Self-Organization and Dissipative Structures: Applications in the Physical and Social Sciences 9781477300312

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Self-Organization and Dissipative Structures

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Self-Organization and Dissipative Structures Applications in the Physical and Social Sciences

Edited by William C. Schieve and Peter M. Allen

University of Texas Press, Austin

Copyright © 1982 by the University of Texas Press All rights reserved Printed in the United States of America First Edition, 1982 Requests for permission to reproduce material from this work should be sent to Permissions, University of Texas Press, Box 7819, Austin, Texas 78712.

Library of Congress Cataloging in Publication Data Main entry under title: Self-organization and dissipative structures. “Based upon the proceedings of the Workshop on Dissipative Structures in the Social and Physical Sciences, held in Austin in September 1978 in honor of Ilya Prigogine.” 1. Self-organizing systems—Addresses, essays, lectures. 2. Chemistry, Physical and theoretical— Addresses, essays, lectures. 3. Biological chemistry—Addresses, essays, lectures. 4. Social structure—Addresses, essays, lectures. 5. Prigo­ gine, Ilya. I. Schieve, W. C. II. Allen, Peter M. (Peter Murray), 1944III. Workshop on Dissipative Structures in the Social and Physical Sciences (1978 : Austin, Tex.) Q325.S44 001.53'3 81-11404 ISBN 0-292-70354-6 AACR2

This volume is dedicated to Ilya Prigogine, whose extraordinary insights in so many fields have had a profound impact on the physical sciences and are now exciting wide interest in the social and biological sciences. During his entire scientific career his focus has been the central issue of far-fromequilibrium phenomena, and from this has come a new paradigm for the evolution of complex systems. It is hoped that this volume will serve both as an introduction to some of these new ideas and as a catalyst in generating further explorations and applications of these general ideas.

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Contents

Preface

ix

1. The Challenge of Complexity /. Prigogine and P. M. Allen

3

2. Self-Organization in Nonequilibrium Chemistry and in Biology JackS. Turner

40

3. On the Dynamics of Technological Evolutions: Phase Transitions 63 Elliott W. Montroll 4. The Many Faces of Scaling: Fractals, Geometry of Nature, and Economics 91 Benoit B. Mandelbrot 5. Successive Reequilibrations as the Mechanism of Cultural Evolution 110 Robert L. Carneiro 6. The Emergence of Hierarchical Social Structure: The Case of Late Victorian England 116 Richard Newbold Adams 7. Self-Organization in the Urban System P. M. Allen

132

8. Criticality and Urban Retail Structure: Aspects of Catastrophe Theory and Bifurcation 159 A. G. Wilson 9. Trip Making and Locational Choice Martin J. Beckmann

174

10. Creation of Order by Environmental Noise in the Volterra-Lotka Model 187 J. W Stucki and W. Horsthemke

Contents

viii 11. Noise-Induced Nonequilibrium Phase Transitions Werner Horsthemke

193

12. Nucléation Paradigm: Survival Threshold in Population Dynamics 203 W. H. Zurek and W. C. Schieve 13. The Efficiency of Oxidative Phosphorylation and the Thermodynamic Buffer Enzymes 225 7. W Stucki 14. Patterns of Nonequilibrium Organization in a Marine Bacterial Population 239 J. Wagensberg and J. Rodellar 15. Boolean Equations with Temporal Delays 245 André de Palma, Isabelle Stengers, and Serge Pahaut 16. Remarks on Traffic Flow Theories and the Characterization of Traffic in Cities 260 Robert Herman 17. Fluctuations in Demand and Transportation Mode Choice J. L. Deneubourg, A. de Palma, and D. Kahn 18. A Calibration of the Boltzmann-Like Theory of Traffic Flow Jenny N. Lam 19. Energy Analysis and Technology Assessment Nicholas Georgescu-Roegen

313

20. Thermodynamic Constraints in Economic Analysis R. Stephen Berry and Bjarne Andresen 21. Economic Dynamics Russell Davidson

323

339

22. From Self-Reference to Self-Transcendence: The Evolution of Self-Organization Dynamics 344 Erich Jantsch 23. Four Different Causal Metatypes in Biological and Social Sciences 354 Magoroh Maruyama

285 303

Preface

This volume is based upon the proceedings of the Workshop on Dissipative Structures in the Social and Physical Sciences held in Austin in September 1978 in honor of Ilya Prigogine. The purpose of the gathering of friends and colleagues was to honor him and his Nobel Prize work in a concrete way by having a “look forward” in the area of science for which that prize suggests so much promise. Its main purpose was to bring together different aspects of recent work in the natural and human sciences which seem to offer new pos­ sibilities of linking the two. The invited speakers came from many different fields and disciplines and their contributions cover, in consequence, a wide range of topics. They are concerned either with the development of mathe­ matical concepts and methods for the description of forms and patterns as they evolve in the living world or with the search to reveal the variables and mech­ anisms necessary to describe adequately the evolution of a system containing many factors responding perhaps to subjective, qualitative stimuli. The introduction of mathematical modeling into the human sciences has, of course, long been a source of controversy, particularly concerning their use as predictive or planning tools for human societies. The representation of hu­ mans as “robots” obeying some behavioral pattern which gives rise to a deter­ ministic trajectory for the system is not only distasteful but also often dis­ agrees with our everyday experience, which tells us not only that adaptations do occur but also that chance plays a considerable role in particular incidents and that individuals can “change the course of history,” whatever that means. Of course, on the other hand, the opposing view is often expressed that these revolutions were in fact inevitable and that the time was ripe for change. The confusion surrounding such views can now be dispelled as a result of recent developments in the physical sciences. When the study of macroscopic systems was extended to situations far from thermodynamic equilibrium, an entirely new type of structure was encountered, whose laws of evolution con­ tain both chance and determinism and whose description assimilates both the quantitative and the qualitative. These new states of matter have been called dissipative structures in order to emphasize their dependence on the flows of

X

William C. Schieve and Peter M. Allen

matter and energy to and from their surroundings. Sometimes they evolve along a stable trajectory of inevitable change, but there are moments of choice or bifurcation when chance plays a vital role and during which a qualitative modification of structure can occur. We find different possible branches of evolution, and the particular path taken by the system through the branching tree of possibilities is determined by the precise history of the system. Suc­ cessive branches may represent increasingly complex structures, and we see that such an evolution is quite the opposite of that of a system moving to ther­ modynamic equilibrium and is instead most evocative of the living world. Dissipative structures, although first discovered and studied in simple nonliving physico-chemical systems, are indeed intimately involved in the biochemistry of living matter and can offer a new basis for the modeling of complex systems, whether they be in the human or the physical sciences. The themes underlying the talks given at the workshop were the develop­ ment of these new methods for the description of living systems and the iden­ tification of the significant variables and mechanisms which, although suffi­ ciently few to allow understanding and modeling, nevertheless capture the richness of the possible evolutions of the system. We have grouped the contri­ butions of the various authors according to the principal interest of each, but, obviously, in such interdisciplinary work, all pigeonholing is destined to fail to some degree. The first paper, by Ilya Prigogine and P. M. Allen, begins appropriately with a description of the ideas underlying dissipative structures and, among other things, the importance of nonequilibrium, fluctuations, and the link be­ tween structure and the flows of energy and matter to and from the system. The role of these new concepts in biological and ecological examples is given. Jack Turner then describes examples of dissipative structures in chemistry and biology, emphasizing the now classic example of the Belousov-Zhabotinskii reaction. The description of the dynamics of technological change, stressing the importance of nonlinearity and discontinuity, is the topic of the next paper, by E. Montroll; after this Benoit Mandelbrot introduces the topic of fractals and discusses their significance for fields as varied as economics and geo­ graphical patterns. The next two contributions come from the human sciences and focus on the identification of vital mechanisms governing the evolution of societies. The paper of R. Carneiro describes how cultural evolution or change occurs in a society, outlining a behavior closely resembling that of a dissipative structure. Richard Adams, who in previous work has explored the relation between energy flow and social structures, pursues this theme further with a case study relating the structural growth of late Victorian England with the flows of energy and goods into and out of the country. Next, the description of a dynamic model of the evolution of urban cen­ ters is considered by P. M. Allen, using the concepts of self-organization

Preface

xi

emerging from dissipative structures; and then, in a similar vein, Alan Wilson describes a model which shows how retailing patterns in space could evolve through discontinuous jumps, or instabilities, as a qualitative change occurs in its structures. Continuing the topic of urban modeling, Martin Beckmann describes the manner in which one could model the locational choices of resi­ dents within an urban center. In the next theme, turning more to the physical sciences, a description is first given of the different aspects of self-organization in chemical and biolog­ ical systems, and then we have several contributions concerning population dynamics that bring out the importance of nonlinear behavior, of the existence of thresholds, and of the effects of fluctuations. In the work of Horsthemke and Stucki it is shown that an entirely new stable state may occur for a system subjected to a certain level of environmental noise, a transition which has no meaning in the absence of the fluctuations. The generality of the idea of nucleation is emphasized by W. Zurek and W. C. Schieve. The next three works, by R. Herman, J. L. Deneubourg, A. de Palma, D. Kahn, and T. Lam, although concerned with the specialized area of traffic flow and travel behavior, in fact raise all the issues vital to the general prob­ lems of modeling social systems. What variables are required to characterize the quality of the road network of a city? How can one describe the behavior of people in traffic? How do they react on each other? By posing these types of questions within a narrow framework, a concrete discussion is made easier, as is the possibility of experimental verification. In attempting to uncover the essential variables for the description of human society, the relation between energy use and the socioeconomic system has often been discussed. Several of the papers are concerned with this point, and Georgescu-Roegen discusses the introduction of thermodynamic or ener­ getic costs into the price system, while Stephen Berry is concerned with deter­ mining the thermodynamic costs of the production of a particular good. The central issue remains, of course, that prices reflect the temporal and informa­ tional horizons of the buyers and sellers at a particular time, and these can be changed by models which try to describe the different possible effects of alter­ native policies. Finally, the contributions of Erich Jantsch and M. Maruyama are con­ cerned with the place of the concepts of self-organizing, dissipative systems within the general scheme of evolutionary thought. Jantsch discusses the rela­ tion between the such concepts as dissipative structures, autopoeisis, coevolu­ tion, and hypercycles, describing the evolution of evolutionary processes in the universe. Summarizing this wide-ranging selection, let us say simply that the cen­ tral issues are the possibility of a creative evolution of complex open systems; the importance of fluctuations and of innovations in this, giving the evolution

xii

William C. Schieve and Peter M. Allen

the dual characteristics of chance and determinism; and the uncovering of the mechanisms governing change in living systems. The workshop was an at­ tempt to focus on these issues and thereby to encourage two necessary and complementary directions of research: that of natural scientists with new mathematical concepts and tools exploring their application and relevance to problems within the human sciences and that of social scientists identifying the crucial elements underlying the tremendous complexities of the living world. We gratefully acknowledge the Office of Graduate Studies and the Col­ lege of Social and Behavioral Sciences of the University of Texas at Austin, General Electric Company, and the U.S. Department of Transportation, with­ out whose financial support the workshop and this volume would not have been possible. In addition we wish to thank all those who have assisted in the realization of these endeavors: Pam Pape, Francine Allen, Lupe Garcia, and Pat Wall. Unfortunately, Erich Jantsch died in December, 1980, and the article in this volume is one of the last that he wrote. The subject will be poorer for the loss of both a remarkable, original mind and a kind and gentle person. W il l ia m

C.

S c h ie v e

Peter A llen

Self-Organization and Dissipative Structures

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1. The Challenge of Complexity

I. Prigogine and P M. Allen

INTRODUCTION The main drive of classical science over a long period was directed toward the reduction of our vision of the complex world to the study of simple basic ob­ jects. This perspective has been enormously successful and has led to the es­ tablishment of a description of physical systems in terms of atoms and mole­ cules and of living systems in terms of the amazingly complex biomolecules that have been identified. Despite this success, however, there are natural lim­ its to these analytical methods, for in many problems involving interacting populations, in neural physiology, for example, additional global concepts are required. Perhaps more than at any other time we are conscious of the extreme complexity of the world, and in response there have been many initiatives and developments, such as, for example, information theory1and systems theory,2 attempting to deal with these complex systems. In mathematics new qualita­ tive methods have been developed, such as the catastrophe theory of Thom,3 that of fractals of Mandelbrot,4 and the renormalization methods of Wilson. Our point of view is to describe how the equations of physics and chemistry can face the challenge of complexity, and the first remark that must be made in this connection is that one may distinguish three basically different methods of describing change in physico-chemical systems. However, the relation be­ tween them has remained far from clear. These three methods are: 1. The phenomenological approach, in which equations are invented which relate successfully the change of macroscopic variables. These vari­ ables are the average values of some fluctuating quantity, but the fluctuations are supposed to be of little consequence for any large system. For example, the equation of Laplace governing the change of temperature at a point, I. Prigogine is with the Center for Statistical Mechanics and Thermodynamics, Uni­ versity of Texas at Austin, and Université Libre de Bruxelles. P. M. Allen is with the Service de Chimie-Physique II, Université Libre de Bruxelles.

I. Prigogine and P. M. Allen

4

= —X V 2T ;

at another example is that of chemical kinetics expressing the law of mass ac­ tion. The equations are both deterministic and exhibit different behavior for positive and negative times. They are not invariant to the change t—>- t. 2. The stochastic approach, an example of which is a description based on a Markov process, in which it is supposed that the evolution of the system can be described by probabilities of transition at any moment and that these probabilities depend only on the state of the system at that moment. Usually, in such an approach for sufficiently large systems the fluctuations merely re­ sult in negligible small corrections since one is considering a probability dis­ tribution to which the law of large numbers applies and for which one may demonstrate an H-theorem guaranteeing the movement toward a unique final distribution. 3. The dynamic laws, which describe matter at the most elementary level, and, therefore, must underlie, in some way, the other approaches. The evolution of a system in these terms is both deterministic and reversible, since when the direction of time is reversed the system simply retraces its previous course in the opposite sense. Let us now briefly outline recent advances that have been made in under­ standing the relation between these three, quite different, approaches. First, the phenomenological description has proved to be far richer and to contain a much greater wealth of phenomena than was at first evident. This richness has even been true for quite simple systems and is the result of the occurrence of bifurcations in the possible behavior of the system. The meaning and impor­ tance of this phenomenon will be discussed later and various examples of such behavior will be given. The very existence of bifurcations implies the neces­ sity of supplementary considerations not included in the macroscopic descrip­ tion, if we are to know the actual path of the system. What we face is the “breakdown” of the macroscopic description during moments of “instabil­ ity,” when small fluctuations are not damped and may carry the system off to some new configuration.5 Similarly, the probabilistic approach also appears to us today in a new light when we describe complex systems, for here again the phenomena of bifurcation introduce into the description distribution functions to which the law of large numbers does not apply and for which the fluctuations, instead of simply being negligible corrections for a sufficiently large system, may be of the same order as the average value, as is the case, for example, for a double­ humped distribution.6 Again, a new and rich description appears which in­ volves the effects of bifurcation, fluctuation, and instability. Finally, how can we understand the prevalence of irreversible processes

The Challenge of Complexity

5

in the universe and the frequent occurrence of stochastic processes, and how are we to reconcile these two processes with the underlying dynamic descrip­ tion which is both reversible and deterministic? Some progress has been made in the understanding of this point, but here we shall not go into any details of such a mathematically complex subject. Let us simply say briefly that the stochastic description may not result, as has often been thought, from our “ig­ norance,” the imprecision of our measurements, but may express some basic characteristic of the deterministic laws of nature. For example, let us take Poincare’s famous example concerning the throwing of dice. Clearly, trajecto­ ries exist which link each initial condition (height, velocity, angular momen­ tum, and direction) with a particular result, but in the immediate neighbor­ hood of any precise initial conditions many trajectories lead to the other five possible results. This complex interweaving of the trajectories, when the path leading to any one face is immediately surrounded by paths leading to all the other five faces, constitutes a dynamic instability. It is in this “instability of motion” that we must look for the reconciliation of dynamics and thermody­ namics.7 What we learn from the irreversible and stochastic basic behavior of chemical and biological systems is that these systems do not belong to the realm of schoolbook dynamics, which discusses such problems as, for exam­ ple, simple harmonic motion. Instead they exhibit these more complex flows, for which we have dynamic instability, whereby the slightest change in the initial condition or the smallest perturbation during the evolution leads to a completely different result, giving rise in turn to an irreversible and proba­ bilistic description of the system.

SELF-ORGANIZING SYSTEMS In this article we are not particularly interested in the description of systems which involve not only such simple physical quantities as temperature, densi­ ties, and so forth but also biological and social systems concerning interacting biomolecules, cells, organisms, human beings, and so forth. For all such sys­ tems we know that the laws of thermodynamics must be valid. However, in considering the physical laws governing such systems, let us be a little more specific. An isolated system, whatever its nature, will approach thermodynamic equilibrium, a state characterized by the maximum value of entropy, that is, of “disorder.” Furthermore, a system subjected only to weak flows of energy or matter, so that it remains only slightly out of equilibrium, adopts a steady state which corresponds to minimum entropy production. So far we find nothing in this description that resembles what we know of social or biological evolution, which is marked by increasing diversification

I. Prigogine and P M. Allen

6

and complexity, rather than by greater uniformity and disorder. However, we must remember that the earth is not in thermodynamic equilibrium and that the flow of solar energy in which it is bathed is sufficient to ensure that for the systems of interest the relevant thermodynamic regime is not that of equi­ librium, or of near equilibrium, but rather that of far from equilibrium. For such systems we know that the internal entropy can spontaneously decrease because the second law of thermodynamics merely states that the change in the entropy of a system dS can be written as the sum of two terms, dfS the internal entropy production and deS the entropy flow across the boundaries of the system, dS = djS + deS\ djS ^ 0 ,

(1)

and that the term dtS is either positive or zero. In systems which are more than a critical distance from equilibrium, however, the flow of entropy through the boundaries can, under certain circumstances, more than compensate the en­ tropy production within the system, in which case a spontaneous self-organization of the system occurs.5 The second important feature of the systems that are of interest to us is that they contain very large numbers of interacting entities and that the range of these interactions is often such that either they involve many individuals directly (for example, the mass media) or the individuals interact through the intermediary of some environmental constraint (drivers interacting because of the finite width of the road,8 species competing for a limited resource, and so on). The third important characteristic is the presence of nonlinearities in the interactions between the elements. In chemical kinetics, for example, we may have present in the reaction scheme steps which are autocatalytic, such as A+X-*2X,

(2)

whereby X stimulates its own production from A. Alternatively, we may have a cross-catalytic series of steps in which X produces a substance Y, which in turn accelerates the production of X. A particular scheme which has been studied in much detail by the group at Brussels and which, in consequence, has become known as the Brusselator, consists of the reaction steps A B + X 2X + Y X

X Y+ D 3X E

(3)

The Challenge of Complexity

7

If these reactions are maintained far from equilibrium by continually supply­ ing A and B and extracting D and E and, in this way, eliminating the back reactions, then a rich variety of self-organizing behaviors are exhibited by the system. Depending on the particular values of the reaction rates, the diffusion parameters, and the concentrations of A and B that are maintained, different types of coherent behavior can appear. This behavior may be a steady tem­ poral oscillation of the reacting concentrations in a well-mixed system or, in a system which is not stirred, may take the form of stationary or moving chemi­ cal waves, of a high concentration of the intermediate X. This simple reaction scheme offers us a glimpse of the richness and complexity of such phenomena. Nonlinearities clearly abound in social phenomena, where a yawn, a de­ sire for an automobile with fins, or a life-style can spread contagiously throughout a population; where a judicious investment can trigger an explo­ sive growth; and where a steady increase in traffic density provokes, at some critical value, a sudden decrease in the speed of vehicles. We see that, in gen­ eral terms, the systems which interest us are large, nonlinear systems operat­ ing far from thermodynamic equilibrium. It is precisely in such systems that coherent self-organization phenomena can occur, characterized by some mac­ roscopic organization or pattern, on a scale much larger than that of the indi­ vidual elements in interaction. It is a structure whose characteristics are a property of the collectivity and cannot be inferred from a study of the individ­ ual elements in isolation. We may say that reductionism, long a strongly crit­ icized attitude in the social sciences, is found to be inadequate even in the physical sciences. The whole is more than the sum of the parts for such systems.

