding to the subpopulations in G 0 , and the eigenvalues of the (k − m) × (k − m) Jacobian corresponding to the subpopulations in G + . Therefore, the stability of E for (4) is determined by the stability of the corresponding equilibria of each subpopulation.

3.1 Extinction and Competitive Exclusion The structure of the Jacobian discussed in the previous section is used to obtain the following stability results. Proposition 1 Consider (4). If there exists j, such that 1 ≤ j ≤ k such that p j (E 0 ) > q j (E 0 ), then E 0 is unstable. If, however, pi (E 0 ) < qi (E 0 ) for all i = 1, 2, . . . , k, then the extinction equilibrium is locally asymptotically stable. Proof By (8),

A Method to Derive Discrete Population Models

det(J (E 0 ) − λIk ) =

479 k 

( f i (E 0 ) − λ),

i=1

identifying the eigenvalues of the Jacobian evaluated at E 0 as f i (E 0 ) = Hence the result follows.

1+ pi (E 0 ) . 1+qi (E 0 )



Proposition 2 Consider (4). Let en , n = 1, 2, . . . , k denote the standard unit-basis vectors in Rk . Let E B = b e j be a boundary equilibrium of (4) for some j ∈ {1, 2, . . . , k} and b > 0. If     1 + b ∂ f j (E B )  < 1   ( j) ∂X

and

pi (E B ) < qi (E B ) for all i = j,

then E B is asymptotically stable. If either one of these inequalities is reversed, then E B is unstable. Proof Here, E B represents an equilibrium in which species X ( j) is the sole survivor. Without loss of generality, label j = k. By (8), det(J (E B ) − λIk ) = (1 + bκk,k − λ)

k−1 

 ( f i (E B ) − λ) ,

i=1

where κkk = ∂ ∂fkX(E(k)B ) . Thus, the eigenvalues are given by f i (E B ) for 1 ≤ i < k and 

1 + bκkk . This completes the proof. Proposition 2 implies that species X ( j) can only be the sole survivor if κ j j < 0, that is, there exists a detrimental density effect, for example, due to intra-specific competition.

3.2 Relationship between the Discrete Model and its Continuous Counterpart The previous results addressed the local asymptotic stability of the trivial equilibrium and specific boundary equilibria that correspond to the survival of only one population. We now discuss the stability of interior equilibria by exploiting the relationship between (1) and (2). In the autonomous case, the continuous analogue of (4) is xi = ( pi (x) − qi (x))xi ,

i = 1, 2, . . . , k.

(10)

We let E Z and E R denote the set of equilibria and J Z (E) and J R (E) the Jacobians evaluated at the equilibria, E, of (4) and (10), respectively. Theorem 1 Consider (4) and (10). Then E Z = E R .

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Proof Clearly, E 0 = (0, 0, . . . , 0) ∈ E Z ∩ E R . Let E = (E (1) , E (2) , . . . , E (k) ) ∈ E Z . pi (E) = 1. Therefore, pi (E) = qi (E) for all i such For each i such that E (i) = 0, 1+ 1+qi (E) (i) that E = 0 and so E ∈ E R . Reversing the argument, if E ∈ E R , then E ∈ E Z . The result follows. 

Remark 1 By Proposition 1, if pi (E 0 ) < qi (E 0 ) for all i = 1, 2, . . . , k, then E 0 is stable for (4). These are the same conditions for the stability of E 0 of the corresponding continuous model (10). Thus, the stability of the extinction equilibrium (if hyperbolic) is the same for both (4) and (10). Theorem 2 Consider (4) and (10). Let E be an equilibrium. Then, J Z (E) = Ik + D J R (E),

(11)

where D = diag(d1 , d2 , . . . , dk ) with di = (1 + qi (E))−1 . Proof Let E = (E (1) , E (2) , . . . , E (k) ) ∈ E Z . By Theorem 1, E Z = E R , and so E ∈ E R . The entries of the Jacobian, J R (E), are given by

(J R (E))i j =

 ⎧ (i) ∂ pi (E) ⎪ E − ⎪ ∂x j ⎨

∂qi (E) ∂x j



, E (i) = 0, E (i) = 0, E (i) = 0,

pi (E) − qi (E), ⎪ ⎪ ⎩ 0,

j = i, j = i,

since pi (E) = qi (E) for E (i) = 0. By (5), for E (i) = 0, the entries of the Jacobian of (4) are given by:

(J Z (E))i j =

⎧ ∂ pi (E) ∂qi (E) pi (E) ⎪ (i) ∂ X (i) (1+qi (E))− ∂ X (i) (1+ pi (E)) ⎨ 1+ + E , j = i, 1+qi (E) (1+qi (E))2 ∂ p (E) ∂q (E) ⎪ ⎩ E (i) ∂ Xi ( j) (1+qi (E))− ∂ Xi ( j) (1+ pi (E)) ,

(1+qi (E))2

=

j = i,

⎧ ∂ pi (E) ∂qi (E) (i) − ∂ X (i) ⎪ ⎨1 + E (i) ∂ X1+q , j = i, i (E) ∂ p (E) ∂q (E) ⎪ ⎩ E (i) ∂ Xi ( j) − ∂ Xi ( j) ,

1+qi (E)

= χ( j=i) + E (i) = χ( j=i) +

j = i,

∂ pi (E) ∂ X ( j)

i (E) − ∂q ∂ X ( j) , 1 + qi (E)

1 (J R (E))i j , 1 + qi (E)

where χ( j=i) = 1 if j = i and 0, otherwise. If instead E (i) = 0, then

A Method to Derive Discrete Population Models

(J Z (E))i j =

⎧ ⎨ 1+ pi (E) , j = i, 1+qi (E) ⎩0,

j = i,

= χ( j=i) +

=

481

 1+

1 ( pi (E) 1+qi (E)

− qi (E)), j = i,

0,

j = i,

1 (J R (E))i j . 1 + qi (E)

Thus, J Z (E) = Ik + D J R (E), where D is a diagonal matrix with entries di = (1 + 

qi (E))−1 . This completes the proof. Theorem 2 establishes a relationship between the stability of equilibria of the discrete model (4) and its continuous analogue (10). For example, it immediately follows that if λ > 0 is an eigenvalue of D J R (E), then E is unstable for the discrete model (4). Other consequences are summarized below. Consider (10). If E = (E (1) , E (2) , . . . , E (k) ) is an interior equilibrium and sii := ∂ pi (E) i (E) − ∂q∂x > 0 for i = 1, 2, . . . , k, then E is unstable, because ∂xi i 0
0. By Theorem 2, μ = 1 + dλ is an eigenvalue of J Z (E), implying that E is unstable for (4). 

If there is only one population, i.e., if k = 1, then Theorem 3 applies and the continuous model (10) is at least as stable as the discrete model (4). Theorem 4 Let D be defined as in Theorem 2. If J R is normal and J R D J RT = J RT D J R , then D J R and J R have the same number of eigenvalues on each open ray from the origin in the complex plane. This implies that E Ru ⊆ E Zu and E Rnh ⊆ E Znh .

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Proof First note that J R and D 2 J R D 2 are congruent, because for P = D − 2 , P T = P, P is invertible, and 1

1

1

1

1

1

1

P T (D 2 AD 2 )P = P D 2 AD 2 P = A. 1

1

If J R D J RT = J RT D J R , then D 2 J R D 2 is normal. Since J R is assumed to be normal, 1 1 it follows by [26], that J R and D 2 J R D 2 share the same number of eigenvalues on each open ray from the origin in the complex plane. 1 1 Since D J R and D 2 J R D 2 share the same eigenvalues, D J R and J R share the same number of eigenvalues on any open ray from the origin. Thus, as in the proof of Theorem 2, if Re(λ) > 0 for an eigenvalue λ of J R (E), then |μ| > 1 is an eigenvalue 

of J Z (E).

3.3 Importance of the Composition of the Model Parameters Let us consider a single species model based on the assumptions: (i) The species has a constant intrinsic growth rate r > 0. (ii) The species is exposed to intra-specific competition at a constant rate c > 0. Based on (i), p(X ) = r and due to (ii), q(X ) = cX , resulting in X t+1 =

1+r Xt . 1 + cX t

(12)

To explore the effect of detailed information on the parameter composition, we assume that more information about the model parameters is available. For example, assume that the intrinsic growth rate r is in fact the difference between the birth and the death rate, i.e., r = b − d > 0, where information about the values of b> 0 and d> 0 is available. Taking this information into account, the growth and decline q (X t ) = processes can be included in the model separately to obtain:  p (X t ) = b and  d + cX t . Thus, taking this more detailed information into account, model (12) could be replaced by: 1+ p (X t ) 1+b Xt = Xt . (13) X t+1 = 1 + q (X t ) 1 + d + cX t In this simple case, the equilibria and their stability is the same for both models (12) and (13). However, this is not always the case. To investigate, more generally, when more information can result in different dynamics, we generalize (12) and (13) to the following multi-species models (14) and (15), respectively:

(i) X t+1

 j



1 + (κi )+ + km=1 Nj=1 (ai,m j )+ X t(m) 1 + pi (Xt ) (i) (i) Xt = =   j X t , (14)



1 + qi (Xt ) (m) k N 1 + (κi )− + m=1 j=1 (ai,m j )− X t

A Method to Derive Discrete Population Models

483

and (i) = X t+1

1+ pi (Xt ) (i) Xt 1 + qi (Xt )

 j bi,m j X t(m) (i) =   j Xt ,

k N (m) 1 + κi,2 + m=1 j=1 ci,m j X t 1 + κi,1 +

k

m=1

N

j=1

(15)

for i = 1, 2, . . . , k, where N is the order of the polynomials in X t(i) in the numerator and denominator,   z, z > 0, −z, z < 0, + − (z) = (z) = , 0, z ≤ 0, 0, z ≤ 0, κi = κi,1 − κi,2 , ai,m j = bi,m j − ci,m j , and κi,1 , κi,2 , bi,m j , ci,m j ≥ 0. +  (i)  j

 Xt In (14), when m = i, the term Nj=1 ai,i j captures the density dependent overall beneficial effects for species X (i) resulting in growth from intra+  (m)  j

 specific species interactions and km=1,m=i Nj=1 ai,m j Xt captures the density dependent mutualistic effects resulting in growth due to the interaction of species X (i) with the other populations. The corresponding summations involving (ai,m j )− , correspond to the corresponding interactions that result in the decline of species X (i) . In (15), additional information is given on the composition of the parameters. In particular, we assume that the net-effect of the model parameters can be written explicitly as the difference of two effects, one that contributes to growth and one that causes a decline of the population. For example, κi = κi,1 − κi,2 , where κi,1 ≥ 0 contributes to growth and κi,2 ≥ 0 to the decline. Similarly, we consider ai,m j = bi,m j − ci,m j , where bi,m j ≥ 0 contribute to the growth and ci,m j ≥ 0 to the decline. We let E Z be the set of equilibria for (14) and E  Z be the set of equilibria for (15) and we denote the Jacobians evaluated at E for (14) and (15) by J Z (E) and J Z (E), respectively. Remark 2 The equilibrium E 0 in which all species are extinct, is an equilibrium for both models (14) and (15). The stability of E 0 is the same for models (14) and (15). To see this, consider the following. The stability of E 0 for (14) is, by Remark 1, the same as for the continuous model xi = ( pi (x) − qi (x))xi and the stability of pi (x) −  qi (x))xi . Since E 0 for (15) is the same as the stability of E 0 for xi = ( qi (x) = pi (x) − qi (x),  pi (x) −  the stability of E 0 is the same for both continuous models, and hence it is the same for both models (14) and (15). Theorem 5 Consider (14) and (15). Then, E Z = E  Z and J Z (E) = Ik − D1 D2−1 + D1 D2−1 J Z (E),

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where D1 = diag((1 + q1 (E))−1 , . . . , (1 + qk (E))−1 ) D2 = diag((1 +  q1 (E))−1 , . . . , (1 +  qk (E))−1 ). Proof When simplified, the right hand side of the continuous version of (15) is the same as the right hand side of the continuous version of (14), since k  N  j  (ai,m j )+ X t(m) − (κi )− pi (Xt ) − qi (Xt ) = (κi ) + +

m=1 j=1

−

k  N  j  (ai,m j )− X t(m) m=1 j=1

= κi +

k  N 



ai,m j X t(m)

j

m=1 j=1

= κi1 +

k  N 

k  N  j  j  bi,m j X t(m) − κi2 − ci,m j X t(m)

m=1 j=1

m=1 j=1

= pi (X t ) −  qi (X t ). Applying Theorem 2, it follows that E Z = E R = E  Z and since J Z (E) = Ik +

 D1 J R (E) and J Z = Ik + D2 J R (E), the result follows. Remark 3 When there is only a single population under consideration, i.e., k = 1, D1 D2−1 = d ∈ R+ , so that the eigenvalues of J Z (E) are given by μ = 1 + d( μ − 1), (E). If  μ > 1, then μ > 1. If  μ < −1, then μ may or where  μ is any eigenvalue of J Z may not be in the unit-circle. Also, if d > 1, then |μ| > 1 for | μ| > 1. This implies that for d > 1, an unstable hyperbolic equilibrium of (15) is also an unstable hyperbolic u u ⊆ E Zu if d > 1. Instead if d < 1, then E Zu ⊆ E  . equilibrium for (14). Thus, E  Z Z Biologically, the above theorem implies that although detailed information about the composition of the model parameters does not change the set of equilibria, it can impact their stability. It will be interesting to investigate this effect and classify population models where more information results in destabilizing (or stabilizing) of the equilibria of the models. This distinction highlights the importance of the available information for such models and reveals their implications.

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4 Examples of Models Obtained Using the Derivation Method In this section, we illustrate our method for deriving discrete population models by determining the corresponding growth and decline functions based on the assumptions for the different models.

4.1 Single Species Models We begin by using the derivation technique described in Sect. 2 to obtain three different single species models based on three different sets of assumptions. Example 1 In our first example, we make the following assumptions on a population, X : (A1) The fitness function includes a constant growth rate. (A2) The fitness function is negatively impacted by intra-specific competition assumed to be proportional to X t . Based on (A1), the growth contribution is determined by a constant rate r > 0, so that p(X t ) = r . Assumption (A2) identifies the decline contribution of the fitness function to be due to intra-specific competition, so that q(X t ) = cX t , where c > 0 represents the strength of the competition. Model (4) then becomes X t+1 =

1+r ν K Xt Xt = , 1 + cX t K + (ν − 1)X t

(16)

which is the Beverton–Holt model or the non-delayed Pielou model, for ν = 1 + r > 1 and c = ν−1 > 0. The dynamics of the Beverton–Holt model are known to be K similar to the dynamics of the continuous logistic growth model, i.e., for positive initial conditions, solutions converge monotonically to the carrying capacity K (e.g., see [8, 10, 41, 42]). Applying model assumptions (A1) and (A2) in a continuous time framework, the per-capita growth rate is identified to be a linear function that is decreasing in the population size. This results in the classical logistic ordinary differential equation, x  (t) = x(r − cx). Thus, the logistic differential equation is the continuous counterpart of the Beverton–Holt model. Example 2 We can also consider a different density dependence with regard to the intra-specific competition, so that assumption (A2) is replaced by (A2∗ ) The fitness is negatively impacted by intra-specific competition assumed to γ be proportional to X t .

