Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (MPB-30) 9780691188362, 0691016534, 0691016526

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Spatial Ecology

M O N O G R A P H S IN POPULATION

B I O L O G Y

E D I T E D B Y S I M O N A . L E V I N A N D H E N R Y S.

Titles available

in the series (by monograph

1. The Theory

of Island

H O R N

number) by Robert H .

Biogeography,

M a c Arthur and E d w a r d O . W i l s o n 6. Stability

and Complexity

10. Geographic

Variation,

14. Some Adaptations

in Model Speciation,

by Robert M . M a y

Ecosystems, and Clines,

of March-Nesting

by John A . Endler

Blackbirds,

by G o r d o n H . Orians 16. Cultural

Transmission

and Evolution:

A Quantitative

Approach,

by L . L . Cavalli-Sforza and M . W . Feldman 17. Resource

Competition

18. The Theory

and Community

of Sex Allocation,

19. Mate Choice

in Plants:

by D a v i d T i l m a n

Structure,

by E r i c L . Charnov

Tactics, Mechanisms,

and

Consequences,

by M a r y Ε W i l s o n and N a n c y B u r l e y 21. Natural

in the Wild, by John A . Endler

Selection

22. Theoretical

Studies

by Samuel K a r l i n

on Sex Ratio Evolution,

and S a b v i n Lessard 23. A Hierarchical

Concept

of Ecosystems,

by R . V . O ' N e i l l , D . L .

D e A n g e l i s , J. B . Waide, and T . F . H . A l l e n 24. Population

Ecology

25. Population

of the Cooperatively

Breeding

Acorn

by Walter D . K o e n i n g and R o n a l d M . M u m m e

Woodpecker,

Ecology

26. Plant Strategies

of Individuals,

and the Dynamics

by A d a m L o m n i c k i and Structure

of

Plant

with

Data,

by D a v i d T i l m a n

Communities, 28. The Ecological

Detective:

Confronting

Models

by R a y H i l b o r n and M a r c M a n g e l 29. Evolutionary Gallmakers,

Ecology

across

and Natural

Three Trophic

Enemies,

Levels:

Goldenrods,

by Warren G . Abrahamson

and Arthur E . Weis 30. Spatial

Ecology:

Interspecific

The Role of Space

Interactions,

in Population

Dynamics

and

edited by D a v i d T i l m a n and Peter K a r e i v a

Spatial Ecology The Role of Space in Population Dynamics and Interspecific Interactions EDITED

BY

DAVID T I L M A N AND PETER KAREIVA

P R I N C E T O N

U N I V E R S I T Y

P R I N C E T O N ,

N E W

1997

PRESS

J E R S E Y

C o p y r i g h t © 1997 by P r i n c e t o n University Press P u b l i s h e d by P r i n c e t o n University Press, 41 W i l l i a m Street, P r i n c e t o n , N e w Jersey 08540 In the U n i t e d K i n g d o m : P r i n c e t o n University Press, Chichester, West Sussex A l l rights reserved

Library of Congress Cataloging-in-Publication Data Spatial ecology: the role o f space i n p o p u l a t i o n dynamics a n d interspecific i n t e r a c t i o n s / e d i t e d by D a v i d T i l m a n a n d Peter Kareiva. p. c m . — ( M o n o g r a p h s i n p o p u l a t i o n biology ; 30) Includes bibliographical references (p.

) a n d index.

I S B N 0-691-01653-4 (alk. paper). — I S B N 0-691-01652-6 (pbk. : alk. paper) 1. Spatial ecology. I. T i l m a n , David, 1949Peter M . , 1 9 5 1 -

. II. Kareiva,

. III. Series.

QH54.15.S62S62

1997

577.8'2—dc21

97-8460 CIP

T h i s b o o k has b e e n c o m p o s e d i n Baskerville P r i n c e t o n University Press books are p r i n t e d o n acid-free paper a n d meet the guidelines for p e r m a n e n c e a n d durability o f the C o m m i t t e e o n P r o d u c t i o n G u i d e l i n e s for B o o k Longevity o f the C o u n c i l o n Library Resources P r i n t e d i n the U n i t e d States o f A m e r i c a 10

9

8

7

6

5

4

3

10

9

8

7

6

5

4

3

(Pbk.)

http://pup.princeton.edu

2

1

Contents Preface List of Contributors

PARTI SINGLE SPECIES DYNAMICS I N SPATIAL HABITATS 1. P o p u l a t i o n D y n a m i c s i n S p a t i a l H a b i t a t s David

Turnan,

Clarence L. Lehman,

and Peter Kareiva

3

2. P r e d i c t i v e a n d P r a c t i c a l M e t a p o p u l a t i o n M o d e l s : The Incidence Function

Approach 21

Ilkka Hanski 3. V a r i a b i l i t y , P a t c h i n e s s , a n d J u m p D i s p e r s a l i n the Spread of an Invading Population

46

Mark A. Lewis

P A R T II PARASITES, P A T H O G E N S , A N D PREDATORS I N A SPATIALLY C O M P L E X W O R L D 4. T h e D y n a m i c s o f Spatially D i s t r i b u t e d H o s t - P a r a s i t o i d Systems Michael P . Hassell and Howard

75

B. Wilson

5. Basic E p i d e m i o l o g i c a l C o n c e p t s i n a S p a t i a l C o n t e x t 111

Elizabeth Eli Holmes 6. Measles: Persistence a n d S y n c h r o n i c i t y i n Disease D y n a m i c s Neil M. Ferguson,

Robert M. May, and Roy M. Anderson

137

7. G e n e t i c s a n d the Spatial E c o l o g y o f Species I n t e r a c t i o n s : T h e Silene-Ustilago Janis Antonovics,

Peter H. Thrall,

System

and Andrew

M.Jarosz

158

CONTENTS P A R T III C O M P E T I T I O N I N A SPATIAL W O R L D 8. C o m p e t i t i o n i n Spatial H a b i t a t s Clarence L. Lehman

and David

185

Tilman

9. B i o l o g i c a l l y G e n e r a t e d Spatial P a t t e r n a n d

the

C o e x i s t e n c e o f C o m p e t i n g Species Stephen W. Pacala and Simon A. Levin

204

10. H a b i t a t D e s t r u c t i o n a n d Species E x t i n c t i o n s David

Tilman

and Clarence L. Lehman

233

11. L o c a l a n d R e g i o n a l Processes as C o n t r o l s o f Species R i c h n e s s Howard

V. Cornell and Ronald

H. Karlson

250

P A R T IV T H E F I N A L ANALYSIS: D O E S SPACE M A T T E R O R N O T ? A N D H O W W I L L W E T E S T O U R IDEAS? 12. T h e o r i e s o f S i m p l i f i c a t i o n a n d S c a l i n g o f Spatially D i s t r i b u t e d Processes Simon A. Levin

and Stephen W. Pacala

271

13. P r o d u c t i o n F u n c t i o n s f r o m E c o l o g i c a l P o p u l a t i o n s : A Survey w i t h E m p h a s i s o n Spatially Implicit Models Jonathan

296

Roughgarden

14. C h a l l e n g e s a n d O p p o r t u n i t i e s for E m p i r i c a l Evaluation of "Spatial T h e o r y " Eleanor K. Steinberg and Peter Kareiva

318

References

333

Index

365

vi

Preface A l t h o u g h the w o r l d is u n a v o i d a b l y spatial, a n d e a c h o r g a n ­ i s m is a discrete entity that exists a n d interacts o n l y w i t h i n its i m m e d i a t e n e i g h b o r h o o d , these realities l o n g have b e e n i g ­ n o r e d by m o s t ecologists because they c a n greatly c o m p l i c a t e field

r e s e a r c h a n d m o d e l i n g . H o w e v e r , several l i n e s o f i n q u i r y

have h i g h l i g h t e d the p o t e n t i a l l y c r i t i c a l r o l e s o f space a n d l e d to g r o w i n g interest i n spatial e c o l o g y . O n e o f the

earliest

threads i n this tapestry c a m e f r o m G a u s e ' s (1935) studies o f predator-prey dinium

dynamics. Even

though

Paramecium

and

Di-

persist i n i n n u m e r a b l e seeps i n n a t u r e , o n e o r b o t h

species i n v a r i a b l y w e n t e x t i n c t i n the l a b o r a t o r y after a p e r i o d o f unstable o s c i l l a t i o n s . G a u s e suggested that p e r i o d i c r e i n v a ­ s i o n by o n e

o r b o t h species, s u c h as m i g h t h a p p e n

in a

s u b d i v i d e d n a t u r a l habitat, was necessary to a l l o w c o e x i s t e n c e . H u f f a k e r (1958) e x t e n d e d this i n a study o f a p h y t o p h a g o u s m i t e a n d its p r e d a t o r .

H e also f o u n d that persistence

was

i m p o s s i b l e i n s m a l l h o m o g e n e o u s habitats b u t was p r o l o n g e d i n m o r e c o m p l e x habitats i n w h i c h t h e r e were b a r r i e r s that e s t a b l i s h e d d i s t i n c t patches l i n k e d by slow d i s p e r s a l . S p a t i a l c o m p l e x i t y was essential f o r the p a r t i a l s t a b i l i z a t i o n a n d persis­ tence o f this p r e d a t o r a n d p r e y i n t e r a c t i o n . F e w p a p e r s have p i q u e d s u c h interest i n spatial e c o l o g y as has H u f f a k e r ' s classic study. M a c A r t h u r a n d W i l s o n ' s (1967) t h e o r y o f i s l a n d b i o g e o g r a p h y a d d e d m o r e threads to the spatial tapestry a n d e s t a b l i s h e d the a b i d i n g interest o f c o n s e r v a t i o n b i o l o g i s t s i n spatial p r o ­ cesses. M a c A r t h u r a n d W i l s o n n o t e d that h u m a n e x p a n s i o n results i n the f r a g m e n t a t i o n o f o n c e c o n t i n u o u s ecosystems, r e s u l t i n g i n species e x t i n c t i o n s . H o w e v e r , the n u m b e r o f species l i k e l y to be t h r e a t e n e d by a g i v e n a m o u n t o f h a b i t a t f r a g m e n t a ­ t i o n s h o u l d d e p e n d o n the spatial p a t t e r n i n g o f the f r a g m e n t a ­ t i o n , i n c l u d i n g s u c h aspects as the sizes o f r e m n a n t

patches,

t h e i r v i c i n i t y to e a c h o t h e r , a n d the existence a n d q u a l i t i e s o f c o r r i d o r s l i n k i n g t h e m . A l t h o u g h n o n s p a t i a l m o d e l s m a y give vii

PREFACE

i n s i g h t s i n t o s u c h issues, e a c h r e a l w o r l d s i t u a t i o n is u n a v o i d ­ ably e x p l i c i t l y spatial, a n d l a n d m a n a g e m e n t a i m e d at h a b i t a t r e s t o r a t i o n o r h a b i t a t p r e s e r v a t i o n m u s t be g r o u n d e d i n space. C o n c e r n s a b o u t i n v a s i o n s b y n o v e l o r g a n i s m s (e.g., 1958;

Elton

M o o n e y a n d D r a k e 1986; D r a k e et a l . 1989; S k e l l a m

1951) a n d i n t e r e s t i n the s p r e a d o f n o v e l g e n o t y p e s also d r e w a t t e n t i o n to s u c h aspects o f space as the r o l e o f g e o g r a p h i c i s o l a t i o n , d i s p e r s a l d y n a m i c s , a n d effective p o p u l a t i o n sizes. Unprecedentedly rapid movements o f m o d e r n humans a m o n g b i o g e o g r a p h i c r e a l m s has p r o d u c e d a crisis o f e x o t i c species invasions with dramatic ecological a n d e c o n o m i c consequences (OTA

1993). M a n y o f these n o v e l o r g a n i s m s have s p r e a d i n a

m a n n e r that E l t o n (1958) c a l l e d " e c o l o g i c a l e x p l o s i o n s . " T h e rate a n d the spatial p a t t e r n i n g o f these i n v a s i o n s has c a p t u r e d the interest o f b o t h r e s o u r c e m a n a g e r s a n d those i n t e r e s t e d i n e c o l o g i c a l patterns. C o n c e p t u a l issues, s u c h as t h e p a r a d o x o f diversity, are also w o v e n i n t o the tapestry o f s p a t i a l e c o l o g y . H u t c h i n s o n (1961) n o t e d that the o p e n , s e e m i n g l y w e l l - m i x e d waters o f lakes a n d o c e a n s m a y c o n t a i n a h u n d r e d o r m o r e species o f p h y t o p l a n k t o n i c algae, a l l o f w h i c h p r e s u m a b l y c o m p e t e w i t h e a c h o t h e r for the same few l i m i t i n g n u t r i e n t s . A v a i l a b l e theory, h o w e v e r , p r e d i c t e d t h a t the n u m b e r o f c o e x i s t i n g species s h o u l d n o t e x c e e d the n u m b e r o f l i m i t i n g resources. T h e same p a r a d o x a p p l i e d to p r a i r i e s , t r o p i c a l r a i n forests, a n d o t h e r terrestrial p l a n t c o m m u n i t i e s i n w h i c h h u n d r e d s o f species c o e x i s t w h i l e c o m p e t i n g f o r o n l y a h a n d f u l o f l i m i t i n g factors. Space s e e m e d to have the p o t e n t i a l o f p r o v i d i n g a s o l u t i o n to this p a r a d o x o f diversity. S u c h spatial s o l u t i o n s have b e e n p r o p o s e d b y L e v i n a n d P a i n e (1974); H o r n a n d M a c A r t h u r (1972); P i a t t a n d W e i s (1977); T i l m a n (1994); a n d others. F i n a l l y , m a t h e m a t i c a l ecologists f o u n d that spatial m o d e l s o f t e n p r e d i c t e d u n e x p e c t e d a n d i n t r i g u i n g spatial p a t t e r n s a n d dynamics. E v e n i n a m o d e l o f a physically h o m o g e n e o u s h a b i ­ tat w i t h e x t r e m e l y s i m p l e r u l e s o f o r g a n i s m a l i n t e r a c t i o n s a n d m o v e m e n t , spatial p a t t e r n s u n a v o i d a b l y d e v e l o p e d . A l t h o u g h the g e n e r a l e x p e c t a t i o n m i g h t be that i n d i v i d u a l s w o u l d b e ­ c o m e u n i f o r m l y s p r e a d across s u c h a h o m o g e n e o u s

habitat,

s i m p l e m o d e l s i n fact r e v e a l a s u r p r i s i n g , r o b u s t , a n d g e n e r a l viii

PREFACE

p r e d i c t i o n — d u m p i n g . A s i n g l e species, g r o w i n g i n a densitydependent manner,

for instance, occupies a

homogeneous

h a b i t a t i n c l u m p s a n d has o t h e r areas i n w h i c h i t d o e s n o t o c c u r . S u c h p a t t e r n s w e r e suggested b y T u r i n g (1952) as a p o s s i b l e e x p l a n a t i o n o f the c o m p a r t m e n t a l i z a t i o n o f a n i m a l s that arises d u r i n g e m b r y o g e n e s i s . M o r e c o m p l e x e x t e n s i o n s o f this a p p r o a c h have r e v e a l e d a t r e m e n d o u s variety o f static a n d d y n a m i c p a t t e r n s r e s u l t i n g f r o m a variety o f l o c a l i n t e r a c t i o n s c o u p l e d with dispersal i n a u n i f o r m environment. T h i s b o o k is d e s i g n e d to h i g h l i g h t the i m p o r t a n c e o f space to a l l five o f the a f o r e m e n t i o n e d topics: stability, p a t t e r n s o f diversity, i n v a s i o n s , c o e x i s t e n c e , a n d p a t t e r n g e n e r a t i o n . W e a i m to i l l u s t r a t e b o t h t h e diversity o f a p p r o a c h e s u s e d to study spatial e c o l o g y a n d the u n d e r l y i n g s i m i l a r i t i e s o f these

ap­

p r o a c h e s . T h e b o o k is n o t i n t e n d e d to b e a t h o r o u g h r e v i e w o f these issues, n o r is it a c o o k b o o k d e s i g n e d to a l l o w the r e a d e r to a c q u i r e the necessary skills f o r p u r s u i n g spatial e c o l o g y . M a n y i m p o r t a n t processes a n d a p p r o a c h e s , e s p e c i a l l y d e t a i l e d a n d e x p l i c i t studies at the l a n d s c a p e l e v e l , are n o t i n c l u d e d . R a t h e r , this b o o k offers a s m o r g a s b o r d o f c o n c e p t s a n d

ap­

p r o a c h e s r e l a t e d to spatial e c o l o g y that, we h o p e , whets

the

appetite

o f the

next

generation

of ecological

researchers

a n d entices t h e m to i n c o r p o r a t e spatial processes i n t o t h e i r thinking. I n a d d i t i o n , this b o o k c o n t a i n s two m a j o r c r o s s c u r r e n t s . T h e strongest o n e c o n c e r n s the trade-off b e t w e e n s i m p l i c i t y a n d r e a l i s m i n d e a l i n g w i t h space. W i t h i n m a n y c h a p t e r s , a n d i n c o m p a r i s o n a m o n g c h a p t e r s , space is d e a l t w i t h o n m a n y levels o f reality. E a c h l e v e l offers i n s i g h t s b u t i m p o s e s r e s t r i c t i o n s o n w h a t c a n b e u n d e r s t o o d . L e v i n a n d P a c a l a ( C h a p t e r 12) a n d S t e i n b e r g a n d K a r e i v a ( C h a p t e r 14) revisit these issues, a n d synthesize t h e m , at the e n d o f the b o o k . T h e r e a l w o r l d has o r g a n i s m s that have specific x-, y-, a n d z - c o o r d i n a t e s , f e c u n d i ­ ties, a n d p r o b a b i l i t i e s o f m o r t a l i t y that m a y c h a n g e w i t h t i m e i n ways that d e p e n d o n t h e i r genotypes, the g e n o t y p e s o f o t h e r o r g a n i s m s i n t h e i r n e i g h b o r h o o d s , the c o o r d i n a t e s o f a l l o t h e r o r g a n i s m s i n t h e i r n e i g h b o r h o o d s , a n d the p o t e n t i a l l y c h a n g ­ i n g p h y s i c a l attributes

o f the

c o m p l e x i t i e s , we

compartmentalized nature (and

have

habitat. T o u n d e r s t a n d

ix

these our

PREFACE

m i n d s ) i n t o the a r t i f i c i a l s u b d i s c i p l i n e s o f e v o l u t i o n a r y , p o p ­ ulation, community, and

ecosystem

e c o l o g y . W e also

focus

o n p a r t i c u l a r i n t e r a c t i o n s s u c h as i n t e r s p e c i f i c c o m p e t i t i o n , predator-prey interactions, a n d host-parasitoid or

host-disease

i n t e r a c t i o n s . S i m i l a r l y , i n s t u d y i n g the spatial aspects o f these s u b d i s i p l i n e s o r b i o t i c i n t e r a c t i o n s , we have u s e d v a r i o u s levels o f r e a l i s m i n the d e s c r i p t i o n o f space. S o m e a p p r o a c h e s , s u c h as L e v i n s - l i k e m e t a p o p u l a t i o n m o d e l s , treat space o n l y i m p l i c ­ itly by c o n s i d e r i n g the p r o p o r t i o n o f p o t e n t i a l sites o c c u p i e d by i n d i v i d u a l s o r p o p u l a t i o n s . O t h e r s ,

s u c h as c e l l u l a r a u ­

t o m a t a m o d e l s , assume that o r g a n i s m s d i v i d e a h a b i t a t i n t o a series o f equal-sized, i d e n t i c a l p a t c h e s that o c c u r i n e i t h e r a square o r h e x a g o n a l g r i d . S t i l l others, s u c h as the a n a l y t i c a l r e a c t i o n - d i f f u s i o n m o d e l s , assume that space is a

continuous

v a r i a b l e , a n d t h a t o r g a n i s m s " d i f f u s e " t h r o u g h space. F i n a l l y , others, s u c h as spatial s i m u l a t o r s , i n c l u d e e x p l i c i t spatial c o o r ­ d i n a t e s f o r e a c h i n d i v i d u a l a n d f o r its r e g i o n o f i n f l u e n c e . A s e c o n d c r o s s c u r r e n t also deals w i t h the s i m p l i c i t y - r e a l i s m trade-off, b u t w i t h r e g a r d to the n e e d to i n c l u d e e v o l u t i o n a r y change i n models o f interspecific interactions. B o t h Antonovics et a l . ( C h a p t e r 7) a n d L e h m a n a n d T i l m a n ( C h a p t e r 8) s h o w t h a t the a d d i t i o n o f e v e n s i m p l e g e n e t i c m e c h a n i s m s c a n l e a d to m a j o r qualitative c h a n g e s i n the p r e d i c t i o n s o f spatial m o d ­ els. E v o l u t i o n a r y m o d e l s have l o n g i n c l u d e d spatial processes, b u t few attempts have

been

made

to synthesize e v o l u t i o n ,

i n t e r s p e c i f i c i n t e r a c t i o n s , a n d space. T h e s e c h a p t e r s suggest that s u c h syntheses m a y offer m a n y insights. A l t h o u g h it is c l e a r that space has b e e n greatly o v e r l o o k e d i n e c o l o g y , it is n o t e v i d e n t h o w space s h o u l d be into theory or

fieldwork.

incorporated

Is it necessary to c o n s i d e r the e x p l i c i t

spatial l o c a t i o n o f a l l i n d i v i d u a l s ? O r are c e l l u l a r a u t o m a t a - l i k e models

reasonable

approximations?