EQUILIBRIUM PHASE TRANSITIONS— EQUILIBRIUM SYSTEMS As we have discussed above, nonlinear interactions can give rise to bifurcat­ ing solutions of the phenomenological equations, such as those of chemical kinetics, for example, and this gives rise to new dynamic, coherent structures, which have been called dissipative structures. However, even in equilibrium systems, bifurcating solutions can occur, but in such cases they correspond to the occurrence of an equilibrium phase transition. Let us consider, for example, the phase transition in a Van der Waals’ gas. In an equilibrium system, there exists a thermodynamic potential that governs the behavior of the system, and in this case it is the Gibbs’ free energy.

I. Prigogine and P. M. Allen

8 F = E -T S ,

(4)

where E is the internal energy, T the temperature, and S the entropy. If we examine the form of this potential, as the transition point is approached and passed, then we find the behavior shown in fig. 1.1. The system is initially in a well-defined potential well, which guarantees that any fluctuation around the average state will be damped. As the transition point is approached, a second well appears in the form of the potential. At a certain value, the two wells are of equal depth, and, since the system may be supposed to occupy either well with equal probability, this point is taken to mark the transition. As we shall see in the next section, the variety of phe­ nomena that can occur in the nonequilibrium phase transitions characteristic of dissipative structures is far greater than the number that can occur for this equilibrium transition. Another important difference is that equilibrium struc­ ture is always characterized by the same length scale as that of molecular in­ teraction, whereas, as we shall see, nonequilibrium structures are charac­ terized by a scale that is much larger than the molecular and involves gigantic numbers of the entities in interaction.

Fig. 1.1. The potential for a Van der Walls’ fluid during a phase transition: (a) a single well; (b) second well appears; (c) moment of transition or equi-occupation, two wells of equal depth; (d) transition complete.

The Challenge of Complexity

9

BIFURCATIONS AND DISSIPATIVE STRUCTURES If we turn now to discuss large, nonlinearly interacting systems that are main­ tained far from thermodynamic equilibrium, then we find that the bifurcating solutions of the phenomenological equations can lead to the appearance of a wealth of possible new structures and organizations. When bifurcation oc­ curs, then the stability of the existing state of the system breaks down, allow­ ing the amplification of some small, random fluctuation to occur and to carry the system off to one of the possible, new branches of solution. As we shall discuss in more detail later, several different types of fluctua­ tions are possible, but here we shall consider the effect of the internal fluctua­ tions, which occur within the system as usually small variations around the average value of each variable. If we consider, for example, a system of equa­ tions expressing the mutual interaction of several populations, x ,9x 2, . . . x„, we have dx, — = G,(x,, x 29 . . . dt

t) ,

(5)

then if there exists a stationary solution, we must have

dt

■= 0 = Gi (jc?,

x°n) .

(6)

We may test the stability of this steady state by studying the behavior of a small deviation from the state. If it decreases in time, then the solution is sta­ ble; but, if it grows in time, then the state is unstable because the least little fluctuation will be amplified in the system and drive it away from the station­ ary state. The small deviation is written in the form e kt, and it is substituted into the equation (5). The resulting values of X are found by solving the equation Det

AG, 3X:

• 4 ) - kSij

o .

(7)

If any of the resulting roots has a positive real part, then it means that in that mode the system is unstable, since it allows the initial small deviation to grow. A similar analysis can be made for a system of equations which depends on space as well as time and the local, or neighborhood, stability of a solution with respect to different spatial perturbations. Again, the necessary condition for stability is that there must be no root having a positive real part.

I. Prigogine and P. M. Allen

10

Having briefly discussed the manner in which the stability of a solution may be studied analytically, with respect to very small perturbations, let us illustrate the occurrence of an instability in a simple nonequilibrium system. The reaction scheme which we shall discuss first is from Schlogl8 and gives rise to a nonequilibrium phase transition which closely resembles that of the van der Waals’ gas that we described above. We have A + 2X —» 3X X ^ B ,

(8)

where X denotes an intermediate variable and the values of A and B are con­ trolled externally. The phenomenological kinetic equation corresponding to this scheme is, dnY — - = k xAn2x — k2n\ — k3nx + k4B . dt

(9)

Introducing the scaled variables and parameters, nx = a( 1 + x) ; k xAlk2 = 3a ; k j k 2 = (3 + 8) ; k4B/k2 = (1 + 8 V ,

( 10)

then the kinetics of the problem take the form

dx — = - x 3 — 8x + 8 '- 8 . dt

(11)

This admits the steady state solution,

8' = x3 4- 8(* + 1) ,

(12)

which is strikingly similar to that of the van der Waals’ gas /? = v3 + T(v + 1) , where v is the specific volume. Equation (12) has a solution which at the point 8 = 8' presents a cusp singularity when there is a triple root x = 0. For 8 = 8' > 0, x = 0 is the only real solution, but for 8 < 0, there are three possible roots,

x = 0 ; x + = V-^~8 ; JC_ = — V ^ 8 .

(13)

The Challenge of Complexity

11

A linear stability analysis, however, reveals that the solution jc = 0 is in fact unstable, for this region 8 < 0. An interesting point that comes out of the full probabilistic analysis of this problem, which we shall outline later, is that the line of coexistence be­ tween the solutions (the moment when transition occurs) is not given correctly by the Maxwell construction, which applied to the van der Waals’ transition. The calculation resulting from a potential function inferred from the kinetic equation (11) by integration, where the transition occurs at the moment of equal well depths, is not in agreement with the calculation that results from regarding the processes (equation (8)) as defining a Markov process and deriv­ ing a stochastic potential which takes takes into account the behavior of the fluctuations. In the case of a chemical system there is no doubt that the latter approach is more correct.9 If we leave the Schlogl model and move on to the more complex set of reactions known as the Brusselator, A ^X B + X -^ 2Y Y + 2X 3X X —> £> ,

(14)

we find a wealth of possible structures. These structures can be temporal os­ cillations of the intermediate concentration, which occur beyond a Hopf bifur­ cation, or we may have stationary or moving chemical waves of concentration.

12

/. Prigogine and P. M. Allen

Fig. 1.3. A cyclic spatiotemporal structure of propagating chemical “waves,” as the concentration of intermediate X follows the sequence indicated in 1—»8.

The scale of this structure is macroscopic, implying the coherent behav­ ior of countless numbers of molecules, and its form depends on the boundary conditions applied to the system or, in certain cases, the dissipative structure can define its own length scale as shown in fig. 1.5. In general, there is no potential from which the kinetics of such systems can be derived because general mechanisms involving two or more variables have crossterms that appear in the differentiation of a potential. These terms only disappear for exceptional cases. Thus, for problems involving two or more variables, in general, catastrophe theory, for example, will be inapplica­ ble. The study of the behavior of such systems can be made by numerical

The Challenge o f Complexity

13

Fig. 1.4. In a well-stirred system the values of X and Y can oscillate around the limit cycle shown here.

Fig. 1.5. If we allow for the diffusion of A into the system from the walls, then we can have the formation of a dissipative structure having its own length scale.

/. Prigogine and P. M. Allen

14

experiments, by the probabilistic approach, and by the study of the kinetic equations, coupled with stability analysis. Here we shall sketch some recent developments in bifurcation theory, first, in the understanding of successive bifurcations in finite systems and, sec­ ond, for the appearance of structure in infinite three-dimensional systems. Let us write the equation for some concentration variable x i9 which is described by a nonlinear kinetic law: ^ = /, (§ Xj §’ M . ot

(15)

where f are the macroscopic state variables and rates and X is a set of param­ eters that may enter the description of the system. For example, for a chemical system in the absence of external forces and thermal effects, we have a reaction-diffusion equation /, = v,(§ xj §, X) + D, V2x,

(16)

Here v, are reaction rates, x the compositions, and D, Fick’s coefficient. At some finite distance from equilibrium, the equilibrium-like solution can reach a bifurcation point and undergo a spontaneous time and/or space symmetry breaking. In fact, as shown in our discussion of the Brusselator, in such systems there exists perhaps a whole series of further instabilities, a cas­ cading self-organization leading to increasingly complex behaviors and ul­ timately to chaotic or turbulent motion. We may say that in some way life is sandwiched between the dangerous uniformity of equilibrium and the dangerous chaos of turbulence. Can order be introduced into this wealth of possibilities? The answer to this question depends, to a great extent, on a judi­ cious choice of the parameters controlling the system. Movement along a oneparameter space, as frequently happens, usually gives a succession of primary bifurcations from the thermodynamic reference state, occurring at a simple eigenvalue. Thus we have the situation as shown in fig. 1.6 with first, second, and third primary bifurcation points. There are, however, two new features that must be added to this tradi­ tional picture. First, bifurcation may be affected by some parasitic parameters acting as impurities on the system. For example, in the Brusselator, the con­ centration of the initial product A may depend on space. This condition may give rise either to a smooth or to a discontinuous passage to a spatial pattern. A second point of great interest is that there usually exists at least one more parameter in the evolution equations which controls the spacing between suc­ cessive primary bifurcations. In the Brusselator reaction scheme, this addi­ tional parameter is the size of the reaction space L. For certain combinations

The Challenge of Complexity

15

Fig. 1.6. Succession of primary bifurcations in terms of parameter X; the solid lines denote stable solutions and the dashed lines denote unstable solutions.

of L and the first parameter X, one may have bifurcation at a double eigen­ value. Thus, slight changes n (L, X) around this point split the bifurcation branches and may give rise to higher bifurcations. This situation is illustrated in fig. 1.7.10 This mechanism generates a great many solutions of increasing complex­ ity some of which have a pronounced spatial asymmetry and will be discussed further in connection with morphogenesis. The above considerations apply to systems of finite dimension, but re­ cently there have been some most interesting advances made in the under­ standing of infinite systems that undergo nonequilibrium phase transitions. In this case, a proper consideration of the fluctuations qualitatively changes the behavior of an infinite system compared to a study based on a mean-field ap­ proximation. By mean-field approximation, we mean a description which takes into account only the average value of the variables, assuming that they are always meaningful and sufficient to describe the behavior of the system. The fluctuations, the deviations from the mean, are completely neglected. For example, if we consider the states predicted by this latter theory for a two- or three-dimensional infinite system, we find in the mean-field approximation the solutions shown in fig. 1.8. We have those corresponding to structures composed of m independent wave vectors and, beginning on the left, solu­ tions involving related wave vectors which correspond to body-centered cubic symmetry or to hexagonal symmetry. The study of the effect of fluctuations on these systems shows that for infinite systems of one or two dimensions all long-range order is destroyed. In three-dimensional systems, structures described by m-independent wave vec­ tors are similarly destroyed by fluctuations, but those specified by a set of related wave vectors (body centered cubic, hexagonal, prismatic, and so forth) may exist in infinite systems.11

/. Prigogine and P. M. Allen

16 iii

Fig. 1.7. Three-dimensional bifurcation diagram in terms of the parameters L, X; the trajectories of the first two primary points 1 and 2 are indicated until the degeneracy point 1 = 2. S is the secondary bifurcation point. The primes indicate projections of bifurcation points in the L, X plane (from G. Nicolis, 1979, in “Far from Equilib­ rium,” eds. Pacault and Vidal, Springer-Verlag, Berlin).

We may briefly summarize the situation revealed by these recent devel­ opments: There are two classes of dissipative structures. The first are sta­ bilized by the finite dimensions of the system and, typically, are of the type shown in figs. 1.3 to 1.5. In the second class, for essentially infinite systems, the instability that occurs is analogous to first-order equilibrium transitions, such as crystallization. We see a whole new morphology arising, one where the size of the central unit is related to kinetic rates, to diffusion, etc.

DISSIPATIVE STRUCTURES IN LIVING SYSTEMS Let us consider, therefore, the link between the simple chemical models we have mentioned, on the one hand, and biological structures and functions, on the other. In other words, let us examine in what manner dissipative structures may be involved in living systems.

The Challenge of Complexity

17

Fig. 1.8. (Schematic): Bifurcation diagram in the mean-field approximation, 'bcc 'bcc 'bcc bcc "o hex m= 1 m= 2 m= 3 r < o q

For example, one of the most striking features of biological systems is the ubiquity of oscillatory phenomena, characterized by periods ranging from seconds to months. In particular, at the subcellular level, for instance, we find a whole series of oscillating enzymatic reactions, which are of two basic types: epigenetic oscillations, in which the periodicity of enzyme synthesis is the result of control mechanisms at the genetic or translational levels and which usually have characteristic times of an hour or more, and metabolic oscillations, on the other hand, in which regulations occur at the level of the enzymes themselves and are characterized by periods of several minutes. All of these oscillations correspond to temporal dissipative structures operating far from thermodynamic equilibrium. One of the best known of these reactions is that of glycolytic oscillation in which the regulatory enzyme, phosphofructokinase, gives rise to oscilla­ tions with periods ranging from two to ninety minutes, in a variety of different test media. A mathematical model, based on data concerning the reaction steps, has been constructed and the experimental oscillations identified with limit cycle type oscillations arising when the uniform state is unstable. Gly­ colysis is a process of great importance in living cells, and it is therefore sig­ nificant that these oscillations should be the manifestation of a temporal dis­ sipative structure.12 Another most interesting metabolic oscillation is that of the periodic syn­ thesis of cyclic AMP in certain species of cellular slime molds (Dyctiostelium discoideum), since it concerns both intra- and intercellular activities. This species exhibits a spectacular transition between two quite different states of

I. Prigogine and P. M. Allen

18

Time (s) Fig. 1.9. Periodic variation of phosphofructokinase activity.

organization. In one, the amoebae are independent, separate entities, while in the other they form a multicellular fruiting body; and the transition from one state of macroscopic organization to the other takes place after starvation. Single cells of D. discoideum, deprived of nutrient, aggregate around centers as a chemotactic response to cyclic AMP. This process of aggregation has a periodicity of several minutes, when concentric waves of amoebae move to­ ward centers, an action suggestive of a periodic release of the chemotactic signal from the centers and of its relay by cells responding to the signal. A model has been proposed for these intracellular periodicities13 and for the periodic release of cyclic AMP by aggregation centers. Cooperative and regulatory properties of adenyl cyclase can give rise to sustained oscillations in the synthesis of cyclic AMP, of the limit cycle type. The model also shows that a D. discoideum cell could become either a center capable of autonomous signalling or a cell capable of relaying the signal, the choice depending on whether the parameters of the adenyl cyclase system correspond to the oscilla­ tory domain or to a stable steady state close to the region of oscillation. At the macroscopic, intercellular level this species of slime mold is also interesting, since the aggregation process itself represents a self-organization occurring beyond instability. In this case the aggregation takes place when the chemotactic substance (cyclic AMP) being emitted from the center overcomes the random diffusive motion of the amoebae and the uniform state becomes unstable; the aggregate then forms where the first fluctuation leading to the emission of pulses of cyclic AMP happened to occur.

The Challenge of Complexity

19

Another example in which these new ideas are important is that of mor­ phogenesis. Pattern formation is a complex process wherein cells of identical genetic makeup become differentiated and give rise to a well-defined spatial structure. This process has been related to the existence of inhomogeneous distributions of chemicals, the so-called morphogens, throughout the morpho­ genetic fields. The working of the morphogenetic system has been explored in a series of experiments in which normal development has been perturbed. Insect eggs have been subjected to ultraviolet radiation, gentle puncturing, and centrifug­ ing, and there have been a series of grafting experiments on the developing bud of chick’s wings and of regenerating amphibian limbs. These experi­ ments have shown either that the normal pattern remained stable or that, along a developmental axis, the polar configuration of the normal pattern may change to inverse polarity (reversal of fore-aft axis) or even may switch to a mirror symmetric configuration, with either a double posterior, for some types of perturbation, or anterior structure, for others. Let us discuss very briefly recent work which explores the relation between such results and the symme­ try properties of the primary and secondary bifurcating solutions of the reaction-diffusion equation of the morphogenetic system.14 The morphogens obey a reaction-diffusion type of equation. For the de­ scription of morphogenesis, it is particularly relevant to study the behavior of the system as it grows in size, and, in this case, it is known that starting from a homogeneous distribution the bifurcating solutions that appear have the form PjCostt r/L P 2c o s 2 tt rIL P 3c o s 3 it r/L

(17)

where r is the spatial coordinate along the axis of a one-dimensional system. Generally the order of appearance of these solutions is respected. We note that the first bifurcation corresponds to a polar symmetry and the second to a mir­ ror symmetry. In a bifurcation analysis involving these primary and secondary bifurcations, Erneux and Hiernaux15 have shown the following two basic sit­ uations as a function of the kinetic bifurcation parameter X. Thus, we see that as X increases from its value at the origin, we pass from the homogene­ ous solution (n = 0) to that of polar symmetry (n = 1), either in case (a) smoothly, or in case (b) discontinuously. In the latter case there are three solu­ tions simultaneously stable, two polar, and one mirror symmetric. A perturba­ tion of a certain size can therefore carry the system from one symmetry to

I. Prigogine and P. M. Allen

20 llll

X \

\

\

Fig. 1.10. Behavior of the variable Jt at some point r, as the bifurcation parameter \ is changed. The point of interest is, in the framework of a theoretical approach of mor­ phogenesis, the possibility of smooth transitions (fig. 1.1a) or discontinuous jumps (fig. 1.1b) between polar patterns (m = 1) and symmetric patterns (m = 2) (5).

regenerates

mirror duplication Fig. 1.11. Behavior of imaginai disc when cut.

another and could reverse the polarity of the existing situations. These three possibilities are just those noted in the experimental evidence. Erneux and Hiernaux have gone one step further and have performed a simple numerical experiment equivalent to experiments that have been per­ formed by sectioning the imaginal disc (the embryo at a very early stage of development). When the disc is cut, the behavior portrayed in fig. 1.11 is always found. The larger section always repairs the damage and regenerates the whole, while the smaller part merely succeeds in replicating itself in a mirror sym­ metric manner. The numerical experiment that has been performed concerns the Brusselator (equation (14)) and consists in setting conditions so that the sys-

The Challenge of Complexity

21

tem is at point A in diagram a of figure 1.10 characterized by multiple steady states. The initial distribution of X is as shown in figure 1.12; (a) shows a polar symmetry on an axis of length L. At t = 0, this system is cut into a large and a small piece, and each of these is then allowed to grow again separately on axes of length L. The result is as shown in (b) and (c): the larger one re­ generates the original pattern, while the smaller one moves to a solution hav­ ing mirror symmetry (n = 2). Again, such a result helps to confirm the intui­ tion that morphogenesis is indeed governed by dissipative sructures on the morphogens. This brief discussion only hints at the complication of the possi­ ble patterns which could evolve, since the very complex structure of the bifur­ cation diagrams (see fig. 1.10) involve only the primary and secondary bi­ furcation points of a one-dimensional system. Clearly, when higher-order bifurcations are involved, and when we have a two- or three-dimensional sys­ tem, the possible complication of structure is enormous, and the exploration of such possibilities theoretically represents a most fertile area of future research. The few examples which we have described confirm the idea that dis­ sipative structures are indeed intimately connected with the fundamental bio­ logical processes which underlie all living matter. The self-organization of a physical system maintained far from thermodynamic equilibrium offers us a bridge of understanding spanning the living and the nonliving, unifying the two realms, and surpassing the simplistic views of reductionists.

(a) ->

L cut

L

->

Fig. 1.12. Computer experiment in which the polar solution of the Brusselator is cut

into a small and a large part. The large part regenerates the original pattern (/?); the small part develops a mirror-image symmetry (c).