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Assuming (A1) and (A2∗ ), the growth contribution remains p(X t ) = r , but the γ decline contribution changes to q(X t ) = cX t . Model (4) then becomes X t+1 =

1+r γ Xt . 1 + cX t

(17)

If γ ∈ (0, 1), then the competition effect is less pronounced compared to values γ > 1. For γ = 1, the model is identical to (16). The model differs from the Sigmoid Beverton–Holt model (e.g., see [15, 24, 27, 29, 34]), a refinement of the classical Beverton–Holt model that also allows the population size to be raised to a power. Applying the same model assumptions (A1) and (A2∗ ) in a continuous framework, the per-capita growth rate is now a power function of the form r − cX γ , so that (10) becomes   x γ  ˜ , (18) = f˜(x(t))x(t) = (r − cx γ )x = r x 1 − x  = F(x(t)) K for c = r/K γ . It is easy to check that the set of equilibria for (18) is E = {E 0 = 0, E 1 = K }. Thus, by Theorem 5, E is also the set of equilibria for (17). Since F˜  (E 0 ) = r > 0 and F˜  (E 1 ) = −γr < 0, E 0 is unstable and E 1 is locally asymptotically stable for (18). By Remark 1, E 0 is also unstable for the corresponding discrete model (17). By Theorem 3, since D −1 = 1 + q(E 1 ) = 1 + cK γ = 1 + r , γr . This implies that E 1 is unstable if we have J Z (E 1 ) = 1 + D J R (E 1 ) = 1 − 1+r γr > 2(1 + r ). Thus, the set of unstable equilibria of the continuous model is a proper subset of the set of unstable equilibria of the discrete model, and so the continuous model is more stable than the discrete model. Example 3 To consider a model that exhibits an Allee effect, we make the following assumptions: (B1) The fitness function contains a constant decline rate to capture a decrease in the population for small population sizes. (B2) The fitness function contains a growth contribution that is proportional to the population size to capture a positive growth effect for intermediate population sizes. (B3) The fitness is negatively impacted when the population size becomes large at a rate assumed to be proportional to the square of the population size. This dominates the positive growth effect assumed in (B2) once the population becomes sufficiently large. From (B2), we determine the growth contribution to be p(X t ) = r X t > 0, while (B1) and (B3) provide the decline processes that result in q(X t ) = κ + δ X t2 , where κ > 0 is only the intrinsic decline rate and δ > 0 describes the intra-specific competition strength. Thus the population is only increasing for intermediate population levels, due to the linear effect in the growth contribution, i.e. 1 + r X t .

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This results in the discrete model with Allee effect X t+1 =

1 + p(X t ) 1 + r Xt Xt . Xt = 1 + q(X t ) 1 + κ + δ X t2

(19)

By Theorem 5, the set of equilibria of (19) is the same as the set of equilibria of X t+1 =

1+ p (X t ) 1 + bX t Xt , Xt = 1 + q (X t ) 1 + κ + d X t + δ X t2

(20)

where r = b − d > 0. This recurrence corresponds to a discrete model proposed in [20, 27] to include an Allee effect. There, a geometrical approach was used to construct a function that satisfies the conditions for an Allee effect. Our method now provides an elementary alternative for deriving models with Allee effects, such as (19) and (20). Based on Theorem 5, we can obtain the local asymptotic stability of (19) using the results originally obtained for (20) in [20]. There exist three equilibria E 0 < E 1 < E 2 for (20), and therefore, by Theorem 5, also for (19). E 0 and E 2 are asymptotically stable and E 1 is unstable for (20), see [20]. We note that D1 D2−1 =

1 + q (E) 1+ p (E) 1 + bE = = > 1, 1 + q(E) 1 + p(E) 1 +rE

u ⊆ E Zu , so that the set of unstable (hypersince r = b − d > 0. Thus, by Remark 3, E  Z bolic) equilibria for (20) are also unstable for (19). Thus, we can immediately conclude that E 1 is unstable for (19). Furthermore, since D1 D2−1 = 1 at E 0 , E 0 is also asymptotically stable for (19). Applying the model assumptions (B1)–(B3) to a species that changes continuously in time, we obtain the corresponding continuous model

x  = x(−κ + r x − δx 2 ). This is a well-known modification of the logistic growth model that also exhibits an Allee effect. By Theorem 2, we immediately know that this continuous model also has the same three equilibria as (19) and (20). The dynamics are consistent with the discrete version (20) as E 0 and E 2 are locally asymptotically stable, while E 1 is unstable for positive model parameters (e.g., see [3]).

4.2 Multi-Species Models The proposed derivation procedure in Sect. 2 can easily be extended to more than one population and we present some examples below for illustrative purposes. Example 4 Consider k > 1 competing populations, where X t(i) represents population i at time t. We make the following model assumptions:

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(C1) The growth contribution in the fitness function of each population is determined by its intrinsic growth rate ri > 0, i = 1, 2, . . . , k. (C2) The decline contribution in the fitness function for each population, X t(i) , ( j) includes terms due to inter-specific competition proportional to X t , j = i with proportionality constant αi j > 0 and a term due to intra-specific competition proportional to its own population size with proportionality constant ri > 0. Ki Based on model assumptions (C1) and (C2), the growth contribution for species X t(i) is then given by i = 1, 2, . . . , k pi (Xt ) = ri , and the decline contributions are given by the intra-specific competition the inter-specific competition between all competitors qi (t, Xt ) =

k  ri (i) Xt + αim X t(m) , Ki m=1,m=i

αim X t(m)

ri Ki

X t(i) and

for m = i. Thus,

i = 1, 2, . . . , k.

According to (4), the system of recurrences describing how each of the populations changes is given by (i) X t+1 =

1 + ri X (i) ,

k (m) t 1 + m=1 βim X t

i = 1, 2, . . . , k,

(21)

where βim = αim for m = i and βii = Krii for i = 1, 2, . . . , k. Constructing a continuous model that is based on the same assumptions yields   k  xi xi = ri xi 1 − − αim xm xi , Ki m=1,m=i

i = 1, 2, . . . , k.

(22)

This model is sometimes called the Gause-Lotka-Volterra equations (see [36]). In the case of two competing species, (21) reduces to X t+1 =

1 + r1 Xt , 1 + X t + α1 Yt r1 K1

Yt+1 =

1 + r2 Yt , 1+ + α2 X t r2 Y K2 t

and has been studied by several authors, (e.g., see [5, 14, 41, 52]). The corresponding continuous model is then     x1 x2 − α1 x2 x1 , − α2 x1 x2 , x2 = r2 x2 1 − (23) x1 = r1 x1 1 − K1 K2

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which is also known as the Lotka–Volterra competition model. For an analysis of this model, see e.g., [18]. As pointed out in [52], the long term global dynamics of both, the discrete and the continuous model, are identical. As the number of competitors increases, the dynamics become increasingly complex. Already, for (22) with k = 3, there are parameter combinations of the competition coefficients such that none of the equilibria for the continuous model are asymptotically stable (see [36, 43]). In [36], additional assumptions are made so that system (22) simplifies to x  = x(1 − x − αy − βz), y  = y(1 − βx − y − αz), z  = z(1 − αx − β y − z), with α, β > 0. The unique interior equilibrium is E = at this equilibrium E is given by (5) as ⎡ 1α 1 ⎣β 1 J R (E) = − 1+α+β α β

1 (1, 1, 1). 1+α+β

(24)

The Jacobian

⎤ β α⎦ . 1

√

−2+α+β 3(α−β) ± i 2(1+α+β) . Thus, for (24), if α + The eigenvalues of J R (E) are −1 and 2(1+α+β) β < 2, E is (locally) asymptotically stable and if α + β > 2, then E is unstable. Recalling the relationship between (4) and (10), the discrete model that corresponds to (24) is

1+1 Xt , 1 + X t + αYt + β Z t 1+1 = Yt , 1 + β X t + Yt + αZ t 1+1 = Zt . 1 + αX t + βYt + Z t

X t+1 = Yt+1 Z t+1

(25)

1 (1, 1, 1) is also an equilibrium of (25). Also, 1 + qi (E) = By Theorem 1, E = 1+α+β 1 + pi (E) = 2, for i = 1, 2, 3. By Theorem 2, we can immediately conclude that if α + β > 2, then E is also unstable for (25). Note however that the stability condition does not carry over from (24) to (25). Take for example α = 23 and β = 14 . Then E is stable for (24) but unstable for (25). Again, (24) is more stable than (25).

Example 5 In [47], a predator-prey model was derived based on the following assumptions: (D1) The fitness function for the prey contains the intrinsic growth, intra-specific competition, and predation rates. (D2) The fitness function for the predator contains its death rate and its growth rate due to consumption of the prey.

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Based on (D1), the growth contribution for the prey, X t , is p1 (X t , Yt ) = r , where r > 0 is the constant growth rate of the prey. Since the prey declines due to intraspecific competition and predation, q1 (X t , Yt ) = Kr X t + αYt , where Kr > 0 is the competition strength, α > 0 is the predation rate, and Yt is the predator level at time t. The model assumption (D2) suggests that the corresponding functions for the predator are given by p2 (X t , Yt ) = γ X t , where γ > 0 is the growth rate of the predator due to its consumption of the prey, and q2 (X t , Yt ) = d, where d > 0 is the constant death rate. Model (4) then becomes X t+1 =

1+r Xt , 1 + X t + αYt r K

Yt+1 =

1 + γ Xt Yt . 1+d

(26)

Model (26) has the trivial equilibrium E 0 , the prey-only equilibrium E K , and a coexistence equilibrium E ∗ = (X ∗ , Y ∗ ) that is in the interior of the first quadrant if γ K > d. E 0 is always unstable for (26). Whenever, γ K < d, E K is locally asymptotically stable and if d < γ K < 1 + 2d, E ∗ is locally asymptotically stable and E K is unstable. For γ K > 1 + 2d, E ∗ is also unstable (see [47]). By Theorem (1), we immediately know that E 0 , E K , and E ∗ are also the equilibria of the continuous model based on the same model assumptions,  x − αx y, x = r x 1 − K

y  = y (−d + γx) .

(27)

By Remark 1, E 0 is always unstable for (27). This is consistent with what is known about the well-studied model (27). It is also known that E K is asymptotically stable for γ K > d. This is the same condition for the stability of E K for (26). However, E ∗ is asymptotically stable for all d < γ K for (27). Thus, although D, as defined in Theorem 2, is in this case not constant, the continuous model (27) is more stable than (26). Example 6 Consider (4) with model assumptions that represent mutualistic relationships between the populations, so that (H1) (H2)

∂ pi (X) ∂ X ( j) ∂ f i (X ) ∂ X (i)

i (X) > 0 and ∂q = 0 for all i = j and i, j ∈ {1, 2, ..., k}. ∂ X ( j) = 0, for all i ∈ {1, 2, ..., k} implying that there is no density dependent intra-specific relation in the fitness function.

Assume there exists a coexistence equilibrium E = (E (1) , E (2) , . . . , E (k) ). Then, (5) is given by ⎧ ⎨1, i = j, (28) (J Z (E))i j = ∂ f (E) ⎩ E (i) i , i = j. ∂ X ( j) Trying to find the characteristic equation and checking if the roots are within the unit-circle is rather unpleasant, especially for k ≥ 4. Here, Theorem 2 provides a more practical choice to investigate the stability of E. The corresponding continuous model is

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xi = f˜i (x)xi = xi ( pi (x) − qi (x)),

with

∂ f˜i ∂xi

= 0,

Then,

∂ pi ∂x j

 J R (E)i j =

> 0,

∂qi ∂x j

= 0,

j = i.

(29)

0, j = i, (i) ∂ pi (E) , i = j. E ∂x j

Since D J R (E) is again a matrix with 0’s on the diagonal, trace(D J R (E)) = 0, so that the sum of all of the eigenvalues is 0. This implies that there exists at least one eigenvalue of D J R (E) with positive real part, resulting in the instability of E for (29). By Theorem 2, the Jacobian of the discrete model has an eigenvalue outside the unit-circle, implying that the interior equilibrium is unstable for the corresponding discrete model.

5 Conclusion In this work, we presented a technique for deriving discrete population models by incorporating growth contributions in the numerator of the fitness function, and decline processes in the denominator of the fitness function. We related the construction of such a fitness function to the formulation of per-capita growth rates for continuous models by exploiting the relation between zero coupon bonds and continuous compounding in finance. We showed that the discrete models derived using this technique satisfy the axiom of parenthood and populations with non-negative initial conditions can never become negative, making them biologically meaningful. Conditions for the asymptotic stability of extinction and competitive exclusion equilibria for these models were provided. We showed that the set of equilibria of a discrete model derived using our technique and the continuous analogue that satisfies the same assumptions is identical and that the Jacobians evaluated at an equilibrium of each model are related by the identity given in (11). We give conditions for which the continuous counterpart is more stable than (or at least as stable as) the discrete model based on the same model assumptions. In either case, we showed that the stability of the extinction equilibrium is the same for both, the discrete model and its continuous counterpart. To encourage a discussion whether more information about the model parameter composition can result in different model behavior, we compared a discrete population model to a family of discrete population models where all parameters were expressed as differences between growth and decline terms. Although the set of equilibria remained the same for all these models, we argued that the stability of these equilibria can generally be different. This suggests that having more information on the parameter composition can indeed impact the dynamics of the population model and should therefore not be dismissed.

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We completed the study by showing how to use the proposed method to derive several single and multi-species models. Acknowledgements The research of Gail S. K. Wolkowicz was partially supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant with accelerator supplement.

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Reproduction Number Versus Turnover Number in Structured Discrete-Time Population Models Horst R. Thieme

Abstract The analysis of the discrete-time dynamics of structured iteroparous populations involves a basic yearly turnover operator B = A + H with a structural transition operator A and a mating and fertility operator H . A and H map a normal complete cone X + of an ordered normed vector space X into itself and are (positively) homogenous and continuous on X + , A is additive and H is order-preserving.  j Assume that r(A) < 1 for the spectral radius of A. Let H R1 with R1 = ∞ j=0 A be the next generation operator and T = r(B), the spectral radius of B, be the (basic) turnover number and R = r(H R1 ) be the (basic) reproduction number. We explore conditions for a turnover/reproduction trichotomy, namely one (and only one) of the following three possibilities to hold: (i) 1 < T ≤ R, (ii) 1 = T = R, (iii) 1 > T ≥ R. In some cases, one may also to consider the lower reproduction  like −(n+1) λ An . R is also useful to study number R = limλ→1+ r(H Rλ ), Rλ = ∞ j=0 the case r(A) = 1 to explore conditions for the dichotomy 1 = T ≥ R or 1 < T ≤ R ≤ ∞. Keywords Integro-difference equations · Integral projection models · Ordered vector spaces · Cones · Homogeneous operators · Continuity of the spectral radius · Stability · Extinction · Population growth factor · Generation growth factor · Net reproductive value · Eigenvector · Resolvent · Feller kernel · Mating function · Pair-formation function · Rank structure

Dedicated to Odo Diekmann on the occasion of his 75th birthday H. R. Thieme (B) School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA e-mail: [email protected] URL: https://horstthieme.weebly.com/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Elaydi et al. (eds.), Advances in Discrete Dynamical Systems, Difference Equations and Applications, Springer Proceedings in Mathematics & Statistics 416, https://doi.org/10.1007/978-3-031-25225-9_23

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1 Introduction The year-to-year dynamics of a structured population are described by a difference equation (or recursive equation) xn = F(xn−1 ),

n ∈ N.