Or

might

metapopula-

t i o n - l i k e p s e u d o s p a t i a l m o d e l s suffice? A r e t h e r e a d d e d i n ­ sights, a n d w h a t are these insights, that c o m e f r o m i n c r e a s i n g l y m o r e realistic c o n s i d e r a t i o n s o f space? S i m p l e m o d e l s , s u c h as metapopulation

a n a l y t i c a l l y tractable

be­

cause o f t h e i r a s s u m p t i o n o f g l o b a l d i s p e r s a l . I n contrast,

m o d e l s , are

often

the

m o r e realistic r e a c t i o n - d i f f u s i o n a n d o t h e r p a r t i a l d i f f e r e n t i a l e q u a t i o n m o d e l s are tractable o n l y f o r cases that are often too χ

PREFACE

s i m p l e to b e o f g r e a t e c o l o g i c a l interest. C e l l u l a r models, and

more

spatially c o m p l e x c o m p u t e r

automata

simulations,

offer m u c h g r e a t e r e c o l o g i c a l reality b u t suffer f r o m the dif­ ficulty

of generalization.

I n total, this b o o k d e m o n s t r a t e s that spatial perspectives c a n provide significant insights into n u m e r o u s ecological questions i n c l u d i n g the d y n a m i c s a n d c o n s e r v a t i o n o f a single species; the

dynamics

and

outcome

of

competitive,

predator-prey,

host-parasite, a n d host-disease i n t e r a c t i o n s ; a n d the

mainte­

n a n c e o f b i o d i v e r s i t y i n f r a g m e n t e d landscapes. It is o u r h o p e that s u c h i n s i g h t s w i l l i n f o r m a n d i n s p i r e f u t u r e

ecological

research. T h i s b o o k w o u l d n o t have b e e n p o s s i b l e w i t h o u t the and

encouragement of Bill

Murdoch and

o f the

support National

C e n t e r f o r E c o l o g i c a l A n a l y s i s a n d Synthesis i n S a n t a B a r b a r a , w h o m we gratefully a c k n o w l e d g e . O u r N C E A S w o r k s h o p o n spatial e c o l o g y was d e e p l y s t i m u l a t i n g . I n d e e d , o u r discussions o f the m a t e r i a l that b e c a m e c h a p t e r s i n this b o o k

reminded

m a n y o f us o f a d o c t o r a l o r a l e x a m i n a t i o n . W e t h a n k M a r i l y n S n o w b a l l , S h a r i S t a u f e n b e r g , a n d the rest o f the N C E A S staff w h o m a d e o u r g a t h e r i n g p o s s i b l e a n d enjoyable. F i n a l l y , we t h a n k N a n c y L a r s o n f o r h e r i m m e n s e assistance i n p r e p a r i n g this m a n u s c r i p t . H e r h e l p was p a r t i a l l y s u p p o r t e d by the N a ­ t i o n a l S c i e n c e F o u n d a t i o n a n d the A n d r e w M e l l o n

Founda­

tion. We

d e d i c a t e this b o o k to o u r teachers a n d m e n t o r s .

F a r n u m , a h i g h school biology teacher i n B e n t o n

Don

Harbor,

M i c h i g a n , s h a r e d his love o f b i o l o g y w i t h D a v i d T i l m a n , a n d , at the U n i v e r s i t y o f M i c h i g a n , Steve H u b b e l l a n d the late P e t e r K i l h a m p r o v i d e d i n s p i r a t i o n a n d n u r t u r e to a n e o p h y t e e c o l o gist. J a n i s A n t o n o v i c s s h o w e d

Peter

Kareiva

the

b i o l o g y over p o l i t i c a l science, R i c h a r d M a c M i l l e n the v a l u e o f d o i n g and

Simon Levin

fieldwork and

virtues

of

emphasized

i n s t e a d o f j u s t t a l k i n g a b o u t it,

R i c h a r d R o o t were

perfect " c o u p l e " as a P h . D . advisory t e a m .

xi

an

unlikely

but

Contributors Roy M. Anderson

is f r o m the D e p a r t m e n t o f Z o o l o g y at O x f o r d

U n i v e r s i t y , C e n t r e for the E p i d e m i o l o g y o f I n f e c t i o u s D i s ­ ease, S o u t h P a r k s R o a d , O x f o r d , O X 1 3 P S , U n i t e d K i n g d o m . Janis

Antonovics

is f r o m

the D e p a r t m e n t o f B o t a n y at

Duke

U n i v e r s i t y , D u r h a m , N o r t h C a r o l i n a 27708. Howard

is f r o m the D e p a r t m e n t o f B i o l o g i c a l S c i ­

V. Cornell

ences at the U n i v e r s i t y o f D e l a w a r e , 117 W o l f H a l l , N e w a r k , D e l a w a r e 19716. Neil M. Ferguson

is f r o m the D e p a r t m e n t o f Z o o l o g y at O x f o r d

U n i v e r s i t y , C e n t r e for the E p i d e m i o l o g y o f I n f e c t i o u s D i s ­ ease, S o u t h P a r k s R o a d , O x f o r d , O X 1 3 P S , U n i t e d K i n g d o m . Ilkka Hanski ics,

is f r o m the D e p a r t m e n t o f E c o l o g y a n d Systemat-

Division

of

Population

Biology,

P.O.

Box

17

( A r k a d i a n k a t u 7), F I N - 0 0 0 1 4 , at the U n i v e r s i t y o f H e l s i n k i , i n Finland. Michael

P. Hassell

is f r o m the D e p a r t m e n t o f B i o l o g y at I m p e ­

rial College, S i l w o o d Park, Ascot, Berks, S L 5 7PY, U n i t e d Kingdom. Elizabeth

Eli

Holmes

is f r o m

the

D e p a r t m e n t o f B i o l o g y at

C o l o r a d o State U n i v e r s i t y , F o r t C o l l i n s , C o l o r a d o 80523. Andrew

M. Jarosz

is f r o m the D e p a r t m e n t o f B o t a n y a n d

Plant

P a t h o l o g y at M i c h i g a n State U n i v e r s i t y , East L a n s i n g , M i c h i ­ g a n 48824. Peter Kareiva

is f r o m the D e p a r t m e n t o f Z o o l o g y at the U n i v e r ­

sity o f W a s h i n g t o n ,

NJ-15, B o x 351800, 24 K i n c a i d H a l l ,

Seattle, W a s h i n g t o n 98195-1800. Ronald

H. Karlson

is f r o m the D e p a r t m e n t o f B i o l o g i c a l S c i ­

ences at the U n i v e r s i t y o f D e l a w a r e , 117 W o l f H a l l , N e w a r k , D e l a w a r e 19716. xiii

C O N T R I B U T O R S

is f r o m the D e p a r t m e n t o f E c o l o g y , E v o l u ­

Clarence L. Lehman

t i o n , a n d B e h a v i o r at the U n i v e r s i t y o f M i n n e s o t a , 100 E c o l ­ ogy B u i l d i n g , 1987 U p p e r B u f o r d C i r c l e , St. P a u l , M i n n e s o t a 55108-6097. Simon A . Levin tionary

is f r o m the D e p a r t m e n t o f E c o l o g y a n d E v o l u ­

B i o l o g y at

Princeton

University,

203

Eno

Hall,

P r i n c e t o n , N e w J e r s e y 08544-1003. Mark

A. Lewis

is f r o m the D e p a r t m e n t o f M a t h e m a t i c s at

the

U n i v e r s i t y o f U t a h , J W B 233, Salt L a k e City, U t a h 84112. is f r o m the D e p a r t m e n t o f Z o o l o g y at O x f o r d

Robert M. May

U n i v e r s i t y , C e n t r e for the E p i d e m i o l o g y o f Infectious D i s ­ ease, S o u t h P a r k s R o a d , O x f o r d , O X 1 3PS, U n i t e d K i n g d o m . Stephen

is f r o m

W. Pacala

the

Department of Ecology

and

E v o l u t i o n a r y B i o l o g y at P r i n c e t o n U n i v e r s i t y , 101 E n o H a l l , P r i n c e t o n , N e w Jersey 08544-1003. Jonathan

Roughgarden

Sciences a n d

the

is f r o m

the

Department of Biological

Department o f Geophysics

at

Stanford

U n i v e r s i t y , S t a n f o r d , C a l i f o r n i a 94305. Eleanor

K. Steinberg

is f r o m the D e p a r t m e n t o f Z o o l o g y at

University of Washington,

the

N J - 1 5 , B o x 351800, 24 K i n c a i d

H a l l , Seattle, W a s h i n g t o n 98195-1800. Peter H.

is f r o m

Thrall

the

D e p a r t m e n t o f B o t a n y at

Duke

U n i v e r s i t y , D u r h a m , N o r t h C a r o l i n a 27708. David

Tilman

is f r o m the D e p a r t m e n t o f E c o l o g y , E v o l u t i o n ,

a n d B e h a v i o r at the U n i v e r s i t y o f M i n n e s o t a , 100 E c o l o g y B u i l d i n g , 1987 U p p e r

B u f o r d C i r c l e , St. P a u l ,

Minnesota

55108-6097. Howard

B. Wilson

is f r o m the D e p a r t m e n t o f B i o l o g y at I m p e ­

r i a l C o l l e g e , S i l w o o d P a r k , A s c o t , B e r k s , S L 5 7PY, U n i t e d Kingdom.

xiv

C H A P T E R

O N E

Population Dynamics in Spatial Habitats David Tilman, Clarence L. Lehman, and Peter Kareiva

A l l o r g a n i s m s are discrete entities that m a i n l y i n t e r a c t w i t h n e i g h b o r i n g i n d i v i d u a l s o f t h e i r o w n o r o t h e r species. T h i s discrete n a t u r e a n d spatial c o n f i n e m e n t is m o s t e v i d e n t f o r sessile o r g a n i s m s s u c h as terrestrial plants, m a r i n e m a c r o phytes, corals, a n d o t h e r o r g a n i s m s that live a t t a c h e d to sur­ faces. H o w e v e r , e v e n m o t i l e o r g a n i s m s have impacts i n a rather

confined region—the

their

greatest

region

through

w h i c h they m o v e . T h e s e s i m p l e observations have p r o f o u n d i m p l i c a t i o n s f o r the

dynamics a n d outcome o f both intra-

specific a n d i n t e r s p e c i f i c i n t e r a c t i o n s . I n p a r t i c u l a r , l o c a l i n t e r ­ a c t i o n s a n d l o c a l m o v e m e n t / d i s p e r s a l m e a n that p o p u l a t i o n densities d o n o t c h a n g e

i n response

to average

conditions

across a large habitat, as is a s s u m e d i n classical n o n s p a t i a l m o d e l s , b u t r a t h e r i n response to the l o c a l c o n d i t i o n s e x p e r i ­ e n c e d by e a c h i n d i v i d u a l . T h i s c h a p t e r presents t h r e e s i m p l e a p p r o a c h e s f o r d e a l i n g w i t h this spatial aspect o f species i n t e r ­ actions a n d p o p u l a t i o n b i o l o g y . T h e t h r e e a p p r o a c h e s that we discuss

are

Levins's

metapopulation-like

model,

cellular-

automaton-like models, a n d reaction-diffusion models.

The

first a n d last a p p r o a c h e s c a n be analytically tractable, whereas the m i d d l e o n e is best a p p r o a c h e d v i a c o m p u t e r s i m u l a t i o n . T h e first two c o n s i d e r i n t e r a c t i o n s i n a spatially s u b d i v i d e d habitat, whereas the last c o n s i d e r s i n t e r a c t i o n s i n a spatially c o n t i n u o u s habitat. T h e s e a p p r o a c h e s are the b u i l d i n g b l o c k s u p o n w h i c h m u c h o f this b o o k is based. O u r g o a l is to abstract the essential features

o f space that c o m e f r o m 3

the

discrete

TILMAN,

L E H M A N ,

A N D

KAREIVA

nature o f individual organisms. T h e r e

are

many additional

c o m p l e x i t i e s to space that are n o t c o n s i d e r e d i n this c h a p t e r b u t are d i s c u s s e d i n s u b s e q u e n t

chapters.

L E V I N S ' S M O D E L F O R A S I N G L E SPECIES L e v i n s ' s (1969) m o d e l p r o v i d e s a s i m p l e d e s c r i p t i o n o f the d y n a m i c s o f a single species l i v i n g i n a h a b i t a t c o m p o s e d o f d i s t i n c t sites. A l t h o u g h i t has g i v e n m a n y i n s i g h t s i n t o c o m m u ­ nities that are s u b d i v i d e d i n t o l o c a l p o p u l a t i o n s , its use as a l i t e r a l m e t a p o p u l a t i o n m o d e l has b e e n c r i t i c i z e d (e.g., H a n s k i 1991,

chap.

2;

Harrison, Thomas,

and

Lewinsohn

1995;

H a r r i s o n a n d T a y l o r 1997). F o r i n s t a n c e , it is v i e w e d as a p o o r d e s c r i p t o r o f m e t a p o p u l a t i o n s because it assumes that a single p r o p a g u l e c a n i n s t a n t l y t r a n s f o r m a n e m p t y site i n t o a l o c a l p o p u l a t i o n that is at its c a r r y i n g capacity, because l o c a l sites c a n n o t differ i n t h e i r c a r r y i n g capacities o r o t h e r m e a s u r e s o f t h e i r quality, a n d because t h e r e is n o possibility o f l o c a l c o n d i ­ t i o n s " r e s c u i n g " a l o c a l site f r o m e x t i n c t i o n . An

alternative use o f L e v i n s ' s f r a m e w o r k is to view it as a

m o d e l f o r the o c c u p a n c y o f sites b y s i n g l e i n d i v i d u a l s as o p ­ p o s e d to p o p u l a t i o n s . T o u n d e r s t a n d

this i n t e r p r e t a t i o n o f

L e v i n s ' s m o d e l , c o n s i d e r a h a b i t a t that is d i v i d e d i n t o sites that are j u s t large e n o u g h to c o n t a i n a single a d u l t i n d i v i d u a l o f a sessile species. A p r o p a g u l e o f this species, u p o n e n t e r i n g a n e m p t y site, w o u l d o c c u p y it. I n d i v i d u a l s o c c u p y i n g sites c o u l d p r o d u c e p r o p a g u l e s that w o u l d be d i s p e r s e d t h r o u g h o u t

the

habitat. T h e rate o f p r o p a g u l e p r o d u c t i o n by a site is c (for c o l o n i z a t i o n rate). I n a d d i t i o n , t h e r e is s o m e m o r t a l i t y rate, m, that is the c h a n c e o f a c u r r e n t l y o c c u p i e d site b e c o m i n g e m p t y . T o k e e p track o f the d y n a m i c s o f this species, it is necessary to k n o w p, the p r o p o r t i o n o f a l l p o s s i b l e sites that are o c c u p i e d at a g i v e n i n s t a n t i n t i m e . I f it is a s s u m e d that p r o p a g u l e s disperse r a n d o m l y a m o n g a l l p o s s i b l e sites, these a s s u m p t i o n s c a n be converted into a simple model:

dp — dt

= cp(l

- p) 4

- mp.

(1.1)

P O P U L A T I O N

DYNAMICS

T h i s states t h a t the rate o f c h a n g e i n site o c c u p a n c y d e p e n d s o n the rate o f p r o p a g u l e p r o d u c t i o n (cp)

(dp/dt)

multiplied

by the p r o p o r t i o n o f c u r r e n t l y o p e n sites (1 — p). T h e rate o f m o r t a l i t y (mp) is s u b t r a c t e d f r o m this to get the n e t c h a n g e i n site o c c u p a n c y . T h i s s i m p l e m o d e l o f site o c c u p a n c y has several i n t e r e s t i n g features.

First, a species c a n persist i n a h a b i t a t i f c > m.

P o p u l a t i o n s g r o w i n a logistic f a s h i o n . W h e n c > m, t h e p r o ­ p o r t i o n o f o c c u p i e d sites w i l l a p p r o a c h a n e q u i l i b r i u m , p (at w h i c h dp /dt

= 0), w i t h m p = l - - . c

(1.2)

T h i s e q u i l i b r i a l d e n s i t y is g l o b a l l y stable ( H a s t i n g s 1980), m e a n ­ i n g that p w i l l a p p r o a c h p f r o m a n y s t a r t i n g density, a n d a n y p e r t u r b a t i o n , as l o n g as p > 0. T h e m o s t i m p o r t a n t a n d i n t e r ­ e s t i n g aspect o f this m o d e l is that n o species is c a p a b l e o f completely

filling

its h a b i t a t at e q u i l i b r i u m . T h e p r o p o r t i o n

o f a l l v i a b l e sites that are left u n o c c u p i e d at e q u i l i b r i u m i n a h a b i t a t o c c u p i e d b y a s i n g l e species is 5, w h e r e m (1.3)

s = 1 - p = c

E q u a t i o n 1.3 shows that a n u n a v o i d a b l e r e s u l t o f l i v i n g i n a spatial h a b i t a t is that a p r o p o r t i o n o f sites w i l l b e e m p t y . T h e g r e a t e r the m o r t a l i t y rate o f a species relative to its c o l o n i z a ­ t i o n rate, the greater w o u l d b e the a m o u n t o f o p e n space. T h i s m o d e l is m a t h e m a t i c a l l y s i m p l e a n d analytically tractable because

o f the

s i m p l i f y i n g a s s u m p t i o n that it m a k e s

about

d i s p e r s a l . B y a s s u m i n g that a l l p r o p a g u l e s are r a n d o m l y dis­ p e r s e d across the

e n t i r e habitat,

this m o d e l e l i m i n a t e s the

effects o f l o c a l d i s p e r s a l . H o w e v e r , by a s s u m i n g t h a t the h a b i t a t is s u b d i v i d e d i n t o sites the size o f a n a d u l t , the m o d e l i m p l i c i t l y addresses the discreteness o f i n d i v i d u a l s a n d the i d e a o f oc­ c u p y i n g space. E v e n t h o u g h space is n o t t r e a t e d e x p l i c i t l y , this m o d e l a n d its derivatives have b e e n r e m a r k a b l y versatile i n u n c o v e r i n g c e r t a i n effects o f space b o t h o n s i n g l e species a n d 5

TILMAN,

on

L E H M A N ,

A N D

KAREIVA

m u l t i s p e c i e s i n t e r a c t i o n s (e.g., S k e l l a m 1951; L e v i n s a n d

Culver

1971; H o r n

and

MacArthur

1972; A r m s t r o n g 1976;

H a s t i n g s 1980; S h m i d a a n d E l m e r 1984; B e n g t s s o n 1989, 1991; T i l m a n 1994; C h a p t e r s 8 a n d 10).