I. Prigogine and P. M. Allen

22

This analysis discussed above deals then with the behavior of a system near instability under the influence of the internal fluctuations affecting vari­ ables and defined by the basic birth and death process. However, as we have mentioned above, behavior of the system may be quite different from that which occurs when a nonlinear system is subjected to external fluctuations. It has been shown recently that the presence of such fluctuations can qualitatively change the results of deterministic analysis. A further example is afforded by a recent tumor immunology model de­ veloped by Lefever and Garay.16 The kinetics of the different cellular populations involved in tumor im­ munology in a locality are written in a manner entirely comparable to that of chemical kinetics: A ------------ (a) X ------------>2X K K E0 + X ------- ¡---- » E ] ------- -----> Eq + p

(b)

(18)

(c)

where x, E0, and E x are the population of tumor cells, free effector cells, and effector cells having recognized and bound a target tumor cell. A, X, K x and K2 are rate constants. The first two steps represent two ways in which the X population can grow: spontaneous conversion of a normal cell (a) or replica­ tion (b). The effector cells E0 destroy the tumor cells X and after a certain delay become free again to attack other X. This process leads to a simple equation which can be written in the form: dx — = a + x(l — 0x) - (3x/(l + x) dt

(19)

where X* X

+

The steady states of equation (19) are solutions of - 0 jc3 + (1 - 0)jc2 + (1 + a - 0)* + a = 0 .

(20)

As we see from fig. 1.13, for a range of values of B, and X, there are two stable branches of solution, one corresponding to low values of X , the micro­

The Challenge of Complexity

23

cancer branch, and the other to high values of X, the cancer branch. From this point of view, then, the onset of a runaway growth of tumor cells corresponds to the system jumping from branch one to branch three. The problem of the onset of such a growth is therefore related more to the exact stability proper­ ties of this bistable situation than to the particular spontaneous origin of the cells X in process (a) in equation (18). The problem of the stability, in general terms, of a nonlinear system, such as this, depends, however, on the origin of the fluctuations to which it is subjected, as we saw in the Schlogl equation transition where the line of coex­ istence was not given correctly by the Maxwell construction. Thus, in the presence of internal fluctuation only, or of additive external noise (where we simply add a term to the differential equations), the basic conditions under which instability occurs are not changed significantly; fluc­ tuations have no effect on the range of parameter values for which bistability occurs. On the contrary, however, noise corresponding to the fluctuation of such a parameter as B, for example, can lead to a phase transition for values of the parameters for which the deterministic picture predicts a single branch

Fig. 1.13. Solutions of equation (20) for different values of a (from R. Lefever and R. Garay, 1977, in Theoretical Immunology, ed. M. Dekker, New York).

24

/. Prigogine and P. M. Allen

of solution. In fig. 1.14 we see that the behavior of the probability distribution presents such a noise-induced phase transition.17 For a 2 = 0.25 the distribu­ tion is single peaked, with a value nearer the upper deterministic steady state solution; when a 2 increases, however, a second peak appears, favoring low values of X and the upper peak disappears. This brief example shows us that the conditions under which a phase transition may occur, leading to the rapid growth of a tumor, cannot be understood strictly in deterministic terms. This whole area of research into fluctuation-induced transitions and their possible relevance to chemical, electrical, and ecological problems is a rapidly grow­ ing field, and it is worth noting that experimental evidence supporting the ex­ istence of such transitions has been reported by Kabashima et al.18

THE RELATION WITH PROBABILISTIC THEORY In the preceding sections we discussed the structure and stabilities associated with the bifurcations that characterize nonlinear, phenomenological kinetic equations. As mentioned in connection with the Schlögl model, with the con­ sideration of infinite systems, and with the tumor immunological example, in order to describe correctly the moment and type of transition that will occur it is necessary to consider specifically the behavior of fluctuations in the system. This step constitutes the link between the phenomenological and the proba­ bilistic methods. In order to establish briefly this link, let us suppose that the basic pro­ cesses in our system are Markovian and that we have established an equation for the evolution of the distribution of probability of the stated variables, the master equation. This equation describes the evolution of the probability of finding the different values of a variable. Instead of discussing the rate of change of X, we discuss the evolution of P(x, t), the probability of X having various different possible discrete values. Changes are supposed to occur in the system by discrete jumps, which have probabilities depending on the state of the system at that precise mo­ ment, and the master equation expresses the net evolution due to the proba­ bility of processes leading to the increase of a particular value of X minus those tending to decrease this value. For example, for the simple processes,

In the first line, the transformation of A into X occurring with rate k, A changes P(X - 1) -» (P(X), whereas P(X) becomes (P(X + 7); and in the second line the rate at which X becomes B, is k (X + 1) if there are X + 1

The Challenge of Complexity

25

Fig. 1.14. The effect of increased noise in the parameter is shown. The probability peak corresponding to the deterministic solution (high x) disappears and the low x solution is stabilized. We have a noise-induced transition (from R. Lefever and W. Horsthemke, 1979, Bull. Math. Biol. Vol. 41, pp. 469-490).

particles of X present, and k X if there are X present. The master equation describing the evolution of P(X), is, therefore, dP(X) — = k,AP(X - 1) - k,AP(X) + k2(X + 1)P(X + 1) dt - k2XP(X) ,

(22)

or in general for a set of variables §*,§, and processes p, dP (§*,§) =

dt

W(§jc,—r„§ / § x ,§ )/>(§ a:,—r,v§ , r) -

W(§x,§/§*,+/■„§)?(§*,§)

(23)

.

This is a difference equation with large nonlinear coefficients, and it has re­ cently proved possible to find some new ways of studying large systems. The first step is to transform these finite difference equations into continuous dif­ ferential equations using the transformation f ( s , t ) = S î xP(jî:,î) , x

(24)

26

/. Prigogine and P. M. Allen

and in particular for large systems to make a systematic expansion in terms of N, representing the size of the system, by a further transformation: em s 'f) = f ( s , t )

yV-» oc .

(25)

By studying the resulting scheme of differential equations, we can resolve the full, probabilistic description of the Schlögl equation and of the Brusselator. Here, we will merely illustrate the result for the homogeneous Schlögl prob­ lem, but we should stress the interest of the inhomogeneous problems, since they raise the question of the spatial extension of a fluctuation, and the whole problem of the nucleation of an instability, which is of great importance and for which much progress has been made over recent years. In order to explore the relation between this approach and that of chemi­ cal kinetics, let us return to the Schlögl equation that we have discussed be­ fore. The reaction scheme is defined as A + 2X —» 3X (26) X -> B and these processes give rise to a master equation, — (X) = *,A (X -1 )(X -2 )P (X -1 ) - k,AX{X-\)P(X) dt + k2(X+l)X(X~l)P(X+\) - k2X (X -l)(X -2)P (X ) + k,(X+l)P(X+l) - kJCP(X) + ktB P (X -l) - k,BP{X) .

(

’

From this one may deduce an equation for the rate of change of the average value of X, (X) = X XP(X,t), which is known as the first-moment equation, and an equation governing the variance of X around this value (§x2) = X (X — (X))2P(X), known as the second-moment equation, and so on, replacing the single equation (27) with an infinite hierarchy of moment equations. Clearly, the first-moment equation is very closely linked to the phenomenological equation for a given variable. Indeed, for linear reaction schemes near or far from equilibrium or for nonlinear reaction schemes near equilibrium, the first moment equation may be safely identified with the phenomenological equa­ tion, because the distribution P(X) is Poissonian and obeys the law of large numbers, by which the variance (hx2) is proportional to (x), which ensures that the fluctuations in large systems are negligible. When, however, we study nonlinear systems far from thermodynamic

The Challenge of Complexity

27

equilibrium, in which the kinetic equations give rise to bifurcations, then the identification between these macroscopic equations and first moment equations may become completely incorrect, as the probability distribution may cease to be sharply peaked, in which case the law of large numbers no longer applies. The probability distribution may, for example, become double humped, in which case the average value of a variable and its variance be­ come meaningless quantities, since they completely fail to define the double­ humped curve. In the analysis of the Schlögl equations, Nicolis and Turner19 show that in the region of the single solution, 8 > 0, the probability is peaked around a single value, and the variance is given by:

lim ----= - 4+ 1 ,, a —>oc a 8

(28)

where a is a measure of the size of the system. Thus, we may say that the law of large numbers holds for finite 8, since (hx2) N, size of system. As the bifurcation point is approached, the variance diverges, that is, the distribution spreads. Beyond the transition point, 8 < 0, the probability distribution is double peaked, but for large systems, there is always one maximum that is dominant. It turns out to be that associated with = - V-8^. The variance of this be­ haves as oo a ~~8 V- 8

7

(29)

We find that for finite negative 8 the law of large numbers again holds since (hX2) oo N, but as the bifurcation point is approached, the law breaks down. At the bifurcation point, the variance diverges according to Dm

2 * 5 . !IM > oo a3'2

,30,

r ( l/4 )

Thus, we have a new type of law of large numbers at this point, (hX2) °c N 3'2', implying that the probability distribution is abnormally flattened here. By finding the line along which the Maxima of the probability distribution are of equal heights in the region 8 < 0, we can trace the line of coexistence between the two solutions on the basis of this stochastic potential. As mentioned earlier (fig. 1.2) it is not the same as that obtained by assuming a deterministic poten­ tial which generates the phenomenological equations.

28

/. Prigogine and P. M. Allen

This simple example illustrates how the probabilistic and phenomeno­ logical approaches are linked and how in the description of a nonlinear system whose basic mechanisms define a Markov process, the detailed study of the behavior of fluctuations can give an important new insight into the phenomena encountered. This was also the case for external fluctuations in the parameters of a nonlinear system, as was illustrated in our example concerning cancer immunology, where the stochastic study revealed that external fluctuations could induce the stabilization of states which were not even solutions, stable or unstable, of the corresponding macroscopic phenomenological equations. These various developments permit us to understand more clearly the relation between the phenomenological equations and the probabilistic description.

SELF-ORGANIZATION IN ECOSYSTEMS When we turn our attention to the study of ecosystems, in which populations of different types interact with their environment and with each other, then the first important remark that must be made concerns the existence of a dramatic new possibility in such systems—that of innovation. Thus, in addition to the fluctuations that we have already discussed, that is the internal fluctuations of variables and the exogenous fluctuations of the environment, when we discuss ecosystems, made up of interacting entities which are themselves already complex organized systems, we face the possibility that their exact form may not be completely stable. Thus, following some internal event, a new type of individual may appear (a mutant, an innovation), one having new behavioral characteristics. Its impact on the system as a whole will be determined by the structural stability of the system to such an event. In other words, if it occurs will the interactions of the system lead to its amplification or will they, on the contrary, repress the new type of individual? Such an evolutionary step occurs in two phases. First, a mutant will ap­ pear as the result of some accident or low-probability event, as we have men­ tioned, can be seen as a fluctuation in reproduction. Furthermore, chance con­ tinues to be all important, while only a few such mutants exist, since the time change of their numbers will obey stochastic dynamics: an individual either is born, lives, or dies; he does not do a little of each; here also the size of the fluctuation will play an important role. The second stage starts, however, when and if a mutant manages to mul­ tiply sufficiently to constitute a population whose growth or decay can be de­ scribed by an equation for the average behavior, which then adds to the mac­ roscopic equations already existing. The question we now ask is will the new population grow to some finite value or will it be rejected by the ecosystems? Mathematically, this question corresponds to studying the stability of the pre-

The Challenge of Complexity

29

existing situation before the appearance of a new population type— a mutant or an innovation. Again we have the two aspects discussed before—the ar­ rival of mutations, which is governed by chance, and the response of the sys­ tem, which is deterministic. Thus, in an ecosystem with n interacting popula­ tions we will have a set of equations,

— 1= dt

G,(x„ x2 . . . x„) ,

(31)

where 1 ^ i ^ n and where G, (x l9 x2. . . xn) is some polynomial function of the different interacting populations. After some time a stable situation will be attained, with some stationary average values of the populations x?, x°, . . . x°. If now some small quantity of mutant populations xn+], xn+2, . . . Xn+A appear spontaneously in the system, then an evolutionary step can only occur if the state x?, x°, . . . x°, xn+, = x n+2 = . . . = xn+A = 0 is unstable. The condition for this is that the equation

,20

8¡jK

Det — ' dXj

=0 for i,j = n + 1, n+2, . . . n+A at \°2 . . . ; x°, xn+l = x . . . xn+ =

n+2

0,

(32)

must give rise to a root with a positive real part. If not then the mutant popula­ tion will not grow. For a simple ecosystem with a single population obeying a logistic equation, for example, we have d x, / X,\ — = a , x ,I 1 ------ I — mx, .

dt

\

(33)

N i/

This equation has a stable steady state: x,° = N, (1 - m,/a,) . If now some mutant population x 2 appears, which also obeys a logistic equa­ tion and has a fraction of niche overlap with x,, p, such that dx 2

—

dt

=

^*2 /\ 1 -

x2

0 xl \

N2

M2 /

— - ß—

- m2x 2 .

Then equation (32) becomes, for a positive root X.,

(34)

I. Prigogine and P. M. Allen

30

(35) N 2 (l - m2/ a2) > PJV, (1 - m j a x)

In the simplest case when we discuss mutations x2within the same niche as x u then p = 1, and the only change that can occur is the successive replacement of populations by mutations which exploit more fully the niche resources as shown in fig. 1.15. This particular example is, of course, extremely simple but nevertheless shows us how the above considerations lead to the interpretation of evolution as a dialogue between fluctuations leading to innovations and the determinis­ tic response of the interacting species already existing in the ecosystem. By refining the arguments briefly stated above, we may determine the evolution­ ary strategy favored by a given environment. Thus an ecosystem rich in re­ sources will tend to be occupied by a large number of specialized species, while a system with sparsely scattered resources will be filled with generalists. These results are born out in a very general sense by the change in the diversity of species from the pole to the equator. The solar energy spectrum lies in the same wavelength range everywhere, but what differs is the quantity of energy available at each wavelength. The diversity of flora and fauna in­ creases markedly with decreasing latitude culminating in the extraordinary richness and diversity of the equatorial forests. In a detailed example of the distribution of finch species (Darwin’s finches) on the Galapagos Islands, one finds results in agreement with these predictions.21 As we shall see, the question of the origin and regulation of the division of labor within insect colonies can be studied using these techniques. For evo­ lutionary discussion, the unit that will be subject to selection is not the single ant or bee but the collectivity, the colony. We find that the appearance of a division of labor, of castes, within the insect society is the result of evolution of large colonies existing in a rich medium, where the relative numbers in

Fig. 1.15. Evolutionary succession of species in the same niche.

The Challenge of Complexity

31

each caste will be regulated by the action of chemical substances which re­ press or accelerate the formation of soldiers, for example.22 The evolution of a predator-prey ecosystem, described by the equations of Volterra and Lotka, has also been studied.23 The equations of x, the prey, and y, the predator, are dx dt dy dt

- * H )

sxy (36)

sxy — dy ,

and this gives rise to the steady states, x° = d/s y° = — (1 - d/s). By usd ing our criterion of evolution (32) on this ecosystem, we can study its evolu­ tion. The prey evolves so as to exploit more fully the available resources and to increase its rate of reproduction, and, in addition, it evolves in such a way as to avoid capture by the predator. The predator, on the other hand, evolves so as to increase the frequency of capture of prey and to decrease its own death rate. The resulting evolution of the ecosystem, where both predator and prey evolve, has two aspects. First, there is an “arms race” between the predator and the prey, concerning the changes in the parameters whereby successive improvements in the hunting techniques of the predator are countered by im­ provements in the avoidance techniques of the prey. The second effect of evo­ lution is a result of the increasing resource exploitation of the prey and the declining death rate of the predator. This effect concerns the ratio of predator to prey populations present in the system as evolution proceeds and is given by y° K ^ = - M l - d/s) , d

(37)

and since K/s increases, and d/s decreases, this ratio y°/x° increases slowly with evolution. The evolutionary process favors the predator, because it uses the prey to exploit the primary resources for it. Thus the evolution of the means by which the prey does this favors the predator just as the improvement of a tool favors the user. Indeed, these ideas help us to understand the appearance of agri­ cultural society. Because of an ability to learn, humans can change their co­ efficients of interaction with the environment (K , s, N) faster than nature can

/. Prigogine and P. M. Allen

32

(genetically) evolve countermeasures. Thus a human society that devotes all its inventiveness to improving hunting techniques and avoiding death from unnatural causes simply diminishes the quantity d/s, which is the prey popu­ lation. This course will lead to the rapid extinction of both predator and prey. Agriculture, on the other hand, corresponds to the use of human abilities to increase K , the prey birthrate, and N. The benefits of this action, as we have seen, are passed on to the consumer, and human population increases as a result. In biological systems in which the behavior of individuals is governed uniquely by their genetic makeup, the amplification of a new type of individ­ ual corresponds to Darwinian evolution by the natural selection of mutants which appear spontaneously in the system. In the case of higher animals, however, we have the possibility of behavioral change and its adoption by im­ itation. Of course, this form of order by fluctuation is particularly pertinent to the social system, for which both new and important techniques characterized perhaps by enormously increased productivity, as well as entirely frivolous fashions, can both invade and conquer the system. Our point of view is that evolution results from the individual trial and error of different strategies by the entities composing the system, rather than from some “global optimiza­ tion” where in some way the good of the species exerts a direct influence. So far, however, we have not considered the first stage of the evolution­ ary process outlined above, that is, the mutation pressure, and the initially hazardous multiplication from a single individual of any new population type. This problem is essentially stochastic. Let us describe briefly some recent work, performed by one of us in col­ laboration with Werner Ebeling, that shows that this stochastic phase is indeed of importance for evolutionary theories.24 Consider some ecosystem which, for a steady environment, has adopted a stable average value of the various populations in interaction, x?, x°, . . . x°. If a mutant type x/ of one of these x, appears, then we can write a master equation that describes how the proba­ bility distribution of x/ will evolve initially while x/ < x°]9 x°, . . . x°„. That is, we may write a linearized master equation valid around the state x°i9 x °2 . . . X°n,

=

0

dP(xr,t) ~7 dt

A r [ ( x - l ) P ( x r- l ) - x /P (x r)] + D v [(*,..+1)/>(jc,. + 1) - x rP(xr)\ ,

K }

where Av contain the first order effects of all the processes tending to increase xv and D, all those tending to decrease xv. In general, Ar and D, will be func­ tions of JC,°, x2° . . . x„°. Let us write Ar and Dr in terms of the values of A, and £>„ that is, of

The Challenge of Complexity

33

Fig. 1.16. The effect of evolution on the predator-prey ecosystem of equations.

the parameters corresponding to the population from which the mutant has sprung. Let us suppose that Ar —

Ax = - '

Dr

(1 + 8) .

D,

Then we see that 8 represents the fractional change in the selective advantage of the mutant type. We can solve the master equation and find that the proba­ bility of a mutant, initially a single individual, surviving a time t is: 5 P survival ( t ) = - Dj'A:

—Ô— *— Lt 1

+

Ô

-

£

°i

The probability of surviving a time long enough, on average, to reproduce is t ^ reprod

4 -1 «"

8

P surviving to reproduce = -----------------g 1+ 8

-

e

~

h

This result is very general, and the function has the form indicated in fig. 1.17 for survival for a time of n generations. It is interesting and informative to compare the form resulting from the more correct stochastic analysis with that coming from the deterministic anal­ ysis expressed by the equation (32). For the latter, there was an absolutely sharp distinction between the behavior of favorable mutations, whose proba­ bility of survival was unity, and that of lethal mutations, which was zero. This means that any discussion of the behavior of neutral mutations was restricted to a very narrow band of possible types having exactly the same value of A/D.

/. Prigogine and P. M. Allen

34

Fig. 1.17. The probability of a mutant surviving a varying number of generations.