(1)

Here, F : X + → X + is an operator on a subset X + of a normed vector space X , 0 ∈ X + and F(0) = 0. The initial point x0 ∈ X + for (1) is given. The vectors in X + represent the structural distributions of the population; the zero vector represents the extinction state of the population. F maps this year’s structural distribution to next year’s structural distribution and is called the population turnover operator. We also speak of (1) as the (year to year) population turnover equation. Assume that X + is a homogeneous subset of X , x ∈ X + , α ∈ R+ =⇒ αx ∈ X + .

(2)

Notice that 0 ∈ X + if X + is a homogeneous subset of X . We use the notation X˙ + = X + \ {0}.

(3)

Further assume that F is Gateaux differentiable at 0, i.e., the directional derivatives at 0 exist in all directions x ∈ X + , 1 F(αx) = ∂ F(0, x) =: B(x), α→0+ α lim

x ∈ X +.

(4)

A short calculation [44, Theorem 1.4] [25, Theorem 3.1] shows that B is a homogenous map on X + , B(αx) = αB(x),

x ∈ X + , α ∈ R+ .

(5)

B is called the basic turnover operator associated with the turnover operator F. Another short calculation shows the following: Lemma 1 Assume that there is c ∈ (0, ∞) such that lim sup x→0

F(x) ≤ c, x

x ∈ X +.

Then the Gateaux derivative B = ∂ F(0, ·) of F at 0 is bounded: B(x) ≤ c x ,

x ∈ X +.

The operator norm of a bounded homogeneous map B : X + → X + is defined by

Reproduction Number Versus Turnover Number …

497

B := sup{ B(x) ; x ∈ X + , x ≤ 1}.

(6)

Since B is homogenous, x ∈ X +.

B(x) ≤ B x ,

(7)

The powers (iterates) B n : X + → X + of a bounded homogeneous operator are also bounded homogeneous operators and B n+m ≤ B n B m ,

m, n ∈ N.

(8)

The spectral radius of B is defined by the Gelfand formula r(B) = inf B n 1/n = lim B n 1/n . n∈N

n→∞

(9)

The last equality follows in the same way as for bounded linear everywhere defined maps (Theorem 3 in [54, Sect. VIII.2]). We call T = r(B) the basic turnover number; it has also been called inherent population growth rate [8] or (population) growth factor [34]. T acts as a threshold parameter under additional properties of X + and F [25, 48, 49, 51]: • If T < 1, then 0 ∈ X + is locally asymptotically stable. • If T > 1, then 0 ∈ X + is unstable in various ways the strength of which depends on the extra assumptions being made. One of these additional assumptions is that X + is additive, x, y ∈ X + =⇒ x + y ∈ X + .

(10)

Together with the homogeneity, (2), this assumption makes X + a wedge. A wedge X + is called a cone if x = 0 is the only element with x ∈ X + and −x ∈ X + . A wedge X + is called normal if there is some c ≥ 1 such that x ≤ c x + y ,

x, y ∈ X + .

(11)

Any normal wedge automatically is a cone (Choose y = −x if x, −x ∈ X + ). While some linear operators are best studied in cones that are not normal [39], normal cones seem a natural framework for population dynamics. If the population is iteroparous, i.e., reproduction occurs several times in a lifetime and there are overlapping generations, the basic turnover operator is of the form B = A+H (12) with bounded homogeneous operators A, H : X + → X + . A describes adult survival and structural development, and H describes reproduction and neonate survival and

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structural development. In one of its earlier incarnations [11] in the framework of matrix population models [7], A is represented by the “transition” matrix and H by the “production” or “fertility” matrix and B by the “population projection” matrix. A is typically assumed to be additive, i.e., x, y ∈ X + =⇒ A(x + y) = A(x) + A(y).

(13)

If mating is not involved or is ignored in reproduction, typically H is additive as well. If mating is taken into account, H is no longer additive (see Sect. 2). If there is no reproduction, the population is expected to die out. Mathematically, this can be expressed by the condition r(A) < 1.

(14)

Let us assume that the cone X + is closed and serially complete. The latter means that a series ∞  xj, x j ∈ X +, (15) j=0

 converges in X + whenever the series ∞ j=0 x j converges in R. An example for a cone that is serially complete, but not complete is the cone of the vector space of real measures on a non-complete metric space under the flat norm [20, 48]. The operators Rλ = RλA =

∞ 

λ− j−1 A j ,

λ ∈ (r(A), ∞),

(16)

j=0

exist and are additive, and the operators H Rλ are homogeneous. RλA acts like a resolvent for A. See Sect. 3.7. If r(A) < 1, the operator H R1 = H

∞ 

Aj

(17)

j=0

exists and is called the next generation operator and R = r(H R1 )

(18)

 j the reproduction number. In fact, the convergence of the series ∞ j=0 A x , x ∈ X + , is equivalent to r(A) < 1 [48, Theorem 7.5]. Since the emphasis of this paper is not on the turnover operator F but on its first order approximation, the basic turnover operator B, we do not use the specifications “basic” [14] or “inherent net” [8, 11] for brevity and write R rather than R0 for a leaner notation. R has also been called the generation growth factor [13, p.182].

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Theorem 1 ([48, Theorem 7.16]) Let X be an ordered normed vector space and let its cone X + be closed, normal and serially complete. Let A and H be bounded additive homogeneous operators on X + , r(A) < 1. Then T − 1 and R − 1 have the same sign. This formulation hides a trichotomy, namely that one (and only one) of the following three statements is satisfied: (i) T < 1 and R < 1, (ii) T = 1 = R, (iii) T > 1 and R > 1. This theorem was shown for finite space dimensions (matrix population models) in [11]. We emphasize that Theorem 1 does not make any compactness or irreducibility assumptions. It is the aim of this paper to explore under which assumptions a result like in Theorem 1 can be shown if H is continuous, homogeneous and order-preserving but not additive on X + . Mathematically, such an extension is quite suggestive because the directional derivatives at the zero vector in directions of the cone are automatically homogeneous but not additive. Biologically, motivation is provided by the fact that, in many species, reproduction requires a mating process, often between females and males that are quite different in their demographic traits (mortality, fecundity, age at maturity, dispersal; see [7, 28, 38] and the references therein). The mating (or pair formation) function is often assumed to be homogeneous; see Assumption 3.1 and the references mentioned there. The mathematical methods we will use will be quite different from those used for Theorem 1, which ultimately rely on a paper by Tosio Kato [27], and will be closer to some used in [34] and lead to a more detailed trichotomy very similar to that in [34]. The interest in such a trichotomy is nourished by the hope that R may be easier to determine or estimate than T . This has been the case for certain matrix population models or discrete-time epidemic models [3, 4, 8–11, 24, 52], but may be less the case for models involving mating (see the Discussion).

2 Preview: Turnover/Reproduction Trichotomy A closed cone X + of an normed vector space X induces an order on X which is compatible with the vector space operations and the topology: For x, z ∈ X , x ≤ z if

z − x ∈ X +.

(19)

X with a closed cone X + is therefore also called an ordered normed vector space. The notion of an order leads to a couple of other concepts. An element u ∈ X˙ + is called a uniform order unit for X + if there is some c > 0 such that (20) x ≤ c x u, x ∈ X +.

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We say that X + has a uniform order unit if there exists a uniform order unit in X + for X + . X + is called a sup-semilattice if, for any x, y ∈ X + , x ∨ y = sup{x, y} exists for any x, y ∈ X + .

(21)

Let X, Y be ordered normed vector spaces with respective cones X + and Y+ and B : X B → Y be an operator defined on a subset X B of X . B is called order-preserving if B(x) ≤ B(z) i f

x, z ∈ X B and x ≤ z.

(22)

For instance, if A : X + → Y+ is additive, then A is order-preserving. Properties of the spectral radius of homogeneous bounded order-preserving operators on cones are collected in Sect. 3. If mating is included in the population model, the basic turnover operator has the form B = A + H where A and H are homogeneous maps from X + to X + and A is additive and H is order-preserving. Recall r(A) < 1 and Rλ in (16). We assume that X + is also normal and serially complete. See (11) and (15). Under these assumptions, B, A, H are  bounded  homogeneous maps [48, Theorem 7.4]. We define the function ξ : r(A), ∞ → R+ by ξ(λ) = r(H Rλ ) = r(Rλ H ),

λ ∈ (r(A), ∞).

(23)

The last equality follows from Theorem 6. The function ξ is decreasing (Proposition 6). If H is just order-preserving but not additive, we face the problem of showing that the function ξ is continuous. So far, we have not been able to prove the continuity or find a counterexample. That lower semi-continuity of the spectral radius (in this case, the right-continuity of ξ) may be a problem is underpinned by an example in [32, Sect. 3]. To work around this problem, we study the full next generation operator family {H Rλ ; λ > r(A)} rather than just H R1 . This leads to some productive concepts (Definition 3) which are not intuitive enough to be included in this preview. In particular, we introduce the lower reproduction number R = lim ξ(λ) ≤ ξ(1) = R. λ→1+

(24)

This right-hand limit exists because ξ is decreasing. Recall that T = r(B) with B = A + H. Definition 1 We say that the turnover/reproduction trichotomy holds weakly if one (and only one) of the following three possibilities holds: (ii) R ≤ 1 = T ≤ R. (iii) 1 > T ≥ R ≥ R . (i) 1 < T ≤ R ≤ R. Notice that T − 1 and R − 1 and R − 1 have the same sign if T = 1, but not necessarily if T = 1.

Reproduction Number Versus Turnover Number …

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Definition 2 We say that the turnover/reproduction trichotomy holds strongly if one (and only one) of the following three possibilities holds: (i) 1 < T ≤ R.

(ii) 1 = T = R.

(iii) 1 > T ≥ R.

Here, T − 1 and R − 1 always have the same sign and, under suitable assumptions for the turnover operator F, R is a sharp threshold parameter for stability versus instability of the zero vector, the extinction state. The strong trichotomy follows from the weak trichotomy provided that R = R if R ≥ 1. Conditions for the last are given in Proposition 10. Theorem 2 Let X be an ordered normed vector space and let its cone X + be closed, normal and serially complete. Let A and H be continuous homogeneous operators on X + and A be additive and H be order-preserving on X + , r(A) < 1. Then the turnover/reproduction trichotomy holds weakly if at least one of the following conditions is satisfied: (a) A and H are compact. (b) X + is complete and H is compact. (c) X + is a sup-semilattice and there are , m ∈ N such that B m and (H Rλ ) are compact for all λ ∈ (r(A), ∞). (d) X + has a uniform order unit. Remark 1 If one of the conditions in Theorem 2 holds, ξ acts as a characteristic function for B = A + H and its spectral radius T in the following sense: (i) ξ(λ) < 1 for all λ > T . (ii) If T > r(A), then ξ(λ) > 1 for all λ ∈ (r(A), T ) and ξ(T ) ≥ 1. Notice that T is uniquely determined by these properties, and   T = inf λ > r(A); ξ(λ) ≤ 1 .

(25)

If T > r(A) and ξ is continuous, then T is the unique solution of the equation ξ(λ) = 1. We combine Theorems 1 and 2 to obtain the following result if H is additive as well. Corollary 1 Let X be an ordered normed vector space and let its cone X + be closed, normal and serially complete. Let A and H be continuous additive homogeneous operators on X + , r(A) < 1, and let at least one of the conditions (a), . . .,(d) in Theorem 2 be satisfied. Then the turnover/reproduction trichotomy holds strongly.

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Remark 2 Let X be an ordered normed vector space and let its cone X + be closed, normal and serially complete. Let A and H be continuous homogeneous operators on X + and A be additive and H be order-preserving on X + , r(A) = 1. Assume at least one of the conditions conditions (a), . . ., (d) in Theorem 2. Then the dichotomy 1 = T ≥ R or 1 < T ≤ R ≤ ∞ holds. Other, more general (and more technical), conditions for the weak trichotomy and Remarks 1 and 2 will be given in Sect. 4.3. The central result is Theorem 10. The strong turnover/reproduction trichotomy, with R = R , has been established for matrix population models in [34]. Any of the conditions (a),..., (d) in The1 H ) = 1 in [34], from which the orem 1 works in this case. The result r(A + R turnover/reproduction trichotomy is derived in [34, Theorem 3.1] and which is of some interest of its own, seems to require stronger conditions for nonadditive homogeneous H in infinite space dimensions and will not be established in this paper. Notice that no irreducibility assumptions are made in Corollary 1. Theorem 2 (a), (b), (c) will be proved in Sect. 4.3, after partial results have been proved in Sects. 4.1 and 4.2. Theorem 2 (d) will be proved in Sect. 4.4. Remark 1 follows from Proposition 8 (f). In Theorem 2, one would suspect that R = R , but we have not been able to prove that in general or find a counterexample. A sufficient condition for this equality will be given in Proposition 10. The equality R = R also holds if B is the additive perturbation of a homogeneous rank one operator (Sect. 5). The dynamics of populations with mating are most easily studied in Banach spaces of bounded continuous functions or their Cartesian products (Sect. 6.1) [51]. These have uniform order units, so Case (d) of the assumptions in Theorem 2 applies well. See Sect. 6.1. Populations without mating also have interesting models in spaces of measures of spaces [48, 49]. Here, the cases (a), (b), (c) in Corollary 1 may apply and provide interesting additional information compared to Theorem 1. For a semelparous population with mating, one can model the dynamics in terms of neonates and does not need separate recursive equations for the two sexes. See [51]. For iteroparous populations, this is no longer possible.

2.1 Iteroparous Populations with Mating In general, one may even have separate ordered normed vector spaces for females, males, and neonates, X, Y, Z , with closed cones X + , Y+ , Z + (Sect. 6). In many cases, though, they will be equal (Sect. 2.2). The cones X + and Y+ are assumed to be normal and serially complete. See (15). The first order approximation of the population turnover equation is given by xn+1 = A• xn + K • φ(xn , yn ) yn+1 = A yn + K  φ(xn , yn )

 n ∈ N.