C E L L U L A R A U T O M A T A A N D INDIVIDUAL-BASED MODELS T h e c r u c i a l s i m p l i f y i n g a s s u m p t i o n above was that o f g l o b a l , r a n d o m d i s p e r s a l o f p r o p a g u l e s . T o m o d e l l o c a l d i s p e r s a l we still e n v i s i o n a p h y s i c a l l y h o m o g e n e o u s h a b i t a t that is s u b d i ­ v i d e d i n t o sites, e a c h o f the size j u s t c a p a b l e o f s u p p o r t i n g a s i n g l e a d u l t . E a c h i n d i v i d u a l has a p r o b a b i l i t y , m, o f m o r t a l i t y p e r u n i t t i m e . E a c h p r o d u c e s p r o p a g u l e s at a rate o f c. H o w ­ ever, the

propagules

p r o d u c e d by e a c h

individual

disperse

l o c a l l y . L o c a l d i s p e r s a l c a n b e as s i m p l e as e q u i p r o b a b l e m o v e ­ m e n t to a l l adjacent

cells ( f o u r adjacent

cells f o r a

habitat

d i v i d e d i n t o a s q u a r e g r i d , o r six f o r a h e x a g o n a l g r i d ) o r a more c o m p l e x pattern

o f d i s p e r s a l to a g r e a t e r n u m b e r

of

n e a r b y sites. A m o d e l i n w h i c h p r o p a g u l e s are r a n d o m l y d i s ­ p e r s e d to adjacent sites is c a l l e d a stochastic c e l l u l a r a u t o m a t o n m o d e l . I n the

l i m i t , as the

number

o f sites that f o r m

the

n e i g h b o r h o o d o f a n i n d i v i d u a l is i n c r e a s e d , the b e h a v i o r o f this m o d e l a p p r o a c h e s that o f L e v i n s ' s m o d e l ( E q . 1.1). In

our

simulations, individual

organisms

are

distributed

across a t w o - d i m e n s i o n a l array o f h e x a g o n a l sites ( F i g u r e 1.1). A t any i n s t a n t o f t i m e ( o f l e n g t h dt), e a c h o c c u p i e d site sends p r o p a g u l e s to a set o f n e i g h b o r i n g sites w i t h a p r o b a b i l i t y o f cdt.

I f a p r o p a g u l e l a n d s o n a n o c c u p i e d site, the p r o p a g u l e is

lost. I f it l a n d s o n a n e m p t y site, the site is c o l o n i z e d a n d is n o w fully o c c u p i e d . T h i s process is r e p e a t e d every t i m e step, dt, to d e t e r m i n e the d y n a m i c s o f g r o w t h o f the species. B e c a u s e this is a s i m u l a t i o n m o d e l , t h e r e is n o c l o s e d - f o r m m a t h e m a t i c a l s o l u t i o n f o r the g e n e r a l case. H o w e v e r , the m o d e l shares m a n y features w i t h L e v i n s ' s m o d e l a n d has the a d v a n ­ tage o f b e i n g easily m o d i f i e d . T h i s allows its u n d e r l y i n g as­ s u m p t i o n s to be m a n i p u l a t e d i n a n e x p l o r a t i o n o f the i m p o r ­ tance

of

these

assumptions

to

the

dynamics, e q u i l i b r i u m

densities, a n d spatial p a t t e r n s p r e d i c t e d by the m o d e l . B e l o w 6

P O P U L A T I O N

F I G U R E 1.1.

D Y N A M I C S

T h e hexagonal grid o f sites used i n o u r simulations o f a

spatially explicit habitat (see

also Chapters 8 a n d

10).

The

shading

illustrates the various rings o f neighbors to w h i c h the propagules p r o ­ d u c e d by the b l a c k e n e d site may be dispersed. T h e lightest shading is the fourth ring. E a c h hexagonal cell is the size o f the area o c c u p i e d by a single adult individual of this species.

we h i g h l i g h t s o m e o f the u n i q u e features that arise w h e n L e v i n s ' s m o d e l is m o d i f i e d i n t o a spatial s i m u l a t i o n that i n ­ cludes n e i g h b o r h o o d dispersal. A n u n a v o i d a b l e o u t c o m e o f l o c a l i n t e r a c t i o n s (for a single species the

i n t e r a c t i o n is that a n o c c u p i e d site c a n n o t

be

i n v a d e d by a p r o p a g u l e ) a n d l o c a l d i s p e r s a l is c l u m p i n g (see D u r r e t a n d L e v i n 1994a,b f o r a t h o r o u g h analysis). T h i s is easily s h o w n i n c e l l u l a r a u t o m a t o n m o d e l s ( F i g u r e 1.2). E a c h o f the six cases we illustrate is f o r a p h y s i c a l l y u n i f o r m h a b i t a t containing 10

4

sites (i.e., a 100 X 100 habitat). I n e a c h case the

species b e c o m e s c l u m p e d , b u t the e x t e n t o f this c l u m p i n g , the 7

Α.

Β.

* fr' ν

α.

m

-

* Dispersai Range = 1

c.• Γ

D. í

) are g i v e n by the

Nicholson-Bailey

t e r m ( E q . 4.2a), a n d the p r i m e s i n d i c a t e the densities

after

dispersal. W e first c o n s i d e r i f a h y p e r p a r a s i t o i d c a n i n v a d e a hostparasitoid

community

and

then 104

whether

the

resultant

H O S T - P A RA S I T O I D

S Y S T E M S

c o m m u n i t y is persistent o r n o t . W e k n o w f r o m the

five-species

i n t e r a c t i o n s i n the p r e v i o u s s e c t i o n that a c o m m u n i t y c a n be i n v a d a b l e yet the r e s u l t a n t system fail to persist. O n this l o c a l scale, persistence d e p e n d s o n the relative c o m p e t i t i v e a d v a n ­ tage o f the p a r a s i t o i d species: T h e h y p e r p a r a s i t o i d s c a n o n l y i n v a d e i f t h e i r s e a r c h i n g efficiency relative to the parasitoid Hammond

(a^/dp

= a)

is h i g h

1977; H a s s e l l

enough

1979; M a y a n d

primary

(Beddington Hassell

and

1981). A

s i m i l a r c o n d i t i o n is also true f o r h y p e r p a r a s i t o i d s i n v a d i n g a host-parasitoid m e t a p o p u l a t i o n : H i g h e r attack rates are always better able to i n v a d e (a

a p p r o x . > 1.5). H o w e v e r , w h i l e i n v a ­

s i o n o f a s i n g l e l o c a l p o p u l a t i o n p a t c h case is a p u r e l y d e t e r ­ m i n i s t i c process, i n v a s i o n o f a m e t a p o p u l a t i o n has a stochastic c o m p o n e n t that stems f r o m the h e t e r o g e n e i t y i n the

abun­

d a n c e o f l o c a l p o p u l a t i o n s i n space a n d the p a r t i c u l a r l o c a l p o p u l a t i o n w h e r e the i n v a s i o n o c c u r s . A n i n v a s i o n m u s t start i n a h a b i t a t w h e r e the l o c a l c o n d i t i o n s are suitable, otherwise the i n v a d i n g p o p u l a t i o n very q u i c k l y dies out. I n v a s i o n success thus d e p e n d s o n the specific l o c a l c o n d i t i o n s w h e r e a n i n v a ­ s i o n is o c c u r r i n g . G l o b a l c o n d i t i o n s , s u c h as the n u m b e r o f l o c a l habitats i n the m e t a p o p u l a t i o n , have n o i n f l u e n c e o n the success o r n o t o f a n i n v a s i o n . Persistence o f the r e s u l t i n g H - P - Q c o m m u n i t y , h o w e v e r , does d e p e n d o n the n u m b e r o f patches. A s f o r the s i m p l e o n e h o s t - o n e p a r a s i t o i d m e t a p o p u l a t i o n , persistence d e p e n d s

on

the h e t e r o g e n e i t y o f p a r a s i t i s m b e t w e e n habitats b e i n g m a i n ­ t a i n e d by s e l f - o r g a n i z i n g spatial patterns,

which

themselves

d e p e n d o n t h e r e b e i n g sufficient space available. T h e spatial patterns o f a n H - P - Q c o m m u n i t y are o n a l a r g e r scale t h a n w h e n the h y p e r p a r a s i t o i d is absent. F o r e x a m p l e , i n o n e series of simulations with μ

= μ

Ρ

= 0.1, a

7 /

P

= 0.3, a n d λ = 2, the

H - P c o m m u n i t y n e e d e d a p p r o x i m a t e l y sixty-four patches f o r a 5 0 % p r o b a b i l i t y o f persistence, whereas w i t h a h y p e r p a r a s i t o i d o f s i m i l a r d i s p e r s a l rate the H - P - Q c o m m u n i t y n e e d e d a m i n i ­ mum

o f 121 patches f o r a 5 0 % p r o b a b i l i t y o f

persistence

( W i l s o n et a l . , i n p r e p a r a t i o n ) . Interesting persistence

trade-offs

between

tend

to

occur

d i s p e r s a l rates a n d 105

in

metapopulation

c o m p e t i t i v e ability

H A S S E L L

measured

A N D

W I L S O N

i n parasitoids by t h e i r s e a r c h i n g efficiencies (see

T i l m a n 1994, T i l m a n a n d D o w n i n g 1994, a n d Rees 1995 f o r c o m p a r a b l e trade-offs i n p l a n t c o m m u n i t i e s ) . T h u s , a l t h o u g h h y p e r p a r a s i t o i d s w i t h h i g h attack rates or d i s p e r s a l rates are b e t t e r able to i n v a d e a host-parasitoid m e t a p o p u l a t i o n , i f they have b o t h h i g h attack and d i s p e r s a l rates they t e n d to o v e r e x p l o i t t h e i r hosts (i.e., p a r a s i t o i d P) a n d c o n s e q u e n t l y fail to persist. S u c h a trade-off is i l l u s t r a t e d i n F i g u r e 4.7. H y p e r p a r a ­ sitoids that have l o w attack rates are too u n c o m p e t i t i v e — t h e y c a n n o t e v e n i n v a d e ( r e g i o n B ) . H y p e r p a r a s i t o i d s that b o t h are g o o d dispersers a n d have h i g h attack rates are able to i n v a d e , b u t they t h e n o v e r e x p l o i t t h e i r hosts, a n d the r e s u l t a n t c o m ­ m u n i t y breaks u p w i t h e i t h e r the e x t i n c t i o n o f b o t h the h y p e r ­ p a r a s i t o i d a n d p a r a s i t o i d o r j u s t the h y p e r p a r a s i t o i d ( r e g i o n A ) , m u c h as i n the r e g i o n o f c o m p l e x d y n a m i c i n s t a b i l i t y s h o w n i n F i g u r e 4.6. I n b e t w e e n the two r e g i o n s A a n d Β the b a l a n c e b e t w e e n attack rates a n d d i s p e r s a l is s u c h to a l l o w c o e x i s t e n c e o f the t h r e e species ( r e g i o n C ) . C o e x i s t e n c e o f c o m p e t i n g parasitoids (H-P-P) o r c o m p e t i n g hosts ( H - H - P ) is greatly r e s t r i c t e d c o m p a r e d

to the H - P - Q

system above ( H a s s e l l et a l . 1994; C o m i n s a n d H a s s e l l 1996). F o r a single l o c a l p o p u l a t i o n system, a n d w i t h o u t a d d i t i o n a l features p r o m o t i n g i n t r a s p e c i f i c stability, o n e p a r a s i t o i d o r o n e host w i l l always d r i v e the Competitive

exclusion

c o m p e t i n g species to e x t i n c t i o n .

also

occurs

within

the

equivalent

m e t a p o p u l a t i o n s , e x c e p t i n relatively s m a l l r e g i o n s o f p a r a m e ­ ter space. F o r e x a m p l e , w i t h two c o m p e t i n g parasitoids, this r e g i o n o f c o e x i s t e n c e o c c u r s w h e r e the d i s p e r s a l rates o f the two p a r a s i t o i d species differ by a n o r d e r o f m a g n i t u d e , a n d w h e n the slower d i s p e r s i n g species also has a greater attack rate. T h u s w h e n c o e x i s t e n c e does o c c u r i n it

seems to

(Hutchinson

depend 1951;

upon Levins

a

kind

and

metapopulations

o f fugitive

Culver

1971;

coexistence Horn

and

M a c A r t h u r 1972; H a n s k i a n d R a n t a 1983; N e e a n d M a y 1992; H a n s k i a n d Z h a n g 1993), b a l a n c i n g d i s p e r s a l rates a n d c o m ­ petitive ability. Interestingly, a n d best seen w h e r e t h e r e

are

clear-cut spirals, this c o e x i s t e n c e tends to be associated w i t h

106

H O S T - P A RA S I T O I D

S Y S T E M S

Β (hyperparasitoid too uncompetitive) 1

1

ι

1

2

3

4

Relative dispersal rate^ F I G U R E 4.7. Persistence plot for the host-parasitoid-hyperparasitoid c o m ­ munity. B o t h axes plot the dispersal a n d attack rates o f the hyperpara­ sitoid relative to the parasitoid, where a = a /ctp, q

/%

=

when

f^p

=

μ —μ^/μ^, a

P

= 0.3,

0.1, a n d λ = 2. T h e lines plot the boundaries o f persistence

η = 20 (four h u n d r e d patches). T h e regions are explained

in

the text.

the slowly d i s p e r s i n g p a r a s i t o i d s b e i n g c o n f i n e d to the c e n t r a l f o c i o f the spirals ( F i g u r e 4.8). H e r e the slow dispersers have a n attack rate advantage i n the areas w h e r e h o s t d e n s i t y fluctuates the least, a n d the fast dispersers have a c o l o n i z a t i o n advantage i n the areas w h e r e h o s t d e n s i t y is fluctuating m o s t s t r o n g l y (the areas swept by the s p i r a l arms). P e r s i s t e n t s p a t i a l s e p a r a t i o n o f the two c o m p e t i n g p a r a s i t o i d species is thus m a i n t a i n e d b y this m e c h a n i s m . S i m i l a r l y , f o r persistence i n the H - H - P c o m m u n i t y the less dispersive h o s t has e i t h e r a l a r g e r i n t r i n s i c rate o f i n c r e a s e o r a s m a l l e r p a r a s i t o i d attack rate, a n d , w i t h c e r t a i n parameter

c o m b i n a t i o n s , the

less dispersive species h o s t is

r e s t r i c t e d to s p i r a l f o c i , i n e x a c t l y the same way as f o r the less dispersive p a r a s i t o i d s . S i n c e these f o c i are relatively static, the

107

H A S S E L L

A N D

W I L S O N

F I G U R E 4.8. Maps o f the spatial density distribution o f hosts (a), highly dispersive parasitoids (b),

a n d sedentary parasitoids (c),

i n a snapshot

f r o m the dynamics o f a persistent P-H-P system with λ = 2, μ

Ν

μ

ρ ι

= 0.5,

μ

Ρ2

=

0.5,

= 0.05, a n d a = 1.3. T h e grids must be mentally super­

i m p o s e d i n order to perceive the relationships between the densities o f the

various species. Spiral foci

exist at the

ends o f the

"mountain

ranges" i n the leftmost figure (excluding ends at the edges o f the grid). In the time evolution o f the system the " m o u n t a i n ranges" are the peaks o f p o p u l a t i o n density waves a n d are thus i n continuous m o t i o n . T h e foci, by contrast, remain i n almost exactly the same place, for indefinitely l o n g times. ( F r o m C o m i n s a n d Hassell 1996.)

less m o b i l e species a p p e a r s to o c c u r o n l y i n i s o l a t e d , s m a l l " i s l a n d s " w i t h i n the habitat, m u c h as i f these were p o c k e t s o f favorable habitat. A s the d i s p e r s a l rates b e c o m e less d i v e r g e n t between

the species, a n d p r o v i d i n g that c o e x i s t e n c e is still

feasible, the n i c h e o f the less dispersive species spreads f u r t h e r i n t o the a r m o f the spirals. A t t e m p t i n g to d e t e c t the patterns p r e d i c t e d f r o m m e t a p o p u ­ l a t i o n m o d e l s f r o m r e a l p o p u l a t i o n s poses e n o r m o u s l o g i s t i c a l (and

financial)

p r o b l e m s , j u s t f r o m the i m p o s e d scale o f the

studies (but see H a n s k i , C h a p t e r 2). I f it e m e r g e s that m e t a p o p ­ u l a t i o n effects

are

a key p a r t to u n d e r s t a n d i n g

d y n a m i c s , ecologists w i l l be f o r c e d to r e a p p r a i s e p r o a c h to

fieldwork.

population their

ap­

H o p e f u l l y , useful tricks w i l l e m e r g e ; f o r

i n s t a n c e , it m a y p r o v e possible to p i e c e t o g e t h e r f r o m l i m i t e d d a t a o n the p o p u l a t i o n density d i s t r i b u t i o n s p a r t i c u l a r patterns o f d e l a y e d c o v a r i a n c e that are d i a g n o s t i c o f d i f f e r e n t k i n d s o f spatial, a n d h e n c e

metapopulation, dynamics (Comins and

H a s s e l l 1996).

108

H O S T - P A RA S I T O I D

S Y S T E M S

SUMMARY

T h i s c h a p t e r has r e v i e w e d s o m e d y n a m i c a l affects o f spatial structure i n host-parasitoid systems at b o t h l o c a l a n d m e t a p o p ­ ulation is m o s t

scales. H e t e r o g e n e i t y conveniently

at

expressed

i n d i v i d u a l v a r i a b i l i t y i n the

the in

local population terms

level

of individual-to-

r i s k o f hosts b e i n g

parasitized,

w h i c h m a y be e i t h e r d e p e n d e n t ( H D D ) o r i n d e p e n d e n t ( H D I ) o f l o c a l h o s t density. T h e s e c a n be i d e n t i f i e d a n d q u a n t i f i e d f r o m s t r a i g h t f o r w a r d d a t a o n the spatial patterns o f p a r a s i t i s m i n r e l a t i o n to the h o s t d i s t r i b u t i o n . D e s p i t e the level o f d e t a i l i n m a n y o f these studies, it is e n c o u r a g i n g that fairly s w e e p i n g c r i t e r i a c a n b e f o u n d that are d e s i g n e d to c a p t u r e the o v e r a l l effects o f spatial h e t e r o g e n e i t y w i t h i n relatively s i m p l e c r i t e r i a f o r e s t i m a t i n g the v a r i a t i o n i n the r i s k o f p a r a s i t i s m w i t h i n the h o s t p o p u l a t i o n . T h e o v e r a l l c o n c l u s i o n is that l o c a l p o p u l a ­ tions c a n be s t a b i l i z e d by sufficiently h e t e r o g e n e o u s d i s t r i b u ­ tions o f parasitism, p a r t i c u l a r l y i f this is i n d e p e n d e n t

o f the

host distribution ( H D I ) . M e t a p o p u l a t i o n s o f hosts a n d parasitoids i n t r o d u c e a n o t h e r d i m e n s i o n to the d y n a m i c s . A l t h o u g h m e t a p o p u l a t i o n s b r o a d l y the

share

same c o n d i t i o n s f o r l o c a l stability as the

con­

stituent p o p u l a t i o n s , the two b e c o m e very d i f f e r e n t w h e n the l o c a l p o p u l a t i o n s are i n d i v i d u a l l y u n s t a b l e . N o w a variety o f s t r i k i n g spatial patterns o f a b u n d a n c e c a n be o b s e r v e d associ­ ated w i t h different d y n a m i c s o f the m e t a p o p u l a t i o n as a w h o l e . It is these spatial p a t t e r n s that m a i n t a i n h e t e r o g e n e i t y i n the h o s t risk o f p a r a s i t i s m a n d so p r o m o t e

persistence.