However, the more correct stochastic analysis shows us that the power of se­ lection is finite and limited. We may speak of a very wide band of quasineutral mutations, since over a considerable range the probability of surviving long enough to reproduce is given by the rather flat curve, D

_ _____ ^_____ ~ J.

surviving to reproduce

g

2

^ Si 8

/ • • •

1 + 8 - e~'T ~s

^(surviving n generations) “

^

1 + 8 - £ 1+ 8 We see then that there are two important forces at work in biological evo­ lution, mutation pressure and Darwinian selection. That is to say, the charac­ ter of evolution will be determined by the probability that a mutation of a given efficiency (8) will occur multiplied by the probability that it will survive long enough to reproduce. Thus if we accept the idea that the most frequent mutations will be those of small 8 (more probably slightly deleterious) and that only occasionally will dramatically advantageous mutants be produced, then we see that we arrive at a view which resolves the apparent conflict be­ tween those who have discussed evolution almost exclusively in terms of one or the other of these two factors: selective advantage (Goodman,25 Maynard-

The Challenge of Complexity

35

Smith,26 Dawkins27); neutral mutational pressure (Kimura, Ohta,28 King, Jukes29). From our analysis we can see that when a strongly advantageous muta­ tion does occur it will be amplified and spread, deterministically, throughout the system, sweeping away the preexisting types and, at the same time, fixing some of the various neutral mutations that characterize its particular lineage and making them standard throughout the system, despite their quasineutral­ ity. This picture also leads to the idea of an evolutionary clock, with an ap­ proximately constant rate of quasineutral drift, and to the fact that speciation, resulting from some ecological isolation, will be marked by a bifurcation in this drift and give rise quite naturally to the evolutionary “trees” of, for exam­ ple, Margloliash,30 for the protein cytochrome-c. Our point of view than encompasses both a stochastic drift, where some­ times, of course, neutral mutations may well be fixed in the population by a purely stochastic elimination, such as in the famous surname elimination problem,31 but will also be fixed uniquely by the spread of an advantageous mutation, amplified deterministically according to our criterion (32), which carries a particular heritage of quasineutral mutations with it. Let us add that we should also consider, the fact that in any real system the environment is fluctuating, and, therefore, evolutionary change may not require the appear­ ance of a particular mutant but may give to previously neutral types quite dif­ ferent efficiencies. Evolution is the result of this very complex interplay between stochastic factors (mutation pressure, environment, small numbers) and deterministic factors (selection pressure, steady environment), and both aspects are in fact crucial. In view of all these interactions, the very existence of such complex systems as a tropical forest or a modern society poses an interesting problem from the start. Is there a limit to complexity? The more elements that enter into interaction, the higher the degree of the secular equation determining the characteristic frequencies of the system and the greater the chance, therefore, of having at least one positive root and hence instability.32 Several authors have suggested that ecological evolution selects certain particular types of systems that are stable. It is nevertheless difficult to give a quantitative form to such a suggestion. Our approach leads to a different an­ swer. A sufficiently complex system will generally be in a metastable state. The value of the threshold of metastability depends on the competition be­ tween growth and damping through surface effects.33 Many complex systems are also systems in which the interactions with the surroundings (which in social problems correspond to such mechanisms as the flow of information) are also strong. Certainly, our present society, when compared to primitive societies, is characterized both by a high degree of complexity and a rapid dissemination of information. The question “is there a limit to complexity?”

/. Prigogine and P. M. Allen

36

1/1

»—

zLU 2

LU

ä

U ^

LU

¿

g

LU ^

o¿ OC

UJ ^

og “ xX U O

3

Z uj D Ï UJ

t* 3 O UJ oc

Fig. 1.18. Evolutionary tree based on cytochrome-c (after E. Margoliash et al., 1972, “The Molecular Basis of Electron Transport,” Miami Winter Symposium, Vol. 4, Academic Press, New York and London).

may have a less clear-cut answer than those that have been considered up to the present. According to our results, an important aspect of the answer would be that complexity is limited by stability, which, in turn, is limited by the strength of the system-environment coupling. We cannot go into more de­ tail here, but we see that the idea of progress or continuous increase of com­ plexity is far from a simple one.34 Let us now turn to structural stability. Carneiro (see also his contribu­ tions to this volume), following Herbert Spencer, has emphasized the dif­ ference between quantitative and qualitative changes in a culture. He dis-

The Challenge of Complexity

37

tinguishes cultural development, in which new cultural traits are coming into being, from cultural growth. In our terminology, cultural development would correspond to instabilities in which stochastic effects play a basic role, while cultural growth corresponds to deterministic developments. Our point of view is that a complex system, such as the social system, is characterized by equations expressing the interdependence of the various ac­ tors of the system and that these intrinsic nonlinearities, in dialogue with fluc­ tuations, result in the self-organization of the system, so that its structures, articulations, and hierarchies are the result, not of the operation of some “global optimiser,” some “collective utility function,” but of successive in­ stabilities near bifurcation points. Such a view takes into account the collec­ tive dimension of individual actions and emphasizes the possibility that indi­ viduals acting according to their own particular criteria may find that the resulting collective vector may sweep them in an entirely unexpected direc­ tion, perhaps involving qualitative changes in the state of the system. It is not surprising then that many attempts at modeling such complex systems have been largely unsuccessful, particularly in the medium and long term. For the short term, if the basic qualitative nature of the system is assumed not to change, then it is possible, not only to write deterministic relations for the system’s evolution, but also to infer a potential function from them— a utility function— which appears to have an objective existence behind the phenom­ ena. However, as we have seen above, near an instability, the behavior of the system may only be correctly ascertained by studying the effects of the fluc­ tuations. This fact becomes clear when we think of a situation when, for ex­ ample, a city with a single business and commercial center and with circular symmetry stretching out to diffuse suburbs suffers an increase in transport costs. While it is possible that the city may simply shrink, without undergoing any basic structural reorganization, it is also possible that the simple sym­ metry may break down and subcenters appear leading to the formation of a polynuclear city. A simple utility function approach can describe the first pos­ sibility but not the second. This type of creative evolution, this morphogene­ sis, requires the active participation of fluctuations as we have seen; it is pre­ cisely an example of order by fluctuation. (The description of urban structure in these terms is dealt with in detail in this volume in the separate article by Allen. Adams35 has studied the relation between energy flow and social struc­ ture in a great many systems and suggests that energy flow reflects the state of organization of a given social system, just as the level of entropy production characterizes a particular dissipative structure. This question is developed fur­ ther in his accompanying article. The evolutionary mechanism that we have outlined suggests why the nat­ ural evolution of the system does not necessarily correspond to the optimistic

38

I. Prigogine and P. M. Allen

implication of the word progress. The collective structure of the system and its evolution do not necessarily correspond to the individual desires of the ele­ ments of the system. The collective structure simply reflects the nonlinearities of the interactions and, therefore, can give rise to unexpected, and often un­ desirable, aspects of progress. Evolution has acquired in some quarters something of the status of a the­ ological principle. As Leach has said,33 a century ago Darwin and his friends were thought to be dangerous atheists, but their heresy simply replaced a be­ nevolent personal deity called God with a benevolent impersonal deity called Evolution. In their different ways, Bishop Wilberforce and T. H. Huxley both believed in fate. It is this religious attitude which still dominates scientific thinking about future development. Darwin’s ideas belong to the same phase of nineteenth-century thought as laissez faire economics—the doctrine that in a free-for-all competition the best will always win out anyway. Leach, however, does not share this view­ point, and neither do we. The evolution characteristic of complex systems, such as we encounter in biology, ecology, or sociology, as we have described here, presents real choices and real freedoms. In consequence, we have the responsibility of trying to understand the dynamics of change in order both to formulate realistic objectives and to discover which actions and decisions should be taken in order to move closer to them.

ACKNOWLEDGMENTS This work has been partially supported by the Actions de Recherche Concer­ tées of the Belgian Government, under convention n°76/81 II.3.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

L. von Bertalanffy, 1968, “General System Theory, Foundation, Development Applications,” G. Braziller, New York. C. Shannon and W. Weaver, 1949, “Mathematical Theory of Communication,” Univ. of Illinois Press, Urbana, 111. R. Thom, 1972, “Stabilité Structurelle et Morphogénèse,” Benjamin, New York. B. Mandelbrot, 1977, “Fractals,” W. H. Freeman, San Francisco. G. Nicolis and I. Prigogine, 1977, “Self-Organization in Nonequilibrium Sys­ tems,” Wiley Interscience, New York, London. G. Nicolis and J. Turner, 1977, Physica, 89A, p326. I. Prigogine, B. Misra and M. Courbage, To Appear. I. Prigogine and R. Herman, 1971, “Kinetic Theory of Vehicular Traffic,” Else­ vier, Amsterdam.

The Challenge of Complexity

39

9. F. Schlogl, 1971, Z. Physik, 248, p446. 10. G. Nicolis, 1979, in “Far from Equilibrium,” eds. Pacault and Vidal, SpringerVerlag, Berlin. 11. M. Herschkowitz-Kaufman and T. Erneux, 1979, Annals of the New York Acad­ emy, of Sciences, Vol. 316, p296. 12. D. Walgraef, G. Dewel and P. Borkmans, Physical Review A, To appear. 13. A. Goldbeter and S. Caplan, 1976, Annual Reviews of Biophysics and Bioen­ gineering, Vol. 5. 14. A. Goldbeter and G. Nicolis, 1976, Prog, in Theor. Biol. 4. 15. T. Erneux and J. Hiernaux, J. of Math. Biol. To Appear. 16. R. Lefever and R. Garay, 1977, in Theoretical Immunology, Ed. M. Dekker, New York. 17. R. Lefever and W. Horsthemke, 1979, Bull. Math. Biol., Vol. 41, pp. 469-490. 18. S. Kabashima et al., 1978, “Oscillatory to Non-oscillatory Transition due to noise in a Parametric Oscillation,” To Appear, J. Phys. Soc. Jap. 19. G. Nicolis and J. Turner, 1977, Physica 89A, p326. 20. P. M. Allen, 1975, Proc. Nat. Acad. Sci. USA, 73, p665. 21. P. M. Allen, 1976, Ecologie Quantitative, E4, Summer School, University of Venice, Ed. A. Orio, and J. Vigneron. 22. J. L. Deneubourg and P. M. Allen, 1976, Bull. Class. Sci. Acad. Roy. Belg. Tome LXII 5-6. 23. P. M. Allen, 1975, Bull. Math. Biol. 37, No4, p389. 24. P. M. Allen and W. Ebeling, To Appear. 25. M. Goodman et al. 1971, Nature, 233; p604. 26. J. Maynard-Smith, “Evolution and the Theory of Games,” Amer. Sci. 64, p. 44-5. 27. J. Maynard-Smith 1974, “The Theory of Games and the Evolution of Animal Conflict,” J. Theor. Biol. 47, p209. 27. R. Dawkins, 1976, “The Selfish Gene,” Oxford Univ. Press, New York. 28. M. Kimura and T. Ohta, 1974, Proc. Nat. Acad. Sci. USA, Vol. 71, No7, p2848. 29. J. L. King and T. H. Jukes, 1969, Science, 164, p788. 30. E. Margoliash et al. 1972, “The Molecular Basis of Electron Transport,” Miami Winter Symposium, Vol. 4, Academic Press, New York and London. 31. M. Bartlett, 1960, “Stochastic Population Models,” Methuen Monograph on Ap­ plied Probability and Statistics, London. 32. R. May, 1973, “Model Ecosystems,” Princeton University Press; Princeton, N.J. 33. G. Nicolis and I. Prigogine, 1977, “Self-Organization in Nonequilibrium Sys­ tems,” Wiley, New York, Chap. 12. 34. I. Prigogine and R. Herman, G. Nicolis and T. Lam, 1974, Collective Phenomena. 35. R. N. Adams, 1975, “Energy and Structure,” University of Texas Press, Austin and London. 36. E. Leach, 1967, “A Runaway World,” The Reith Lectures, BBC.

2. Self-Organization in Nonequilibrium Chemistry and in Biology Jack S. Turner

INTRODUCTION The enormous complexity and organization of living things have long held great fascination for scientist and nonscientist alike. Despite longstanding in­ terest—from the time of the ancient Greeks, at least—even so basic a ques­ tion as establishing the compatibility of biological order and evolution with physical law remained open well into the present century. Indeed, it was not until the mid-1960s, with the development by Glansdorff and Prigogine of a new thermodynamics applicable far from equilibrium,1 that it was possible even to address this vital question within a proper theoretical framework. The main conclusion of the new theory which settles the issue is that there exists a class of thermodynamic systems which exhibit two completely different types of behavior: in some circumstances the expected tendency toward maximum disorder, but in others the spontaneous appearance of qualitatively new states of matter which exhibit a high degree of organization in space, time, or func­ tion. The coherent behavior of such macroscopic states is precisely that which characterizes living systems, and the then surprising prediction of their ap­ pearance in purely dissipative systems, such as catalytic chemical reactions, under suitable nonequilibrium conditions, provides the final connection. The first systematic investigation of self-organization in nonequilibrium chemistry was initiated over a decade ago by Ilya Prigogine and his coworkers at the Free University of Brussels.2From modest beginnings involving a sim­ ple model of a hypothetical chemical reaction, the study of nonequilibrium instabilities, and of the “dissipative structures” 24 which emerge beyond them, has literally exploded into a dynamic new discipline with far-reaching implications for self-organization phenomena in such diverse fields as phys­ ics, chemistry, biology, sociology, and economics. Indeed, the contributions to the present symposium reflect in part the efforts of many to identify general Jack S. Turner is with the Center for Statistical Mechanics and Thermodynamics, Uni­ versity of Texas at Austin.

Nonequilibrium Chemistry and Biology

41

features and common elements which emerge in a variety of disciplines and to learn from each other’s experience how to better understand their own. Since my own experience is mainly in the area of instabilities and transitions to dis­ sipative structures in nonequilibrium chemistry, I shall concentrate on several aspects of this specific area which may be of wider interest. In the context of nonequilibrium chemistry, the general ideas will be in­ troduced in the following sections, together with an overview of the variety of possibilities for nonequilibrium order. Here I will focus on the BelousovZhabotinskii reaction, a remarkable system which is the best-studied and most versatile laboratory example of chemical dissipative structures. In discussing chemical instabilities, I will emphasize wherever possible the connection to biology. This will provide an appropriate setting for examining evolutionary aspects of dissipative structures, which may well be of greater interest in the disciplines in this collection which focus on higher levels of organization. In discussing the mechanism by which instabilities appear, some questions con­ cerning fluctuations and nucleation will also be considered.

DISSIPATIVE STRUCTURES, FLUCTUATIONS, AND NUCLEATION It is now well established that open systems undergoing nonlinear chemical reactions and transport far from thermodynamic equilibrium are subject to in­ stabilities in the thermodynamic branch of solutions and may evolve subse­ quently to other operating regimes characterized by new types of behavior The class of chemical systems that may exhibit such behavior is characterized by two essential features. On one hand, certain types of nonlinearities are re­ quired both to permit the existence of more than one macroscopic state and to provide an appropriate chemical mechanism by which instabilities may ap­ pear. On the other hand, it is necessary that the system operate in the farfrom-equilibrium (i.e. nonlinear) domain of irreversible thermodynamics. That is, a minimum level of dissipation is essential to both creation and main­ tenance of the “dissipative structures” which may arise beyond instability. In the coupling of chemical kinetics with transport, it is well understood now that the types of nonlinearities which can lead to instabilities and dissipa­ tive structures are provided by auto- and cross-catalytic processes, by chemi­ cal steps which by positive or negative feedback influence their own rates of reaction. The surprising consequences of even a simple catalytic nonlinearity are probably best illustrated by the Brusselator, a mathematical model of a hypothetical chemical system. Since the first studies over a decade ago, the Brusselator has been the theoretical prototype for chemical dissipative struc­ tures. Some of its many remarkable features are discussed in these proceed­

.3

Jack S. Turner

42

ings by Prigogine and Allen. As is well known, in biological systems there exists an essentially limitless variety of mechanisms and pathways which sat­ isfy both requirements for the emergence of dissipative structures. In con­ trast, comparatively few examples have been documented in ordinary (i.e., nonbiological) chemistry which possess the essential qualifications. Never­ theless, it is in fact a nonbiological chemical system, the Belousov-Zhabotinskii (BZ) reaction which has become the experimental prototype for self­ organization phenomena and dissipative structures in all of nonequilibrium chemistry. Because this single reaction has been found to exhibit an almost incredible variety of instabilities, I will focus in the next sections on this spe­ cial system in discussing some of the phenomena which may occur in self­ organizing systems. Let me begin here by reviewing some of the general as­ pects of the theory of dissipative structures. Consider an open, multicomponent mixture of species Snamong which a number of reactions R^ may occur. If X„ (r,0 denotes the concentration of species Sn at position r and time t, then the usual deterministic description of chemical kinetics leads to a set of coupled partial differential equations of the form

,5

Y t (r,t) = F » ( M 'V ) } ) + D„V2X„(r,t)

(1)

The first term gives the change in X n(r, t) that results from chemical reactions and, as noted above, will be a nonlinear function of the composition variable for systems of interest here. The second term gives the change in X„ (r, t) due to diffusion (the only transport mechanism treated here), which obeys a linear phenomenological law of the Fick type. Because of their nonlinearity, equations (1) may admit several time-independent solutions X°n(r), one of which is termed the thermodynamic branch since it is the continuous extension away from thermodynamic equilibrium of the unique equilibrium state. As external parameters of the system are varied to displace it away from equilibrium, the observed behavior will depend cru­ cially on the stability properties of the various states which are accessible un­ der the changing conditions. This situation is indicated schematically in fig. 2.1.