(26)

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Here, xn ∈ X + and yn ∈ Y+ are the respective structural distributions of females and males in the n th year shortly before the reproductive season. The right hand side of (26) provides B(xn , yn ) for the basic turnover map B on X + × Y+ . We assume that K • and K  are continuous homogeneous order-preserving operators from Z + to X + and from Z + to Y+ , respectively; they represent the survival, structural development, per mated pair fertility, and the respective sex ratios of female and male neonates. A• and A are continuous, homogeneous, additive operators on X + and Y+ , respectively, r(A• ) < 1 and r(A ) < 1; they represent survival and structural development of female and male adults (one year and older). φ is a homogeneous continuous and order-preserving operator from X + × Y+ to Z + and represents mating. The characteristic function ξ in (23) takes the form ξ(λ) = r(Bλ ),

λ > max{r(A• ), r(A )},

(27)

where Bλ : Z + → Z + is the operator given by   Bλ (z) = φ Rλ• K • z, Rλ K  z ,

z ∈ Z+.

(28)

The resolvents exist because X + and Y+ are serially complete, Rλ•, =

∞ 

λ− j−1 A˜ j ,

A˜ = A• , A .

(29)

j=0

Theorem 3 The turnover/reproduction trichotomy holds weakly (Definition 1), with ξ given by (27) and (28), if φ is continuous and one of the following conditions is satisfied. (a) X + and Y+ are serially complete and A• , A , K • and K  are compact. (b) X + and Y+ are complete and K • and K  are compact. (c) X + and Y+ are serially complete and have uniform order units. Remark 3 It is worth mentioning that T is the spectral radius of an operator on X + × Y+ while R can be calculated as the spectral radius of an operator on Z + .

2.2 Individual Development Modeled by Feller Kernels More concretely, let X = Y = Z = C b (S) be the Banach space of bounded continuous real-valued functions on a metric space S. S is the state space of individual characteristics [12] (Sect. 6.1). If the structure of the population is given by spatial distribution, S represents the habitat of the population.

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The Cartesian product C+b (S) × C+b (S) is the state state of female and male population densities. The function x ∈ C+b (S) is to be interpreted as the structural (measureindependent) density of females [51]. Let μ be a nonnegative finite measure on B, the σ-algebra of Borel subsets of S, and T ∈ B. Then T x dμ is the amount of females in the set B measured by μ. Depending on μ, this could be the number or the biomass of females in T or the food consumed by them in a day. That x is the density of females can be seen as follows. Let s ∈ S and Bμ = {T ∈ B; μ(T ) > 0}. Since x is continuous at s,

1 xdμ → x(s), s ∈ T ∈ Bμ , Δ(T ) → 0, μ(T ) T where Δ(T ) = sup{d(t1 , t2 ); t1 , t2 ∈ T } is the diameter of T . Cf. [51]. Any of the operators Q = A• , A , R • , R  in (26) are integral operators of the form

f (t)κ(dt, s), s ∈ S. (30) (Q f )(s) = S

This makes (26) an integro-difference equation [26, 30, 33, 35, 43, 53] or an integral projection model (see [15–17] and the references therein). More precisely, κ (which depends on Q) is a Feller kernel. This means that κ : B × S → R+ where B is the σ-algebra of Borel sets in S and, for each s ∈ S, κ(·, s) is a non-negative finite measure on B; moreover, the Feller property is satisfied, namely, that Q maps C b (S) into itself. If the population structure is given by spatial distribution, κ is somehow related to the spatial movement though a precise interpretation is not possible in this generality, but only in various special cases. See the discussions in [50, 51]. The operator φ : (C+b (S))2 → C+b (S) is a Nemytskii (or superposition) operator φ(x, y)(s) = ϕ(s, x(s), y(s)),

s ∈ S, x, y ∈ C+b (S),

where the mating function (pair formation function) ϕ : S × R2+ → R has the following additional properties. Assumption 3.1 (a) ϕ : S × R2+ → R+ is continuous. (b) For each s ∈ S, ϕ(s, ·) : R2+ → R+ is homogeneous and order-preserving. (c) ϕ(s, 1, 0) = 0 = ϕ(s, 0, 1), s ∈ S. (d) ϕ(s, 1, 1) > 0, s ∈ S. (e) ϕ(·, 1, 1) bounded on S. Here ϕ(s, x, y) is the number of mated pairs that, at location s ∈ S, are formed from x females and y males [21–23, 47]. Examples are given by variants of the harmonic mean and of the minimum function

Reproduction Number Versus Turnover Number …

505

 xy β1 (s)x  + β2 (s)y  s ∈ S, x, y ∈ R+ . ϕ(s, x, y) = min β1 (s)x, β2 (s)y

ϕ(s, x, y) =

See [18, Sect. 7.3] for the construction of mating functions (reinterpreted as transmission functions) with other desired properties. Since all positive constant functions are uniform order units for C+b (S), Theorem 3 applies with condition (c) and the turnover/reproduction trichotomy holds weakly (Definition 1). It is worth mentioning that T is the spectral radius of an operator on C+b (S) × b C+ (S) while R can be calculated as the spectral radius of an operator on C+b (S). (See Remark 3.) At the end of Sects. 6 and 6.1, we present two examples in which R = R and semi-explicit formulas can be given for the reproduction number.

3 More About the Spectral Radius We start with an example which shows what may happen to the spectral radius if the cone is not normal.

3.1 An Example by Bonsall [6, Sect. 2(iv)] Let X be the vector space over R of all complex-valued functions defined on the closed complex unit disk D = {ζ ∈ C; |ζ| ≤ 1} that are continuous on D, real-valued on the interval [−1, 1] of the real axis, and regular (holomorhic) on the interior of D. X is an ordered Banach space with the supremum norm and the closed cone   X + = f ∈ X ; f (t) ≥ 0, −1 ≤ t ≤ −1/2 . Then X = X + − X + and X + has the constant function 1 with value one as interior point and uniform order unit. The spectral radius of any bounded linear operator A on X with AX + ⊆ X + equals to the spectral radius of the restriction of A to X + [47, Theorem 3.4, Rem.3.5]. X + is not a normal cone. We consider the bounded linear operator 1 f (ζ), ζ ∈ D, f ∈ X, (31) (A f )(ζ) = − ζ + 2 and, for α ∈ R+ , the operators Aα on X with Aα f = α f + A f,

f ∈ X.

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We have Aα (X + ) ⊆ X + for all α ∈ R+ . For all f ∈ X , 1 n (Anα f )(ζ) = α − ζ − f (ζ), ζ ∈ D. 2   This implies r(Aα ) = maxζ∈D α − ζ − 21 , where the inequality ≥ follows by considering f = 1. After some calculations, r(Aα ) =

3

− α, 0 ≤ α ≤ 21 , α + 21 , α ≥ 21 . 2

(32)

We notice that r(Aα ) is a strictly decreasing function of α ∈ [0, 1/2] though Aβ f − Aα f ∈ X + for f ∈ X + and 0 ≤ α ≤ β. This shows that monotonicity of the spectral radius (Sect. 3.4) may fail if the cone is not normal. We also notice that r(Aα ) < α + r(A0 ) if α > 0. So part (iii) of the subsequent result, which is proved at the end of Sect. 4.4, may fail if the cone is not normal. This is related to r(A) not being in the spectrum of A. For a Hilbert space example where the spectral radius of a bounded linear operator mapping a cone in itself is not in its spectrum see [40]. Proposition 1 Let X + be a closed, normal and serially complete cone. Let α ∈ (0, ∞) and H : X + → X + be homogeneous, continuous and order-preserving. Define B : X + → X + by B(x) = αx + H (x), x ∈ X + . Then r(B) = α + r(H ) if one of the following is satisfied in addition: (i) X + is complete and H is compact, (ii) X + has a uniform order unit, (iii) H is additive. The proof of Proposition 1 can be found at the end of Sect. 4.

3.2 Lower and Upper Bounds for the Spectral Radius Let X be an ordered normed vector space with closed cone X + , X B a homogeneous subset of X + , and B : X B → X B a homogeneous operator. See (2) and (5). Theorem 4 ([47]) Let B : X B → X B be an order-preserving bounded homogeneous operator. If x ∈ X˙ B , λ ∈ R+ and B(x) ≥ λx, then r(B) ≥ λ. To establish upper bounds for r(B), we need some more terminology. For u ∈ X˙ + , a homogeneous map B : X B → X B is called uniformly u-bounded [29] if there exists some c > 0 such that B(x) ≤ c x u,

x ∈ X B.

(33)

Reproduction Number Versus Turnover Number …

507

If u is a uniform order unit for X B (see (20), then all homogeneous bounded B : X B → X B are uniformly u-bounded. A homogeneous map B is called uniformly order-bounded if it is uniformly u-bounded for some u ∈ X˙ + . Such a point u is called an order bound for B. An element u ∈ X + is called a normal point for X B if the set { x ∈ X B ; x ≤ u} is a bounded subset of R+ . If the cone X + is normal, then any u ∈ X˙ + is a normal point for any subset X B of X + . The next result shows how upper bounds for the spectral radius can be obtained [47]. Theorem 5 Let B : X B → X B be an order-preserving bounded homogeneous operator. Assume that u ∈ X˙ B is a normal point for X B and some power of B is uniformly u-bounded. If μ ∈ R+ and B(u) ≤ μu, then r(B) ≤ μ. Corollary 2 Let B : X B → X B be an order-preserving bounded homogeneous operator. Assume that u ∈ X˙ B is a normal point for X B and that some power of B is uniformly u-bounded. Then, if μ ∈ R+ and B(u) = μu, we have r(B) = μ. More detailed results can be found in [44] for the case X B = X + .

3.3 Commutation Rules We consider two normed vector spaces X and Y and two homogeneous subsets X + of X and Y+ of Y and bounded homogeneous maps A : X + → Y+ and B : Y+ → X + . Then the compositions B A and AB are bounded homogeneous maps on X + and Y+ , respectively. Theorem 6 r(AB) = r(B A) under just-mentioned assumptions. Proof Let · denote the norms on both X and Y and also the respective operator norms. By induction, (AB)n+1 = A(B A)n B for all n ∈ N. Since A and B are bounded, (AB)n+1 ≤ A (B A)n B for all n ∈ N. We can assume that A > 0 and B > 0; otherwise the claim is trivial. In order to establish r(AB) ≤ r(B A), we can also assume that r(AB) > 0. By (9), r(AB) = lim (AB)n+1 1/n ≤ lim sup A 1/n (B A)n 1/n B 1/n = r(B A). n→∞

By symmetry, equality holds.

n→∞



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3.4 Monotonicity of the Spectral Radius Let X be an ordered normed vector space with closed cone X + . Let A : X A → X A and B : X B → X B be bounded homogeneous operators on homogeneous sets X A and X B contained in X + . Theorem 7 Let X + be a normal cone. We assume that X A ⊆ X B ⊆ X + and A(x) ≤ B(y),

x ∈ X A , y ∈ X B , x ≤ y.

(34)

Then r(A) ≤ r(B). Proof By induction, for all n ∈ N, An (x) ≤ B n (x),

x ∈ X A.

(35)

By assumption, (35) holds for n = 1. For the induction step, let n ∈ N and (35) hold for n. Let x ∈ X A . By (34), since An (x) ∈ X A and B n (x) ∈ X B , An+1 (x) = A(An (x)) ≤ B(B n (x)) = B n+1 (x). This proves (35) for all n ∈ N. Since X + is a normal cone (see (11)), there exists some c ≥ 1 such that x ≤ c y for all x, y ∈ X + with x ≤ y. By (35), An (x) ≤ c B n (x) ,

n ∈ N, x ∈ X A .

By (6), since X A ⊆ X B , An (x) ≤ c B n x ,

n ∈ N, x ∈ X A ,

and An ≤ c B n for all n ∈ N. By (9),   r(A) = lim An 1/n ≤ lim sup c1/n B n 1/n = r(B). n→∞

n→∞

 Corollary 3 Let X + be a normal cone. We assume that X A ⊆ X B ⊆ X + , B is orderpreserving, and (36) A(x) ≤ B(x), x ∈ X A. Then r(A) ≤ r(B). Proof Let x ∈ X A and y ∈ X B and x ≤ y. By (36), A(x) ≤ B(x). Since B is orderpreserving, B(x) ≤ B(y) and so A(x) ≤ B(y). Apply Theorem 7. 

Reproduction Number Versus Turnover Number …

509

In a similar way, Theorem 7 implies the next result. Corollary 4 Let X + be a normal cone. We assume that X A = X B ⊆ X + , A is orderpreserving, and A(x) ≤ B(x) for all x ∈ X A . Then r(A) ≤ r(B).

3.5 Cartesian Products Proposition 2 For j = 1, 2, let X j be normed vector spaces and X j+ be homogeneous subsets of X and B j : X j+ → X j+ be bounded homogeneous operators. Define B : X 1+ × X 2+ → X 1+ × X 2+ by   B(x1 , x2 ) = B1 (x1 ), B2 (x2 ) ,

xj ∈ X j,

j = 1, 2.

Let | · | be any norm on R2 and   (x1 , x2 ) =  x1 , x2 ,

xj ∈ X j.

Then X 1 × X 2 is normed vector space, X 1+ × X 2+ is a homogeneous subset of X 1 × X 2 and B is a bounded homogeneous operator on X 1+ × X 2+ and   r(B) = max r(B1 ), r(B2 ) . Proof For all n ∈ N,   B n (x1 , x2 ) = B1n (x1 ), B2n (x2 ) , and

x j ∈ X j+ ,

  B n (x1 , x2 ) =  B1n (x1 ) , B2n (x2 ) .

Since the spectral radius of a bounded homogeneous operator does not depend on equivalent norms, we can choose   (x1 , x2 ) = max x1 , x2 . Then

  B n (x1 , x2 ) = max B1n (x1 ) , B2n (x2 ) .

(37)

This implies     B n (x1 , x2 ) ≤ max B1n x1 , B2n x2 ≤ max B1n , B2n (x1 , x2 ) and B n ≤ max{ B1n , B2n }. Further,

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  B n 1/n ≤ max B1n 1/n , B2n 1/n .   By (9, r(B) ≤ max r(B1 ), r(B2 ) . By (37), B n (x1 , 0) = B1n (x1 ) ,

x1 ∈ X 1 ,

and B1n (x1 ) ≤ B n x1 r(B2 ) ≤ r(B). and B1n ≤ B n . By (9), r(B1 ) ≤ r(B). Similarly,  In combination, r(B) = max r(B1 ), r(B2 ) .



If, for j = 1, 2, X j+ are closed normal (serially) complete cones of X j , then X 1+ × X 2+ is a closed normal (serially) complete cones of X 1 × X 2 .