I n v a s i o n a n d persistence are n o t always l i n k e d i n spatially distributed

host-parasitoid

communities,

particularly

for

m e t a p o p u l a t i o n systems. T h u s , a p a r a s i t o i d o r h y p e r p a r a s i t o i d m a y be able to i n v a d e a c o m m u n i t y f r o m w h i c h i t was absent, but

the

resultant

c o m m u n i t y may then

break

up with

the

e x t i n c t i o n o f o n e o r m o r e species. A l t e r n a t i v e l y , a p a r a s i t o i d o r h y p e r p a r a s i t o i d species c o u l d have a h i g h p r o b a b i l i t y o f persis­ tence i n a n e s t a b l i s h e d c o m m u n i t y b u t have a l o w p r o b a b i l i t y

109

H A S S E L L

A N D

W I L S O N

o f i n v a d i n g a c o m m u n i t y f r o m w h i c h it was absent. T h e p a r t i c ­ u l a r h i s t o r y o f invasions o f a c o m m u n i t y c a n thus have a d r a m a t i c effect o n the r e s u l t i n g c o m m u n i t y structure. C o e x i s ­ tence o f c o m p e t i n g species i n these m e t a p o p u l a t i o n s , w h e n i t does o c c u r , n o t o n l y d e p e n d s u p o n a b a l a n c i n g o f d i s p e r s a l rates a n d c o m p e t i t i v e ability b u t also seems to be

associated

with some degree o f self-organizing n i c h e separation the

competing

species. T h i s is best seen w h e n

the

between spatial

d y n a m i c s show c l e a r spirals, i n w h i c h case the relatively i m m o ­ b i l e species tends to be c o n f i n e d to the c e n t r a l f o c i o f the spirals a n d the h i g h l y dispersive species to the r e m a i n d e r o f the " t r a i l i n g a r m " o f the spirals.

110

C H A P T E R

FIVE

Basic Epidemiological Concepts in a Spatial Context Elizabeth Eli Holmes

INTRODUCTION It is o b v i o u s to any observer o f e p i d e m i c s that the s p r e a d o f disease is u n a v o i d a b l y spatial. Disease m o v e s f r o m i n d i v i d u a l to i n d i v i d u a l f o l l o w i n g the n e t w o r k o f contacts b e t w e e n i n d i v i d u ­ als w i t h i n a p o p u l a t i o n . F o r m a n y h o s t - p a t h o g e n systems,

the

p a t t e r n o f s p r e a d is a c o m b i n a t i o n o f l o c a l t r a n s m i s s i o n o u t f r o m a focus o f i n f e c t i o n a n d l o n g - d i s t a n c e t r a n s m i s s i o n , w h i c h establishes n e w f o c i . Y e t m o s t classical e p i d e m i o l o g i c a l t h e o r y glosses over the spatial d i m e n s i o n o f disease t r a n s m i s s i o n a n d i n s t e a d assumes that every i n d i v i d u a l is e q u a l l y l i k e l y to c o n t a c t every o t h e r . A key q u e s t i o n is to w h a t e x t e n t d o we lose i n s i g h t o r are quantitatively m i s l e d by m o d e l i n g the i n t r i n s i c a l l y spatial process o f disease s p r e a d w i t h n o n s p a t i a l theory. O b v i o u s l y , spatial m o d e l s are necessary to address spatial questions, s u c h as the v e l o c i t y at w h i c h disease spreads over a l a n d s c a p e o r the spatial p a t t e r n o f disease p r e v a l e n c e (see M u r r a y 1990 a n d C l i f f et a l . 1981 f o r reviews). H o w e v e r , m a n y o f the m o s t basic a n d i m p o r t a n t e p i d e m i o l o g i c a l questions are

n o t spatial: W i l l

a

p a t h o g e n cause a n e p i d e m i c ? C a n it i n v a d e a n d persist i n a p o p u l a t i o n ? W h a t f r a c t i o n o f the p o p u l a t i o n w i l l be infected? W h e n will epidemics occur a n d reoccur? Early

epidemiological theory

addressed

these

sorts

of

questions u s i n g s i m p l e n o n s p a t i a l m o d e l s o f c o m m u n i c a b l e disease s u c h as the K e r m a c k - M c K e n d r i c k m o d e l ( K e r m a c k a n d

111

H O L M E S

M c K e n d r i c k 1927):

dS —

=

-/3SJ -

bS + b

dt dl dt dR —

= β$Ι - μΐ

-

bl

(5.1)

= μΐ - bR.

T h e s e m o d e l s d i v i d e the p o p u l a t i o n i n t o susceptible, i n f e c t e d , a n d resistant (i.e., i m m u n e ) classes, S, I, a n d R. A l l i n d i v i d u a l s are

e q u a l l y l i k e l y to c o n t a c t

every o t h e r

individual i n

the

p o p u l a t i o n , a n d t h e r e are n o differences a m o n g i n d i v i d u a l s e x c e p t f o r t h e i r i n f e c t i o n status. I n d i v i d u a l s r e c o v e r a n d b e ­ c o m e i m m u n e at rate μ a n d d i e at rate b. T h e p o p u l a t i o n is a s s u m e d to be stable so that a l l i n d i v i d u a l s that d i e are

re­

p l a c e d by b i r t h s o f n e w susceptible i n d i v i d u a l s . S u s c e p t i b l e i n d i v i d u a l s b e c o m e i n f e c t e d at a rate βΐ that is s i m p l y p r o p o r ­ t i o n a l to the f r a c t i o n i n f e c t e d . F r o m such simple nonspatial theory c o m e some o f o u r most f u n d a m e n t a l p r i n c i p l e s a b o u t the d y n a m i c s o f disease w i t h i n p o p u l a t i o n s , a n d a l t h o u g h m o d e r n e p i d e m i o l o g i c a l m o d e l s are far m o r e realistic a n d i n t r o d u c e a variety o f c o m p l e x i t i e s (see A n d e r s o n a n d M a y 1991 f o r a review), these basic c o n c e p t s are still u s e d to u n d e r s t a n d a n d t h i n k a b o u t r e a l disease systems. T h e p u r p o s e o f this c h a p t e r is to discuss h o w epidemiological concepts

change,

fundamental

qualitatively o r

quantita­

tively, w h e n we m o v e f r o m a n o n s p a t i a l to a spatial m o d e l o f c o m m u n i c a b l e disease. Basic Epidemiological

Concepts

1. T h e r e p r o d u c t i v e rate o f a disease. T h e basic r e p r o d u c ­ tive rate o f a disease, R , 0

is the e x p e c t e d n u m b e r

of new

i n f e c t i o n s c a u s e d by o n e i n f e c t e d i n d i v i d u a l i n a sea o f suscep­ tible i n d i v i d u a l s ( M a c D o n a l d 1957). 112

S P A T I A L

2. D e t e r m i n i s t i c

E P I D E M I O L O G Y

threshold

theorem.

Kermack

and

M c K e n d r i c k (1927) s h o w e d that i n a s i m p l e m o d e l w i t h

no

i n f l u x o f susceptible i n d i v i d u a l s , t h e r e exists a t h r e s h o l d d e n ­ sity o f susceptibles b e l o w w h i c h a n e p i d e m i c c a n n o t o c c u r . 3. Stochastic t h r e s h o l d t h e o r e m . T h e K e r m a c k - M c K e n d r i c k m o d e l assumes that the

p o p u l a t i o n is large e n o u g h

to

be

c o n s i d e r e d i n f i n i t e a n d that densities c a n be c o n s i d e r e d c o n ­ t i n u o u s variables. B a i l e y (1975) s t u d i e d stochastic versions o f the K e r m a c k - M c K e n d r i c k m o d e l to e x p l o r e the d y n a m i c s o f disease i n finite c o m m u n i t i e s a n d s h o w e d that disease become

established i n a p o p u l a t i o n unless the

cannot

size o f

the

c o m m u n i t y is above the stochastic t h r e s h o l d . 4. T h r e s h o l d f o r a disease to b e c o m e e n d e m i c . T h e p r e v i o u s two p o i n t s c o n s i d e r the d y n a m i c s o f a n e p i d e m i c . I n this case, n a t u r a l m o r t a l i t y , b, is n e g l i g i b l e . I f we c o n s i d e r i n s t e a d

the

r e i n t r o d u c t i o n o f susceptible i n d i v i d u a l s d u e to b i r t h s , t h e n disease m a y persist w i t h i n the p o p u l a t i o n a n d b e c o m e

en­

d e m i c . A n a l o g o u s to the e p i d e m i c t h r e s h o l d , t h e r e exists a n e n d e m i c t h r e s h o l d f o r the disease to establish a n d persist. 5. E q u i l i b r i u m levels o f disease. K e r m a c k a n d M c K e n d r i c k (1932) s h o w e d that E q u a t i o n 5.1 p r e d i c t s stable e q u i l i b r i u m levels o f susceptible, i n f e c t e d , a n d resistant i n d i v i d u a l s . T h e s e levels are a f u n c t i o n o f the rates μ , β, a n d b. 6. P e r i o d i c i t y o f

epidemics.

The

long-term

records

of

i n f e c t i o u s diseases p r o v i d e some s t r i k i n g e x a m p l e s o f cyclic predator-prey

dynamics.

The

Kermack-McKendrick

model

offers a n e x p l a n a t i o n f o r these cycles as a n i n t r i n s i c p r o d u c t o f the

interaction between

hosts a n d p a t h o g e n s ( S o p e r

1929;

A n d e r s o n a n d M a y 1983).

SPATIAL VERSUS How

NONSPATIAL

does the spatial d i m e n s i o n c h a n g e

these basic c o n ­

cepts? T o tackle this q u e s t i o n , I discuss w h a t two types o f spatial m o d e l s p r e d i c t f o r c o n c e p t s (1) t h r o u g h (6) above. T h e focus o f the d i s c u s s i o n w i l l be o n a c e l l u l a r a u t o m a t a v e r s i o n o f the K e r m a c k - M c K e n d r i c k m o d e l . T h i s m o d e l is i n d i v i d u a l b a s e d a n d e x p l i c i t l y captures

the

n o t i o n o f a spatial n e t w o r k o f

contacts b e t w e e n i n d i v i d u a l s . H o w e v e r , d i s c u s s i o n o f o n l y o n e 113

H O L M E S

type o f m o d e l leaves o n e b l i n d to the d e g r e e to w h i c h the results d e p e n d

o n a p a r t i c u l a r m o d e l structure,

i n s i g h t i n t o results f r o m o n e

type

and

much

o f m o d e l is g a i n e d

c o m p a r i n g results w i t h o t h e r types o f m o d e l s . T h u s , i n

by the

c o n c l u d i n g sections, the results f o r the c e l l u l a r a u t o m a t a m o d e l are c o m p a r e d to a p a r t i a l d i f f e r e n t i a l e q u a t i o n m o d e l that also incorporates local transmission. Kermack -McKendnck Before

i n t r o d u c i n g the

Model

cellular automata

m o d e l , let

me

b r i e f l y discuss the classical K e r m a c k - M c K e n d r i c k m o d e l ( E q . 5.1). T h e key feature o f this m o d e l is the i d e a that disease t r a n s m i s s i o n is d e s c r i b e d by the βΙΞ t e r m , w h i c h is k n o w n as the mass-action a s s u m p t i o n . I n this case, the rate at w h i c h susceptibles b e c o m e i n f e c t e d is d i r e c t l y p r o p o r t i o n a l to

the

f r a c t i o n i n f e c t e d ; d o u b l i n g the n u m b e r i n f e c t e d d o u b l e s

the

rate o f disease t r a n s m i s s i o n . F o r this m o d e l , the

reproductive

rate o f the disease is

R

0

+b

μ

0

is a n a l o g o u s to the i n t r i n s i c rate o f increase, r , i n s i m p l e

population growth models. R

0

is n o t g e n e r a l l y o b s e r v e d b e ­

cause i t is the m a x i m u m possible g r o w t h rate, w h i c h o c c u r s at the very b e g i n n i n g o f a n e p i d e m i c w h e n the p o p u l a t i o n is 1 0 0 % susceptible. See B a i l e y (1975) f o r a m o d e r n d i s c u s s i o n o f this m o d e l a n d S e r f l i n g (1952) f o r a h i s t o r i c a l review. Cellular Automata

Model of an Infectious

Disease

I n the basic lattice m o d e l , sites are d i s t r i b u t e d o n a square lattice o n w h i c h e a c h site has a set o f p h y s i c a l l y n e i g h b o r i n g sites. E a c h site r e p r e s e n t s a n i n d i v i d u a l that c a n have o n e o f t h r e e states: susceptible, i n f e c t e d , o r resistant. Disease trans­ m i s s i o n is m o d e l e d as a p r o b a b i l i s t i c process. E a c h i n f e c t e d site has a n e q u a l a n d i n d e p e n d e n t p r o b a b i l i t y , q, o f i n f e c t i n g a susceptible n e i g h b o r . T h u s the p r o b a b i l i t y that a susceptible 114

SPATIAL

EPIDEMIOLOGY

site b e c o m e s i n f e c t e d is 1 — (the p r o b a b i l i t y o f n o t b e i n g i n f e c t e d ) \ n u m b e r of infected neighbors

= 1 At

each

time

step,

a

(1 - q) site

/~ ~\

.

changes

state

based

(5.2)

on

the

probabilities: £

I

^

^ j

R

^

^ ^ n u m b e r of infected neighbors

μ

sites d i e a n d are r e b o r n susceptible In

b.

the s i m u l a t i o n s d i s c u s s e d h e r e , t h r e e d i f f e r e n t

neighbor­

h o o d s are c o m p a r e d : the f o u r d i r e c t l y adjacent sites, the e i g h t nearest n e i g h b o r s , a n d the twenty-four nearest n e i g h b o r s . Sites are u p d a t e d s y n c h r o n o u s l y , m e a n i n g t i m e is discrete. D i s c r e t e t i m e c a n affect the d y n a m i c s o f c e l l u l a r a u t o m a t a

(Ingerson

a n d B u v e l 1984; N o w a k , B o n h o e f e r , a n d M a y 1994), a n d m i n i m i z e these effects,

the s i m u l a t i o n s w e r e r u n w i t h

to

small

t r a n s i t i o n p r o b a b i l i t i e s to a p p r o x i m a t e c o n t i n u o u s t i m e . I n this d i s c u s s i o n , the m o d e l w i t h l o c a l t r a n s m i s s i o n is o f t e n c o m p a r e d to the a n a l o g o u s m o d e l w i t h g l o b a l t r a n s m i s s i o n . I n the g l o b a l m o d e l , the " n e i g h b o r h o o d " is the e n t i r e p o p u l a ­ t i o n , a n d every i n f e c t e d site is e q u a l l y l i k e l y to i n f e c t

any

susceptible site i n the e n t i r e p o p u l a t i o n . T h e rate o f disease t r a n s m i s s i o n is: (the p r o p o r t i o n susceptible) X (the p r o b a b i l i t y that a susceptible site b e c o m e s infected), that is,

S[l for

q

g

small. T h e

-

(1 - q ) g

parameter

I N

]

«

q

is the

g

(q N)lS g

p r o b a b i l i t y that

an

i n f e c t e d i n d i v i d u a l infects a susceptible n e i g h b o r f o r the g l o b a l m o d e l . S i n c e t r a n s m i s s i o n c a n b e d e s c r i b e d as a l i n e a r f u n c ­ t i o n o f IS f o r q

g

s m a l l , the c e l l u l a r a u t o m a t a m o d e l w i t h g l o b a l

t r a n s m i s s i o n is the a n a l o g u e to the K e r m a c k - M c K e n d r i c k m o d e l with β = q

Ν. 115

H O L M E S

T h e d e f i n i t i o n o f the disease r e p r o d u c t i v e rate i n the c e l l u ­ lar

automata

m o d e l is a n a l o g o u s

to the

d e f i n i t i o n for

the

K e r m a c k - M c K e n d r i c k m o d e l . It is the m e a n i n f e c t i o u s p e r i o d times

the

rate at w h i c h n e w i n f e c t i o n s are

caused

by

an

i n f e c t e d site that is s u r r o u n d e d by susceptible sites. T h u s ,

+b

μ where N

n

is the n u m b e r o f n e i g h b o r s (4, 8, 24, o r N). W h e n a

s i m u l a t i o n o f the m o d e l w i t h l o c a l t r a n s m i s s i o n was c o m p a r e d to o n e w i t h g l o b a l t r a n s m i s s i o n , R

was k e p t i d e n t i c a l i n the

0

s i m u l a t i o n s , b u t the

n e w i n f e c t i o n s were

distributed

l o c a l l y o r g l o b a l l y . Specifically, a n d to m a k e the R 's

either equiva­

0

lent, p o p u l a t i o n size ?»

=

9 g 6

where q

n

and q

g

Ü

f

—

^

T

Z

—

(

5

*

3

)

n u m b e r or n e i g h b o r s

are the t r a n s m i s s i o n p r o b a b i l i t i e s f o r the l o c a l

and global models.

EPIDEMIOLOGICAL PRINCIPLES IN A SPATIAL C O N T E X T Epidemic

Threshold

T h e K e r m a c k - M c K e n d r i c k t h r e s h o l d f o r a n e p i d e m i c to o c ­ c u r i n a p o p u l a t i o n is R

0

t h r e s h o l d is the

> 1/S.

The

motivation behind

Kermack-McKendrick

the

concept

of

"herd

i m m u n i t y . " T h i s i d e a says that to p r o t e c t a p o p u l a t i o n f r o m a disease, it is n o t necessary to vaccinate the e n t i r e p o p u l a t i o n ; i t is e n o u g h to vaccinate a p r o p o r t i o n (1 — 1/R ). 0

basic results

from

the

transmission

is that the

cellular automata

O n e o f the

m o d e l with

Kermack-McKendrick

local

threshold

is

overly conservative. I n the c e l l u l a r a u t o m a t a m o d e l w i t h l o c a l t r a n s m i s s i o n , e a c h i n f e c t e d i n d i v i d u a l interacts w i t h a relatively s m a l l n e i g h b o r h o o d . F r o m the i n f e c t e d i n d i v i d u a l ' s p e r s p e c ­ tive, its " w o r l d " q u i c k l y fills w i t h o t h e r i n f e c t e d i n d i v i d u a l s , 116

S P A T I A L

E P I D E M I O L O G Y

a n d the rate at w h i c h it causes n e w i n f e c t i o n s r a p i d l y d e c l i n e s . To

o v e r c o m e this severe

d e p l e t i o n o f the l o c a l

p o o l , the disease m u s t have a h i g h e r R

0

e p i d e m i c (e.g., R

= 1.29 f o r f o u r n e i g h b o r s a n d R

0

for

susceptible

i n o r d e r to cause a n =

0

1.13

e i g h t n e i g h b o r s ) . T h i s m e a n s that w h e n o n e takes i n t o

a c c o u n t l o c a l i z e d t r a n s m i s s i o n , o n e n e e d s to v a c c i n a t e a s m a l l e r p r o p o r t i o n o f the p o p u l a t i o n . A s a n aside, the stochastic n a ­ ture o f t r a n s m i s s i o n is often m u c h m o r e i m p o r t a n t t h a n

the

l o c a l n a t u r e o f t r a n s m i s s i o n i f o n l y a few i n f e c t e d i n d i v i d u a l s are i n t r o d u c e d i n t o a susceptible p o p u l a t i o n . I n a stochastic m o d e l , t h e r e is s o m e p r o b a b i l i t y that the disease w i l l g o e x t i n c t by c h a n c e a l o n e e v e n t h o u g h R

0

> 1. I n a stochastic lattice

m o d e l , this p r o b a b i l i t y o f c h a n c e e x t i n c t i o n is (1 /R Y

where a

Q

is the i n i t i a l n u m b e r i n f e c t e d ( W h i t t l e 1955; K e n d a l l 1965). W h e n a = 1 and R The

is s m a l l , this p r o b a b i l i t y is q u i t e h i g h .