To assess the stability of the thermodynamic branch or other steady state, one considers the response of the system to small perturbations about that state. If perturbed states near a steady state X°(r) are written in the form X n(r,t) = X M + xa(r,t)

(2)

Nonequilibrium Chemistry and Biology lin e a r

43

n o n lin e a r

thenhodynamic branch

d is t a n c e from e q u ilib riu m

Fig. 2.1. Macroscopic states accessible to a nonlinear thermodynamic system (schematic).

then, in the limit of small perturbations (i.e., \xJX°„\ < 1), the nonlinear chem­ ical balance equations (1) yield a set of linear equations in the xn. These equa­ tions are then analyzed in terms of normal modes of the form x n(r,t) = x°n exp (cor + ikr)

(3)

where for simplicity a single spatial coordinate r is considered, with w and \=2ir/k the frequency and wavelength of the mode. With periodic boundary conditions, the linearized equations possess a nontrivial solution of the form (3) provided the characteristic equation (4) is satisfied. The subscript zero implies evaluation at the steady state, and bnj denotes the Kronecker delta (&„,• = 1 if n—j , and zero otherwise). The eigenfrequencies are in general complex, w = cor + /a),, the stability properties of the linearized equations and, ultimately, of the steady state solu­ tion {X°(r)} of the full nonlinear system, depending on the sign of the real part, o)r. Without going further into mathematical detail,3 we can identify three basic situations from the point of view of stability: (1) All cor < 0: The

Jack S. Turner

44

steady state is asymptotically stable, whatever the nonlinear terms in equation (1). Following a small perturbation, all normal modes decay. (2) At least one wr > 0: The steady state is unstable; the mode with oor > 0 will grow. In either case ( ) or ( ), the behavior of individual modes will be oscillatory if the cor­ responding o),^ 0 and monotonic otherwise. (3) At least one (or = 0, all other (or < 0: The steady state is marginally stable with respect to the mode having cor = 0. As a solution of the linearized equations that mode will neither decay nor grow but may oscillate if it has in addition co, ^ 0. The marginally stable steady state of the nonlinear system may be stable or unstable depending on the explicit form of the nonlinear terms. To this point nothing has been said about the effect of diffusion on the stability of steady states. In general one would expect diffusion to be a sta­ bilizing influence because of the tendency of transport to damp inhomogenei­ ties. This tendency notwithstanding, the explicit presence of diffusion in the equations permits the existence of spatially inhomogeneous perturbations and therefore increases greatly the variety of situations that may arise once an in­ stability occurs. The many types of behavior that may appear beyond instability include time-independent states, spatially uniform or nonuniform, as well as states that are organized in time (homogeneous chemical oscillations) or in time and space (travelling chemical waves). All these features are found in the Brusselator, for example (see the article by Prigogine and Allen in these proceed­ ings). As has been emphasized throughout the theoretical development which state will arise under given conditions depends crucially on the specific type of perturbation that destabilizes the original state. In terms of sponta­ neous molecular fluctuations that are always present, both the final state and the evolution toward that state will be determined by the space-time charac­ teristics of the fastest-growing unstable mode. Moreover, when spatially lo­ calized fluctuations are too “small” to provide a stable nucleus for a transi­ tion, then states beyond the instability point may be realized. Such states are termed metastable, since they are stable only until a large enough fluctuation appears to form a supercritical nucleus. The most important consequences of these nondeterministic aspects of dissipative structures will be summarized in the following paragraphs. In recognition of the dominant role of fluctuations at nonequilibrium in­ stabilities and transitions, Prigogine and his coworkers have introduced the concept of “order through fluctuations” to distinguish transitions to dissipa­ tive structures from those which obey the ordering principle of thermody­ namic equilibrium Specifically, this new ordering principle for nonequilib­ rium systems refers to the amplification of fluctuations beyond instability and to their ultimate stabilization by continuous exchange of matter or energy with the environment. In order to study this basic mechanism underlying all such

1

2

,4

.4

Nonequilibrium Chemistry and Biology

45

macroscopic instability phenomena, it is obvious that one must go beyond the deterministic description and formulate the problerii of chemical kinetics at the molecular level. From a practical point of view, however, it is convenient to adopt a stochastic approach which is intermediate between the microscopic and macroscopic levels of description. In a deterministic description attention is focused on the average con­ centration of species Sn in volume V at time t, and fluctuations about such average values are neglected in deriving differential equations which give their time evolution. For a system undergoing chemical reactions the simplest theory which takes fluctuations into account assumes that the macroscopic mass-balance equations (1) define a Markov process in the number-of-particles space of the constituent species If X„ denotes the number of molecules of type S„, then the state of the system is specified by the probability P(X,, . . . 9XN9 t) of having X, molecules of type Sl9 X 2 molecules of type Sl9 etc., in volume V at time t. Within the framework of the birth-and-death formula­ tion of stochastic theory, the evolution of P({X„};t) is given by a stochastic master equation of the following general form:

.6

—— dt

=

2 [WJ{X„ - vJ^{X„})P({X„ - v j ' t ) |x =

1

(5)

w ,( R } ->{*„ + v j ) P ( { x ny9t ) ] . For every reaction R^ there are separate gain (“birth”) and loss (“death”) contributions to the evolution of t). Each such contribution is a prod­ uct of WJ^iX?} —> {X° + v„J), the transition probability (per unit time) for reaction R ^ to occur given that the system is in the state {X°}9and P({X°}\t)9 the probability that the system is in the state {X°} at the time t. Here is the stoichiometric coefficient for species S„ in reaction R ^ and in this context de­ notes the net increase in x n for a single occurrence of reaction R ^ Transport processes such as diffusion are taken into account by dividing the system into a number K of spatially uniform “cells.” The chemical reac­ tions going on in cell k are described by a reactive transition operator R k de­ fined by the right-hand side of equation (5), and additional gain-loss terms account for diffusion between cells. The result is a multivariate master equa­ tion which has the general form (for a single composition variable X)9

— dt

=

2

k=\

+

d (Xk + 1)P(Xk. i - 1, X, + 1) - X kP(Xk. t,Xk) + d (Xk + 1)P(Xk + 1, X k+I - 1) - X kP(Xk, X k+I)]

6

( )

Jack S. Turner

46

Here only those arguments which change in a process are written explicitly. Changes or additional terms must be introduced to account for the boundary conditions. The quantity d is a stochastic diffusion coefficient which is related to the usual Fick-type constant by a relation obtained from simple kinetic the­ ory arguments for ideal mixtures

,4

d ~ D /U r

(7)

where / is the length of a cell and lr is the mean free path. Despite the fact that the multivariate master equation is a linear equation, analysis of it is difficult, even for simple chemical systems. Consequently a number of approximate an­ alytical methods have been developed One remarkable result that has been obtained analytically is that one finds long-range correlations in the neighborhood of instabilities Far from instability each cell is correlated only with its immediate neighbors, as one would expect. Near the instability point, however, long-range correlations appear, much like those found near equilib­ rium critical points. Indeed, there is a deep analogy between nonequilibrium instabilities and equilibrium phase transitions and critical phenomena In the last few years several computer algorithms have been developed for simulating the stochastic evolution of chemical systems These new methods complement the analytical approaches and, in addition, permit the study of complex systems beyond the reach of even approximate analytical techniques. In view of the complexity of the multicell stochastic descriptions, it is of great interest to seek a reduced formulation which retains the essential fea­ tures of the microscopic chemical dynamics and yet is more tractable in appli­ cation. To this end a kind of “mean-field” theory of chemical instabilities has been developed in recent years by Prigogine, Nicolis, Malek-Mansour, and coworkers In this approach the detailed space dependence is simplified by focusing on a single homogeneous subvolume AV V and treating its contact (via transport) with its surroundings in a self-consistent (mean-field) manner. In this way the concept of fluctuation size (AV) is made explicit, so that ques­ tions of nucléation can be addressed in a tractable stochastic theory. The result of reducing the stochastic description to a completely local theory is a nonlinear master equation for the probability P AV(X,, X2% . . . \t) of finding X, molecules of type Su X of type S2, . . .in the subvolume AV at time t. For a single chemical species this equation has the form

.4,678

.8

.4’9,10,11

.111215

.1415

2

^ PUX\t) = R , vP ,v(X;t) dt

Nonequilibrium Chemistry and Biology + D(X)[PAV( X - U t ) ~ PAV(X;t)\

47

8

( )

+ D [(X+l)PAV(X+l;i) - XP U X 'M . Here /?AVis again the reactive transition operator of equation (5), and D is an effective diffusion coefficient of the form (7). Self-consistency is ensured, for a macroscopically homogeneous system, by the appearance in the first diffu­ sion term of the average value (X) = 2 x

(9)

P AV(XU)

This term reflects the influence of the surroundings on the fluctuating sub­ volume a v . Because of assumptions made in deriving the nonlinear master equation ( it is useful mainly to provide a stochastic stability theory for macro­ scopically homogeneous systems. As has been emphasized already, stability or instability of a given state is determined by the behavior of spatially lo­ calized fluctuations about that state. Since the range or coherence length of fluctuations appears explicitly, the nonlinear master equation is well suited to the problem of characterizing localized fluctuations around globally homoge­ neous steady states. For the same reasons, it may be viewed as the first step toward a nucleation theory of nonequilibrium instabilities. The nonlinear master equation has been applied to several types of chem­ ical instabilities, including transitions to spatially homogeneous oscillations and transitions between multiple steady and spatial dissipative structures states In each case, the competition between diverging local chemical re­ action and damping by diffusive contact with the surroundings leads to a crit­ ical fluctuation size necessary to provide a stable nucleus for transition. The predictions of chemical nucleation have been verified recently, for transitions between multiple steady states, by microscopic computer simulations based on reactive molecular dynamics It is important to realize that this is a new kind of nucleation phenomena arising solely from chemical rather than physical forces between atoms and molecules. In contrast to the more familiar nucleation processes associated with equilibrium first-order phase transitions, here it is the chemical state of matter; and not matter itself which is spatially localized to form the nucleus for transition.

8),14

.1611

,414

.11

JackS. Turner

48 THE BELOUSOV-ZHABOTINSKII REACTION

Having outlined the main theoretical aspects of dissipative structures, let me now turn to the interface between theory and experiment in a much-studied chemical system. The great interest of the Belousov-Zhabotinskii (BZ) reac­ tion lies in its unique ability to exhibit a wide variety of characteristic non­ linear behavior. Ranging from homogeneous chemical oscillations in stirred reactors to travelling chemical waves and stationary spatial structures in un­ stirred mixtures, the phenomena observed in this system include all the types of nonequilibrium coherent structures that have'been studied theoreti­ cally in the past decade. In the last few years, moreover, new experiments have suggested that the same chemical processes which give rise to coherent phenomena may also yield incoherent or chaotic behavior as well It is gen­ erally acknowledged that each of the above types of nonequilibrium order plays an important role in the organization of life processes the idea that some aspects of biological disorder may be understood in the same way presents fascinating new possibilities which are just beginning to be explored. In contrast to the vast majority of laboratory and theoretical model sys­ tems which exhibit similar behavior the BZ reaction is not biological and can be studied with relative ease under carefully controlled laboratory conditions. Although the BZ reaction is highly complex, involving the cerium-catalyzed bromination and oxidation of malonic acid by bromate ion, its detailed mech­ anism is now reasonably well understood Therefore this remarkable system affords the unique possibility for parallel theoretical and experimental inves­ tigations into the nature both of the mechanism of macroscopic chemical in­ stabilities and of the dissipative structures which may arise beyond them. Experiments on the BZ system can be of several types, depending on the particular behavior to be investigated. In batch (i.e., closed system) studies, the system is prepared in an initial condition far from equilibrium by mixing known concentrations of the primary chemical ingredients. A schematic rep­ resentation of the subsequent evolution of the system toward a unique equi­ librium state is shown in fig. 2.2. An ordinary chemical system would proceed monotonically to equilibrium, and for some initial conditions this ex­ traordinary chemical system will too (e.g., path A of fig. 2.2). Under other conditions the system enters a region of its “state space” in which it organizes spontaneously in time or space. For a period of time thereafter, an ordered behavior will be superposed on the otherwise monotonic approach to equi­ librium. As the initial chemical reactants are consumed, this coherent behav­ ior may change, qualitatively as well as quantitatively, until it ceases entirely and the system proceeds on toward equilibrium. If the mixture is stirred so as to remain spatially uniform, then, as we shall see, several types of simple and not-so-simple patterns of homogeneous

517

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;4’10,19,20

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.21

Nonequilibrium Chemistry and Biology

49

INITIAL CONDITIONS

Fig. 2.2. Schematic evolution of a closed system initially far from thermodynamic equilibrium, based on observations of homogeneous oscillations in the stirred Belousov-Zhabotinskii reaction.

chemical oscillation may arise. If the mixture is only weakly stirred or is not stirred at all, then oscillations in space as well as in time are possible. Although in many ways the resulting chemical waves and spatial struc­ tures are the most spectacular manifestations of the underlying interplay be­ tween chemistry and transport, these phenomena are also much less tractable from the theoretical point of view, and so will not be discussed further here. In addition, biological systems are characterized by existing spatial nonuni­ formities in the form of membranes, macromolecules, metabolite gradients, and so forth. Hence, spontaneous transitions leading to spatial symmetry breaking in an initially homogeneous system are most relevant biologically to special situations such as first stages of morphogenesis. Perhaps of greater direct importance to existing biological (and social) structures are the multi­ plicity of spatial and spatiotemporal dissipative structures which are accessi­ ble under the same conditions and the possibility of transitions to new struc­

Jack S. Turner

50

tures as their relative (and absolute) stability properties alter with changing conditions. In the BZ reaction spatial uniformity is maintained by continuous stir­ ring. In a particular biological context spatial coherence may be achieved on a macroscopic scale despite detailed microscopic heterogeneity. In such cases an appropriate first step in a theoretical analysis will be the study of homoge­ neous models which may serve to discriminate among candidate microscopic mechanisms and to assess the need for treating explicitly the spatial aspects of the problem. Examples of the latter approach will be identified in the follow­ ing paragraphs. From both theoretical and experimental viewpoints, the study of tran­ sient phenomena occurring in a closed system evolving toward equilibrium presents formidable problems. Not only can one not know in advance which path a system will take (cf., fig. . ), but it is even difficult (if not impossible) to characterize the path at all quantitatively. Moreover, general questions re­ garding regions of stability for various states can hardly be considered at all. The kind of control needed to overcome these problems is provided by a flow system in which initial reactants are supplied continuously and effluent re­ moved. In such an open-system configuration, it is possible to examine in a precise way a system’s phase space. In the present context this means that any part of a particular trajectory in the closed system schematic of fig. can be studied for a period of time that is entirely at the observer’s discretion. In particular, steady states may be established, and transitions to new states stud­ ied as any desired external variable is changed. In this way a direct interplay between experiment and theory is made possible, as we shall see in the next section.

22

2.2

PERIODIC AND NONPERIODIC CHEMICAL OSCILLATIONS In closed-system studies of the BZ reaction, three principal modes of homo­ geneous oscillation have been observed: (a) low-frequency, large amplitude, highly nonlinear (i.e., nonharmonic) limit cycle oscillations (fig. . d); (b) higher-frequency, small amplitude, quasiharmonic oscillations (fig. 2.3a); and (c) double-frequency oscillations containing variable numbers of each of the two previous types (fig. 2.4b). All three types are shown in fig. 2.4. By far the most familiar feature of the BZ reaction, the relaxation oscillations of type a were explained by Field, Koros, and Noyes in their pioneering study of the detailed BZ reaction mechanism Much less well known experimentally are type b oscillations although they are more easily analyzed mathematically in terms of bifurcation theory The double-frequency mode (c), first reported by Vavillin and coworkers has been studied also by the present author and

23

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.24 22

Nonequilibrium Chemistry and Biology

51

VVVWWY ÛÛ cn

•W V W V Y V V . d

Ti me Fig. 2.3. Experimental traces of bromide ion concentration in closed-system studies of the Belousov-Zhabotinskii reaction, showing (a) quasiharmonic (i.e., sinusoidal) os­ cillations; (b) and (c) increasingly nonlinear oscillations; and (d) large-amplitude re­ laxation oscillations. The vertical bars at left represent equal concentration ranges; the horizontal bars represent equal time intervals of one minute.

coworkers, who explained the phenomenon qualitatively on the basis of the FN models of the BZ reaction In closed-system experiments, the observed oscillations represent tran­ sients in the overall monotonie approach to equilibrium (cf., tig. 2.2). Al­ though the main features of oscillation may not change significantly during an observation period, it is nevertheless not possible to characterize the state of the system, even approximately, in terms of the concentrations of the initial reactants at any time, for example. Under open-system conditions, however, the kind of control needed to maintain an invariant stationary or oscillating regime, and to vary selectively the principal reactants, is readily attained. As would be expected, therefore, oscillations of types a and b above can be main­ tained indefinitely in a stirred flow reactor. In addition to the two simplest sit­ uations, however, two qualitatively new modes of oscillation that were not anticipated have also been observed under these conditions: (d) intermittent bursts of type a oscillations separated by periods of quiet (homogeneous quasi-steady state) behavior (cf., fig. 2.5) and (e) mixed frequency os­ cillations with single occurrence of type a separated by a generally variable number of type b (cf., figs. 2.6 and 2.7b). For example, in a recent experi-

.23

,2526

18

Jack S. Turner

52

*—

TIME

(minutes)

*•—

TIME

(minutes)

Closed-system experiments showing (a) continuous and (b) discontinuous transitions between two types of chemical oscillation in variations of the BelousovZhabotinskii reaction (reproduced from reference 23, with permission).

Fig. 2.4.

-5

10

S

1Ó6 -7

10

0.

400.

8 0 0 . 1200. 1600. 2000. Tine

(Soo)

Fig. 2.5. Bursts of oscillation observed in a computer simulation of the open Be-

lousov-Zhabotinskii reaction.

Nonequilibrium Chemistry and Biology

53

mental study of the latter phenomenon in the Ferrion-catalyzed version of the BZ reaction, Schmitz and coworkers reported mixed-frequency oscillations ranging from periodic or quasi-periodic to apparently chaotic as the flow rate was varied. In order to understand such complex oscillatory modes in terms of chem­ ical mechanism, it is convenient to begin with a simplified version of the full BZ mechanism which retains the essential features of experimental obser­ vations. To that end I consider a reduced three-variable model, the Oregonator, which was prepared by Field and Noyes and focus on a reversible ver­ sion by Field

18

21

,27

.28

A + F = S ’D + sf">

Benoit B. Mandelbrot

106

The symmetric solutions of (S) and (AD) have the characteristic function

7

exp(iuz)dPr(U u], (2a) Fr[logeZ(t + one day) - log,Z(i) < —u], both for the daily closing prices of cotton in New York, 1900-1905 (communicated by the United States Department of Agriculture). (lb) Fr[logeZ(t + one day) - logeZ(0 > u], (2b) Fr[logeZ(t + one day) - log,Z(0 < —u], both for an index of daily closing prices of cotton on vari­ ous exchanges in the United States, 1944-1958 (communicated by Hendrik S. Houthakker). (lc) Fr[logeZ(t + one month) - logeZ(t) > u], (2c) Fr[logeZ(t + one month) - log,Z(0 < - u] , both for the closing prices of cotton on the fif­ teenth of each month in New York City, 1880-1940 (communicated by the United States Department of Agriculture). The reader is advised to copy the horizontal axis and the theoretical dis­ tribution on a transparency and to move both horizontally. The theoretical curve will then be superimposed on either of the empirical graphs with slight discrepancies of general shape. This is precisely what my scaling criterion postulates. (The slight asymmetry can be handled too; it requires skew vari­ ants of the stable distribution.) The curves (la) and (lb), (2a) and (2b) would be identical if the pro-

2

Fig. 4.5

cesses ruling cotton price change were stationary, but in fact they differ by a horizontal translation. Since translation on doubly logarithmic coordinates corresponds to a change of scale in natural coordinates, this discrepancy led me to concur in my 1963 paper with the economists’ opinion that price change distributions around 1950 differed from their counterparts around 1900. I thought the distribution preserved the same shape but had a smaller scale. More recently, however, I found that this concession to opinion was beyond necessity. I became aware that the data on which curves (la) and (2a) were based had been read incorrectly (Mandelbrot 1972). Once this error is corrected, one is led to curves (la*) and ( a*) that are nearly identical to the curves (lb) and (2b). In other words, the process that rules the changes in the price of cotton seems, in a first approximation, to have remained stationary over the very long period under study. There is no disputing the major changes in the value of currency and similar events. But such overall long trends are negligible in comparison with the fluctuations with which we deal here. One cannot deny that the data give at casual glance the impression of being grossly nonstationary, but this impression is the result of casual impressions formed against the background of a belief that the underlying process is Gaussian. My alternative to the nonstationary Gaussian process is a stationary non-Gaussian stable process. I believe the latter gives a much greater hope of eventually being related to a good economic theory and of yielding a realistic statistical algorithm. To appreciate the nature of this achievement, it is vital to look carefully at the scales. In this instance, scaling implies that the distribution based on a record of daily price changes over a period Tjf five years of average economic variability extrapolated to monthly price changes goes right through the data

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Benoit B. Mandelbrot

from the various recessions, the depression, and so forth. It accounts for all the most extreme events of nearly a century in the history of an essential and most volatile commodity. I do not believe there is any other comparably suc­ cessful prediction in economics. It warrants being explored further.

OTHER FORMS OF SCALING The impression that I devoted much of my life work to diverse facets of scal­ ing is correct, even though I was late to recognize this fact myself and did not adopt this term wholeheartedly until a few years ago. The turning point was when I went from nongeometric examples, like those in the second part of this paper, to geometric examples, like those in the first part. Examples of both kinds abound in my 1977 book Fractals and in my 1982 book, The Fractal Geometry of Nature. Even my earliest work—which concerned socalled Zipf rank size rule of word frequencies— is now best understood and appreciated if it is not centered on the subject of statistical linguistics, but on the method of scaling. Scaling is of course an ancient idea, thoroughly familiar to Leibniz; and the scientific application of scaling is the work of many hands. One facet has been familiar to zoologists since Julian Huxley under the term allometry. An­ other facet occurs in the urbanists’ central place theory. In the theory of tur­ bulence, scaling has been basic since the work of Lewis Fry Richardson.