3.6 Existence of (Lower) Eigenvectors Proposition 3 Let X + be the closed cone of an ordered normed vector space and B : X + → X + be continuous, homogeneous and order-preserving, T := r(B). Then there exists an eigenvector v ∈ X˙ + such that B(v) = T v if one of the following is satisfied: (a) B is compact and T > 0. [48, Theorem 7.8] (b) X + is complete and normal and B = A + H with A and H being continuous, homogeneous and order-preserving, A additive and H compact, r(A) < r(B) =: T . [36, Theorem 4.9]. Recall the concept of a sup-semilattice (21). Proposition 4 Let X + be a closed cone of an ordered normed vector space and a supsemilattice. Let B : X + → X + be bounded, homogeneous, and order-preserving, m ∈ N, r ∈ (0, ∞) and w ∈ X˙ + such that B m (w) ≥ r m w. Then there exists some v ∈ X˙ + , v ≥ w, such that B(v) ≥ r v. The proof uses a construction in [1, Theorem 5.1].  −j j Proof We can assume that m ≥ 2. Set v = m−1 B (w). j=0 r For j = 0, . . . , m − 1, v ≥ r − j B j w. Since B is order-preserving and homogeneous, Bv ≥ r − j B j+1 (w), j = 0, . . . , m − 1. In particular, Bv ≥ r −(m−1) B m (w) ≥ r w. By definition of a supremum as a least upper bound,

Reproduction Number Versus Turnover Number …

Bv ≥

m−2 

511

r − j B j+1 (w) ∨ (r w) = r

j=0

m−1 

r − j B j (w) = r v.

j=0

 We combine Proposition 3 and Theorem 4. Theorem 8 Let X + be a closed cone of an ordered normed vector space and a supsemilattice. Let B : X + → X + be continuous, homogeneous, and order-preserving and B m be compact for some m ∈ N, T := r(B) > 0. Then there exists some v ∈ X˙ + such that B(v) ≥ T v. Proof By (9), r(B m ) = T m > 0. By Proposition 3, since B m is compact, there is  some w ∈ X˙ + such that B m (w) = T m w. Apply Proposition 4. More examples for the existence of (lower) eigenvectors associated with the spectral radius can be found in [44, 45].

3.7 Left Resolvents Let X + be the closed cone of an ordered normed vector space X and let X + be serially complete. See (15). Further let B : X + → X + be homogeneous and bounded. B is called superadditive if B(x + y) ≥ B(x) + B(y),

x, y ∈ X + ,

(38)

B(x + y) ≤ B(x) + B(y),

x, y ∈ X + .

(39)

and subadditive if

If B is superadditive, it is order-preserving. B is additive if it is both superadditive and subadditive, (13). Since X + is serially complete, the series RλB (x)

=

∞ 

λ− j−1 B j (x),

λ ∈ (r(B), ∞), x ∈ X + ,

(40)

j=0

converge in X + with the convergence being uniform for x in bounded subsets of X + . We call RλB the left resolvent of B because of the following property. Proposition 5 ([46, Sect. 5]) For λ ∈ (r(B), ∞), x = λRλB (x) − (RλB B)(x) = RλB (λx) − RλB (B(x)), If B is superadditive,

x ≥ λRλB (x) − (B RλB )(x),

x ∈ X +.

x ∈ X +.

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B(RλB x) ≤ RλB (B(x)) = λRλ (x) − x.   If B is additive, equality holds in the last formula, I+ = λI+ − B RλB , where I+ is the identity map on X + and, for any x ∈ X + , y = Rλ x is the unique solution of λy − By = x.

If B is subadditive,

We repeat the short proof because this result is fundamental for our discussion. Proof Let λ > r(B) and x ∈ X + . By (40), RλB (B(x)) =

∞ 

λ− j−1 B j+1 (x) =

j=0

∞ 

λ− j B j (x) = λRλB (x) − x.

j=1

This proves the first equality. The second follows from the homogeneity of B. Now assume that B is superadditive. Let x ∈ X + . For all  ∈ N, B RλB (x) ≥B

 



 λ− j−1 B j (x) ≥ λ− j−1 B j+1 (x)

j=0

=

+1 

j=0

λ− j B j (x) = λ

j=1

Since X + is closed,

+1 

λ− j−1 B j (x) − x.

j=0

B RλB (x) ≥ λRλ (x) − x.

Now assume that B is subadditive. For x ∈ X + , B(RλB x) ≤

n 

λ− j−1 B j+1 (x) + z n ,

zn = B

j=0

∞ 

λ− j−1 B j (x) .

j=n+1

Since B is bounded, z n → 0 as n → ∞ and B(RλB x) ≤ RλB (B(x)).



Example 1 (Rank-one operators and their resolvents) Let B : X + → X + be a rankone operator given by B(x) = θ(x)u with some u ∈ X˙ + and θ : X + → R+ bounded homogeneous. By induction, for all n ∈ N, B n (x) = θ(x)(θ(u))n−1 u, x ∈ X + , and r(B) = θ(u) by (9). θ(u) is an eigenvalue of B associated with the eigenvector u and the eigenfunctional θ. See [47, Example 2.1]. By (40), for λ > θ(u), RλB (x) = λ−1 x + θ(x)

∞  j=1

λ− j−1 (θ(u)) j−1 u =

θ(x) 1 x+ u. λ λ(λ − θ(u))

Reproduction Number Versus Turnover Number …

513

4 Turnover Versus Reproduction Number Let X + be the normal closed cone of the ordered normed vector space X . Assume throughout this section that X + is serially complete and A, H : X + → X + are homogeneous and order-preserving. As before, define B = A + H. By [48, Theorem 7.4], A, H, B are bounded homogeneous operators. By Corollary 4, T := r(B) ≥ r(A).

(41)

Recall the resolvent of A, (40), Rλ = RλA =

∞ 

λ−n−1 An ,

λ > r(A).

(42)

n=0

The series converges pointwise because X + is serially complete. Recall the definition of the characteristic function of the operator B = A + H in (23). Notice that additivity of A is not needed yet. Proposition 6 For all x ∈ X + , λRλ x and λH (Rλ x) are decreasing functions of λ ∈ (r(A), ∞) (with values in X + ) and λξ(λ) is a decreasing function of λ ∈ (r(A), ∞) (with values in R+ ). In particular, ξ is decreasing and ξ(λ) → 0 as λ → ∞. Moreover, there is some ˇ and ξ is zero on (λ, ˇ ∞) λˇ ∈ [r(A), ∞] such that ξ is strictly decreasing on (r(A), λ) with the possibility that one of these intervals is empty. Proof Let λ˜ > λ > r(A) and x ∈ X + . Then, ˜ ˜x = λR λ

∞  n=0

λ˜ −n An x ≤

∞ 

λ−n An x = λRλ x.

n=0

˜ R ˜ x ≤ λH Rλ x. Since X + is Since H is homogeneous and order-preserving, λH λ normal and λH Rλ is an order-preserving operator, r(λH Rλ ) = λξ(λ) is a decreasing function of λ by Corollary 3.  Definition 3 We say that B = A + H is next-generation admissible if λ > r(A) and ξ(λ) ≥ 1 =⇒ T ≥ λ.

(43)

We say that B = A + H is turnover admissible if and   λ ∈ r(A), T =⇒ ξ(λ) ≥ T /λ.

(44)

Remark 4 Let α ∈ R+ and H : X + → X + be homogeneous and order-preserving. Define B : X + → X + by B(x) = αx + H (x) for x ∈ X + . Then B is homogeneous

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and order-reserving. One would assume that r(B) = α + r(H ), but we have not been able to verify this in general or find a counterexample. However, the following holds: (a) If B is next-generation admissible, then r(B) ≥ α + r(H ). (b) If B is turnover admissible, then r(B) ≤ α + r(H ). (c) If B is both next-generation and turnover admissible, then r(B) = α + r(H ). Proof Define A : X + → X + by Ax = αx for x ∈ X + . Then A is homogeneous and additive and r(A) = α. ) 1 x and ξ(λ) = r(H . By (42), Rλ x = λ−α λ−α

(a) Let B be next-generation admissible. If r(H ) = 0, the statement is valid. Let r(H ) > 0 and set λ = r(H ) + α. Then λ > r(A) and ξ(λ) = 1. By (43), T = r(B) ≥ λ = α + r(H ).

(b) Let B be turnover admissible. If r(B) ≤ α, the statement is valid. If T = r(B) > r(A), by (44), ξ(T ) ≥ 1 and T − α ≤ r(H ).  See Proposition 1 for a more concrete consequence of this result.

4.1 Starting Point: Next Generation Operator We consider the next-generation operator-family {H Rλ ; λ > r(A)}. Assumption 8.1 For all λ > r(A), there exists a lower eigenvector v ∈ X˙ + (that may depend on λ) such that H Rλ (v) ≥ ξ(λ)v. Remark 5 Assumption 8.1 is satisfied if A and H are continuous and one of the following holds: (a) H is compact (Proposition 3 a). (b) X + is a sup-semilattice and for any λ > r(A) with ξ(λ) > 0 there exists some m ∈ N such that (H Rλ )m is compact (Theorem 8). If r(A) < 1, we define the reproduction number by R = ξ(1),

(45)

If r(A) ≤ 1, Proposition 6 allows us to define the lower reproduction number by R = lim ξ(λ).

(46)

R ≤ R.

(47)

λ→1+

and, if r(A) < 1, If r(A) = 1, R may be infinite.

Reproduction Number Versus Turnover Number …

515

We also can and do extend ξ to [r(A), ∞) by setting ξ(r(A)) =

lim

λ→r(A)+

ξ(λ).

(48)

It is possible that ξ(r(A)) = ∞. The next result motivates why we call ξ the characteristic function of B as ξ plays a role for B that is somewhat analogous to the role of the characteristic polynomial for a matrix. Lemma 2 Assume that A be superadditive and Assumption 8.1 is satisfied. Let λ > r(A) and ξ(λ) ≥ 1. Then there exists some w ∈ X˙ + with B(w) ≥ λw and T = r(B) ≥ λ.

(49)

In particular, B = A + H is next-generation admissible. Proof Let λ ∈ (r(A), ∞), ξ(λ) = r(Rλ H ) = r(H Rλ ) = r ≥ 1. H Rλ is bounded homogeneous and order-preserving. By Assumption 8.1, there exists some v ∈ X˙ + such that H Rλ (v) ≥ r v ≥ v. Set w = Rλ (v). Then w ∈ X˙ + . By Proposition 5, since A is assumed to be superadditive, v ≥ λw − Aw and H w ≥ λw − Aw. We reorganize and obtain Bw ≥ λw. By Theorem 4, T = r(B) ≥ λ. By Definition 3, B = A + H is next-generation-admissible.  Lemma 3 If B = A + H is next-generation admissible, the following hold: (a) (b) (c) (d) (e) (f)

If r(A) < 1 and R ≥ 1, then T ≥ 1. If r(A) ≤ 1 and R > 1, then T > 1. If T < 1, then r(A) < 1 and R ≤ R < 1. If ξ(r(A)) > 1, then T > r(A). ξ(λ) < 1 for all λ > T . If r(A) < 1 and R ≤ 1, then T ≥ R.

Proof Let B = A + H be next-generation admissible: λ > r(A) and ξ(λ) ≥ 1 =⇒ T ≥ λ.

(50)

Statement (a) is immediate from the definition of next-generation admissibility with λ = R. (b) Let R > 1. By (46), there exists some λ > 1 such that ξ(λ) > 1 and T ≥ λ > 1 by (50). Statement (c) is the contrapositive of (a). (d) By (48), there exists some λ > r(A) such that ξ(λ) > 1. By (50), T ≥ λ > r(A). (e) This is the contraposition of (50). (f) Assume that r(A) < R ≤ 1. By Proposition 6, R = ξ(1) ≤ Rξ(R) and  ξ(R) ≥ 1 and T ≥ R by (50). If R ≤ r(A), then T ≥ r(A) ≥ R by (41).

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4.2 Starting Point: Basic Turnover Operator In this section, we make an assumption for the basic turnover operator B = A + H . Assumption 8.2 If r(A) < r(B) =: T , then there exists a lower eigenvector v ∈ X˙ + such that B(v) ≥ T v. Remark 6 Assumption 8.2 holds if A and H are continuous and one of the following hold (Proposition 3 and 4): (a) B is compact. (b) X + is complete and H is compact and A is additive. (c) X + is a sup-semilattice (see (21)) and a power of B is compact. Proposition 7 Let Assumption 8.2 be satisfied and A be subadditive and orderpreserving. Then B = A + H is turnover admissible. Proof Recall Definition 3. Let r(A) < λ ≤ T . By Assumption 8.2, there exists some v ∈ X + such that T v ≤ Bv = A(v) + H (v). Since Rλ is homogeneous, orderpreserving and subadditive, T Rλ v = Rλ (T v) ≤ Rλ A(v) + Rλ H (v). In detail, by (40), ∞ 

T λ−n−1 An (v) ≤

n=0

∞ 

λ−n−1 An+1 (v) + Rλ H (v).

n=0

We solve for Rλ H (v), change the index of summation in the second sum, and use λ≤T, ∞ ∞   T λ−n−1 An (v) − λ−n An (v) ≥ (T /λ)v. Rλ H (v) ≥ n=0

n=1

By Theorem 4, ξ(λ) = r(Rλ H ) ≥ T /λ. By Definition 3, B = A + H is turnover admissible. Lemma 4 If B = A + H is turnover admissible, the following hold. (a) (b) (c) (d) (e)

If T > 1 ≥ r(A), then ξ(λ) > 1 for all λ ∈ (1, T ), and R ≥ T > 1. If T = 1 > r(A), then R ≥ T = 1. If r(A) < 1 and R < 1, then T < 1. If r(A) ≤ 1 and R ≤ 1, then T ≤ 1. If r(A) < T , then ξ(T ) ≥ 1. If r(A) < λ < T , then ξ(λ) > 1.



Reproduction Number Versus Turnover Number …

517

Proof Since B = A + H is turnover admissible, by Definition 3, r(A) < λ ≤ T =⇒ ξ(λ) ≥ T /λ. Statements (a) and (b) are immediate from (51). Statement (c) follow from statements (a) and (b) by contraposition. (d) Set λ = T in (51). (e) This is immediate from (51).

(51)



Remark 7 Assume that A is additive, T > r(A) and B has an eigenvector v ∈ X˙ + with Bv = T v. (See Proposition 3 for sufficient conditions.) Further assume that, for any u ∈ X˙ + and λ ∈ (r(A), ∞) with wλ := H RλA u = 0, wλ is a uniform order unit for H . Then λ = T = r(B) is the unique solution of ξ(λ) = 1. Proof If A is additive and Bv = T v ∈ X˙ + , the same proof as for Proposition 7 shows that RT H (v) = v. Set w = H (v). Then w ∈ X˙ + and H RT w = w. By assumption, w is a uniform order unit for H and thus for H RT . By Theorem 4 and 5, ξ(T ) = 1. Uniqueness of this solution follows from Proposition 6. 

4.3 Synopsis Throughout this subsection, we assume the overall assumptions listed at the beginning of the section and that B = A + H is next-generation and turnover admissible (Definition 3). The latter are the case, for instance, if Assumptions 8.1 and 8.2 are satisfied. We also assume that A is additive. We combine Lemmas 3(d) and 4(e). Recall (48). Theorem 9 r(B) > r(A) if and only if ξ(r(A)) ≥ 1. We combine Lemmas 4 and 3. Proposition 8 If r(A) < 1, the following hold: (a) T > 1 if and only R > 1. If one and then also the other is true, 1 < T ≤ R ≤ R. (b) T ≥ 1 if and only if R ≥ 1. If one and then also the other is true, R ≥ T ≥ 1. (c) T < 1 if and only if R < 1. If one and than also the others hold, 1 > T ≥ R ≥ R . (d) T = 1 if and only if R ≤ 1 ≤ R. (e) T > r(A) if and only if ξ(r(A)) > 1. (f) ξ(λ) < 1 for all λ > T . If T > r(A), then ξ(λ) > 1 for all λ ∈ (r(A), T ) and ξ(T ) ≥ 1. T is uniquely determined by property (f) in Proposition 8.