0

traditional Kermack-McKendrick threshold

described

above is the t h r e s h o l d f o r the disease to increase w h e n it enters a p o p u l a t i o n . H o w e v e r , spatial m o d e l s o f disease i n t r o d u c e a n e w type o f the e p i d e m i c t h r e s h o l d , w h i c h is the

threshold

r e p r o d u c t i v e rate for a p a n d e m i c . A p a n d e m i c is a n e p i d e m i c that affects

the e n t i r e p o p u l a t i o n . T h e B l a c k P l a g u e i n the

1300s a n d 1500s, the flu e p i d e m i c o f the early 1920s, a n d the s m a l l p o x e p i d e m i c s that d e c i m a t e d the A m e r i c a s are r e a l - w o r l d e x a m p l e s o f p a n d e m i c s that s p r e a d across e n t i r e The R

0

continents.

c e l l u l a r a u t o m a t a m o d e l p r e d i c t s that t h e r e is a t h r e s h o l d f o r a p a n d e m i c to o c c u r . A t the r e p r o d u c t i v e rates n e a r the

t h r e s h o l d f o r a s m a l l i n n o c u l u m o f i n f e c t i o n to increase i n a p o p u l a t i o n , the disease w i l l b e g i n to s p r e a d f r o m the i n i t i a l site o f i n f e c t i o n , b u t t h e n d i e out. W i t h a h i g h e r basic r e p r o d u c t i v e rate, the disease w i l l s p r e a d farther a n d farther u n t i l , w i t h a high enough R ,

it is able to s p r e a d t h r o u g h o u t a very large

0

p o p u l a t i o n ( F i g u r e 5.1). T h e p a n d e m i c t h r e s h o l d f o r the c e l l u ­ lar

automata

m o d e l with

n e i g h b o r s is 4 < R

0

t r a n s m i s s i o n to the

< 6 ( K u u l a s m a a 1982). I f R

0

four

nearest

is b e l o w this,

the disease causes a s m a l l , spatially l i m i t e d e p i d e m i c . I f it is above, it c a n s p r e a d t h r o u g h o u t even a n i n f i n i t e p o p u l a t i o n . The

p r o b a b i l i t y o f a p a n d e m i c also increases as R

above

Q

the

pandemic

threshold. F r o m

increases

analogy with

similar

m o d e l s ( B a l l 1983; B r a m s o n , D u r r e t t , a n d S w i n d l e 1989) the 117

o O X o o

S P A T I A L

E P I D E M I O L O G Y

p a n d e m i c t h r e s h o l d d e p e n d s o n the size o f the c o n t a c t n e i g h ­ b o r h o o d a n d s h o u l d t e n d to R

= 1.0 as the

0

neighborhood

size increases. Threshold The

stochastic

Community

threshold

Size

theorem

states that t h e r e is a

t h r e s h o l d c o m m u n i t y size f o r a p a n d e m i c to o c c u r i n a p o p u l a ­ t i o n . T h i s i d e a stems f r o m w o r k o n a m o d e l e q u i v a l e n t to the c e l l u l a r a u t o m a t a m o d e l w i t h g l o b a l t r a n s m i s s i o n . I n this m o d e l , s m a l l e r p o p u l a t i o n size leads to a n i n c r e a s e d p r o b a b i l i t y that the i n c i p i e n t e p i d e m i c w i l l g o e x t i n c t i n the early stages m e r e l y by c h a n c e . I f we assume that t h e r e are n o b i r t h s a n d n o loss o f i m m u n i t y , the t h r e s h o l d p o p u l a t i o n size is N = μ/β t

1975).

Below

a n d above 1 — N/N

the

this,

the

epidemic

dies

threshold, pandemics

out

with

(Bailey certainty,

occur with probability

w h e r e Ν is p o p u l a t i o n size.

t

o f this i d e a i n the

cellular

a u t o m a t a m o d e l w i t h l o c a l t r a n s m i s s i o n . F r o m the

There

is n o

real equivalent

infected

i n d i v i d u a l ' s perspective, the w o r l d is the same w h e t h e r it is i n a n i n f i n i t e p o p u l a t i o n o r i n a p o p u l a t i o n b a r e l y l a r g e r t h a n its neighborhood however,

size. I n s t e a d o f a t h r e s h o l d c o m m u n i t y size,

t h e r e exists a t h r e s h o l d n e i g h b o r h o o d

size b e l o w

w h i c h the disease dies out. T h e existence o f this t h r e s h o l d c a n be s u r m i s e d by n o t i n g that the p a n d e m i c t h r e s h o l d is 4
c*-"

Ε

•·φ ·Γ4^·

\

: η

€S^7r-

·

Ou»

jrFΜ · "

,5

F I G U R E 8.4. Aggregation developing with time. (A, Β , C ) R a n d o m distribu­ tions at t = 0 for the

best three competitors i n an explicitly spatial

metapopulation m o d e l with local dispersal. T h e habitat contains 2,950 individual sites arranged i n a hexagonal lattice. Dots show sites o c c u p i e d by individual plants. (D, E , F ) Distribution at t = 5,000. T h e best c o m ­ petitor (D) is clustered c o m p a r e d to the initial r a n d o m distribution. T h e second competitor (E) a n d the third competitor ( F ) are m o r e clustered still, because they are restricted to the already-clustered sites not

occu­

p i e d by better competitors. Level o f clustering approaches an asymptote.

n e o u s , b u t r e p r o d u c t i o n is c o n t i n u o u s . H o w e v e r , f o r a vast array o f p l a n t a n d a n i m a l species, seed p r o d u c t i o n o r b i r t h is a n e p i s o d i c event often s y n c h r o n i z e d w i t h the seasons, whereas growth a n d competitive displacement occur gradually through­ o u t the year. T h u s , i n reality, r e p r o d u c t i o n is b e t t e r m o d e l e d as i n s t a n t a n e o u s a n d discrete, w h i l e g r o w t h a n d c o m p e t i t i v e d i s p l a c e m e n t are b e t t e r m o d e l e d as c o n t i n u o u s . We

c a n i n s t a l l discrete r e p r o d u c t i o n w i t h g r a d u a l g r o w t h

a n d c o m p e t i t i o n by c o n s i d e r i n g a c o n t i n u o u s - t i m e d i f f e r e n t i a l equation

at

each

site. I n d i v i d u a l s increase

or

decrease i n

b i o m a s s a c c o r d i n g to the a m o u n t o f l i m i t i n g r e s o u r c e

avail­

able, a n d the r e s o u r c e increases o r decreases a c c o r d i n g to the a m o u n t c o n s u m e d a n d the a m o u n t r e s u p p l i e d . T o m a k e this 195

L E H M A N

A N D

T I L M A N

1.0 ~ τ 0.8 —

Competitive Rank (1 = best competitor) F I G U R E 8.5. R a n d o m spatial distribution versus competitive rank. Spatial distribution of c o m p e t i n g species i n an explicitly spatial metapopulation m o d e l with local dispersal in a habitat o f 100 X 99 hexagonal sites. T h e horizontal axis shows the position o n the competitive hierarchy, with the best competitor n u m b e r e d

1. N u m b e r 0 represents u n i n h a b i t e d

T h e vertical axis is a simple index o f randomness (μ/σ ) 2

all

spatial scales f r o m 2 X 2

through

33 X 33.

indicate n o n r a n d o m spatial distributions.

sites.

averaged over

Deviations f r o m

unity

(Note that values o f unity i n

this index d o not imply randomness; H u r l b e r t 1990.) Randomness de­ creases for the first few competitors, then reaches a noisy asympotote (see Figure 8.4).

p r e c i s e , we a p p l y a specific m o d e l o f r e s o u r c e c o m p e t i t i o n ( A p p e n d i x 8.1) to e a c h h e x a g o n a l site, k e e p i n g m o r t a l i t y a n d c o l o n i z a t i o n as b e f o r e . T h e c o m p e t i t o r s at a site i n t e r a c t o n l y t h r o u g h the r e s o u r c e . The

equations

establish a n

implicit

competitive hierarchy

t h r o u g h r e s o u r c e values. E a c h species i has a c h a r a c t e r i s t i c level o f r e s o u r c e c a l l e d R* b e l o w w h i c h its p o p u l a t i o n c a n n o t survive, b u t above w h i c h its p o p u l a t i o n grows. T h e species w i t h the lowest i ? * displaces a l l o t h e r s by d r i v i n g the n u t r i e n t level at the site b e l o w the l e v e l at w h i c h o t h e r species c a n survive o r i n v a d e ( T i l m a n 1982). E q u a t i o n 8.2 entails two key a s s u m p t i o n s : (1) A species is displaced immediately w h e n propagules f r o m a better c o m p e t i ­ t o r arrive, a n d (2) a species is u n a b l e to survive o r i n v a d e a g i v e n site i f a species h i g h e r i n the c o m p e t i t i v e h i e r a r c h y is a l r e a d y present. I n contrast, f o r a m o r e realistic m o d e l , the 196

C O M P E T I T I O N

IN

S P A T I A L

H A B I T A T S

effect o f i n v a s i o n by a better c o m p e t i t o r is d e l a y e d . A b e t t e r competitor

c a n i n v a d e i m m e d i a t e l y , f o r the

resource

level

m a i n t a i n e d at the site b y the p o o r e r c o m p e t i t o r is sufficient f o r the b e t t e r c o m p e t i t o r to g r o w . H o w e v e r , the b e t t e r c o m p e t i t o r displaces the p o o r e r c o m p e t i t o r o n l y by r e d u c i n g the r e s o u r c e , f o l l o w i n g E q u a t i o n 8.4. T h i s takes t i m e . T h u s , i n c r e a s e d l o c a l d y n a m i c a l r e a l i s m m e a n s a delay i n c o m p e t i t i v e d i s p l a c e m e n t . S e c o n d , w h e n a b e t t e r c o m p e t i t o r dies, the r e s o u r c e

level

c a n r e c o v e r a n d a l l o w p o o r e r c o m p e t i t o r s to i n v a d e . B u t i t recovers as a d y n a m i c a l v a r i a b l e , f o l l o w i n g E q u a t i o n 8.3. T h i s also takes t i m e . T h u s , i n c r e a s e d l o c a l d y n a m i c a l r e a l i s m also m e a n s a delay i n r e c o l o n i z a t i o n , at least f o r species l o w o n the c o m p e t i t i v e h i e r a r c h y (i.e., the p o o r e r c o m p e t i t o r s ) . T h e n e t effect o f the first o f these alterations is to favor p o o r e r c o m p e t i t o r s , w h i l e the n e t effect o f the s e c o n d is to favor better c o m p e t i t o r s . H o w the two b a l a n c e d e p e n d s o n the r e c o v e r y rate o f the r e s o u r c e , the r e s o u r c e s u p p l y p o i n t , a n d the difference

a m o n g species i n m i n i m a l r e s o u r c e

require­

ments. F i g u r e 8.6 shows a n u m e r i c a l result f o r a n e n v i r o n m e n t a p p r o x i m a t e d by discrete cells i n a h e x a g o n a l g r i d . T h e two species coexist, b u t species 2 is m o r e w i d e s p r e a d t h a n it w o u l d be i n the c o r r e s p o n d i n g n o n r e s o u r c e m o d e l .

EVOLUTION A N D A LONGER TIMESCALE T h e s l o w i n g o f the timescale i n c o m p e t i t i v e systems w i t h spatial structure m a y m a k e i t i m p o r t a n t to c o n s i d e r e v o l u t i o n ­ ary effects.

V a n d e r l a a n a n d H o g e w e g (1995), f o r

example,

f o u n d that c e r t a i n t h e o r e t i c a l p r e d a t o r a n d p r e y species c a n persist i n d e f i n i t e l y w h e n a l l o w e d to evolve, b u t b e c o m e e x t i n c t i f e v o l u t i o n is b l o c k e d . I n a l l the c o m p e t i t i v e m o d e l s d e s c r i b e d thus far, e v o l u t i o n has b e e n b l o c k e d — c h a r a c t e r i s t i c s o f e a c h species are

fixed

p a r a m e t e r s . W h a t c a n be e x p e c t e d at l o n g

timescales w h e n p a r a m e t e r s are a l l o w e d to evolve? T o address this q u e s t i o n , we e x a m i n e a s i m p l e f o r m u l a t i o n o f p h e n o t y p i c e v o l u t i o n . O u r p h e n o t y p i c trait o f interest is the p o s i t i o n o f e a c h species a l o n g a n R*-dispersal

axis, w h i c h is d e f i n e d by

a s s u m i n g a positive c o r r e l a t i o n b e t w e e n R* a n d d i s p e r s a l a b i l ­ ity ( a n d a trade-off b e t w e e n d i s p e r s a l a n d c o m p e t i t i v e ability, 197

L E H M A N

A N D

T I L M A N

0.6 0.5 —

Sp. 2

0.4 - :

,

Pi

Ί

1.2 — \

0.1 —

Sp. 1

0.0 0

1000

2000

3000

4000

5000

t F I G U R E 8 . 6 . Coexistence o f two species growing continuously a n d r e p r o ­ d u c i n g discretely while explicitly c o m p e t i n g for a single resource. In each hexagonal

cell,

species compete for a single resource

according

to

E q u a t i o n 8 . 3 . T h e horizontal axis is time; the vertical axis is p o r t i o n of sites o c c u p i e d . C o l o n i z a t i o n a n d mortality are such that, i n the explicitly spatial analog o f E q u a t i o n 8 . 2 , species 1 w o u l d occupy m o r e sites than species 2 . However, because o f increased dynamical realism, i n c l u d i n g delayed displacement o f species 2 by species 1, species 2 is m o r e preva­ lent. W i t h o u t space, species 2 w o u l d be extinct.

since l o w R*

c o r r e s p o n d s to c o m p e t i t i v e superiority).

This

c o m p o s i t e trait c a n b e d e s c r i b e d by the p a r a m e t e r x¿. W e assume that a large p r o p o r t i o n α « 1 o f o f f s p r i n g possess the parental parameter value x

i9

b u t a s m a l l p r o p o r t i o n (1 —

a)/2

possess p a r a m e t e r v a l u e x + e, a n d the same p r o p o r t i o n (1 — i

a)/2

possess p a r a m e t e r v a l u e x — 6 , w h e r e i

6 is a

small

i n c r e m e n t . I n the i n f i n i t e s i m a l l i m i t , this k i n d o f p h e n o t y p i c evolution becomes simply diffusion

i n the p a r a m e t e r space

( A p p e n d i x 8.2). Solutions

through

Time

F i g u r e 8.7 shows a p a r t i c u l a r n u m e r i c a l s o l u t i o n w h e n p a ­ r a m e t e r s are able to evolve. T h e p a r a m e t e r

χ that evolves

r e p r e s e n t s R*, w h i c h establishes the c o m p e t i t i v e h i e r a r c h y — the l o w e r the R*,

the b e t t e r the c o m p e t i t o r . C o l o n i z a t i o n

increases

which

with

x,

corresponds

to

a

competition-

c o l o n i z a t i o n trade-off. M o r t a l i t y is t h e same f o r a l l p h e n o t y p e s . 198

C O M P E T I T I O N

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H A B I T A T S

Β

0.25

t = 102,000

¿ = 0

0.20 — 0.15 — 0.10 0.05 — 0.00 0.25

= 500,000

t = 200,000

0.20 0.15 0.10 0.05

A Λ [i ΑΑΛ

0.00 0

F I G U R E 8.7.

50

100

150

200

0

Patterns i n phenotype space

50

1

100

200

150

at intermediate times. T h e

horizontal axis is the phenotypic variable x, w h i c h represents R*,

or

competitive rank, a n d w h i c h is positively correlated with dispersal ability. T h e vertical axis represents a b u n d a n c e o f the c o r r e s p o n d i n g phenotype in

the

region. (A) A t t = 0,

a single

phenotype

is seeded.

(B) By

t = 102,000, original phenotype has evolved to lower R*, its abundance has decreased, a n d a weedy species has risen to h i g h levels. (C) A t t = 200,000, six distinct peaks appear along the phenotype axis, repre­ senting six different coexisting species. (D) By t = 500,000, species have b e c o m e m o r e tightly packed, with about twenty species coexisting. T h i s process continues until species are p a c k e d to the limits allowed by the dynamics o f the system.