CONCLUSION It is an ancient observation that the variety of complexities in the real world is boundless, while the number of workable mathematical techniques to tame them is astonishingly small. Two most promising newcomers in that com­ pany are geometric scaling and the nonstandard geometric scaling shape, the fractals.

REFERENCES Alexander, S. S. (1961) Price movements in speculative markets: trends on random walks. Industrial Management Review ofM.I.T., II, Part 2, 7-26. Reproduced in Cootner (1964) 199-218. ---------. (1964) Price movements in speculative markets: No. 2. Industrial Man­ agement Review of M.I.T., IV» Part 2, 25-46. Reproduced in Cootner (1964) 338-372.

Fractals, Geometry of Nature, and Economics

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Bachelier, L. (1900) Théorie de la spéculation. (Thesis for the doctorate in mathemati­ cal sciences, March 29, 1900.) Annales de l’ecole normale supérieure, série III, XVIII, 21-86. English translation in Cootner (1964). Cootner, P. H. (1964) The Random Character of Stock Market Prices. Cambridge, Mass.: M.I.T. Press. Fama, E. F. (1963) Mandelbrot and the stable Paretian hypothesis. Journal of Busi­ ness, XXXVI (October) 420-429. Reproduced in Cootner (1964). ---------. (1965) The behavior of stock-market prices. Journal of Business, XXXVIII (January) 34-105. (Based on a Ph.D. thesis, University of Chicago: The Distri­ bution of Daily Differences of Stock Prices: a Test of Mandelbrot’s Stable Paretian Hypothesis.) ---------, and Blume, M. (1966) Filter rules and stock-market trading. Journal of Busi­ ness, XXXIX (January) 226-241. Feller, W. (1968-1971) An introduction to the Theory of Probability and Its Appli­ cations. Vol. I (3d ed.) and Vol. II (2d ed.). New York: John Wiley & Sons. Lamperti, J. (1966) Probability, a Survey of the Mathematical Theory. Reading, Mass.: W.A. Benjamin. Mandelbrot, B. B. (1963) The variation of certain speculative prices. Journal of Busi­ ness, XXXVI (October) 294-419. Reproduced in Cootner (1964) 307-332. ---------. (1966) Forecasts of future prices, unbiased markets, and ‘martingale’ mod­ els. Journal of Business, XXXIX (January) 242-255 [important errata in a sub­ sequent number of the journal]. ---------. (1967) The variation of some other speculative prices. Journal of Business, XL (October) 393-413. ---------. (1971) When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models. Review of Economics and Statistics, LIII (August) 225-236. ---------. (1972) Correction of an error in “The variation of certain speculative prices.” (1963) Journal of Business, XL (October) 542-543. ---------. (1973a) Formes nouvelles du hasard dans les sciences. Economie Appliquée, XXVI, 307-319. ---------. (1973b) Le syndrome de la variance infinie et ses rapports avec la discon­ tinuité des prix. Economie Appliquée, XXVI, 321-348. ---------. (1973c) Le problème de la réalité des cycles lents et et le syndrome de Joseph. Economie Appliquée, XXVI, 349-365. ---------. (1975) Les objets fractals: forme, hasard et dimension. Paris and Montreal: Flammarion. ---------. (1977) Fractals: Form, Chance, and Dimension. San Francisco and Reading: W. H. Freeman and Company. ---------. (1978) Les objects fractals. La Recherche 9, No. 85 (January) 1-13. ---------. (1982) The Fractal Geometry of Nature. San Francisco and Oxford: W. H. Freeman and Company. Roll, R. (1970) Behavior of Interest Rates: the Application of the Efficient Market Model to U.S. Treasury Bills. New York: Basic Books. (Based on a Ph.D. thesis of the same title, University of Chicago, 1968.)

5. Successive Reequilibrations as the Mechanism of Cultural Evolution Robert L. Carneiro

For more than a century, anthropologists have regarded human societies as social systems. To view a society as a system is to say that it consists of a number of parts, structurally differentiated and functionally specialized, that are articulated into a working whole. Although the concept of a society as a system was first put forward by an evolutionist (Spencer 1860), it was most strongly taken up by a school of an­ thropologists known as functionalists who generally ignore, if they do not deny, social evolution. Yet there is no necessary antagonism between func­ tionalism and evolutionism (White 1945). Indeed, I have argued that evolution presupposes function and function provides a mechanism for evolution (Car­ neiro 1970:ii). Let us explore this relation further. Anthropology has often drawn on concepts derived from physics, con­ cepts which have proved fruitful because genuine parallels exist between physical systems and social systems. One such concept is that of equilibrium. An equilibrium is a condition of balance among the forces to which a system is subjected. These forces are of two types: external and internal. Ex­ ternal forces include such things as the impinging of the natural environment and the influence of neighboring societies, whether friendly or hostile. Inter­ nal forces include the competing interests of clans, castes, classes, occupa­ tions, industries, institutions, and ideologies within a society. If a serious im­ balance occurs among these forces, the stability, unity, and even survival of the society may be threatened. Thus, a viable social system is one capable of accommodating, reconciling, or adapting to the internal strains and stresses created within it by the operation of opposing forces. Functionalists who study isolated primitive societies intensively but only over a brief span of time seldom see much change taking place within them. This relative stability of simple societies, coupled with the functionalists’ the­ oretical inclinations, tends to make them describe societies as social systems in a state of equilibrium. If such a society undergoes a perturbation of some

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Robert L. Carneiro is with the American Museum of Natural History, New York City.

Mechanism of Cultural Evolution

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sort— a raid, a shortage of women, a disputed succession, a depletion of game—the perturbation will eventually even out and the society will return to its original condition. In describing how social systems accomplish this, some functionalists go beyond equilibrium theory in physics and make use of the biological notion of homeostasis. Homeostasis is the capacity of living systems for self-regulation or, more precisely, for self-equilibration. Animal organisms have the ability to respond to impinging forces in such a way as to restore the conditions tem­ porarily disrupted by the operation of these forces. The example most often used in illustrating homeostasis is that of a thermostat. A thermostat is a regulatory device that keeps the temperature of a sys­ tem at or near an acceptable level. In doing so, it performs four functions: ( ) it detects when the temperature has fallen below the allowable limit, ( ) it turns on a heating mechanism, (3) it determines when the heat has returned to an acceptable level, and (4) it turns off the heating mechanism. To the extent that human societies are homeostatic, they behave in the same way. They as­ sess when some value of the social system has been pushed beyond acceptable limits and undertake some action to counteract this displacement until the sys­ tem is restored to its previous condition. Much has been said, especially in recent years, about societies as self­ regulating systems. Indeed, whole volumes have been written presenting so­ cieties as such (e.g., Rappaport 1968). To be sure, over a certain range of circumstances, societies do operate in this way, but a homeostatic model of society gives us too narrow a perspective. It tells us why societies stay the same but not why they change. Yet, if we look at the full sweep of human history, what strikes us most is not that equilibria were maintained but that they were overthrown. Had social homeostasis always worked, we would all still be living in Paleolithic bands. The most salient feature of human history, then, is not stasis, but evolution. To account for evolution we must leave equi­ librium theory behind and enter the realm of disequilibrium theory. We must learn why equilibria are overthrown, how they are reestablished, and what effect this has on the structure of societies. In attempting to explain this process, let us first return to physics and borrow another of its concepts, that of elastic limits.2Elasticity is the capacity of a body to undergo deformation and, when the deforming forces are with­ drawn, to regain its original shape. Of course, there are limits to a body’s elasticity. If it is deformed beyond a certain point, it will no longer return to its normal shape. This point of no return is called its elastic limit. If the amount of deformation to which a body is subjected exceeds this limit, the body either ruptures or else is permanently changed in shape. For example, if a force of a certain magnitude is applied to the free end of a metal rod and then withdrawn, the rod will spring back into place. If the force is increased so that

1

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Robert L. Carneiro

it pushes the rod beyond its elastic limits, the rod will take a permanent set. So it is with human societies. Every social system has a margin of elas­ ticity. It can be subjected to certain forces—wars, floods, famines, riots, plagues, strikes, inflation, unemployment— and, as long as the magnitude of these forces is not excessive, the system will essentially return to its original conditions once the impinging forces abate. If it is not pressed beyond this margin, the society will be able to reestablish its old equilibrium. Thus, a functionalist studying a society within this range of perturbing forces is justi­ fied in applying a homeostatic model to it and in considering it to be a system in stable equilibrium. But if the society is subjected to forces that exceed this margin of elasticity, its existing institutions will not be able to cope with these forces. Under heavy stress, the society will be permanently deformed, that is, it will be forced to change its structure. How this process operates can be better understood with the help of the graph in fig. 5.1 The ordinate of this graph represents some variable quantity of a society, while the abscissa represents time. As a concrete example, let us take the case of an autonomous Neolithic village and say that the variable involved is its food supply. When the village granary is full, the line representing the food supply on the graph will be at L0. While the quantity of stored food ranges somewhere between L0 and L,, it is deemed adequate by the society and no steps are taken to augment it. But as soon as the food supply falls below the threshold represented by L,, the subsistence system of the society comes into play. The subsistence system produces more food until it has restored the food supply to L0, or at least above L,. As long as the food supply does not fall below L2, only the subsistence system of the society will be invoked in returning the magnitude of this vari­ able to L0. No other system will be involved. However, if the crisis in subsis­ tence is more severe and the food supply falls below L2, then certain auxiliary systems, not ordinarily involved in food production, come into play. For in­ stance, if the dearth of food is the result of a drought, the village shaman or priest may perform certain rituals to promote rain. If the food supply does not fall below L3, the crisis will be met solely through the normal functioning of existing institutions, and, after the food supply returns to L0, the social system will be found to be structurally un­ affected. Although the existing institutions may have been forced to exercise their functions very actively, no changes will have been induced in the basic structure or function of these institutions, and no new institutions will have come into being. Suppose, though, that the food shortage in our Neolithic village becomes so severe that the supply falls below L3. The elastic limits of the society will then have been exceeded. Thereafter, the society will not return to its previous condition. A permanent change or elaboration will have occurred in its struc-

Mechanism of Cultural Evolution

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Time Fig. 5.1. Graph representing different degrees of fluctuation in some variable of a so­ cial system. As the magnitude of the deflection in the variable increases, the response of the society becomes correspondingly greater, taking qualitatively different forms as it passes through the successive levels L,, L2, and L3.

ture as the social system sought to correct the major disequilibrium to which it had been subjected. For our Neolithic village, this change might entail, say, a shift from dry farming to irrigation From a hypothetical example of the operation of this process, let us turn to an actual one drawn from American economic history. The depression of 1922 was relatively mild and short-lived. Plotted on the graph (fig. 5.1), its effect on American society, in whatever units it might be measured, would no doubt show the line representing it as falling somewhere between L, and L2. Accordingly, when the forces creating this depression abated, the economic system returned to its previous state. No structural changes had been produced. The depression that began with the stock market crash of 1929 was an altogether different story. The deflection it produced in the American eco­ nomic system— indeed, in American society as a whole—was much greater than that of 1922. The line we would draw to represent it on the graph would surely fall below L3. Thus, we can say that the elastic limits of the American economic system were exceeded by the Depression of the 1930s and that, as a result, qualitative changes took place in the society. New structural units—the Securities and Exchange Commission, the Federal Deposit Insurance Corpo­

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Robert L. Carneiro

114

ration, the Social Security Administration, the Public Works Administration, and others—came into being. Their function was not only to try to overcome the effects of the depression, but also to prevent future fluctuations in the busi­ ness cycle of so great a magnitude. Indeed, much New Deal legislation can be seen as an attempt by a badly disequilibrated society to reequilibrate itself by undergoing certain structural transformations

.4

DISEQUILIBRATION, REEQUILIBRATION, AND EVOLUTION The examples given above represent the kinds of events that have recurred in human history. During the last 3 million years, and especially during the last , , social systems have been subjected to increasingly greater perturba­ tions. The smaller perturbations could be handled successfully by existing so­ cial institutions; the larger could not. It was these that induced qualitative changes in the structure of the societies involved. How does the process of serious perturbation followed by reequilibra­ tion, so often manifested by societies, relate to cultural evolution? Evolution is this process writ large. As we have seen, the successful reequilibration of a society often requires the elaboration of its parts. Successive reequilibrations would thus serve to increase a society’s structural complexity. If we follow Herbert Spencer in regarding increased complexity as the hallmark of evolu­ tion in general, we can say that sociocultural evolution is the natural outcome of societies undergoing successive reequilibrations as they seek to adapt to the changing conditions of existence. Or, as Spencer (1896:527) himself put it, “Evolution . . . is an increase in complexity of structure and function . . . incidental to the . . . process of equilibration. . . .” (see Carneiro 1973). Evolution, then, is no transcendant process, remote, abstract, and ob­ scure. Instead, it is recurring changes of structure accommodating to func­ tion, accumulated and projected over an extended period of time. Simple as the process is in its essentials, it is all we need to explain the major features of that long development that began with tiny nomadic bands in the Paleolithic and has led with ever-quickening steps to the vast and complex nations of today.

10000

NOTES 1. In recent years, many functionalists have begun calling themselves structuralfunctionalists, or even structuralists, in recognition of the fact that in the workings of human societies, structure and function are indissolubly linked. 2. As useful as this concept is, it surprises me that other anthropologists have so

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far failed to make any use of it in explaining the process of social change; see my “On the Relationship between Size of Population and Complexity of Social Organization.” 3. Of course, successful modification of structure does not have to occur. Instead, a society may break apart, and its component units may join other societies or a so­ ciety may be exterminated altogether. But if faced with a crisis, a society that man­ ages to adapt and survive will generally do so by structural elaboration. 4. The continuing debate between capitalists and socialists is really one about the ade­ quacy of structures to carry out functions. The capitalist argues that the economic perturbations to which capitalist society is periodically subjected can be satisfac­ torily accommodated through the operation of existing economic institutions or, at most, with only slight and occasional modifications in them. The socialist, on the other hand, contends that a thoroughgoing transformation of the social system is needed if the magnitude of the economic perturbations is to be kept within accept­ able limits.

REFERENCES Carneiro, Robert L. 1967 “On the Relationship Between Size of Population and Complexity of Social Organization.” Southwestern Journal of Anthropology 23, pp. 234-243. 1970 “Foreword,” The Evolution of War, by Keith F. Otterbein, pp. i-vi. HRAF Press, New Haven. 1973 “Structure, Function, and Equilibrium in the Evolutionism of Herbert Spen­ cer.” Journal of Anthropological Research 29, pp. 77-95. Rappaport, Roy A. 1968 Pigs for the Ancestors; Ritual in the Ecology of a New Guinea People. Yale University Press, New Haven. Spencer, Herbert 1860 “The Social Organism.” The Westminster Review 73, pp. 90-121. 1896 First Principles. 4th ed. D. Appleton and Company, New York. White, Leslie A. 1945 “History, Evolutionism and Functionalism: Three Types of Interpretation of Culture.” Southwestern Journal of Anthropology, 1, pp. 221-248.

6. The Emergence of Hierarchical Social Structure: The Case of Late Victorian England Richard Newbold Adams

This essay will suggest some concepts for the analysis of complex societies. Since contemporary sociology, political science, and economics have devel­ oped something of a terminological jungle for this subject matter, any venture that promises to increase the verbiage needs a rationale. My concern is that the wide range of concepts developed in these social sciences do not provide dynamics that allow a meaningful system that itself meets the criterion of being energetic. An energetic criterion requires that the elements involved op­ erate in a manner that permits the analysis in terms of energetic processes. An energetic process is one that can do or does work. Digging coal is work, so is chairing a committee, and so is solving an equation. Obviously the scale or amount of work varies greatly, but all are measurable in energetic terms. In meeting the energetic criterion we are also describing a system that will con­ form to the laws of thermodynamics and, therefore, can benefit from the dynamics implicit therein, as well as the theoretical potential of far-fromequilibrium dynamics as currently developed by Prigogine and others Besides meeting the energetic criterion, the set of concepts must also be framed in a meaningful system. They must describe a structure wherein the various parts stand in certain deterministic relations to one another or in rela­ tions that are explicitly indeterministic. In order to formulate such a set of concepts that meet these two criteria, I have examined a period of complex society history—the late Victorian era of Great Britain. I felt it important to work in the context of historical materials since a great deal of the theorizing to date has been much more abstract, deal­ ing with general systems rather than specifically societal systems. There is obviously much to be gained from this more abstract approach, particularly in mathematical treatment. The problem that I am addressing, however, requires that we look to the particular kinds of structures that come into being in human societies since they necessarily may have particulars that make them

.1

Richard Newbold Adams is with the Department of Anthropology, University of Texas at Austin.

Hierarchical Social Structure

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special cases of, or special departures from, the more abstract models. A failure of general systems approaches has been that they have skipped this step and have built models that do not incorporate elements that are central to social systems but are not necessarily taken into account in more general models. Two such processes will occupy us here. The first is that human societies have an intrinsic expansive tendency. I will not try to provide an explanation for this since it involves special difficulties that would lead us far from what can be handled here. The second is that one form this expansion takes is in the appearance of new social structures; in thermodynamic terms, new dissipative structures. Again, I will not try to delineate a rigorous explanation of the par­ ticular circumstances under which this usually occurs. In the following, I will first give a general description of the process of expansion that was accompanied by a simultaneous internal deceleration of Britain in this epoch. What is presented here is part of a larger model that cannot be presented in the space available. This partial presentation must, therefore, leave some questions unexplored.