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Proof (a) Lemmas 4 (a) and 3 (b). (b) Lemmas 4 (a), (b) and 3 (a). (c) T < 1 ⇐⇒ R < 1 is the contraposition of (b). The inequalities in (c) follow from Lemma 3 (f). (d) Contraposition of part (a) provides that T ≤ 1 if and only if R ≤ 1. Combination with part (b) provides the statement. (e) Apply Lemma 4, (51) and 3 (d). (f) Apply Lemma 4, (51) and part (d), and Lemma 3 (e).  Proposition 8 can be reformulated like this. Theorem 10 Let B = A + H be next-generation and turnover admissible (Definition 3), for instance if Assumption 8.1 and 8.2 are satisfied, and let r(A) < 1. Then the turnover/reproduction trichotomy holds weakly (Definition 1). Proof of Theorem 2 (a), (b), (c). Apply Theorem 10. (a) Let A and H be continuous and compact. Then B is continuous and compact and the operators H Rλ are continuous and compact and Assumption 8.1 and 8.2 are satisfied by Proposition 3 (a). (b) Let X + be complete and H be complete. Then the operators H Rλ are continuous and compact and Assumption 8.1 is satisfied by Proposition 3 (a). Assumption 8.2 is satisfied by Proposition 3 (a). (c) Let X + be a sup-semilattice and powers of B and of the operators H Rλ be compact. Then Assumption 8.1 and 8.2 are satisfied by Theorem 8.  Combining Lemmas 4 and 3, we can also prove the following result for r(A) = 1. Theorem 11 Let B = A + H be next-generation and turnover admissible (Definition 3), for instance if Assumption 8.1 and 8.2 are satisfied, and let r(A) = 1. Then the following dichotomy holds: (ii) 1 < T ≤ R ≤ ∞. (i) 1 = T ≥ R Proof By (41), T ≥ r(A) = 1. If T = 1, then R ≤ 1 by contraposition of Lemma  3 (b). If T > 1, then 1 < T ≤ R ≤ ∞ by Lemma 4 (a). Proof of Remark 2, conditions (a), (b), (c). Apply Theorem 11 similarly as Theorem 10 in the proof of Theorem 2 (a), (b), (c).  Recall (48). Proposition 9 The case 1 > T ≥ R ≥ R has the following subcases. (i) ξ(r(A)) > 1 > T > r(A). (ii) 1 > T = r(A) and ξ(r(A)) ≤ 1 and R ≤ T ξ(r(A)). Proof Lemma 3 (d) and its converse, Lemma 4, (51). As for the last inequality in (ii), by Proposition 6, for all λ ∈ (T , 1), R = ξ(1) ≤ λξ(λ). Now take the limit as λ → T = ξ(r(A)).



Reproduction Number Versus Turnover Number …

519

There is the possibility that T = 1 = R and R > 1. This cannot occur if R = R , actually if R ≤ R . Following [32], we define the cone spectrum of a bounded homogeneous operator A by (52) σ+ (A) = {r ≥ 0; ∃v ∈ X˙ + : A(x) = r x}. Proposition 10 Assume that A is continuous and H is compact and continuous and R ≥ 1 and that, for each r ∈ (1, R), there exists some s ∈ (r, R) such that s∈ / σ+ (R1 H ). Then R = R ≥ 1 and T > 1. Proof It is sufficient to prove that R ≤ R . Suppose R < R. Then there exists a sequence (λk ) in (0, 1) such that λk → 1 as k → ∞ and lim inf k→∞ ξ(λk ) < ξ(1) = R. We have Rλk H − R1 H → 0, k → ∞. Let r < R. Then there exists some s ∈ (r, R) such that s ∈ / σ+ (R1 H ). By [32, Theorem 2.4], lim inf ξ(λk ) ≥ s ≥ r. k→∞

Since this hold for any r < ξ(1) = R, we have lim inf k→∞ ξ(λk ) ≥ R, a contradiction. 

4.4 Cones with Uniform Order Units To prove Theorem 2 (d), let X be an ordered normed vector space with a serially complete normal closed cone X + that has a uniform order unit u ∈ X˙ + . See (11), (15), and (20). The subsequent result connects us to Sects. 5 and 7 of [45], which we will use in the following. Lemma 5 (a) Every increasing Cauchy sequence in X + converges. (b) Every decreasing sequence in X converges. Proof (a) Let (ym ) be an increasing Cauchy sequence in X + . For convergence, it is sufficient that (ym ) has a convergent subsequence with limit in X + . Since (ym ) is a Cauchy sequence, we can find a strictly increasing sequence of numbers (m j ) in N  such that ym j+1 − ym j ≤ 2− j . Set z k = kj=1 [ym j+1 − ym j ]. Since (ym ) is increasing, ym j+1 − ym j ∈ X + . By construction and the serial completeness of X + , (z k ) converges to some z ∈ X + . By telescoping, z k = ym k+1 − ym 1 . So (ym k ) converges with limit z + ym 1 ∈ X + and (ym ) has a converging subsequence in X + . (b) Let (xn ) be a decreasing Cauchy sequence in X . Set yn = x1 − xn , n ∈ N. Then (yn ) is an increasing Cauchy sequence in X + and converges to some y ∈ X + by part (a). So, xn → x1 − y as n → ∞. 

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Let H : X + → X + be homogeneous, continuous and order-preserving and A : X + → X + be homogeneous, continuous and additive. Define B : X + → X + by B = H + A.

(53)

Then B is homogeneous, continuous and order-preserving. Since u is a uniform order unit for X + , all operators H, A, B are uniformly u-bounded. See (33). Let ψ : X → R+ be the monotone companion half-norm on X , ψ(x) = d(x, −X + ) = inf{ y ; x ≤ y ∈ X },

x ∈ X.

(54)

Since X + is normal, there is some δ0 ∈ (0, 1] such that δ0 x ≤ ψ(x) ≤ x ,

x ∈ X +.

The functional ψ is order-preserving and subadditive on X and ψ(αx) = αψ for all x ∈ X and α ∈ R+ . See [44, Sect. 4]. For n ∈ N, we define Hn : X + → X + by Hn (x) = H (x) +

ψ(x) u, n

x ∈ X +,

(55)

and Bn = A + Hn . Then, for n ∈ N, we have the perturbed basic turnover operators Bn (x) = B(x) +

ψ(x) u, n

x ∈ X +,

(56)

and the perturbed next generation operator families Hn (Rλ x) = H (Rλ x) +

ψ(Rλ x) u, n

x ∈ X + , λ ∈ (r(A), ∞).

(57)

The characteristic functions of Bn are given by   ξn (λ) = r Hn Rλ ,

λ ∈ (r(A), ∞).

(58)

By Sects. 5 and 7 of [45], Bn and Hn Rλ satisfy Assumptions 8.2 and 8.1. Actually, r(Bn ) > 0 and ξn (λ) = r (Hn Rλ ) > 0 and there exist vn , wn ∈ X˙ + such that Bn (vn ) = r(Bn )vn ,

Hn Rλ (wn ) = ξn (λ)wn ,

n ∈ N.

By Lemmas 2 and 7, for each n ∈ N, Bn = A + Hn is both next-generation and turnover admissible. We now show that B = A + H is both next-generation and turnover admissible. By construction,

Reproduction Number Versus Turnover Number …

521

Bn (x) ≥Bn+1 (x) ≥ B(x), x ∈ X + , n ∈ N, Hn Rλ (x) ≥Hn+1 Rλ (x) ≥ H Rλ (x). By Corollary 3, Tn =r(Bn ) ≥ r(B) = T , ξn (λ) ≥ ξn+1 (λ) ≥ ξ(λ),

n ∈ N, λ ∈ (r(A), ∞).

(59)

Since u is a uniform order unit for X + and B is homogeneous and order-preserving, there is some c ∈ (0, ∞) such that, for all n ∈ N and x ∈ X + , Bn (x) ≤Bn (c x u) ≤ c x Bn (u) ≤ c x (B(u) + ψ(u)u)   ≤c x c B(u) + ψ(u) u. Similarly, for all n ∈ N and x ∈ X + ,     Hn (Rλ x) ≤Hn Rλ c x u = c x Hn (Rλ u) ≤ c x H (Rλ u) + ψ(Rλ u)u   ≤c x c H (Rλ u) + ψ(Rλ u) u. By [45, Theorem 8.2], in combination with (59), Tn  T ,

ξn (λ)  ξ(λ),

n → ∞, λ ∈ (r(A), ∞).

(60)

Notice that in [45, Theorem 8.2] we have used r+ instead of r for the spectral radius. Since u is a uniform order unit of X + , Bn is uniformly u-bounded and η u (Bn ) = r(Bn ) by [45, Theorem 2.8] or [44, Theorem 12.10]. Proposition 11 (a) B = A + H is next-generation admissible: λ > r(A) and ξ(λ) ≥ 1 =⇒ T ≥ λ. (b) B = A + H is turnover admissible: r(A) < λ ≤ T =⇒ ξ(λ) ≥ T /λ. Proof (a) Let λ > r(A) with ξ(λ) ≥ 1. By (59), ξn (λ) ≥ 1 for all n ∈ N. Since Bn = A + Hn is next-generation admissible, Tn ≥ λ for all n ∈ N and T ≥ λ by (60). This implies that B = A + H is next-generation admissible (b) Let r(A < λ ≤ T . Let n ∈ N. Then Tn ≥ T by (59) and Bn = A + Hn is turnover admissible. So ξn (λ) ≥ Tn /λ. By (60), ξ(λ) ≥ T /λ. This implies that B = A + H is turnover admissible.  Proof of Theorem 2 (d). Combine Theorem 10 and Proposition 11. Proof of Remark 2 (d). Combine Theorem 11 and Proposition 11.

 

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Proof of Proposition 1. For (i) and (ii), apply Remark 4. B is both next-generation and turnover admissible. If X + has a uniform order unit, this follows from Proposition 11. If X + is complete and H is compact, this follows from Remark 5 and Lemma 2 and from Remark 6 and Proposition 7. (iii) We proceed as in [48, Sect. 7.5] built on the references given there. Set X˜ = X + − X + . There exists a norm · ∼ on X˜ such that x = x ∼ for all x ∈ X + and x ≤ x ∼ for all x ∈ X˜ . Since X + is serially complete under · , X˜ is complete under · ∼ [48, Prop.7.12]. Since X + is a normal cone of X under · , it is a normal cone of X˜ under · ∼ . H can be extended to a linear operator H˜ on X˜ that is bounded with respect to · ∼ and H = H˜ ∼ for the respective operator norms and r(H ) = r( H˜ ). If X is a real normed vector space, we consider its complexification and extend H˜ to the complexification with the operator norm and the spectral radius remaining the same [5], [41, p. 261]. Let H˜ also denote this last extension. Similarly, we extend ˜ B(x) ˜ B to B, = αx + H˜ (x) for x in the complexification of X˜ . Recall that the resolvent set ρ( H˜ ) are those λ ∈ C such that λI − H˜ has a bounded inverse and that ˜ the spectrum of H˜ is C \ ρ( H˜ ) and r( H˜ ) = maxλ∈σ( H˜ ) |λ|. The same holds for B. ˜ Since X˜ is an ordered Banach space Obviously, λ ∈ σ( H˜ ) if and only λ + α ∈ σ( B). ˜ and X = X + − X + and X + is normal, r(H ) ∈ σ( H˜ ) [5, Theorem 1] [41, App.2.2]. ˜ = α + r( H˜ ) and so r(B) = α + r(H ). One now readily shows that r( B) 

5 Additive Perturbations of Rank-One Operators Let X be an ordered normed vector space with closed normal complete cone X + and B : X + → X + be given as B = A + H;

H (x) =θ(x)u,

x ∈ X +,

(61)

where A : X + → X + is an additive homogeneous operator, u ∈ X˙ + , and θ : X + → R+ a homogeneous order-preserving functional. Recall the resolvents Rλ = RλA , λ > r(A), (40). The operators H Rλ (x) = θ(Rλ x)u,

x ∈ X +,

are of rank one and the characteristic function (23) takes the form ξ(λ) = r(H Rλ ) = θ(Rλ u),

λ ∈ (r(A), ∞).

(62)

The vector u is an eigenvector of H Rλ associated with ξ(λ) and θ is an associated eigenfunction. See Example 1. Since θ is order-preserving and homogeneous, λξ(λ) is a decreasing function of λ > r(A) even if X + is not a normal cone (Proof of Proposition 6). Further, Assumption 8.1 is satisfied.

Reproduction Number Versus Turnover Number …

523

By Theorem 9, the condition r(B) > r(A) is equivalent to lim

λ→r(A)+

ξ(λ) > 1.

The following result improves [47, Corollary 5.4]. Theorem 12 Let A and θ be continuous, r(A) < 1. Then the turnover/reproduction trichotomy holds strongly so that we have one of the following three possibilities: (ii) 1 = T = θ(R1 u), (ii) 1 > T ≥ θ(R1 u). (i) 1 < T ≤ θ(R1 u), If lim θ(Rλ u) > 1, then λ = T is the unique solution of 1 = θ(Rλ u). λ→r(A)+

Proof We apply Theorem 10 and Remark 1. Since H is compact and continuous and X + is assumed to be complete, the Assumptions 8.1 and 8.2 are satisfied. Since θ is continuous by assumption and Rλ u is a continuous function of λ > r(A), the characteristic function ξ is continuous on (r(A), ∞).  Theorem 13 Assume that A and θ are continuous, r(A) = 1. Then the following dichotomy holds: (i) 1 = T ≥ R or 1 < T ≤ R ≤ ∞ with R = limλ→1+ θ(Rλ u). 

Proof Apply Theorem 11.

Example 2 We consider the case that the additive homogeneous operator A is also a rank-one operator, Ax = ϑ(x)w for x ∈ X + , where w ∈ X˙ + and ϑ : X + → R+ is additive and homogeneous and ϑ(w) < 1. By Example 1, r(A) = ϑ(w) and Rλ x = λ−1 x +

ϑ(x) w, λ(λ − ϑ(w))

λ > ϑ(w).

By (62), ξ(λ) = θ(Rλ u) = λ−1 θ u +

ϑ(u) w , λ − ϑ(w)

λ > ϑ(w).