T h e system starts w i t h a single late-successional p h e n o t y p e o f low, b u t n o t m i n i m a l , R*. P h e n o t y p i c d i f f u s i o n generates a l l p h e n o t y p e s at e x c e e d i n g l y l o w p o p u l a t i o n levels. I n d i v i d u a l s o f the i n i t i a l p h e n o t y p e spawn o f f s p r i n g o f h i g h e r R* a n d h i g h e r c o l o n i z a t i o n rate, w h i c h

t e n d to b e o u t c o m p e t e d by t h e i r

parents, a n d also o f f s p r i n g o f l o w e r R* a n d l o w e r c o l o n i z a t i o n rate, w h i c h t e n d to d i s p l a c e t h e i r p a r e n t s . T h u s the c o l o n i z a ­ t i o n rate t e n d s to decrease. A f t e r sufficient t i m e , the i n i t i a l species a p p r o a c h e s its p h y s i o l o g i c a l l i m i t s , a n d its e v o l u t i o n 199

L E H M A N

A N D

T I L M A N

slows. T h e r e is a p e a k i n the p h e n o t y p e space at the l o w e r l i m i t o f the c o l o n i z a t i o n rate ( F i g u r e 8.7A). N o w a w e e d y species w i t h h i g h R* a n d h i g h c o l o n i z a t i o n rate arises s y m p a t r i c a l l y ( F i g u r e 8.7B) a n d , w i t h p a s s i n g t i m e , evolves to be less weedy by the same m e c h a n i s m that r e d u c e d the c o l o n i z a t i o n rate o f the i n i t i a l species. T h i s process c o n t i n u e s , w i t h m o r e a n d m o r e species a r i s i n g , e a c h w e e d y at first b u t e a c h g r o w i n g less so with time. E a r l y i n the process, even t h o u g h any p h e n o t y p e is possible, the p h e n o t y p e s p r e s e n t are separated

by large gaps i n

p h e n o t y p e space. C o n t i n u o u s p h e n o t y p i c v a r i a t i o n has organized under

the

force

the been

o f c o m p e t i t i o n i n t o clusters o f

p h e n o t y p e s . W i t h the passage o f t i m e , the clusters

become

m o r e a n d m o r e closely p a c k e d . Y e t f o r m u c h o f the t i m e , the clusters r e m a i n u n s a t u r a t e d , as r e a l c o m m u n i t i e s m a y be (e.g., C o r n e l l a n d L a w t o n 1992). W h a t exactly is the f o r c e that f o r m s the p h e n o t y p i c peaks i n this e v o l v i n g c o m p e t i t i v e system? A c l u e lies i n the

limiting

s i m i l a r i t y already k n o w n i n c o m p e t i t i v e systems ( M a y 1981; P a c a l a a n d T i l m a n 1994; H u r t t a n d P a c a l a 1995), a n d specifi­ cally k n o w n f o r the i m m u t a b l e v e r s i o n o f this system ( T i l m a n 1994). A s s u m i n g m o r t a l i t y does n o t vary a m o n g species, t h e n to coexist, the s e c o n d c o m p e t i t o r m u s t have a h i g h e r c o l o n i z a t i o n rate t h a n the best c o m p e t i t o r . B u t a n a r b i t r a r i l y h i g h e r c o l o ­ n i z a t i o n rate w i l l n o t d o . T h e c o l o n i z a t i o n rate m u s t be h i g h e r by a m i n i m u m a m o u n t , that a m o u n t d e t e r m i n e d by c h a r a c t e r ­ istics o f the best c o m p e t i t o r ( T i l m a n 1994). W h e n p a r a m e t e r s are i m m u t a b l e , l i m i t i n g s i m i l a r i t y e x c l u d e s c e r t a i n species f r o m the r e g i o n . B u t w h e n p a r a m e t e r s c a n evolve, l i m i t i n g s i m i l a r i t y leads to the o r i g i n o f species, that is, to s y m p a t r i c s p e c i a t i o n o f h u n d r e d s o f species. W e have e x p l o r e d e v o l u t i o n o n l y i n i m p l i c i t l y spatial c o m p e ­ t i t i o n m o d e l s , a n d o n l y u n d e r the s i m p l e m e c h a n i s m o f p h e n o ­ typic i n h e r i t a n c e w i t h a d i r e c t trade-off b e t w e e n c o m p e t i t i v e ability a n d d i s p e r s a l . T h e results raise m a n y questions,

and

a d d i t i o n a l w o r k is n e e d e d to d e t e r m i n e t h e i r g e n e r a l i t y a n d robustness, especially the effect o f r a n d o m walks to e x t i n c t i o n o f r a r e p h e n o t y p e s , o f e x p l i c i t genetics, a n d o f e x p l i c i t space. 200

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SUMMARY

I m p l i c i t spatial structure, c o u p l e d w i t h trade-offs i n species dispersal a n d c o m p e t i t i v e traits, p r o m o t e s c o e x i s t e n c e

among

c o m p e t i n g species that w o u l d otherwise be subject to c o m p e t i ­ tive e x c l u s i o n . C o e x i s t e n c e c o n t i n u e s as spatial s t r u c t u r e is m a d e m o r e e x p l i c i t a n d as l o c a l p o p u l a t i o n d y n a m i c s are m a d e m o r e realistic. A d d e d r e a l i s m leads to n e w p h e n o m e n a , s u c h as spatial p a t t e r n i n g a n d i n c r e a s i n g stochasticity d o w n the c o m ­ petitive

hierarchy.

Explicit

spatial

structure

coupled

stochasticity h e l p s e x p l a i n the a p p a r e n t d i s c r e p a n c y

with

between

observations o f s t r o n g c o m p e t i t i o n at the fine scale a n d r a n ­ d o m assemblages o f species at the c o m m u n i t y scale. I m p l i c i t spatial structure,

interacting with phenotypic mutation

competitive dynamics, can

organize

a continuous

and

array

of

p h e n o t y p e s i n t o discrete species-like s y m p a t r i c clusters. I n a l l these cases, the d y n a m i c s o f c o m p e t i t i o n over a spatial h a b i t a t c a n be d r a m a t i c a l l y d i f f e r e n t f r o m the c o r r e s p o n d i n g d y n a m ­ ics i n e a c h l o c a l site.

APPENDIXES Appendix

8.1 : Resource

Competition

T h e f o l l o w i n g single-resource m o d e l is f r o m T i l m a n (1990): dR

n

— dt

= a(R

0

- R ) -

Σ

QifiWBi

..j

dB: - ¿ = [f¿R)

-

mjB,.

' ¿ - « x

x

)

=

^- ~ e

Q

x

~

X

n

W

M

a

k

(

)

·

9

is the l o c a t i o n o f a species k n e i g h b o r a n d M

Cik

3

)

is

the spatial scale f o r the c o m p e t i t i v e effects o f species k n e i g h b o r s o n species i f o c a l plants ( a n a l o g o u s to the m e a n d i s p e r s a l distance i n E q u a t i o n 9.2). It is a s i m p l e m a t t e r to w r i t e a c o m p u t e r p r o g r a m that w i l l s i m u l a t e the d y n a m i c s o f o u r two p l a n t species. B e g i n n i n g w i t h a n i n i t i a l n u m b e r a n d spatial d i s t r i b u t i o n a n d t a k i n g At to be a very s m a l l t i m e i n t e r v a l , o n e cycles r e p e a t e d l y t h r o u g h f o u r steps: (1) k i l l e a c h i n d i v i d u a l w i t h p r o b a b i l i t y μ At, ί

e a c h i n d i v i d u a l , c a l c u l a t e the

(2) f o r

con- a n d heterospecific

local

densities by s u m m i n g the distance weights ( E q . 9.3), (3) have e a c h i n d i v i d u a l give b i r t h w i t h p r o b a b i l i t y AtF

i9

a n d (4) dis­

perse e a c h n e w o f f s p r i n g to its n e w l o c a t i o n by d r a w i n g a p s e u d o r a n d o m n u m b e r f r o m the e x p o n e n t i a l p r o b a b i l i t y d e n ­ sity ( E q . 9.2). B y c y c l i n g r e p e a t e d l y t h r o u g h these f o u r steps, o n e c a n p r e d i c t the n u m b e r a n d spatial d i s t r i b u t i o n o f the species at any t i m e i n the future. W i t h the a d d i t i o n o f two spatial d i m e n s i o n s ( w h i c h c o m p l i ­ cates the d i s c u s s i o n b e l o w o n l y a little), this m o d e l is a service­ able d e s c r i p t i o n o f the c o m p e t i t i v e process that d o e s a p p l y to some

field

experiments

systems a n d c a n be e s t i m a t e d u s i n g s i m p l e (see

field

P a c a l a 1986b; P a c a l a a n d S i l a n d e r 1990). 208

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

H o w e v e r , p r e c i s e l y because o f its fidelity to the finite scales o f r e a l c o m m u n i t i e s , i t is stochastic, n o n l i n e a r , a n d spatial a n d thus m a t h e m a t i c a l l y difficult. I n the l i m i t as e i t h e r the M 's

o r M ' s t e n d to i n f i n i t y , the c

D

m o d e l b e c o m e s m a t h e m a t i c a l l y i d e n t i c a l to the L o t k a - V o l t e r r a c o m p e t i t i o n equations. W i t h

i n f i n i t e M ' s , the c

m o d e l is a

m e a n - f i e l d m o d e l because e a c h i n d i v i d u a l interacts w i t h every o t h e r

equally

i n d i v i d u a l . T h e p a r a m e t e r s o f the

V o l t e r r a e q u a t i o n s are t h e n r- = / • — μ · , r /K i

p e t i t i o n coefficients g i v e n by the a-

Lotka-

= /3¿, w i t h c o m -

i

i n E q u a t i o n 9.1.

W i t h i n f i n i t e M ' s b u t finite M ' s , we get a P o i s s o n d i s t r i b u D

c

t i o n o f i n d i v i d u a l s . T h e m a c r o s c o p i c p o p u l a t i o n - l e v e l m o d e l is t h e n f o u n d by c a l c u l a t i n g the e x p e c t a t i o n o f f e c u n d i t y times s u r v i v o r s h i p f o r a r a n d o m l y c h o s e n i n d i v i d u a l . T h i s gives a slightly d i f f e r e n t set o f L o t k a - V o l t e r r a e q u a t i o n s w i t h r- = / · — μ — ί

tween

a n d o t h e r p a r a m e t e r s as b e f o r e . T h e d i f f e r e n c e the

two L o t k a - V o l t e r r a l i m i t s is that i n the

be­

case o f

i n f i n i t e d i s p e r s a l a n d finite M ' s , n e i g h b o r h o o d s are b i a s e d by c

the p r e s e n c e o f the f o c a l i n d i v i d u a l , c a u s i n g e x t r a within-species density d e p e n d e n c e (the e x t r a t e r m o f m i n u s β

ί

s i o n f o r r ). {

T h e l o n g - d i s p e r s a l l i m i t is the

i n the e x p r e s ­ basis f o r

most

p r e v i o u s l y p u b l i s h e d analytically tractable n e i g h b o r h o o d m o d ­ els (i.e., P a c a l a a n d S i l a n d e r 1985; P a c a l a 1986a,b), the hawks a n d doves m o d e l w i t h m i g r a t i o n o n a fast timescale i n C h a p t e r 12 ( E q . 9.9; see also D u r r e t t a n d L e v i n 1994b), a n d

many

discrete-cell m o d e l s ( S k e l l a m 1951; H a s t i n g s 1980; C h e s s o n 1983; S h m i d a a n d E l m e r 1984; C r a w l e y a n d M a y 1987; T i l m a n 1994; P a c a l a a n d T i l m a n 1994). T h e l o n g - d i s p e r s a l l i m i t (with finite scales o f c o m p e t i t i o n ) yields a n average o f the r i g h t - h a n d side o f the c o r r e s p o n d i n g classical m e a n - f i e l d l i m i t , w i t h the

average

taken

over

the

P o i s s o n v a r i a t i o n i n l o c a l c r o w d i n g . It is i m p o r t a n t to u n d e r ­ stand that, a l t h o u g h the P o i s s o n l i m i t o f o u r s i m p l e n e i g h b o r ­ h o o d m o d e l differs little f r o m the classical m e a n - f i e l d l i m i t , the l o n g - d i s p e r s a l l i m i t m a y differ s t r i k i n g l y f r o m m e a n - f i e l d l i m i t s i f density d e p e n d e n c e is n o n l i n e a r (e.g., r e p l a c e E q u a t i o n 9.1 w i t h a n e x p o n e n t i a l decay; see P a c a l a a n d S i l a n d e r 1985). H a w k s a n d doves c o e x i s t i n the P o i s s o n l i m i t ( C h a p t e r 12), b u t n o t i n the classical m e a n - f i e l d l i m i t . 209

P A C A L A

A N D

L E V I N

R e t u r n i n g to the s i m p l e n e i g h b o r h o o d m o d e l w i t h

finite

scales, we b e g i n by s t u d y i n g the system e x p e r i m e n t a l l y . T y p i ­ cally, the first e x p e r i m e n t that o n e w o u l d p e r f o r m i n the

field

is a r e m o v a l e x p e r i m e n t to m e a s u r e the s t r e n g t h o f inter­ specific c o m p e t i t i o n . H e r e we m e a s u r e c o m p e t i t i o n as i t has b e e n m e a s u r e d i n l i t e r a l l y h u n d r e d s o f field e x p e r i m e n t s (see G u r e v i t c h 1992), by r e m o v i n g a s m a l l n u m b e r o f i n d i v i d u a l s o f e a c h species i n separate plots, a n d t h e n m e a s u r i n g

the

r e s u l t i n g c h a n g e s i n p o p u l a t i o n size. I n t e r s p e c i f i c c o m p e t i t i o n is q u a n t i f i e d as the p e r - c a p i t a c h a n g e f o l l o w i n g h e t e r o s p e c i f i c r e m o v a l d i v i d e d by the p e r - c a p i t a c h a n g e f o l l o w i n g c o n s p e c i f i c r e m o v a l . W e l a b e l this q u a n t i t y a^ . st

community-level strength

It is easy to s h o w that the

of competition, a^ ,

is e q u a l

st

the i n d i v i d u a l - l e v e l s t r e n g t h o f c o m p e t i t i o n , a^,

to

i n e i t h e r the

mean-field or Poisson limit. F i g u r e 9.1 shows the results o f o n e set o f r e m o v a l e x p e r i ­ m e n t s , first r e p o r t e d i n P a c a l a (1997). T h i s e x a m p l e is f o r the s y m m e t r i c case i n w h i c h f

Y

β, a n d

=f

2

= f,

μ

1

= μ

2

= μ, β\ = /3 = 2

a n d i n w h i c h a l l scales (the M ' s a n d c

M p ' s ) are e q u a l to M . I n e a c h r e m o v a l e x p e r i m e n t , the m o d e l was first i t e r a t e d a h u n d r e d t i m e u n i t s to a p p r o x i m a t e e q u i l i b ­ r i u m (on a toroidal habitat one thousand units long). Twenty p e r c e n t o f i n d i v i d u a l s o f o n e species were t h e n r e m o v e d at r a n d o m , a n d the p o p u l a t i o n size c h a n g e s w e r e m e a s u r e d over 0.2 f u r t h e r t i m e u n i t s . A l l e x p e r i m e n t s w e r e r e p l i c a t e d n i n e times. N o t e that as e x p e c t e d , a

e s t

a p p r o x i m a t e l y equals a i f the

scales are large (to p r o d u c e the triangles s h o w n , every i n d i v i d ­ ual

interacted

equally with

every o t h e r

and

Poisson). H o w e v e r , w i t h finite scales (circles) a function

o f a.

For a

d i s p e r s a l was e s t

is a

humped

sufficiently close to o n e , i n t e r s p e c i f i c

c o m p e t i t i o n at the c o m m u n i t y level (a ) est

actually b e c o m e s

w e a k e r as i n t e r s p e c i f i c c o m p e t i t i o n at the i n d i v i d u a l level

(a)

b e c o m e s stronger. T h e p a r a m e t e r values u s e d to p r o d u c e the circles i n F i g u r e 9.1 y i e l d a n average o f a p p r o x i m a t e l y ten to twenty-five i m p o r t a n t n e i g h b o r s p e r - c a p i t a (ten to twenty-five i n d i v i d u a l s w i t h i n the c e n t r a l 9 5 % o f the d i s p e r s a l a n d c o m p e ­ t i t i o n f u n c t i o n s ) . F i g u r e 9.2 shows that the effects o f

finite

scales are substantial even w i t h 5 0 - 1 2 5 i m p o r t a n t n e i g h b o r s 210

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

spatial non-spatial analytic

0.0

0.2

0.4

0.6

0.8

F I G U R E 9.1. T h e strength o f interspecific competition (a ) est

1.0

as a function

of the strength o f interindividual interference ( a ) as d e t e r m i n e d using removal experiments within the p o i n t process. T h e m e a n o f each o f the nine replicates is shown ±

one standard deviation. Circles depict runs

with M = 0.2, a n d triangles depict runs with spatially global competition and infinite dispersal (Poisson scatter o f offspring). T h e smooth curve is a plot o f E q u a t i o n 9.13. Parameter values were Δ * = 0 . 1 , / = 3.2, β = 0.28, μ = 0.4, M = 0.2, a n d the habitat was a thousand units long.

per

p l a n t ( M = 1 ) a n d that the s t r e n g t h o f c o m p e t i t i o n d e ­

creases as M

decreases. M o r e o v e r , the

a-a

e s t

relationship

appears to b e h u m p e d f o r a l l finite values o f M . F o r e x a m p l e , a r u n w i t h M = 5 a n d a = 0.99 ( a p p r o x i m a t e l y t h r e e h u n d r e d i m p o r t a n t n e i g h b o r s p e r p l a n t o n average) p r o d u c e d a n a c t u a l s t r e n g t h o f c o m p e t i t i o n o f o n l y 0.04. 211

P A C A L A

A N D

L E V I N

Τ

τ 1

τ 1

in

Τ 1

ο ο 2

0

τ 1

4

3

5

1/Μ

F I G U R E 9.2. T h e strength o f interspecific competition ( a

e s t

) as a function

o f the scales of competition a n d dispersal ( M ) . T h e horizontal axis gives the reciprocal a =

The

of M . Parameter values were as i n Figure 9.1, but with

0.96.

results i n F i g u r e s 9 . 1 - 9 . 2

questions. W h y are

leave us w i t h a series o f

these c o u n t e r i n t u i t i v e results

obtained?

W h a t e l i m i n a t e s c o m p e t i t i o n as the species b e c o m e i d e n t i c a l (as a a p p r o a c h e s 1)? A r e these p h e n o m e n a i m p o r t a n t i n r e a l ecosystems? W h y are the m e a n - f i e l d a n d P o i s s o n l i m i t s that d o m i n a t e the t h e o r e t i c a l l i t e r a t u r e so m i s l e a d i n g a n d i n c o m ­ plete? T h e answers l i e i n the i n t e r p l a y b e t w e e n stochastically s e e d e d spatial s t r u c t u r e a n d d y n a m i c s .

MOMENT

CLOSURE

A Pseudospatial

Model

B e f o r e t u r n i n g to the analysis o f the p o i n t process itself, we first c o n s i d e r a s i m p l e r system that shares its l o c a l l y stochastic d y n a m i c s . W e m a k e two c h a n g e s i n the p o i n t process. First, the e n v i r o n m e n t is d i v i d e d i n t o a g r i d w o r k o f i d e n t i c a l cells, a n d 212

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

the l o c a l densities that affect f e c u n d i t y (the L ' s i n E q u a t i o n 9.1) are s i m p l y the w i t h i n - c e l l densities. S e c o n d , w h e n a n o f f s p r i n g o f species i is p r o d u c e d , it stays w i t h i n its m o t h e r ' s c e l l w i t h p r o b a b i l i t y m a n d moves to a r a n d o m l y c h o s e n c e l l w i t h p r o b a b i l i t y 1 — m (as i n m a n y discrete-cell m o d e l s s u c h as S h m i d a a n d E l m e r 1984 a n d P a c a l a a n d T i l m a n 1994). T h e s e c h a n g e s to the p o i n t process i m p o s e b o t h a p a t c h structure that obviates i n f o r m a t i o n a b o u t w i t h i n - p a t c h l o c a t i o n s a n d a dispersal p a t t e r n that obviates i n f o r m a t i o n a b o u t the spatial a r r a n g e m e n t o f patches. i

i

L e t P(n , n ) be the p r o b a b i l i t y at t i m e t that a r a n d o m l y c h o s e n p a t c h c o n t a i n s n i n d i v i d u a l s o f species 1 a n d n i n d i v i d u a l s o f species 2. T h e p r o b a b i l i t y that a p a t c h c o n t a i n i n g n i n d i v i d u a l s o f species i loses a n i n d i v i d u a l d u r i n g the s m a l l t i m e i n t e r v a l At is s i m p l y the p r o b a b i l i t y that o n e o f the n i n d i v i d u a l s dies: μ η At. S i m i l a r l y , the p r o b a b i l i t y that the p a t c h gains a n i n d i v i d u a l is the p r o b a b i l i t y that a n i n d i v i d u a l is b o r n w i t h i n the p a t c h a n d does n o t disperse: (f — β η — α^β η )η At(l - ra ) = F^n η^At(\ - m¿), p l u s the p r o b a b i l ity that a r e c r u i t disperses i n t o the p a t c h f r o m outside it: l

2

1

2

i

i

ί

ί

i

ί

]

¿

ί

00

n ^ A t m ^ l

ί

oo

Σ

Σ n

ί

{

= 0

n

P(n ,n )n F (n ,n ) 1

2

i

i

1

(9.4)

Atm^

2

= 0

2

T h e above e x p r e s s i o n is s i m p l y the m e a n p r o d u c t i o n o f n e w species i across a l l patches, times the p r o b a b i l i t y o f dispersal. W i t h these d e f i n i t i o n s , we m a y write a n e x p r e s s i o n for t e m ­ p o r a l changes i n the p r o b a b i l i t y d i s t r i b u t i o n P(n ,n ): l

dP(n ,n ) l

2

r = -P(n ,n )\n F (n ,n )(l l

2

1

l

l

-

2

2

πι ) λ

at

+ n F (n , 2

2

— m)

2

2

+ μη

+ μη ι

n )(l

1

ι

2

+ Ρ(η

λ

-

+ nFm

2

l

l

l

l,w )[(w 2

213

1

+ l)F (n 1

nFm\ 2

l

-

2

2

l,w ) 2

P A C A L A

A N D

Χ (1 — m )

+ nF

x

l

+ P(n ,n 1

X (1 — m ) + 2

x

-

2

l)F ( η , 2

λ

n

1)

2

nFm 2

2

2

+ Ι , η ^ ί η ! + 1)/^

+ Ρ(η

λ

+ P(n

m ]

l

l)[(w

-

2

L E V I N

?2

1 ?