GREAT BRITAIN, 1870 TO 19142 Whether or not one approves of what happened there, nineteenth century Brit­ ain clearly had tremendous consequences for much of the rest of the world. Because it had destroyed a good portion of its forests in earlier centuries, Britain had early shifted its dependence to coal. It was the world’s major man­ ufacturer, had the world’s largest merchant marine, and had been the leader of an industrial revolution that was ramifying around the globe. By 1850 laissez faire was a major plank of British foreign economic policy and held, among other things, that free trade was crucial to assuring markets for exports, cheap imports, and the working of Adam Smith’s invisible hand in bringing de­ served benefits to Her Majesty’s kingdom. Free trade was indisputably advan­ tageous to those who had the capital to exploit pioneer environments and worked to British advantage for most of the nineteenth century. Laissez faire paralleled other British characteristics. One was an evan­ gelical fervor which gave the British way of doing things a veneer of spiritual righteousness. It allowed the principles of evolution to be twisted into a doc­ trine that simultaneously coopted religion to the side of those who achieved success and explained away cruelties as natural selection. There was also strong antipathy to government intervention, though it was recognized even by classical theorists that if the state were to do nothing in such areas as edu­ cation and public health nothing was likely to be done. For most other things, the policy for the government was to keep hands off. This circumspect role of

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the government meant that the sufferings of the poor were considered to be a necessary, if disagreeable, part of the natural order. Free trade found favor not merely with capitalists. The working class provided strong support. The repeal of the Corn Laws earlier in the century coupled with cheap foreign grain brought down the price of food. Indeed the devotion to free trade remained part of British working-class ideology down to the 1930s Depression. The preference for this policy— which on the sur­ face would seem to benefit only the able and wealthy—was deep. Since laissez faire gave the capitalist and entrepreneur a free hand, one might think that labor would have welcomed governmental measures to promote their own interests against the employers and entrepreneurs. But this was not the case; the working class had learned through bitter experience to be intensely sus­ picious of existing institutions that tried to change social policy. The work­ house showed clearly enough how government intervention worked. More­ over, this was also an era of declining prices and a steady rise in real wages. It was the era of the beginning of production for the mass market. A wage increase that in 1870 really could be used for little more than increased consumption of beer or gin was by 1900 buying an increasing array of cheap consumer products. With essentially all but total national support, the free trade policy proved to be a capital success until the last quarter of the century. Then the halcyon era felt a shudder. The greater export successes were threatened by the rapid economic growth of Germany and the United States. A continuing fall in prices did not destroy profits, but obviously cut into them. A series of droughts accompanied by an avalanche of cheap foreign grains reduced the British wheat industry by half and converted almost a quarter of the arable land into pasture. This change in agriculture in turn led to the final economic weakening of the provincial artistocracy and a consistent spread of the fran­ chise. The stern evangelical morality, that vocal backbone of mid-Victorian righteousness, rapidly broke down. Social studies in London and elsewhere revealed appalling poverty; easily a quarter of the population in the heart of the world’s richest and most successful nation did not have basic subsistence. Many recruits for the South African War were rejected as undernourished and developmentally malformed. Poverty that could be ignored as long as it re­ mained among the poor now affected the ability of the wealthy to defend themselves. As one historian has suggested, it was an era in which fear took hold: one could not predict when the next economic crash or failure would occur or the ability of the government to do anything—the midcentury se­ curity faded. Doubts naturally emerged about the benefits of free trade. The increase of cheap imported grain was an economic disaster to many farmers, while the encroachment of German and U.S. industrial products on former British mar­

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kets posed an increasing problem, especially since other nations erected tariff barriers to protect their own industries at the same time that they were con­ stricting the British markets. It is not surprising that a protectionist movement emerged. Arguments favoring protection were clear: other European coun­ tries were building their own competitive industries by virtue of it. British exports were not finding markets and, above all, foreign imports were con­ tributing to depression in British industry. The wheat industry had all but dis­ appeared, and now iron and steel were clearly suffering. “Made in Germany” labels were crowding the retail markets in London and other cities, and the lack of expanding industry was producing unemployment and misery. Politi­ cally, however, no British government could introduce protection against the combination of profits for the wealthy and cheap food for the poor. There was another reason, however, that free trade remained in effect. From the 1820s on, Britain suffered a negative balance of merchandise trade. Yet, over the same period the net balance of payments was positive. This posi­ tive balance of payments was achieved through so-called invisibles, the in­ come from British shipping, banking, insurance services, and, of even greater importance in the latter years, foreign investment. Many London banking houses were originally firms from other countries with a basic interest in for­ eign investments and did not provide fixed capital for domestic investment. The Bank of England policies were consonant with this orientation. In the forty years before World War I, the component of the total British capital in­ vested abroad increased from 18 percent to 30 percent. While domestic in­ vestment increased 80 percent during this era, foreign investment increased 250 percent. Quite clearly, the development of foreign nations, both through direct investment and through services for the expanding international trade, was a profitable venture for the investing sector of British society. Although British industry was still expanding in some sectors, its relative international position was deteriorating. This period of British history has naturally been of extraordinary interest to historians, especially those concerned with the economic and social pro­ cesses involved in development. It has long been recognized as having been in some sense a decline, a loss of British preeminence. This characterization has been difficult, however, since what marked decline and disadvantage for some sectors of the society had more beneficial consequences elsewhere. It is inter­ esting, therefore, to find that one index, albeit unidimensional, seems to re­ flect in a straightforward way something of what was happening. This is the index of the change in flow of per capita use of commercial energy. Ideally what is needed here is data on the total use of energy over time. By total use I mean the amount of energy used in all the various forms in which it is found. This term would include not only the obvious macroflows of commercial energy, but also the biochemical energy used in nutrition, loss of

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ability to do work through the wearing out of instruments, tools, and ma­ chinery, and so forth. Both conceptually and technically, however, this com­ plete inclusion is not possible. No one has yet mapped out a total national energy flux over time. While such a mapping may occur in the future, but, even if that becomes possible, it will be difficult to make comparisons with the past. Probably of greater importance would be the relations between flows. Moreover, social scientists have been slow to realize that they must have models that simultaneously include relevant microflows, such as mental processes and individual behavior, along with the more obvious macroflows of mass behavior and extrasomatic conversions. We simply lack a model that can handle the extraordinary diversity of energy flows and forms that constitute a human society; but, even without satisfactory detail, there is evidence that there were important changes in the general energy flux of Great Britain dur­ ing this era. For this evidence we may look at two gross areas of energy ex­ change: commercial energy consumption and imports and exports of goods. Commercial energy consumption per capita in the United Kingdom con­ sistently expanded during the Industrial Revolution until the period of 1850 to 1880 when it began to show a slight deceleration. Between 1880 and 1910 it actually leveled off. Fig. .1 shows this trend for ten-year intervals and, while it is impressive, it is also misleading since it does not allow for the fluctuations that took place. Fig. 6.2 shows annual figures for 1900-1964 and suggests that the important variations occurred with respect to the two world wars and the depressions. It is important to note there that we are specifically using per capita figures in both figs. and . . Imports and exports pose a special problem since an immense variety of energy forms is involved and the potential energy of each can vary consider­ ably with different uses. At present there seems to be no ready way of han­ dling these in simple energetic terms. (I would like to do so in terms of energy cost, but it is quite impossible to obtain satisfactory figures.) As a result, we must revert to economic statistics which are, for many reasons, extremely un­ satisfactory. Nevertheless, on a per capita basis, and corrected for changing prices, fig. 6.3 shows two interesting things. First, imports minus exports per capita show a fairly straight increase until 1896, at which time their rate of increase slows. If this is seen as part of the total energy flow, it means that the ratio of input to output declines in a manner that tends to parallel that of com­ mercial energy. Second, the activity of imports and exports separately also make sense in this context. Imports, that is, inputs into Britain, decreased visibly just before the turn of the century. Exports, while always somewhat more irregular than imports, began a notable increase a short time later. While not sufficient in themselves, these figures suggest that Britain was also undergoing a deceleration in energetic goods during this era.

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Fig. 6.1. U.K. energy consumption per capita (data from William S. Humphrey and Joe Stanislaw, “Economic Growth and Energy Consumption in the U.K., 1700-1975,” Energy Policy 1 [March 1979]; and United Nations Statistical Papers, Series J).

THE ABSENCE OF STEADY STATES In the identification of dissipative structures in physical chemistry the steady state is an important feature. Similarly, in many organisms and in classical models of the ecosystems the steady state is a manifestation of maturity. The same cannot be said, however, for societies. A true steady state is not com­ mon for human species as a whole. Moreover, since life reproduces itself in terms of specific populations, it must be argued that a tendency toward social expansion is much more common than is one toward a steady state. A society that fails to overreproduce is aiming for eventual extinction. Assuming con­ tinuing action of natural selection, we have to assume that by and large living populations are programmed to overexpand. I do not think we can be satisfied if we seek to understand the dynamics of this macroprocess solely by looking at the mechanisms involved. We note that there are progressively different patterns in the steady state when we compare the organism, the society, and the species. Human organisms clearly display a long adulthood which is a steady state. The human species, in con­ trast, has never, so far as we know, manifested a steady state but rather has been in a state of constant expansion. Between the two are societies which

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Fig. 6.2. U.K. domestic fuel consumption per capita 1900-1964 (data from London and Cambridge Economic Service, Key Statistics of the British Economy, 1900-1962 [London: Times Publishing Company, 1962]; William S. Humphrey and Joe Stanislaw, “Economic Growth and Energy Consumption in the U.K., 1700-1975,” Energy Pol­ icy 1 [March 1979]; and Joel Darmstadter, Joy Dunkerley, and Jack Alterman, How Industrial Societies Use Energy: A Comparative Analysis, published for Resources for the Future [Baltimore: Johns Hopkins University Press, 1977]).

may or may not manifest a steady state. Human societies particularly manifest cases of control by internal mechanisms that give evidence of having devel­ oped in response to the consequences of outside constrictions. While individ­ ual human societies vary in this, they have shown progressively less inclina­ tion to exercise such controls as they have become more complex. Recently a good deal of evidence has been adduced to show that not only have many human societies achieved a steady state but also human history has consisted of an irregular oscillation from a condition of ecological equilibrium to one of intensification of exploitation and depletion of resources. Since the appearance of Ester Bosrup’s seminal work on agricultural development, this line of thought has become increasingly dominant in anthropology The propositions in these theories vary in detail. One is whether ecologi­ cal equilibrium (a steady state of a society and its environment) is in fact a condition that has reoccurred with some regularity, only to be set aside by intensification and subsequent depletion, or whether it has been much more

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Fig. 6.3. Value of imports and exports per capita deflated by price and using the Sauerbeck-5to/j'f price index (data from B. R. Mitchell and Phyllis Deane, Abstract of British Historical Statistics [Cambridge: Cambridge University Press, 1962]).

irregular, with some branches or lines of growth showing few phases of equi­ librium and others enjoying such a state for thousands of years. There is cer­ tainly good reason to think that surviving gathering societies, such as the nineteenth century Australian aborigines, had achieved repeatedly, if not con­ tinually, a steady-state condition. Even at a higher degree of societal complex­ ity, the chiefdom, and with the emergence of the archaic state, there were eras of apparent steady states. M. Harris has recently argued that in the ancient civilizations of Egypt, Mesopotamia, India, and China, “Stationary popula­ tions were as much the rule . . . as they were during the paleolithic era While they may have been as steady as in the paleolithic, they lasted by no means as long and they certainly experienced marked fluctuations. Over the past four hundred years it becomes increasingly difficult to speak of steady states for any very extended period as societies have become successively and progressively involved in the spread of science, capitalism, and industrial technology. Great Britain, of course, found its great role in world history during this recent period. It became the scene of the cutting edge of the Industrial Revo­ lution, as it became the scene of the Marxist analysis that so penetratingly revealed the two-sided consequences of the expansion. If it was the scene of

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the most accelerated economic growth known to the world until that time, it also exposed the degrading suppression of concern over the survival of the individual worker. The late Victorian era, however, apparently saw a reversal of these trends, both a slowing down of the industrial acceleration and a grad­ ual improvement in the way of life of the lower class.

THE EMERGENCE OF HIERARCHICALLY SUPERIOR STRUCTURES: BRITAIN During this expansive era, Great Britain increasingly incorporated more and more of the environment into itself. It used up much of its forests, then started in on coal and iron. At the same time, it gradually filled the islands more heavily with people and carved the landscape—cluttering it with industrial and urban structures, slicing it with railroads, blotching it with slag heaps. Of equal, if not greater, importance was the reach into foreign areas where it cap­ tured products and began the heavy dependency on imports that marked Brit­ ish survival from the eighteenth century onwards. Great Britain in this era was a great expanding dissipative structure, con­ suming increasing amounts of energy. Dissipative structures are not, as some would have them, merely “castles of order bulwarks or inhibitions against the loss of energy or the increase of entropy. On the contrary, they hasten the expenditure, conversion, and transformation of energy. In the case of late Victorian Great Britain, foreign investment yielded benefits, such as contributing heavily to the foreign railways that helped cheap grain reach Britain, the opening of foreign textile markets, and so forth. Foreign investment obviously was a crucial part of foreign develop­ ment. Not only colonies, but also the United States and European and Latin American sovereign countries were the recipients of services and exported capital. As the first industrial nation, Britain contributed both to the building of the areas that could demand her exports and to developing the foreign ca­ pacity to produce the imports she increasingly needed. Quite naturally, the maturing of foreign markets and producers gradually changed them from de­ pendents to competitors. In a very real way, Britain was a pioneer industrial market builder and foreign developer. If, today, third world nations argue that this process was unequal and uneven, so indeed it was; but, while Britain’s textiles contributed to the underdevelopment of such industries elsewhere, it is hard to imagine that Argentina, Peru, India, or even the United States would have developed their own railway networks without British capital and technology. As foreign markets expanded, Britain’s external constructions moved elsewhere. Between 1870 and World War I, her foreign investments

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gradually shifted from Europe to empire countries, especially to Canada, Australia, and New Zealand. In expansion a very important process involves the structure and its en­ vironment. In a successful structure effective and presumably correct ways to control its output enable the structure’s environment to provide the inputs nec­ essary for the structure. A structure must exercise certain controls over its environment and bring it into a systematic relation to itself for this process to work. The relations between the structure and those portions of the environ­ ment will affect each other. They, together with the structure, begin to form a new larger structure, an arrangement composed of previous structures as they become modified, plus new elements and relations that emerge as parts of the newly emerging whole. In this way a new and larger structure comes into being, and, as this new, larger structure confronts its own environmental prob­ lems, it too undertakes to adapt this environment to meet its input and output needs. Economic historians have been unable to decide clearly whether foreign investment was better for Britain than greater domestic investment would have been, but it seems doubtful that there was a very wide choice. If we look at Britain as a society that was expanding significantly—expanding home production, imports, exports, earnings, credits abroad, and population—we must also look at the outside environment that allowed this. Markets do not just appear when goods arrive, and capital is by definition low in under­ developed areas. In fact, Britain had to underwrite the creation of much of this environment. I would argue that Great Britain underwent a deceleration in energy per capita consumption in the late nineteenth century because as the first indus­ trial nation, she had to create a worldwide, industrially oriented environment, indispensable to continuing as an industrial power. She had to create an en­ vironment, at the international level, which would provide the necessary ex­ ternal inputs and which would be willing to accept outputs. In doing this, she quite wittingly exported machinery, technical expertise, patent rights, and capital— as well as labor—that helped initiate the emergence of industrial or­ ganization elsewhere in the world. This export is very apparent in the early nineteenth century but continued unabated in later years and increased just before World War I. The construction of an environment appropriate to Great Britain’s inter­ ests must have appeared appropriate to the individual investors, for the for­ eign enterprises often paid very well. It also fitted industrial interests needing foreign markets, and it fitted interests of the workers by inhibiting govern­ mental interference in their attempts to obtain a more prominent share of the obviously growing wealth. The dialectic, however, in the process was that this

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very outpouring of British goods made it possible for the social environment to develop outside, and especially for Germany and the United States to hasten their own development so that by the end of the century Great Britain had dropped into second place in many major indices of production and devel­ opment. The environmental construction to which Britain was contributing was the economic development of the world, and this development neces­ sarily included her future competitors, competitors that already had resources that Britain did not have and were developing others. To these native advan­ tages they were able to add that which Britain exported.

EMERGENCE OF HIERARCHICALLY SUPERIOR STRUCTURES: THEORY A problem that now appears is that of identifying when a new structure has come into being. Where dissipative structures can be identified by their steady state there is little problem, but where we deal with continuingly expanding structures we must look to other indicators. One way is to observe when the existing components— in this case we can look at Britain—give evidence of subordinating themselves to the larger whole. In doing this, they are giving up decision making to decisions made beyond their area of control. These new decisions are being made in the larger structure, and in that way we can see that a new level of self-organization and self-maintenance is coming into being. At first these outside decisions will be uncoordinated among themselves; that is, they will be responding separately to different sets of factors. As the various participating structures expand, as the amount of energy being used by the systems increases, as interaction among them intensifies, they will tend to centralize in a few places that will be more highly coordinated among themselves The gradual centralization of this new level of decision making is what allows us to differentiate the emergence of a new structure. What Britain was building was a worldwide social structure that, on the one hand, necessarily led to her own slowdown but, on the other, was unques­ tionably essential for her own survival. She was building survival devices or mechanisms which, if we see them as parts of an emerging whole, we can regard as a kind of survival vehicle; an extensive social, political, and eco­ nomic complex that would both sustain and supply her. At the same time it required that she make important internal readjustments— among which were those resulting in a deceleration in the growth energy consumption per capita. To see this change in a larger, theoretical perspective, we must jump to a totally different social context. Human individuals in their growth and social­ ization spend a great deal of time trying to rearrange the environment, to

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change their own patterns or behaviors, to cope better with their surroundings, to assure that the resources they will need for survival will be available when they need them and that the environment will accept the outputs that they are capable of giving. In doing this they use a number of survival vehicles. Most of those that they use in the early years of life already exist. The first social survival vehicle is the organic womb of the mother. Immediately upon birth, infants enter a more complex social unit, the domestic unit. This unit provides nutrients, affection, physical care, and the training that is to equip them to survive as adults. Later, play groups and social groups teach skills in interper­ sonal relations. Schools and churches incubate further social skills but also specifically try culturally to train the nervous system in prescribed ways, to program the individuals to react in a limited series of ways. Later still, busi­ ness or industrial organizations, governmental or private agencies, provide in­ come. If these seem inadequate, other groups, labor unions or political orga­ nizations, may be sought to find redress or supplemental help. All these social groups or units may be seen as social survival vehicles, created collectively at some time to help in the survival of the individuals who initiated them. A survival vehicle is, then, in the first instance a social assemblage that is brought into being by individuals to better the conditions of their own sur­ vival. It is a larger dissipative structure, within which the individuals may hope to be buffered against harsher elements of the environment and to re­ ceive specific things that they need. Vehicles will, however, follow their own evolution. Some expire rapidly; others may become of special interest to a few of the members. The leaders (for example, the parents in the domestic unit, the teachers and administrators in the schools and universities, religious leaders in churches, the owners and managers of businesses, and the labor and political leaders) all have a particu­ lar interest in the survival of the vehicles. Thus the survival vehicles often incorporate mechanisms leading to their own perpetuation. As such units be­ come self-organizing and self-maintaining, they also will incorporate ele­ ments that can lead them later to launch into a trajectory of expansion or to be limited in what they do. This development, of course, means that the very survival vehicles that individuals have helped to construct may be taken from their control (if they were ever in their control) and may follow a path that is in fact against the best interests of individual members. Thus survival vehicles, once in being, be­ come autonomous and independent of the interests of the individual members. On reflection, it is apparent that the notion of a hierarchical nest of sur­ vival vehicles, dissipative structures that have emerged out of collectivities of lesser dissipative structural components, satisfies the problem of both individ­ uals and social organizations, operations with their own dynamics, within a single conceptual model. With them it is possible to see a social organization

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1

( ) as an outgrowth of the activity of its individual component parts—down to and including the individual human beings and their individual nervous sys­ tems and genetic structures— and ( ) as a system of social parts operating on a level of their own in response to the environment as seen at that level. Reductionism is not a problem here because the model entertains the factors that operate both from the lower levels and from the collateral elements at the higher level. Moreover, it allows us to pursue the emergence of new levels with the same multiple perspective. Although space does not allow me to go into the details here, the hierarchical structuring of survival vehicles can also reveal how counterorganizations or countervehicles emerge and the individual dynamics behind them, as well as the emergent controls from various levels. From the point of view of energetic analysis, each vehicle and each hier­ archical nest of vehicles must be seen as a separate dissipative structure. Each operates through a required input of human energy and material resources; each is an energy conversion mechanism. Each also operates through an orga­ nization of power relations, an organization of nervous systems, communica­ tion systems, triggers, and so on. These microflows of energy act as control devices, catalysts, inhibitors. They direct and constrain the decisions and ac­ tions of the individuals involved. Each survival vehicle must both ensure its own inputs and be constantly concerned with output policy—the effect of its actions and products on the social environment and the physical environment. Survival vehicles are complex social units operating in fluctuating environ­ ments and, as such, experience natural and often unpredictable fluctuations on their own part. Not only can the amount and kinds of energy that are expended and channeled by these survival vehicles be calculated, but we should also ini­ tially assume that the laws of thermodynamics are as applicable to these pro­ cesses as to those observed in nonhuman biological activities. Thus we should expect, following Cottrell that where vehicles are using increasing amounts of energy, the amount required for the maintenance of a survival vehicle will increase more rapidly than the increase in total energy flow of the vehicle. We should also expect, following Margalef that as vehicles of greater scope come into being they will do so by the systematic exploitation of lesser vehi­ cles, including especially those that compose them. They will take more en­ ergy away from the lesser vehicles, especially in the form of controls. They will compete with other units, other vehicles—often of quite different kinds— for resources. The question now is, why should we make something of the emergence of new structures? The answer, in brief, is that the appearance of new struc­ tures implies the appearance of an assemblage of parts that contain their own deterministic order or laws of behavior. A structure can only be explained in terms of a recounting of the parts and the natural behavior of those parts. So