ξ is continuous if θ is continuous, and the turnover/reproduction trichotomy holds strongly (Definition 2) if ϑ(w) < 1. ˜ + ϑ(x)w + θ(x)u, x ∈ X + . Here A˜ : X + → Example 3 We consider B(x) = Ax X + is additive, homogeneous and continuous, ϑ : X + → R+ is additive, homogenous and continuous and θ : X + → R+ is order-preserving, homogenous and con˜ < 1. tinuous. Further, we assume that r( A) ˜ Let Ax = Ax + ϑ(x)w, x ∈ X + . By Theorem 12, we have the trichotomy that one of the following holds: (i) 1 < r(A) ≤ ϑ( R˜ 1 w) (ii) 1 = ϑ( R˜ 1 w) = r(A) (iii) ϑ( R˜ 1 w) ≤ r(A) < 1, ˜ where R˜ λ is the resolvent of A.

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Notice that T = r(B) ≥ r(A) > 1 if (i) holds. Suppose (iii) holds, ϑ( R˜ 1 w) ≤ r(A) ≤ 1. To calculate the resolvent Rλ of A in these two cases, we use that A is additive and so y = Rλ x is the unique solution of ˜ + ϑ(y)w + x, λy = Ay

λ > r(A) ≥ ϑ( R˜ 1 w).

See Proposition 5. By the same proposition, y = ϑ(y) R˜ λ w + R˜ λ x. We apply the additive homogenous functional ϑ to this equation and solve the result for ϑ( R˜ λ x) , λ ≥ 1 ≥ r(A). ϑ(y) = 1 − ϑ( R˜ λ w) Notice that ϑ( R˜ λ w) ≤ ϑ( R˜ 1 w) < 1 for λ ≥ 0 by Lemma 6. We substitute this formula into the previous equation, Rλ x = y =

ϑ( R˜ λ x) ˜ Rλ w + R˜ λ x, 1 − ϑ( R˜ λ w)

λ ≥ 1 > r(A) ≥ ϑ( R˜ 1 w).

We leave it to the reader to show that y is a solution, indeed. By (62), with H (x) = θ(x)u, x ∈ X + , ξ(λ) = θ(Rλ u) = θ



ϑ( R˜ λ u) ˜ Rλ w + R˜ λ u , 1 − ϑ( R˜ λ w)

λ ≥ 1.

Since θ and ϑ are continuous, the characteristic function ξ is continuous. By Theorem 12, the turnover/reproduction trichotomy holds strongly with R=θ



ϑ( R˜ 1 u) ˜ R1 w + R˜ 1 u . 1 − ϑ( R˜ 1 w)

Let (ii) hold: 1 = ϑ( R˜ 1 w) = r(A). We can make the same calculation as before except that it only is valid for λ > 1 which is sufficient to define R = limλ→1+ ξ(λ). By Theorem 13, we have the dichotomy (ii.1) 1 = T ≥ R or (ii.2) 1 < T ≤ R ≤ ∞. In principle, one can use these results to recursively treat B(x) = A0 x +

n  j=1

θ j (x)u j + θn+1 (x)u n+1

Reproduction Number Versus Turnover Number …

525

with additive homogeneous continuous A0 : X + → X + , additive homogeneous continuous θ j : X + → R+ , j = 1, . . . , n, and order-preserving homogeneous continuous θn+1 : X + → R and u j ∈ X˙ + , j = 1, . . . , n + 1. Example 4 As mentioned before, in this paper we are mainly interested in the ordered Banach space X = C b (S) with the supremum norm. Let the operator B on C+b (S) be given by B( f )(s) = g1 (s) f (s) + θ( f )g2 (s),

s ∈ S, f ∈ C+b (S),

(63)

with two given functions g j ∈ C+b (S) and a given homogeneous, order-preserving, continuous functional θ : C+b (S) → R+ . This fits into the framework of this section with A being the multiplication operator (A f )(s) = g1 (s) f (s),

s ∈ S, f ∈ C+b (S).

(64)

One readily determines r(A) = sup g1 (S).

(65)

Further, (An f )(s) = (g1 (s))n f (s), and (RλA f )(s) =

f (s) , λ − g1 (s)

s ∈ S, f ∈ C+b (S), s ∈ S, f ∈ C+b (S).

(66)

(67)

The characteristic function (62) takes the form ξ(λ) = θ(h λ ), λ > sup g1 (S), g2 (s) , s ∈ S. h λ (s) = λ − g1 (s)

(68)

Since θ is order-preserving, we have the two cases that θ(g2 ) = 0 and ξ(λ) = 0 for all λ > sup g1 (S) or θ(g2 ) > 0 and ξ(λ) > 0 for all λ > sup g1 (S). So let us assume that θ(g2 ) > 0. We have r(B) > r(A) if and only if lim

λ→sup g1 (S)+

θ(h λ ) > 1.

If g1 is constant, this is satisfied. Assume that g1 is not constant. Since

(69)

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H. R. Thieme

h λ (s) ≥ we have h λ (s) ≥

g2 (s) , λ − inf g1 (S)

g2 (s) , sup g1 (S) − inf g1 (S)

So r(B) > r(A) if

s ∈ S,

s ∈ S, λ ≥ sup g1 (S).

θ(g2 ) > sup g1 (S) − inf g1 (S).

In special cases, weaker sufficient conditions can certainly be found. For instance, let μ j , j = 1, 2, be nonnegative finite measures. Define θ( f ) = min





 f dμ2 ,

f dμ1 , S

S

f ∈ C+b (S),

and assume that g1 (s) < sup g1 (S) for all s ∈ S. By the monotone convergence theorem, condition (69) becomes

1 < min

j=1,2

f dμ j , S

f (s) =

g2 (s) , s ∈ S, sup g(S1 ) − g1 (s)

with one or both integrals possibly being infinite. Under this condition, T is the unique solution λ of 1 = θ(h λ ). If sup g1 (S) < 1, the following trichotomy holds: (i) 1 < T ≤ θ(h 1 ), with h 1 (s) =

g2 (s) , 1−g1 (s)

(ii) 1 = T = θ(h 1 ),

(iii) 1 > T ≥ θ(h 1 ).

s ∈ S.

6 Iteroparous Populations with Mating In iteroparous populations, individuals reproduce during several years and there are overlapping generations. We analyze the model in Sect. 2.1. Recall that the cones X + , Y+ , and Z + are closed and that X + and Y+ are normal and serially complete. We define A on X + × Y+ by A(x, y) = (A• x, A y),

x ∈ X + , y ∈ Y+ .

(70)

and K : Z + → X + × Y+ by K (z) = (K • z, K  z), z ∈ Z + . Then the basic turnover operator B acts on X + × Y+ ,

(71)

Reproduction Number Versus Turnover Number …

B = A + H,

527

H = K φ.

(72)

By Proposition 2 and (70), r(A) = max{r(A• ), r(A )} and   Rλ (x, y) = Rλ• x, Rλ y ,

x ∈ X + , y ∈ Y+ , λ > r(A). •

(73)



Here Rλ = RλA operates on X + × Y+ , Rλ• = RλA on X + , and Rλ = RλA on Y+ and, for instance, Rλ is given by (40) with the others being given analogously. With H = K φ, the characteristic function ξ in (23) takes the form ξ(λ) = r(K φRλ ) = r(φRλ K ),

λ > r(A).

(74)

The second equality follows from Theorem 6. Notice that the first spectral radius is taken of an operator on X + × Y+ and the second of an operator on Z + . By (73) and (71), the operator φRλ K operates on Z + like   (φRλ K )(z) = φ Rλ• K • z, Rλ K  z ,

z ∈ Z+.

(75)

Proposition 12 The turnover/reproduction trichotomy holds weakly (Definition 1) if φ is continuous and r(A• ) < 1, r(A ) < 1 and if one of the following conditions is satisfied in addition. (a) A• , A , K • φ and K  φ are compact. (b) X + and Y+ are complete and K • φ and K  φ are compact. (c) X + and Y+ have uniform order units. Proof Apply Theorem 2 with (72). Notice that, if A• and A are compact, so is A. Further, if K • φ : X + × Y+ → X + and K  φ : X + × Y+ → Y+ are compact, so is H = K φ : X + × Y+ → X + × Y+ . If u • ∈ X + is a uniform order unit for X + and u  ∈ Y+ is a uniform order unit for  Y+ , then (u • , u  ) is a uniform order unit for X + × Y+ . Proof of Theorem 3. Apply Proposition 12. If K • and K  are compact, so are K • φ  and K  φ. Example 5 Let K • and K  be rank-one operators with the same functional, K  (z) = ζ(z)u  ,

z ∈ Z + ,  = •, ,

(76)

with u • ∈ X˙ + , u  ∈ Y˙+ and a continuous, homogeneous, order-preserving functional ζ : Z + → R+ . Then H = K φ is a rank-one operator, H (x, y) = θ(x, y)(u • , u  ),

x ∈ X +,

y ∈ Y+ ,

(77)

with the homogeneous, continuous, order-preserving functional θ : X + × Y+ → R+ , θ(x, y) = ζ(φ(x, y)),

x ∈ X + , y ∈ Y+ .

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H. R. Thieme

So B is the additive perturbation of a rank-one operator. By the considerations in Sect. 5, the turnover/reproduction trichotomy holds strongly (Definition 2) with R = ξ(1) and the characteristic function   ξ(λ) = ζ φ(Rλ• u • , Rλ u  ) ,

λ > r(A).

Notice that ξ inherits continuity from ζ. Example 6 We assume that X = Y = Z and that X + has a uniform order unit. Further we assume that all operators A• , A , K • , K  are perturbed rank-one operators with the same spanning vector v ∈ X˙ + , A•, (x) = ϑ•, (x)v + α•, x K •, (x) = ζ •, (x)v + β •, x

 x ∈ X +.

(78)

Here α• , α , β • , β  ∈ R+ and ϑ• , ϑ , ζ • , ζ  are homogeneous continuous functionals from X + to R+ and ϑ• and ϑ are additive and ζ • and ζ  are order-preserving. Assume that α• + ϑ• (v) < 1 and α + ϑ (v) < 1. We further assume that there is a homogeneous continuous order-preserving function ϕ˜ : R2+ → R+ such that φ(αx, βx) = ϕ(α, ˜ β)x,

α, β ∈ R+ , x ∈ X + .

(79)

Then v is an eigenvector of φRλ K which operates like (75), and the turnover/ reproduction trichotomy holds strongly (Definition 2) with

R = ϕ˜

ζ • (v) + β • ζ  (v) + β  . , • • 1 − α − ϑ (v) 1 − α − ϑ (v)

Remark 8 An abstract example for (79) can be given if X is a normed vector lattice [42, II. Sect. 5]: A vector lattice is an ordered vector space such that, for any x, y ∈ X , their supremum x ∨ y and their infimum x ∧ y exist. Since the order is induced by a cone, these operations are compatible with the vector space operations. In a normed vector lattice, these operations are continuous from X × X to X . Usually, one chooses φ(x, y) = x ∧ y because then φ(0, x) = φ(x, 0) for all x ∈ X + which makes sense for a mating function. Notice that φ(αx, βx) = min{α, β}x,

x ∈ X + , α, β ∈ R+ ;

so (79) is satisfied. Proof By Proposition 1 and Example 1, r(A• ) = α• + ϑ• (v) < 1 and also r(A ) < 1. By (78), ζ • (v) + β • v. Rλ• K • (v) = (ζ • (v) + β • )Rλ• (v) = λ − α• − ϑ• (v)

Reproduction Number Versus Turnover Number …

529

The same holds for Rλ K  (v). By (75) and (79),

φRλ K (v) = ϕ˜

ζ  (v) + β  ζ • (v) + β • , v. λ − α• − ϑ• (v) λ − α − ϑ (v)

By Theorem 4 and (74),

ξ(λ) ≥ ϕ˜

ζ • (v) + β • ζ  (v) + β  ˜ , =: ξ(λ). λ − α• − ϑ• (v) λ − α − ϑ (v)

(80)

If v is a uniform order unit of X + , equality holds by Theorem 5. If v is not a uniform order unit, we pick a uniform order unit u ∈ X˙ + and set n ∈ N. vn = v + (1/n)u, Let B˜ n be the operator φRλ K with v being replace by vn . Since vn is a uniform order unit for X + , we conclude from our previous considerations that

r( B˜ n ) = ϕ˜

ζ  (vn ) + β  ζ • (vn ) + β • , λ − α• − ϑ• (vn ) λ − α − ϑ (vn )

if n ∈ N is large enough that the denominators are positive. Replacing v by vn in φRλ K affects both Rλ and K via (78) and (70), (71), (73). Since v ≤ vn and φ is order-preserving, (φRλ K )(x) ≤ B˜ n (x),

x ∈ X + , n ∈ N.

Since φRλ K is order preserving, by Theorem 4, ξ(λ) = r(φRλ K ) ≤ r( B˜ n ). We combine the various inequalities and use (80), ˜ ξ(λ) ≤ ξ(λ) ≤ r( B˜ n ). ˜ as n → ∞ and ξ(λ) = Since vn → v as n → ∞ and ϕ˜ is continuous, r( B˜ n ) → ξ(λ) ˜ ξ(λ) and ξ is continuous.  Example 7 In this example, we assume that adult development and survival is the same for males and females, A• = A . Further, we assume that K  (z) = ζ  (z)v,

z ∈ Z + ,  = •, ,

(81)

with some v ∈ Z˙ + and homogeneous, continuous, order-preserving functionals ζ • , ζ  : Z + → R+ . Then 



Rλ K  (z) = ζ  (z)Rλ v,

z ∈ Z+,

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H. R. Thieme

and

  (φRλ K )(z) = φ ζ • (z)Rλ• v, ζ  (z)Rλ• v .

Assume (79). Then   (φRλ K )(z) = ϕ˜ ζ • (z), ζ  (z) Rλ• v,

z ∈ Z+,

provides a rank-one operator (with eigenvector Rλ• v), the spectral radius of which is ϕ(ζ ˜ • (Rλ• v), ζ  (Rλ• v)) = ξ(λ). Recall (74). ξ inherits continuity from ϕ˜ and the turnover/reproduction trichotomy holds strongly with   R = R = ϕ˜ ζ • (R1• v), ζ  (R1• v) .