2

+

l)(w

2

+

(9.5)

1)μ . 2

T h e p r o b l e m w i t h the system E q u a t i o n 9.5 is that it describes a n i n f i n i t e n u m b e r o f e q u a t i o n s , o n e f o r every p o s s i b l e c o m b i ­ nation of η

a n d n . W e seek a s i m p l e r system that i n c l u d e s

λ

2

o n l y s o m e o f the i n f o r m a t i o n i n E q u a t i o n 9.5. Let N habitat

{

be the average a b u n d a n c e o f species i across the (mean

number

per

patch).

The

first

equation

in

E q u a t i o n 9.6 g o v e r n s the d y n a m i c s o f the first m o m e n t , N¡ (see the A p p e n d i x f o r its d e r i v a t i o n ) : dNdt

dt

dC — at

2 α · Α · ( 1 -mjNjC

= (1, 2), (2,1)

0

= 0[-(μ

+

ι

μ) 2

+ (1 -mMN^Nz) -

a

21

β (1 2

+ (1

— τη )Ν σ 2

2

2

—a

-m^F^Nz,^)] 1 2

β (1 λ

—

m^N^^. (9.6)

T h e p r o b l e m w i t h this e q u a t i o n is that i t d e p e n d s o n s e c o n d moments, σ

2

a n d C , as w e l l as o n the first m o m e n t s N

x

214

and

G E N E R A T I O N

O F

S P A T I A L

T h e s e s e c o n d m o m e n t s are σ ,

N.

2

2

P A T T E R N

the v a r i a n c e f r o m p a t c h to

p a t c h i n the a b u n d a n c e o f species i, a n d C , the c o r r e s p o n d i n g c o v a r i a n c e b e t w e e n the a b u n d a n c e s o f the two species: 00

C=

00

Σ

Σ

(nj

-JV )(« -iV )P( 1

2

2

M l

,n ) 2

= 0 n — 0 2

00

00

Σ

=

n

Σ

n n P(n ,n ) l

2

1

-

2

NN. X

2

= 0 n —0 2

1

T h i r d m o m e n t s are d e f i n e d as 00

Σ n

1

= 0 n

00

Σ

( » , · - N,)(nj =

2

- Nj)(n

k

-

N )P( ,n ) k

ni

2

0

w h e r e ¿, / , a n d k m a y b e e q u a l to o n e o r two i n any c o m b i n a t i o n (i.e., i = 1, j = 1, k = 2), a n d so o n to f o u r t h a n d h i g h e r m o m e n t s . A l t h o u g h i n this case the e q u a t i o n s f o r the

means

d e p e n d o n l y o n the first two m o m e n t s , i n o t h e r cases h i g h e r m o m e n t s w i l l also be i n v o l v e d . T h e r e a s o n that t h i r d a n d h i g h e r m o m e n t s d o n o t arise i n the e q u a t i o n s f o r the

means

i n E q u a t i o n 9.6 is that the d e n s i t y - d e p e n d e n t f u n c t i o n s (the are l i n e a r ; h i g h e r m o m e n t s w o u l d result w i t h n o n l i n e a r

F^s)

functions. T h e s e c o n d m o m e n t s i n the first e q u a t i o n i n E q u a t i o n 9.6 a c c o u n t f o r the effects o f b i o l o g i c a l l y g e n e r a t e d spatial structure. T h e t e r m 2V + σ /N¡ 2

¿

is the m e a n l o c a l density o f c o n -

specifics i n the p a t c h o f a r a n d o m l y c h o s e n i n d i v i d u a l o f species i (rather t h a n i n a r a n d o m l y c h o s e n p a t c h ) . T h e m e a n l o c a l density grows w i t h the b e t w e e n - p a t c h v a r i a n c e , σ \ 2

be­

cause i n c r e a s i n g n u m b e r s o f i n d i v i d u a l s are l o c a t e d i n clusters as σ

2

increases. S i m i l a r l y , N- + C/Nj

is the m e a n l o c a l density

o f heterospecifics i n the p a t c h o f a r a n d o m l y c h o s e n species i. T h i s density is elevated above the g l o b a l m e a n , Nj, i f the two species are spatially a g g r e g a t e d ( C > 0) a n d d e p r e s s e d b e n e a t h i t i f the species are spatially segregated ( C < 0). N o t e that i f b o t h the v a r i a n c e a n d c o v a r i a n c e are e q u a l to z e r o , t h e n the 215

P A C A L A

first

A N D

L E V I N

e q u a t i o n i n E q u a t i o n s 9.6 is i d e n t i c a l to the classical

m e a n - f i e l d l i m i t s , whereas i f the spatial d i s t r i b u t i o n is P o i s s o n (cr

2 ¿

= iV- a n d C = 0), the

first

e q u a t i o n is i d e n t i c a l to

the

long-dispersal limit. T h e e q u a t i o n s f o r the m e a n s , Ν

λ

and N, 2

d o n o t constitute a

c l o s e d system o f e q u a t i o n s , because the variances a n d c o v a r i ances are themselves f u l l state variables that w i l l c h a n g e t h r o u g h t i m e because o f l o c a l i n t e r a c t i o n s a n d finite dispersal. T o close the system, we r e q u i r e the e q u a t i o n s g o v e r n i n g the d y n a m i c s of

σ, 2

cr , a n d 2

2

C. T h i s illustrates the

difference

between

E q u a t i o n s 9.5 a n d 9.6 a n d the k i n d o f spatial m o d e l s that d o m i n a t e the t h e o r e t i c a l l i t e r a t u r e . S i m p l y a d d i n g

diffusion

terms to e q u a t i o n s g o v e r n i n g m e a n densities is n o t sufficient to d e s c r i b e stochastic processes w i t h l o c a l i n t e r a c t i o n s a n d

finite

dispersal. I n the A p p e n d i x we d e r i v e the

equations

governing

the

variances a n d covariances. T h e s e e q u a t i o n s c o n t a i n t h i r d m o ­ m e n t s . B e c a u s e the t h i r d m o m e n t s are a g a i n state variables, the e q u a t i o n s g o v e r n i n g the first two m o m e n t s d o n o t c o n s t i ­ tute a c l o s e d system. T h i s illustrates the c r u x o f the p r o b l e m : A n y system o f e q u a t i o n s g o v e r n i n g the first η m o m e n t s c o n ­ tains yet h i g h e r m o m e n t s . W h a t is n e e d e d is s o m e c l o s u r e r u l e that e i t h e r states that s o m e m o m e n t s are n e g l i g i b l e o r e x ­ presses the h i g h e r m o m e n t s i n terms o f l o w e r m o m e n t s . H e r e we a d o p t the s i m p l e s t c l o s u r e r u l e . I n the A p p e n d i x we s h o w that the

terms c o n t a i n i n g the

third moments

in

the

e q u a t i o n s f o r the variances a n d covariances are n e g l i g i b l e i f both mean numbers o f individuals per patch and

movement

rates are n o t t o o s m a l l . B y o m i t t i n g these n e g l i g i b l e terms, we arrive at the a p p r o x i m a t e a n d c l o s e d system i n E q u a t i o n s 9.6. T h e a c c u r a c y o f the a p p r o x i m a t i o n 9.6 is assessed s i m p l y by c o m p a r i n g the m o m e n t s p r e d i c t e d by E q u a t i o n s 9.5 a n d 9.6 ( F i g u r e s 9.3 a n d 9.4). F o r e x a m p l e , c o n s i d e r the

symmetric

case u s e d to p r o d u c e F i g u r e s 9 . 1 - 9 . 2 ( w i t h e q u a l / ' s , /x's, /3's, a ' s , a n d ra's). T h e r e l a t i o n s h i p b e t w e e n the c o m m u n i t y - a n d i n d i v i d u a l - l e v e l strengths o f c o m p e t i t i o n p r e d i c t e d by a p p r o x i ­ m a t i o n 9.6 is a close a p p r o x i m a t i o n o f the r e l a t i o n s h i p p r e ­ d i c t e d by the a c t u a l system ( E q . 9.5) i n m o s t cases ( F i g u r e 9.3). 216

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

Strength of Interference (a) F I G U R E 9.3. E q u a t i o n 9.7 as predicted by the pseudospatial m o d e l ( E q . 9.5) (labeled exact) a n d the m o m e n t Equations 9.6 (labeled appr.). T h e value f r o m the m o m e n t equations with a = 1.0 a n d m = 0.09 is missing because the equations failed to converge for this set of parameter values. O t h e r parameters were / = 2.2, μ = 0.88, a n d β = 0.066.

The

a p p r o x i m a t i o n fails w i t h s m a l l m a n d

a

close to

one

because stochastic drift o f the relative a b u n d a n c e s w i t h i n e a c h p a t c h is t h e n large, a n d this leads to large t h i r d

moments

(because m o s t patches c o n t a i n p r i m a r i l y e i t h e r species 1 o r species 2). N o t e that the a~a

est

r e l a t i o n s h i p g i v e n by E q u a t i o n s 9.5 a n d

9.6 m a y be h u m p e d , b u t o n l y i f the m o v e m e n t rate m is s m a l l . I n contrast, o u r n u m e r i c a l w o r k suggests that the r e l a t i o n s h i p is h u m p e d i n the p o i n t process f o r any finite value o f M (as i n F i g u r e 9.1). B e c a u s e o f the

analytically tractable

a p p r o x i m a t i o n (9.6),

we are n o w i n a p o s i t i o n to e x p l a i n the o c c u r r e n c e o f a n inverse r e l a t i o n s h i p b e t w e e n a a n d a

e s t

. U s i n g the first equa­

t i o n i n E q u a t i o n 9.6, it is easy to s h o w f o r a s m a l l n u m b e r o f 217

G E N E R A T I O N

O F

(c)

S P A T I A L

P A T T E R N

Equilibrium Mean Abundances 20

5 -

0

0

0.2

0.4 0.6 Strength of Interference (a)

0.8

F I G U R E 9.4. T h e moments at e q u i l i b r i u m predicted by the pseudospatial m o d e l ( E q . 9.5) (labeled exact) a n d the m o m e n t Equations 9.6 (labeled appr.).

Parameter

interspecific

values

covariance.

were (b)

as

in

Figure

9.4.

(a)

E q u i l i b r i u m intraspecific

Equilibrium variance,

(c)

E q u i l i b r i u m means.

i n d i v i d u a l s r e m o v e d i n the r e m o v a l e x p e r i m e n t that

a

est

1 +

a

1 +

C/Ñ

(9.7)

σ /Ν 2

w h e r e the hats above the m o m e n t s signify e q u i l i b r i u m ; a

e s t

is

d e p r e s s e d b e n e a t h a because the spatial c o v a r i a n c e is negative at e q u i l i b r i u m . N e g a t i v e c o v a r i a n c e i m p l i e s i n t e r s p e c i f i c spatial s e g r e g a t i o n . I f c o v a r i a n c e is sufficiently negative, t h e n

each

p a t c h c o n t a i n s p r i m a r i l y o n l y o n e o f the two species. T h e cause o f the s p o n t a n e o u s spatial s e g r e g a t i o n i n the m o d e l is a p r o cess d i r e c t l y a n a l o g o u s to g e n e t i c drift, w h i c h r e m o v e s g e n e t i c polymorphism

in

small

populations. Local

populations i n

the m o d e l s are k e p t s m a l l by density d e p e n d e n c e , 219

allowing

P A C A L A

relatively large r a n d o m (demographic

A N D

fluctuations

stochasticity). T h e

L E V I N

i n l o c a l relative a b u n d a n c e random

"drift"

of

either

species t o w a r d h i g h l o c a l relative a b u n d a n c e is r e i n f o r c e d by l o c a l dispersal, w h i c h biases the c o m m u n i t y c o m p o s i t i o n o f n e w r e c r u i t s i n favor o f the l o c a l l y m o s t a b u n d a n t species. N o t e that the d e g r e e o f the spatial s e g r e g a t i o n p r e d i c t e d by the p s e u d o s p a t i a l m o d e l at e q u i l i b r i u m (the c o v a r i a n c e i n E q . 9.7) increases as dispersive c o u p l i n g decreases ( F i g u r e 9.4a). T h i s e x p l a i n s the c o r r e s p o n d i n g result i n F i g u r e 9.2 f r o m the p o i n t process. A g a i n , l o c a l d i s p e r s a l facilitates l o c a l " e c o l o g i c a l d r i f t . " T h e segregation also increases as i n t e r s p e c i f i c c o m p e t i ­ t i o n at the i n d i v i d u a l level s t r e n g t h e n s ( F i g u r e 9.4a) because the d e t e r m i n i s t i c forces o p p o s i n g drift t o w a r d l o c a l m o n o d o m i n a n c e decrease i n s t r e n g t h as a a p p r o a c h e s o n e . A h u m p i n the r e l a t i o n s h i p b e t w e e n a a n d a

e s t

o c c u r s w h e n the i n c r e a s ­

i n g spatial segregation o v e r w h e l m s the i n c r e a s i n g s t r e n g t h o f individual-level competition. C h a n g e s i n m a n d a also affect a

t h r o u g h c h a n g e s i n the

e s t

variances (see E q . 9.7). T h e s e c h a n g e s g e n e r a l l y c o m p l e m e n t those d e s c r i b e d above; within-species spatial a g g r e g a t i o n ( v a r i ­ a n c e to m e a n r a t i o greater t h a n o n e ) d e v e l o p s to a c c o m p a n y the between-species

s e g r e g a t i o n ( F i g u r e 9.4b). H o w e v e r ,

the

effect o f the v a r i a n c e is c o m p l i c a t e d by the fact that w i t h s m a l l a, i n c r e a s e d c o u p l i n g m a y e i t h e r increase c l u s t e r i n g ( σ or overdispersion ( σ

2

< Ν)

>

Ν)

d e p e n d i n g o n the values of f,

β,

2

a n d μ. It is possible to derive f r o m E q u a t i o n 9.6 useful

simple

f o r m u l a e f o r the spatial m o m e n t s at e q u i l i b r i u m . F o r e x a m p l e , i f e q u i l i b r i u m p o p u l a t i o n sizes are large a n d m is close to o n e t h e n the c o v a r i a n c e at e q u i l i b r i u m is a p p r o x i m a t e l y

C « -A^iVcXl and

a

est

αβ •)

—

(9.8)

is a p p r o x i m a t e l y

a

e s t

« a 1 -

(1

220

(9.9)

N o t e that a

e s t

is a h u m p e d f u n c t i o n o f a , b u t that the h u m p

c a n o c c u r f o r a b e t w e e n z e r o a n d o n e o n l y i f m is sufficiently s m a l l ( a n d β/μ>

0.5, F i g u r e 9.5).

F i n a l l y , a l t h o u g h we d o n o t have the space to treat this subject fully, the m o d e l 9 . 5 - 9 . 6 also p r e d i c t s n e w a n d u n e x ­ p l o r e d m e c h a n i s m s o f c o e x i s t e n c e . F o r e x a m p l e , F i g u r e 9.6 shows a n i n t e r e s t i n g case i n w h i c h the two species are u n a b l e to coexist i n e i t h e r the m e a n - f i e l d o r P o i s s o n l i m i t s because a

n

= a2i = 1. I n e i t h e r o f these l i m i t s , species 2 is the supe­

r i o r c o m p e t i t o r , b u t species 1 has a h i g h e r p o p u l a t i o n g r o w t h rate i n u n c r o w d e d c o n d i t i o n s . Species 1 is weedy, w i t h early successional vital rates, whereas species 2 has late-successional vital rates. It is easy to c o n s t r u c t a s i m p l e s u b m o d e l o f r e s o u r c e c o m p e t i t i o n to show that species 2 has the lowest R* Tilman

(sensu

1982). N o t e that the two species c o e x i s t i f we give

species 1 the m o r e r a p i d d i s p e r s a l to c o m p l e m e n t its w e e d y vital

rates ( F i g u r e 9.6,

t h e r e is m o r e

m

l

= 0.9

and

happening here than

ra

2

= 0.5).

c o l o n i z a t i o n trade-off. N o t e that i f we i n c r e a s e 221

However,

a simple competitionτϊΙλ

still f u r t h e r

P A C A L A

A N D

1

L E V I N

Γ

1.4 -

O

20

40

60

time

80

100

F I G U R E 9 . 6 . T w o runs of the pseudospatial m o d e l ( E q . 9 . 5 ) i n which species 1 c o m p e t e d against an e q u i l i b r i u m m o n o c u l t u r e Invasion succeeded if m

= 0 . 9 but failed if m

1

values include f 0.02, m

2

x

= 2.2, μ

x

1

= 0.2, and a

l

2

= 0 . 2 0 , ft = 0 . 2 , f

2

= o¿

21

l

=

1.0

or m

l

=

= 1.05, μ

2

= 0.8, β

2

=

= 1. W i t h these parameter values, species

2 h a d a larger e q u i l i b r i u m i n m o n o c u l t u r e m

o f species 2.

= 1.0. O t h e r parameter

than species 1 with either

0.9.

to 1.0, t h e n species 1 goes e x t i n c t ( F i g u r e 9.6). I f m

x

l a r g e , t h e n species 1 c a n n o t d e v e l o p the d e g r e e

is t o o

o f spatial

s e g r e g a t i o n it r e q u i r e s to c o e x i s t w i t h species 2. F o r s i m p l i c i t y , we n o w restrict o u r a t t e n t i o n to the simplest case, i n v o l v i n g the smallest possible p e r t u r b a t i o n o f the l o n g d i s p e r s a l (Poisson) l i m i t . W e assume that m

2

= 1 so that species

2 always has a P o i s s o n d i s t r i b u t i o n a n d that m

1

is o n l y slightly

less t h a n o n e . W e f u r t h e r stack the d e c k against i n t e r e s t i n g b e h a v i o r b y a s s u m i n g that a coexistence a n d founder

l 2

= a

= 1. T h i s e l i m i n a t e s the

2 l

c o n t r o l i n b o t h the P o i s s o n

and

mean-field limits. T h e a s s u m p t i o n that m

2

= 1 and m

l

is n e a r o n e ensures that

spatial a g g r e g a t i o n a n d s e g r e g a t i o n i n the m o d e l w i l l be s m a l l . T h u s , to observe the effects o f space, we m u s t c o r r e s p o n d i n g l y 222

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

w e a k e n the n o n s p a t i a l forces i n the m o d e l . L e t X

be

pi

the

e i g e n v a l u e g o v e r n i n g the i n v a s i o n o f species i i n a n e q u i l i b ­ r i u m m o n o c u l t u r e o f species j i n the P o i s s o n case (m

l

1). It is easy to s h o w that assume that b o t h X

=f

i

and X

pl

- μ

-

ί

2

We

—

Y

—

=

2

are n e a r z e r o , i n d i c a t i n g near-

p2

c o m p e t i t i v e e q u i v a l e n c e i n the P o i s s o n l i m i t (e.g., (f is close to (f

= m

- μ^/β^

μ )/β λ

λ

μ )/β ). 2

2

R e t u r n i n g to the case i n w h i c h m

< 1, species 1 c a n i n v a d e

l

a n e q u i l i b r i u m m o n o c u l t u r e o f species 2 (first c o n d i t i o n b e l o w ) , a n d species 2 c a n i n v a d e a n e q u i l i b r i u m m o n o c u l t u r e o f species 1 (second condition) i f 0 < λ , ! + (1 -

0 < λ

where

5 = β ((/ χ

ρ 2

+ (1 - πι )β ^Α λ

- μ )/β

2

π ι ^ β ^ - Α )

2

-

2

- S ^J

2

+ μ )

1)/(μ

and

2

λ

(9.10)

Α = 1 +

β / μ . C o n d i t i o n 9.10 is d e r i v e d u s i n g a s t a n d a r d l o c a l stability ι

ι

analysis, a s s u m i n g that a

l 2

terms o f o r d e r (1 - m^ ,

\

2

= a p j

2 l

, X, pi

= m

2

XX, pi

pj

— 1, a n d o m i t t i n g a l l A ( l - mj, p i

X (l

-

pj

nij) o r h i g h e r . T h e q u a n t i t i e s S a n d A are easily i n t e r p r e t a b l e as spatial effects. Q u a n t i t y S describes the b u i l d u p o f spatial s e g r e g a t i o n d u r i n g i n v a s i o n . T o see this, r e p l a c e dC/dt

i n the last e q u a t i o n

i n E q u a t i o n s 9.6 by z e r o a n d solve f o r the e q u i l i b r i u m c o v a r i ­ a n c e C(N N )*. 19

T h e n d i v i d e C(N N )*

2

l9

by N

2

a n d set

l9

N

2

e q u a l to the m o n o c u l t u r e e q u i l i b r i u m f o r species 2 (iV * = 2

(f

2

-

μ )/β 2

2

-

1): C(N NÍ) l9

lim

—

= -5(1 - Wj).

(9.11)

T h u s , S quantifies h o w spatial s e g r e g a t i o n grows relative to the m e a n d u r i n g invasion. S i m i l a r l y , A describes the b u i l d u p o f i n t r a s p e c i f i c aggrega­ t i o n by species 1 d u r i n g i n v a s i o n (species 2 c a n n o t aggregate because m

= 1). R e p l a c i n g da /dt 2

2

w i t h z e r o i n E q u a t i o n s 9.6

a n d s o l v i n g f o r the e q u i l i b r i u m v a r i a n c e σ (Ν 2

ΐ9

223

N )*, 2

we

find

P A C A L A

A N D

L E V I N

that

lim

a*(N N?) l9

=

= A(1

Ν,

-

If A is positive, t h e n species 1 clusters as i t invades because grows faster t h a n N

v

1 becomes

σ

2

I n contrast, i f A is negative, t h e n species

evenly d i s p e r s e d as it invades. P u t t i n g a l l this

t o g e t h e r , we see f r o m E q u a t i o n 9.10 that species 1 w i l l i n v a d e i f i n t e r s p e c i f i c s e g r e g a t i o n grows sufficiently faster t h a n aggrega­ t i o n a n d w i l l fail to i n v a d e i f a g g r e g a t i o n grows sufficiently faster t h a n segregation. T o e x p o s e o n l y the spatial effects, it is c o n v e n i e n t to c o n ­ sider the f u r t h e r r e s t r i c t i o n that λ

= λ

ρλ

species w i l l c o e x i s t i f 1 < S/A

ι

ably e x c l u d e species 2 i f 1 < S/A w i l l e x c l u d e species founder

control will

ρ 2

2

and μ/ λ

1 i f 1 > S/A result i f μ /μ 1

and 2

— 0. T h e n , the two species 1 w i l l i n v a r i ­

< μ /μ ;

μ

2

μ /μ λ

< S/A

< S/A; 2

species 2

< S/A;

and

< 1. W e

have

c h e c k e d a l l o f these o u t c o m e s b o t h i n the exact system 9.5 a n d i n the a p p r o x i m a t i o n 9.6; spatial effects a l o n e are c a p a b l e o f p r o d u c i n g any o f the f o u r d i f f e r e n t o u t c o m e s o f c o m p e t i t i o n . T w o p r i m a r y c o n c l u s i o n s e m e r g e f r o m the stability analysis. First, l o n g d i s p e r s a l is n o t always b e n e f i c i a l . It facilitates per­ sistence i f the t e n d e n c y to segregate is l a r g e r t h a n the

ten­

d e n c y to cluster b u t i m p e d e s persistence otherwise. S e c o n d , the i n t e r p l a y o f l o c a l c o m p e t i t i o n a n d d y n a m i c s is d y n a m i c a l l y r i c h ; it has the p o t e n t i a l to t r a n s f o r m any o f the f o u r q u a n t i t a ­ tively d i f f e r e n t

o u t c o m e s o f c o m p e t i t i o n i n the P o i s s o n o r

mean-field limits (coexistence, f o u n d e r c o n t r o l , exclusion o f species 1, o r e x c l u s i o n o f species 2) i n t o any o t h e r o u t c o m e . The Point

Process

W e n o w r e t u r n to the p o i n t process that p r o d u c e d F i g u r e s 9 . 1 - 9 . 3 . T h e p o i n t process is m o r e difficult t h a n the p s e u d o spatial m o d e l above because t h e r e is n o s i m p l e e x p r e s s i o n a n a l o g o u s to E q u a t i o n 9.5. H o w e v e r , we c a n p r o c e e d d i r e c t l y to the m o m e n t a p p r o x i m a t i o n s a n a l o g o u s to E q u a t i o n s 9.6. 224

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

B o l k e r a n d P a c a l a (1997) s h o w that the e q u a t i o n f o r

the

m e a n a b u n d a n c e o f species i i n the p o i n t process converges to the first e q u a t i o n i n E q u a t i o n s 9.6 w i t h

(9.12) w h e r e the U's are g i v e n by E q u a t i o n 9.3. T h e f u n c t i o n c^iy)

is

the i n t e r s p e c i f i c spatial c o v a r i a n c e f u n c t i o n . C o n s i d e r m a n y pairs o f s m a l l q u a d r a t e s o f a r e a A, w i t h the m e m b e r s o f e a c h p a i r a distance y apart. T h e e x p e c t e d c o v a r i a n c e b e t w e e n

the

o f species 1 i n o n e m e m b e r o f a p a i r a n d species 2 i n

densities

the o t h e r is c^iy)

i n the l i m i t o f s m a l l A. S i m i l a r l y , c (y) u

gives

the within-species spatial c o v a r i a n c e . T o c o m p l e t e the m o d e l , B o l k e r a n d P a c a l a (1997) d e r i v e d i n t e g r o - p a r t i a l d i f f e r e n t i a l e q u a t i o n s f o r the d y n a m i c s o f c (y), 22

a n d c (y). 12

c (y), u

D u r i n g the d e r i v a t i o n , they c l o s e d the sys­

t e m by a s s u m i n g that a l l t h i r d c e n t r a l m o m e n t s were

zero,

exactly as i n the d e r i v a t i o n o f E q u a t i o n s 9.6 ( B o l k e r a n d P a c a l a 1997). P a c a l a (1997) r e p o r t e d

an equation for a

s o l v i n g f o r the e q u i l i b r i u m values o f N

lf

N, 2

e s t

, o b t a i n e d by

c (y), u

c (y), 22

and

c (y): 12

F o r s i m p l i c i t y , this e x p r e s s i o n is p r e s e n t e d f o r the s p e c i a l case i n w h i c h M is relatively large a n d a is close to o n e . T h e fact that a

e s t

i n E q u a t i o n 9.13 m a y be negative f o r very s m a l l M is

a n artifact o f this a p p r o x i m a t i o n . H o w e v e r , the p l o t i n F i g u r e 9.1 shows that E q u a t i o n 9.13 g e n e r a l l y p r o v i d e s a r e a s o n a b l y accurate a p p r o x i m a t i o n ( n o t e that M

is relatively s m a l l i n

F i g u r e 9.1). U n l i k e the p s e u d o s p a t i a l system (Eqs. 9 . 5 - 9 . 6 ) , b u t consist­ ent w i t h the s i m u l a t i o n s o f the p o i n t process, e x p r e s s i o n 9.13 225

P A C A L A

p r e d i c t s that the a - a

A N D

L E V I N

r e l a t i o n s h i p is h u m p e d even f o r large

e s t

values o f M. N o t e i n e x p r e s s i o n 9.13 that the h u m p w i l l always o c c u r at a value o f a b e t w e e n z e r o a n d o n e , w i t h the

hump

closer to o n e the l a r g e r the e c o l o g i c a l scales M . T h e d i f f e r e n c e b e t w e e n the p s e u d o s p a t i a l m o d e l a n d p o i n t process is e x p l a i n e d by the between-species

the

covariance

( r e c a l l e x p r e s s i o n 9.7). I n the same case l e a d i n g to e q u a t i o n 9.13: C(y)~

N e

-(\y\/M /(l-a)N/^ )}

g

(

u

)

2My(l - α)Ν/μ B e c a u s e o f the I — a

t e r m i n the d e n o m i n a t o r o f E q u a t i o n

9.14, the i n t e r s p e c i f i c spatial c o v a r i a n c e b e c o m e s s t r o n g l y n e g ­ ative as a a p p r o a c h e s o n e . T h i s i m p l i e s that the two species s p o n t a n e o u s l y segregate i n t o large m o n o s p e c i f i c p a t c h e s i n the p o i n t process i f a is close to o n e (even i f M is large), t h e r e b y c a u s i n g the o b s e r v e d d r o p i n a

e s t

t o w a r d z e r o ( F i g u r e 9.1).

T h e p h e n o m e n o n is s i m i l a r to the spatial c l u s t e r i n g e x h i b i t e d by s o m e discrete i n t e r a c t i n g p a r t i c l e systems, s u c h as the v o t e r m o d e l ( D u r e t t a n d L e v i n 1994a). T h e difference between

the p s e u d o s p a t i a l m o d e l a n d

the

p o i n t process is that e x t r e m e levels o f spatial s e g r e g a t i o n o c c u r o n l y at very l o w levels o f c o u p l i n g ( s m a l l m) i n the

pseudo-

spatial m o d e l , b u t at large o r s m a l l e c o l o g i c a l scales ( M ) i f a is n e a r o n e i n the p o i n t process ( c o m p a r e e x p r e s s i o n s 9.8 a n d 9.13). B e c a u s e the

spatial d i s t r i b u t i o n i n the p o i n t

process

b r e a k s i n t o ever l a r g e r a n d m o r e m o n o s p e c i f i c p a t c h e s as

a

a p p r o a c h e s o n e , the effective level o f b e t w e e n - p a t c h c o u p l i n g progressively decreases, u l t i m a t e l y c a u s i n g d y n a m i c s a n a l o g o u s to those i n the p s e u d o s p a t i a l m o d e l w i t h very l o w ra's. E x p l i c i t space b r i n g s w i t h it the p o t e n t i a l f o r s e l f - o r g a n i z a t i o n o f a loosely c o u p l e d metapopulation.

CONCLUSIONS T h r e e p r i m a r y results e m e r g e f r o m o u t analysis o f the m o ­ m e n t e q u a t i o n s : (1) W i t h interactions, a

e s t

finite

d i s p e r s a l a n d spatially l o c a l

is a h u m p e d f u n c t i o n o f a. T h e h u m p moves 226

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

to the r i g h t as the scales o f c o m p e t i t i o n a n d d i s p e r s a l i n c r e a s e . A s a result, s t r o n g c o m p e t i t i o n at the i n d i v i d u a l level leads to weak c o m p e t i t i o n at the c o m m u n i t y level. T h i s p h e n o m e n o n is c a u s e d by a n e c o l o g i c a l a n a l o g u e o f g e n e t i c drift. A s a proaches

one,

the

d e t e r m i n i s t i c forces

ap­

o p p o s i n g stochastic

e c o l o g i c a l drift decrease, a n d the t e n d e n c y o f e i t h e r species to drift to l o c a l e x t i n c t i o n ( o r

fixation)

increases, w h i c h gives

d i s t i n c t patches o f e a c h species. It is i m p o r t a n t to u n d e r s t a n d that the d e p r e s s i o n o f c o m m u n i t y - l e v e l c o m p e t i t i o n by this m e c h a n i s m is p o t e n t i a l l y large relative to m o r e w i d e l y r e c o g ­ n i z e d m e c h a n i s m s . (2) B e c a u s e a

e s t

decreases as M

decreases,

s h o r t scales r e d u c e c o m m u n i t y - l e v e l c o m p e t i t i o n . S m a l l c o m ­ petitive n e i g h b o r h o o d s r e d u c e c o m p e t i t i o n because they f a c i l i ­ tate the stochastic d r i f t that leads to spatial s e g r e g a t i o n . S h o r t d i s p e r s a l f u r t h e r isolates l o c a l r e g i o n s f r o m o n e a n o t h e r a n d r e i n f o r c e s drift. (3) I n t e r s p e c i f i c spatial s e g r e g a t i o n c a n cause c o e x i s t e n c e . I n contrast to the advantages o f l o n g d i s p e r s a l i n fugitive-species m o d e l s (e.g., L e v i n s a n d C u l v e r 1971), s h o r t d i s p e r s a l facilitates persistence i f a species' t e n d e n c y to segre­ gate exceeds its t e n d e n c y to aggregate ( c o n d i t i o n 9.10). T h r e e i n d e p e n d e n t lines o f evidence provide e m p i r i c a l sup­ p o r t f o r s o m e o f the above p h e n o m e n a . First, spatial segrega­ t i o n is n e a r l y u b i q u i t o u s a n d easy to observe. I n any p l a c e w h e r e m u l t i s p e c i e s v e g e t a t i o n c a n b e v i e w e d f r o m above, c o u n t i n d i v i d u a l s o f a species close to r a n d o m l y s e l e c t e d p o i n t s (e.g., w i t h i n a c i r c l e w i t h r a d i u s e q u a l to c a n o p y h e i g h t ) a n d t h e n close to r a n d o m l y selected i n d i v i d u a l s o f a n o t h e r species. A first estimate that is (1 — b)% o f the s e c o n d i m p l i e s a p p r o x i ­ m a t e l y a b% r e d u c t i o n i n the s t r e n g t h o f c o m p e t i t i o n ( i f the n e i g h b o r h o o d r a d i u s u s e d is a p p r o p r i a t e ) . I f y o u try this, y o u w i l l c o m m o n l y observe biases i n excess o f 5 0 % . S e c o n d , i n t h r e e o f f o u r instances, o b s e r v e d levels o f l o c a l spatial segrega­ t i o n were s h o w n to have a large effect o n c o m m u n i t y - l e v e l competition

in

field-calibrated

m o d e l s (yes

i n Pacala

and

D e u t s c h m a n 1996; C a i n et a l . 1995; a n d Rees, G r u b b , a n d K e l l y 1996; n o i n P a c a l a a n d S i l a n d e r 1990). M o r e o v e r , spatial segre­ g a t i o n arises s p o n t a n e o u s l y i n d a t a - d e f i n e d m o d e l s (see P a c a l a et a l . 1996; P a c a l a a n d D e u t s c h m a n 1996). T h i r d , as r e p o r t e d i n P a c a l a (1996), K e l l y a n d T r i p l e r ( u n p u b l i s h e d ) r e v i e w e d o v e r 227

P A C A L A

A N D

L E V I N

t h r e e h u n d r e d p u b l i s h e d e x p e r i m e n t a l field slides o f c o m p e t i ­ t i o n . T h e y f o u n d that p l a n t - c e n t e r e d e x p e r i m e n t s i n w h i c h n e i g h b o r s were r e m o v e d f r o m q u a d r a n t s c e n t e r e d o n

focal

p l a n t s r e p o r t e d statistically s i g n i f i c a n t c o m p e t i t i o n at n e a r l y t h r e e times the rate o f e x p e r i m e n t s i n w h i c h plots were estab­ l i s h e d w i t h o u t r e f e r e n c e to the l o c a t i o n s o f p l a n t s ( 7 2 % versus 26%

of

experiments

A p p a r e n t l y , the

showed

fine-scale

competition,

respectively).

spatial s e g r e g a t i o n p r e s e n t i n the

p l o t - c e n t e r e d e x p e r i m e n t s b u t n o t i n the p l a n t - c e n t e r e d e x p e r ­ iments caused

the

t h r e e f o l d r e d u c t i o n i n the

interspecific

competition detected. O b v i o u s l y , the above e v i d e n c e f o r the causes a n d

conse­

q u e n c e s o f spatial s e g r e g a t i o n is i n c o m p l e t e . E x p e r i m e n t s are n e e d e d to test o u r f o u r t h e o r e t i c a l results against alternative hypotheses. T h e m o s t l i k e l y alternative h y p o t h e s i s is that h a b i ­ tats are spatially h e t e r o g e n e o u s at fine scales a n d that species coexist a n d segregate, n o t because o f finite d i s p e r s a l a n d l o c a l i n t e r a c t i o n s a l o n e , b u t because e a c h species is the d o m i n a n t c o m p e t i t o r i n a d i f f e r e n t h a b i t a t type. O n e p a r t i c u l a r l y w o r t h ­ w h i l e e x p e r i m e n t w o u l d be to t r a n s p l a n t plants f r o m a n a t u r a l c o m m u n i t y : (1) to a n e w l o c a t i o n w i t h s c r a m b l e d spatial struc­ ture, (2) to a n e w l o c a t i o n w i t h p r e s e r v e d spatial structure, (3) to the same l o c a t i o n w i t h s c r a m b l e d spatial structure, a n d (4) to the same l o c a t i o n w i t h p r e s e r v e d spatial structure ( o b v i o u s l y w i t h replicates). C o m p a r i s o n s o f the r e s u l t i n g d y n a m i c s i n 1 a n d 3 against 2 a n d 4 w o u l d s h o w the i m p o r t a n c e o f

fine-scale

spatial structure, whereas c o m p a r i s o n o f 1, 2, a n d 3 against 4 w o u l d test the alternative h y p o t h e s i s o f physically

heteroge­

n e o u s habitat. B y f o l l o w i n g the t r e a t m e n t s t h r o u g h t i m e , o n e c o u l d also observe the d y n a m i c s o f spatial p a t t e r n

to g a i n

f u r t h e r i n s i g h t i n t o the genesis o f spatial s e g r e g a t i o n . T h e m e t h o d s p r e s e n t e d h e r e a l l o w o n e to d e r i v e the m a c r o ­ s c o p i c p o p u l a t i o n d y n a m i c e q u a t i o n s i m p l i e d by m e a s u r a b l e m i c r o s c o p i c rules g o v e r n i n g b i r t h s , deaths, a n d m o v e m e n t by i n d i v i d u a l s . T h e s e m e t h o d s m a y be a p p l i e d to m o d e l s o f v i r t u ­ ally any e c o l o g i c a l i n t e r a c t i o n . T h e y p r o m i s e to close the gap i n the c u r r e n t e x p l a n a t o r y t h e o r y c o n t a i n i n g the causes a n d consequences o f biologically generated 228

heterogeneity.

G E N E R A T I O N

O F

S P A T I A L

P A T T E R N

ACKNOWLEDGMENTS We

gratefully

Foundation,

acknowledge

NASA

the

support

(NAGW-3741,

of

the

NAGW-468),

Mellon

the

NSF

( D E B - 9 2 2 1 0 9 7 ) , a n d the D O E ( D E - F G 0 4 - 9 4 E R 6 1 8 1 5 ) .

APPENDIX Derivation Let

of the Moment

the r a n d o m v a r i a b l e q

Equations

9.6

be the n u m b e r o f species i

it

individuals within a patch. By definition N

is the m e a n o f

{

is the v a r i a n c e , a n d C is the c o v a r i a n c e b e t w e e n q

σ

2

lt

D u r i n g the s m a l l t i m e i n t e r v a l At, amount

—1 w i t h p r o b a b i l i t y μ ^

q, it

and

w i l l c h a n g e by the

q

it

a n d by the a m o u n t

At

+1

with probability: [?,·