6.1 Iteroparous Populations More Concrete Let S be a metric space and represent the habitat of a population. We consider the situation that the first order approximation of the population turnover equations has the form

xn+1 (s) =

S

yn+1 (s) =

xn (t)q • (dt, s) + yn (t)q  (dt, s) +

S



S

⎫ ϕ(t, xn (t), yn (t))κ• (dt, s) ⎪ ⎬

s ∈ S, n ⎪ ∈ Z+ . ϕ(t, xn (t), yn (t))κ (dt, s) ⎭

(82)

S

Here we assume that q • , q  , κ• , and κ are Feller kernels. See (30) and the subsequent lines for explanation. The recursive system (82) consists of so-called integro-difference equations (though there is no visible difference) [26, 30, 33, 35, 43, 53]. Sometimes, they are also called integral projection models (see [15–17] and the references therein). In order to bring this model into the form of Sect. 6, we choose X = Y = Z = C b (S), the Banach space of bounded continuous real-valued functions on S with the supremum norm, which is ordered by the closed cone of nonnegative functions in C b (S), C+b (S). Every positive constant function on S is a uniform order unit for C+b (S). We define (A• x)(s) =



x(t)q • (dt, s), S   φ(x, y)(s) = ϕ s, x(s), y(s) ,

(K • x)(s) =



x(t)κ• (dt, s), S

s ∈ S,

and analogous definitions are made for A and K  .

x, y ∈ C (S). b

(83)

Reproduction Number Versus Turnover Number …

531

Lemma 6 Let ϕ : S × R2+ → R+ satisfy Assumption 3.1. Then the following hold: (i) ϕ(s, x, y) = 0 if s ∈ S, x, y ∈ R+ and x = 0 or y = 0. (ii) ϕ(s, x, y) > 0,  s ∈ S, x > 0, y > 0. (iii) If c ∈ (0, ∞), ϕ S × [0, c]2 is bounded. Proof (i) Combine Assumption 3.1 (b) and (c). (ii) Combine Assumption 3.1 (b) and (d). (iii) Combine Assumption 3.1 (b) and (e).  Lemma 7 ([51, L.7.1]) Let ϕ satisfy the Assumptions 3.1 and φ be the Nemytskii operator associated with ϕ via (83) and Y1 and Y2 be equicontinuous bounded subsets of C+b (S). Then φ(Y1 × Y2 ) is an equicontinuous bounded subset of C+b (S). Recall that Yi are equicontinuous if for any s0 ∈ S and any  > 0 there is some δ > 0 such that |h(s) − h(s0 )| <  for all h ∈ Yi and all s ∈ S with d(s, s0 ) < δ. We obtain the following result from Theorem 3 (c). Theorem 14 Let Assumption 3.1 be satisfied and r(A• ) < 1 and r(A ) < 1. Then the turnover/reproduction trichotomy (Definition 1) holds weakly. The next example is a rephrasing of Example 5. Example 8 Let κ• (T, s) = μ(T )γ • (s) and κ (T, s) = μ(T )γ  (s) for T ∈ B and s ∈ S with a nonnegative finite measure μ, μ(S) > 0 and γ • , γ  ∈ C˙ +b (S). Then

R = R = ξ(1) = S

  ϕ s, (R1• γ • )(s), (R1 γ  )(s) μ(ds).

(84)

If, in addition, r(A• ) < 1 and r(A ) < 1, the turnover/reproduction trichotomy (Definition 2) holds strongly. Example 9 Let q • (T, s) =ν • (T )γ(s) + α• χT (s), q  (T, s) =ν  (T )γ(s) + α χT (s),

s ∈ S, T ∈ B,

κ• (T, s) =μ• (T )γ(s) + β • χT (s), κ (T, s) =μ (T )γ(s) + β  χT (s), with measures ν • , ν  , μ• , μ : B → R+ , α• , α , β • , β  ∈ R+ , and γ ∈ C˙ +b (S). Here χT is the indicator function of the Borel subset T of S, χT (s) = 1, s ∈ T, Assume



γdν • + α• < 1, S

χT (s) = 0, s ∈ S \ T.

γdν  + α < 1. S

(85)

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Then γ is an eigenvector of φRλA K if ϕ(s, ·) does not depend on s and   γdμ• + β • S S γdμ + β R = R = ϕ . , 1 − S γdν • − α• 1 − S γdν  − α Further the turnover/reproduction trichotomy (Definition 2) holds strongly. Proof This is a rephrasing of Example 6. Notice that (79) is satisfied by (83) if ϕ(s, ·) does not depend on s.  Example 10 We consider the following special case of (82) in which adult individuals do no longer disperse,

xn+1 (s) = p(s)xn (s) + γ(s)

S

yn+1 (s) = p(s)yn (s) + γ(s)

⎫ ϕ(t, xn (t), yn (t))μ• (dt) ⎪ ⎬

s ∈ S, n ∈ Z+ . ⎪ ϕ(t, xn (t), yn (t))μ (dt) ⎭ 

S

The only difference for the two sexes are the measures μ• and μ which incorporate the probabilities where an arbitrary one-year-old, independently of where it is located at the end of the year, was born at the beginning the year as well as the per pair fertility and the sex ratios at birth. The numbers p(s) describes sex-independent probabilities at which adults (which stay at s ∈ S) survive the year where sup p(S) < 1. We have that Rλ• = Rλ and (Rλ• x)(s) =

x(s) , λ − p(s)

λ > sup p(S).

The operator φRλ K on C+b (S) takes the form



γ(s) γ(s) zdμ• , zdμ λ − p(s) S λ − p(s) S

γ(s)

, = ϕ s, zdμ• , zdμ λ − p(s) S S

(φRλ K )(z)(s) = ϕ s,

where we have used the homogeneity of ϕ(s, ·). We assume that ϕ(s, ·) does not depend on s. Then v ∈ C˙ +b (S) with v(s) = γ(s)/(λ − p(s)) is an eigenfunction of φRλ K and, by Theorem 4,

ξ(λ) ≥ ϕ S

γ(t) μ• (dt), λ − p(t)

S

γ(t) μ (dt) . λ − p(t)

If v is a uniform order unit of C+b (S), equality holds by Theorem 5. Otherwise, we apply a perturbation technique as in Example 6 and replace γ by γn with γn (t) = γ(t) + (1/n), n ∈ N, t ∈ S. Since γn is a uniform order unit for C+b (S), the characteristic function ξn for the perturbed system is

Reproduction Number Versus Turnover Number …

533

γ(t) + (1/n)

γ(t) + (1/n)  μ• (dt), μ (dt) . ξn (λ) = ϕ λ − p(t) λ − p(t) S S Proceeding as in Example 6, we obtain that

ξ(λ) = ϕ

S

γ(t) μ• (dt), λ − p(t)

S

γ(t) μ (dt) , λ − p(t)

even if v is not an order unit for C+b (S). ξ inherits continuity from ϕ and so the turnover/reproduction trichotomy (Definition 2) holds strongly with

R = R = ξ(1) = ϕ

S

γ(t) μ• (dt), 1 − p(t)

S

γ(t) μ (dt) . 1 − p(t)

6.2 A Rank-Structured Population We consider a population in which each individual has a rank j ∈ N. Within a year, an individual (provided that it survives) is either be promoted to the next higher rank or stays at its rank. Let p j denote the probability that an individual of rank j survives the year and is promoted to rank j + 1 and q j be the probability that it survives and is not promoted, j ∈ N. (86) q j , p j ≥ 0, p j + q j < 1, If q j = 0 for all j ∈ N, the model becomes an age-structured population model. A natural ordered normed vector space is the sequence space X = 1 with x = ∞ j=1 |x j | < ∞, x = (x j ), because x is the total number of individuals if x is a nonnegative sequence while x j is the number of individuals at rank j. Define the basic turnover map B : 1 → RN , B(x) = (B j (x)), by B1 (x) = θ(x) + q1 x1 B j (x) = p j−1 x j−1 + q j x j ,

 j ≥2

x = (x j ) ∈ X + .

(87)

Here θ : 1+ → R+ is homogeneous, order-preserving and continuous and describes the mating process between individuals of different ranks. Reproduction is of Leslie type: All neonates have the lowest rank 1 [7, 11, 34]. For instance, θ(x) =

∞ 

β jk ϕ(x j , xk ),

x = (x j ) ∈ 1+ ,

(88)

j,k=1

with β jk ≥ 0 for all j, k ∈ N.

∞  j,k=1

is to be understood as lim

n→∞

n  j,k=1

.

534

H. R. Thieme

This form of a mating function means that mating is assumed to be rank-selective. The numbers β jk combine the probabilities that individuals of rank j and k mate and the fertility of such a pair. The expression ϕ(x j , xk ) represents the number of pairs that result from x j individuals at rank j and xk individuals of rank k. For a two-sex population, one may like to assume a 1:1 sex ratio at each rank. The mating function ϕ : R2+ → R+ is homogeneous and order-preserving, ϕ(x, 0) = 0 = ϕ(0, x) for all x ∈ R+ and ϕ(x, y) > 0 if x, y ∈ (0, ∞). We assume that there is a Lipschitz constant Λ > 0 such that |ϕ(z 1 , z 3 ) − ϕ(z 2 , z 3 )| ≤ Λ|z 1 − z 2 | |ϕ(z 1 , z 2 ) − ϕ(z 1 , z 3 )| ≤ Λ|z 2 − z 3 |

 z 1 , z 2 , z 3 ≥ 0.

(89)

By the triangle inequality, for all x j , xk , y j , yk , |ϕ(x j , xk ) − ϕ(y j , yk )| ≤ Λ(|x j − y j | + |xk − yk |). For all n ∈ N, x = (x j ), y = (y j ) ∈ 1+ , n n      β jk ϕ(x j , xk ) − β jk ϕ(y j , yk )  j,k=1

≤

n 

j,k=1

β jk Λ|x j − y j | +

j,k=1

≤Λ

β jk Λ|xk − yk |

j,k=1

n  n  j=1

n 

n  n

 β jk |x j − y j | + Λ β jk |xk − yk |.

k=1

k=1

j=1

We assume that there is some M > 0 such that ∞  k=1

β jk ≤ M,

∞ 

βk j ≤ M,

j ∈ N.

(90)

k=1

Then, for all n ∈ N, x = (x j ), y = (y j ) ∈ 1+ , n n      β ϕ(x , x ) − β jk ϕ(y j , yk ) ≤ 2ΛM x − y .  jk j k j,k=1

j,k=1

Since this holds for all n ∈ N, θ given by ϕ in (88) satisfying the assumptions above is defined on 1+ and is Lipschitz continuous. B is the additive perturbation of a rank one-operator, B(x) = Ax + θ(x)u with A(x) = (A j (x)),

Reproduction Number Versus Turnover Number …

535



A1 (x) = q1 x1 A j (x) = p j−1 x j−1 + q j x j ,

j ≥2

x = (x j ) ∈ 1+ ,

(91)

the sequence u = (u j ) ∈ ˙1+ with u 1 = 1 and u j = 0 for j ≥ 2 and θ(x) =

∞ 

β jk ϕ(x j , xk ),

x = (x j ) ∈ 1 .

(92)

j,k=1

 ∞ Since A(x) = |q1 x1 | + ∞ j=2 | p j−1 x j−1 + q j x j | ≤ j=1 |q j + p j | |x j |, 1 1 A maps  into  and so does B. It is easy to see that A is a linear map. The previous inequality shows that A is a bounded linear map and that the operator norm of A satisfies A ≤ sup j∈N (q j + p j ). So A < 1 if sup(q j + p j ) < 1,

(93)

j∈N

which we assume from now on. This may be a brute-force assumption, but makes biological sense because q j + p j is the probability at which an individual at rank j survives the year. Since A is a linear operator on 1 , Rλ = (λ − A)−1 . To calculate (λ − A)−1 u, we solve the equation λx − Ax = u, that is, the system λx1 − q1 x1 = 1,

λx j − q j x j = p j−1 x j−1 ,

j ≥ 2.

This gives the recursion x1 =

1 , λ − q1

xj =

p j−1 x j−1 , λ − qj

j ≥ 2.

It is solved by (Rλ u) j = x j =

j−1 1  pi , λ − q j i=1 λ − qi

j ∈ N, λ > sup( pk + qk ), k∈N

0 where i=1 := 1. We conclude that the turnover/reproduction trichotomy of Theorem 12 holds with ξ(λ) = θ(Rλ u), ∞ 

R = ξ(1) =

β jk ϕ(Q j , Q k ),

j,k=1

Q1 =

1 , 1 + q1

Qj =

j−1 1  pi , 1 − q j i=1 1 − qi

j ≥ 2.

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These results generalize reproduction/turnover trichotomies for Leslie agestructured models (which are one-sex models) in several ways [11, 34]. Age-structure (all qi = 0) is replaced by rank-structure where yearly promotion to the next higher rank is not obligatory (qi ≥ 0). There is no a priori maximum rank (or age). Mating is accounted for under the assumption of one-to-one sex ratios and same promotion probabilities for the two sexes but possibly sex-specific mating preferences as far as the rank of the mate is concerned.

7 Discussion The (basic) turnover number T = r(B), the spectral radius of the basic turnover operator B, is related to the question as to whether a population modeled by (1) dies out or survives provided that B is a suitable first order approximation of the turnover map F in (1) [25, 48, 49, 51]. If B = A +H , interest in the (basic) reproduction num−n−1 n A , and in turnover/reproduction ber R = r(H R1A ) = r(R1A H ), RλA = ∞ n=0 λ trichotomies stems from the fact that sometimes R can be more easily determined or estimated than T and, after suitable reorganization, is a combination of terms that have a meaningful biological interpretation [3, 4, 8–11, 24, 52]. Unfortunately, this is less the case when H is only homogeneous and order-preserving but not also additive. If H is just homogeneous and order-preserving, (semi)explicit formulas for R can be given if H is a rank-one operator (Sect. 5) or other rather extreme assumptions are made (Examples 5 and 6). As for numerical approaches, though, R may be the spectral radius of an operator on C+b (S) while T is a spectral radius of an operator on C+b (S) × C+b (S) (see the end of Sect. 2.2). As a trade off, the calculation of R involves the resolvent R1A of A, (16) . The latter is not so severe a drawback, if H has finite rank,   θi (x)u i , x ∈ X +, (94) H (x) = i=1

where u i ∈ X˙ + and θi : X + → R+ are continuous homogeneous order-preserving functionals. Since H is compact, R is associated with an eigenvector v ∈ X˙ + of R1 H (Theorem 3),   Rv = θi (v)R1 u i . (95) i=1

So determining R boils down to determining the spectral radius of a homogeneous order-preserving operator on a finite-dimensional subspace, though this may be still difficult enough: Homogeneous operators on finite-dimensional cones can have infinitely many eigenvalues [31, Theorem 5.2.4, Example 5.2.5]. The situation is much easier if the functionals θi are additive on X + . Applying θ j to (95) and setting z j = θ j (v) yields the system

Reproduction Number Versus Turnover Number …

Rz j =

 

θ j (R1 u i )z i ,

537

j = 1, . . . , n,

i=1

and the problem of finding R is reduced to finding the largest positive eigenvalue of a nonnegative matrix. Another complication caused by considering homogeneous order-preserving instead of homogeneous additive operators on a cone is that one may need to also consider the lower reproduction number R = limλ→1+ r(H RλA ) to have a sharp turnover/reproduction trichotomy (Theorem 1, Corollary 1, Definition 1, and Theorem 2). The question whether R = R is a special case of the question under which conditions the spectral radius is a lower-semicontinuous function of homogeneous continuous order-preserving operators. While the example in [32, Sect. 3] for failure of lower semicontinuity is not a counterexample to r(H R1A ) = limλ→1+ r(H RλA ), it is related to why we could not prove this equality except in special cases (Sect. 5 and Examples 5 and 6). Even the case of (94) is not clear to us for non-additive H ; according to [31] the lower-semicontinuity of the spectral radius has been an open problem even in finite-dimensional cones at the time [31] was written (Problem 5.5.1). Acknowledgements The author thanks Odo Diekmann and Roger Nussbaum for helpful comments and two anonymous referees for their constructive remarks. Special thanks goes to Senada Kalabusic for the extraordinary help in adapting the script to the style demands of the proceedings.

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