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long as each member of the assemblage acts in terms of the behavior of the component parts and each responds to a given set of unstructured conditions, then the total behavior must be couched in reductionistic terms that relate to those circumstances. When the assemblage takes on its own emergent behav­ ior, however, then the new behavior can no longer be explained solely in re­ ductionistic terms. Rather, explanation concerning the whole new structure now must be couched in terms that relate this new behavior to the set of condi­ tions in which it occurs. This fact has been recognized in the larger scope of human knowledge: “Scientific knowledge is organized in levels, not because reductionism in principle is impossible, but because nature is organized in levels, and the pattern at each level is most clearly discerned by abstracting from the detail of the levels far below. (The pattern of a halftone does not become clearer when we magnify it so that the individual spots of ink become visible.) And nature is organized in levels because hierarchic structures— structures of Chinese boxes—provide the most viable form for any system of even moderate complexity.” The argument in this paper has been that expanding social systems are entirely a part of nature and that their ultimate elements are no different from those composing the more easily visible physical world. Their inevitable ex­ pansion carries the emergence of higher levels of hierarchy, and these must be recognized as real dissipative structures that will take on their own behavior. In the present case, the lesser components lose their autonomy of action; they lose the ability to respond individualistically to an unstructured environment because they now confront one that is structured. In the nature of social struc­ ture, they become subordinated to the larger expanding dissipative structure that encompasses them. Thus not only the individual vehicles, but also the expanding hierarchy of vehicles, will grow and decline in conformance with some known princi­ ples. We need to delineate the energy processes involved and identify the con­ trol and work roles of the various component structures as they operate in, take form, and contribute to the larger environment and the emerging struc­ tures therein. If we do this, I believe it may be possible to understand better the alternatives open to them. I am not suggesting that we can eliminate the indeterminism in all complex processes, but I suspect that there are consisten­ cies in these emergent structures that, coupled with the fact that we are the “molecules” that compose them, suggest a potential for understanding emer­ gent social dissipative human structures in a manner that may not be possible in physical, chemical, or genetic structures. There are many problems that confront us in understanding the emer­ gence of these structures; let me mention only two. First, we have little com­ prehension of the circumstances of when and where such higher structures emerge. It may be argued that the process is essentially indeterministic and

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that it can occur through the rechannelling or rearrangement of energy within a system (i.e., it need not require additional inputs from the outside) and, therefore, may be the result of an entirely unpredictable fluctuation within. It may also be argued that the process can only occur with any stability through the increase of the energy output of the substructures and, thus, might be re­ ducible to a deterministic theory. Second, we do not know much about the immediate evolution of an emerging structure, whether its appearance requires some specific sequence of steps or stages of formulation. For example, I would tend to argue that the emergence of the larger industrial, capitalist world structure, to which Britain made such an important contribution in the late nineteenth century, is still an uncentralized, coordinated arrangement As such, it would have to be re­ garded as essentially amoral, incapable for the moment of making decisions as to what was best for itself. I would also argue, however, that the direction was towards higher-level centralization within that structure that would result in its own morality, displacing that of the separate nations and firms operating within it. Thus, the experiments with world organizations during the twen­ tieth century reflect attempts to seek a more centralized order, as do the many regional bloc arrangements. Given these arguments, it follows that, of the two suggested explanations of the emergence, I would be inclined toward the sec­ ond, that the emergence was the result of increasing energy and, therefore, was to some degree deterministic. If society is composed of overlapping layers and sectors of dissipative structures, we obviously must find a clear picture of how these come about; for each one of them not only implies immediately a host of human ideas and feelings, but equally the appearance of new units in a complex environment. The emergence of new survival vehicles, of new social dissipative structures, only makes sense in a thermodynamic perspective. In any event, the appli­ cation of dissipative structure analysis to social organizations requires the de­ velopment of a sociology appropriate to it. It is a task to which I hope an increasing number of sociologists, social scientists, and anthropologists will dedicate themselves.

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NOTES 1.

2.

Ilya Prigogine, Peter Allen, and Robert Herman, “The Evolution of Complexity and the Laws of Nature,” in Goals in a Global Community, ed., Ervin Laszlo and Judah Bierman, vol. 1. New York: Pergamon Press, 1977. I make no further attempt here to document the material on which this summary description of late Victorian Britain rests. It is set forth in Richard N. Adams, Energy and the Relationship between Great Britain and the Development of the

Hierarchical Social Structure

3.

4. 5.

6. 7. 8. 9. 10. 11.

131

World Structure, 1870-1914 (Cambridge and New York: Cambridge University Press, in press). The materials for this paper were developed while I was a fellow at the Center for Advanced Studies in the Behavioral Studies, Palo Alto, Califor­ nia, and was supported there by grants from the National Endowment for the Hu­ manities (Number FC-26278-76-1030) and the University Research Institute of the University of Texas at Austin. Cf. Ester Boserup, The Conditions of Agricultural Development (Chicago: Aldine, 1965); Robert Carneiro, “A Theory of the Origin of the State,” Science 169:733-738, 1970; Brian Spooner, ed., Population Growth: Anthropological Implications (Cambridge, Mass.: MIT Press, 1972); Richard Wilkinson, Progress and Poverty (New York: Praeger Publishers, 1973); Mark Cohen, The Food Crisis in Prehistory (New Haven: Yale University Press, 1977); Richard Adams, Energy and Structure (Austin: University of Texas Press, 1975); and Marvin Harris, Can­ nibals and Kings (New York: Random House, 1977). Harris, Cannibals and Kings, p. 155. Kenneth Boulding, “Expecting the Unexpected: The Uncertain Future of Knowl­ edge and Technology,” in Beyond Economics (Ann Arbor: University of Michigan Press, 1970), p. 162. Adams, Energy and Structure. Peter Dawkins, The Selfish Gene (London: Oxford University Press, 1976), p. 25. Fred Cottrell, Energy and Society (New York: McGraw-Hill, 1955). Ramon Margalef, Perspectives in Ecological Theory (Chicago: University of Chi­ cago Press, 1968). Herbert Simon, “The Organization of Complex Systems,” in Hierarchy Theory, Howard H. Pattee, ed. (New York: George Braziller, 1973), pp. 26-27. Adams, Energy and Structure.

7. Self-Organization in the Urban System

P. M. Allen

INTRODUCTION There is an urgent need for a better understanding of the mechanisms of change in social and biological systems. The strong interdependence that characterize modern societies, together with the impact of such societies on their ecosystems demand a better comprehension of the working of such com­ plex systems as a necessary precursor to the formulation of social and eco­ nomic policy. Of course, when we realize the complexity of the social system or of a biological organism, then it seems immediately obvious why such modeling will be difficult. Yet, physical systems too are amazingly complex, involving billions of atoms and molecules colliding, reacting, combining and so on. However, the natural sciences have nevertheless managed to understand and successfully model the behavior of many such systems, and we can discern two bases for this understanding. First, we have the dynamical laws that for a simple system comprised of a very few particles in interaction tell exactly what will be the state of the system at some later time. Second, there is the concept of thermodynamic equilibrium for which the state of the system, containing many billions of atoms and molecules, in ceaseless interaction and movement, can nevetheless be characterized by only a few macroscopic variables, such as temperature, pressure, concentration. The laws of equilibrium thermodynamics, such as the equation of state for example, permit the prediction of the characteris­ tics of some new equilibrium state, despite the enormous complexity of the system. For living systems, the first idea in order to model situations and make predictions was therefore to use one of these two analogies: a trajectory ac­ cording to the laws of some social mechanics or the idea of equilibrium and of an equation of state. While there may be certain situations in which these P. M. Allen is with the Service de Chimie-Physique II, Université Libre de Bruxelles.

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analogies are useful, it must be said that there is a fundamental conflict be­ tween the use of such methods and everyday experience. These methods de­ scribe an entirely deterministic evolution from any given initial state, whereas in the real world of human affairs we are conscious of choice and of the importapce of history. Indeed the whole purpose of developing mathematical models of human systems is to guide choices, a process that implies that such choices do in fact exist. The whole concept of equilibrium and of the movement to equilibrium of complex systems, which comes from the physical sciences, is quite at vari­ ance with the observed evolution of living systems. The former are marked by the destruction of order and the disorganization of the system, while the latter evolve towards greater diversification and functional complexity. This basic dissimilarity was the root of very real fears which some human scientists ex­ pressed concerning the reductionist attitude that the application of such mod­ els implied and the impossibility of describing qualitative features of evolu­ tionary change. We may caricature the situation by saying that there were, broadly speaking, two schools of thought. One believed in the supremacy of the qual­ itative, holding that human and social history was a series of events and anec­ dotes, of great men and crucial battles, defying mechanistic interpretation. The other saw history as a sequence of successive inevitabilities, reflecting equally inevitable technological and economic changes, where great men were the product of their time rather than the contrary and were in a sense replaceable. These two views can now, we feel, be seen as the complementary aspects of a wider perspective. This results from recent developments in the physical sciences, for, when the study of macroscopic systems was extended to situa­ tions far from thermodynamic equilibrium involving nonlinear interactions (feedback) an entirely new type of structure was encountered, whose laws of evolution are quite different from that of equilibrium systems. These new states of matter have been called dissipative structures (see fig. 7.1) to em­ phasize their dependence on the flows of matter or energy from their sur­ roundings. They obey laws of evolution which are both deterministic and stochastic and their description assimilates both the quantitative and the qual­ itative. Thus, there are times when the system follows a stable trajectory of inevitable change and other moments of choice or bifurcation when chance plays a vital role and when a qualitative modification of structure can occur (see fig. 7.2). We have different possible branches of solution and the particu­ lar path taken by the system through the branching type of possibilities is de­ termined by the precise history of the system ( , ). Successive branches may represent more and more complex structures and we see that such an evolution is quite the opposite of that of a system

12

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Fig. 7.1. A dissipative structure: this figure shows a chemical dissipative structure with the reaction pattern undergoing a self-organization in which the stable structure is maintained by flows of the reacting components. Such a phenomenon can only arise in a nonlinear system maintained far from thermodynamic equilibrium.

moving to thermodynamic equilibrium and is most evocative of that of the living world. Dissipative structures, although first discovered and studied in simple nonliving physicochemical systems, are indeed intimately involved in the biochemistry of all living matter, and the self-organization of a physical system maintained far from thermodynamic equilibrium offers us a bridge of understanding spanning the living and the nonliving, unifying the two realms, and surpassing the simplistic views of reductionists.

THE URBAN HIERARCHY, A COMPLEX, EVOLVING SYSTEM When we study the different urban centers in a region, we find a great variety of sizes, forms, and characters from very large cities, with densities perhaps approaching 50x103 hab/Km2, to small villages, having perhaps only 50 in­ habitants. The first questions that must be asked are what is the reason for the spatial distribution of these different centers and what will be their evolution? If we cannot answer this question at least partially, then any plans or designs we may have within any particular center will most probably be futile. Several answers are possible. Perhaps urban centers evolve quasi-independently, and their spatial and size distribution are the results simply of their exponential growth from an initially random condition. Or, do these distribu-

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Near a bifurcation point; see fig. 7.2b

Parameter of the system

a

\

Fluctuations are crucial in deciding the average trajectory

/Fluctuations do

lF xm v not disturb

average trajectory

(a)

v A

/ '

(b)

Fig. 7.2. Chance and determinism in the evolution of a complex system.

tions result as some long-term equilibrium pattern expressing some optimiza­ tion, such as, for example, the least transportation for the consumer popula­ tion? The point of view adopted here is, however, that the spatial distribution of the different urban centers within a region reflects a dynamic interaction process involving the growth and decay of centers as a result of the action of economic and social forces which express the conflicts and common interests of the individuals, families, firms, administrators, and others who are the fac­ tors of the system (3). The existence and maintenance of a town or city, and of its internal struc­ ture as well (commercial and banking area, industrial zone, poor neighbor­ hoods, rich suburbs) depend on the flows of goods and services in, out, and throughout the city as well as of those of commuters traveling daily to their places of work. Flowing into the city we have for example food, building ma­ terials, energy, raw materials and flowing out we find finished products, ser­ vices, pollutants, wastes, and these flows reflect the fact that the town or city

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is the seat of economic functions. The essential power that maintains the flows, and thereby the urban center and its internal structure, is that of the economic exchanges with the outside world. The existence of a town or city at a particular place implies a past history during which economic innovations have appeared and prospered there, lead­ ing to a production of goods or services which were exported to the surround­ ing region (4). If we examine carefully the basic mechanisms which lead to the concentration of many economic functions at a single point, then we see that in essence the concentration arises from the fact that the cost or difficulty of introducing a new function or of extending an old one does not increase in a simple, proportional manner with size but that instead unit costs decrease be­ cause of economies of scale and externalities and this process favors the con­ centration of economic functions at a point. At the same time it inhibits the installation of a similar unit over some distance around the center, a distance which will depend on the costs of transportation. The structure corresponding to the separation between cities and towns is therefore characterized by the nonlinearities in the costs of production, re­ flecting interdependence, and also by the costs of transportation. This struc­ ture is analogous to a dissipative structure in a physical system whose spatial regularities reflect the nonlinear elementary interactions and the coefficient of spatial diffusion. The point of view that emerges from these considerations is that the ur­ ban hierarchy will be described by equations expressing the behavior of the different actors of the system. These equations will express the interdepen­ dence of the various actors, and these intrinsic nonlinearities result in the self­ organization of the system, so that its structures, articulations, and hierar­ chies are the result, not of the operation of some global optimizer, some collective utility function, but of successive instabilities near bifurcation points. The self-organization of an urban hierarchy corresponds to the elabora­ tion of a stable pattern of coexisting centers. Thus our equations may per­ fectly well permit the existence of two similar centers close to one another, but it may not correspond to a stable situation. In such a case, then a small difference in size at some time, of random origin perhaps, will be amplified by the nonlinearities of the interactions, and one of the centers will be elimi­ nated. We should note also here that the measure of explanation accorded by such a theory is not as strong as the usual causal explanations of classical physics, since chance is involved together with deterministic equations in de­ scribing the evolution, and the role played by any particular center is therefore partly decided by historical accidents. The view we have advanced of a self-organizing system takes into ac­

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count the collective dimension of individual actions, and emphasizes the pos­ sibility that individuals acting according to their own particular criteria may find that the resulting collective vector may sweep them in an entirely unex­ pected direction, involving perhaps qualitative changes in the state of the system. In a nonlinear system the whole is not given trivially by the sum of the parts. As we described above, the urban center is the seat of economic func­ tions, and its size depends essentially on the number and variety of economic functions situated at that point (5,6). If we locate a new activity in a particu­ lar town, then as the extra jobs are filled, the increased local population aug­ ments the demand for all the functions located in the vicinity, which in turn causes more employment offers and a further increase of the number of local residents. The effects spiral round and round until a new stationary state is attained. Clearly, the larger the number of local functions, the larger the effect of this urban multiplier, and this effect is clearly important in determining the evolution of the sizes of the different centers in a region. As the basic variables of our model let us take the number of residents, and the number of jobs in each locality. Our basic actors are individuals and entrepreneurs, and the simple mechanisms we suppose can be summarized by saying that individuals tend to migrate under the pressure of the distribution of employment, and entrepreneurs offer or take away employment depending on the market available, taking into account the competition from other centers for the sale of a particular good or service. The interaction diagram is shown in fig. 7.3. The local population and the local capacity for employment are linked by the urban multiplier—a posi­ tive feedback. The employment concentration offers externalities and com­ mon infrastructure, which again gives rise to a positive feedback, while the residents and entrepreneurs together compete for space in a center that pro­ vides a negative feedback. These basic mechanisms can be expressed by two very simple equations describing the change of the population xf of the point /, and the evolution of the employment Et offered at that point (7). We have fix

— 1 = kxiE, - x ¡) - mxl - ?} which arises from Max z = — 'XSjj log Sij {Sij}

( 10)

ij

subject to 2 5 , = e,p,

( 11)

j

2 5 , log Wj = B

( 12)

2 Svcv = C

(13)

ij

where B and C are total benefits (assumed to be measured on a logarithmic scale) and costs. This model assumes a set of fixed and given structural vari­ ables {W}. This problem can be extended (Coelho and Wilson, 1976) to include {W} as variables by changing the range of variables in the objective function: Max z = - 2 S ij log S/j

i V.IV}

ij

(14)

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subject to equations (11) to (13) as before and the additional constraint on total shopping facility size, W/. XWS = W j

(15)

The new problem can then be formulated in Lagrangian form as Max L = - h S y log Sfj {•VHy

ij

+a(ZSu log W - B ) ij + p (C -2 S ,c iy) ij

+7 ( W - Z W j ) j ^ a s - e f ,) . l ij

(16)

7

This equation gives some insights into the parameters: a , (3 and are the Lagrangian multipliers associated with constraints (12), (13) and (15) respec­ tively; the {A,} in equation ( ) are transformations of the multipliers |x, associ­ ated with equation ( ). It can be easily checked that, with { W } now varying, which are the solutions of

11

8

dL — =0 , dS ¡j

(17)

still satisfy equation (7) with {A,}, obtained by solving for |j l , from equation (11) by equation ( ). The equations for {W,} are

8

aXStj d

L

i

-----= ------dWj Wj

-7 = 0 ,

(18)

which can be written l S y = — Wj / a or as

(19)

A. G. Wilson

164 Dj = — Wj a

(20)

using equation (9). Equation (19) can be solved for {W7}, but only numerically, as the {Sj, through equations (7) and ( ), are complicated nonlinear functions of {W7}. If we set

8

21

1 =k

( )

20

then equation ( ) can be written as Dj = kWj ,

(22)

and this will play a significant role in our explorations shortly. We have now provided a procedure for calculating equilibrium values of {Wj\ and, because of the nonlinearities in equation (19) (or (equation 22)), we can expect the surface of such values, traced out as the parameters a, (3 and (and also {e}, {P} and {cÿ}, as these are also assumed to be given), to vary to exhibit structural singularities. We explore the nature of these in the next sec­ tion. Meanwhile, as a final preliminary, we explore the nonequilibrium dy­ namics of the structural variables. Equation (22) suggests that if Dj > kWj shops at j are profitable and should expand and if making a loss should contract. Thus, a suitable differen­ tial equation is

7

Wj = e(Dj-kWj)

(23)

for some suitable constant e. Here we assume that, after a disturbance, con­ sumers move into their equilibrium so rapidly that we retain the equilibrium equations (7) and ( ) rather than use differential equations in In the context of our earlier brief review of bifurcation theory, it is of interest to see whether the dynamical system represented by equation (23) (to­ gether with associated equations for {D7}, {5ÿ} and {A,}) is a gradient system. It can easily be checked that it is not the most obvious gradient system associdL ated with equation (16), since we have already calculated— in equation dWj (18) and this is not equal to W). In fact,

8

dL Wj = Wj — . dWi

7

(24)

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It can be shown, however, that there is another Lagrangian that generates equation (23) and involves replacing the log Wj term in the second term of L by Wj and the constraint (15) by XWj2 = W2 .

(25)

There would be a corresponding change in the {Sj equations: Wf would be replaced by eaWj. Neither this result nor the constraint (25) appear helpful: the first modification implies that the benefits of shopping center size increase lin­ early rather than logarithmically (or something less than linearly) and the con­ straint (25) seems impossible to interpret. However, we will shortly make brief use of the fact that this dynamical system is a gradient system but will basically consider bifurcation in the nongradient system given by the equi­ librium formulation (16) and the differential equation (23). Because the two systems have essentially the same equilibrium points—the solutions of equa­ tion ( )—we can assume they have similar bifurcation properties. It may be useful to conclude this section by writing both the equilibrium conditions for W} and the differential equations in full to emphasize the nonlinearity. Take {Dy} from equation (9), {Sj from equation (7), {A,} from equa­ tion ( ) and substitute these in turn in equations (22) and (23). This gives the equilibrium condition as

22 8

= kWj

(26)

k

and the differential equations as 'e ^

-kW,

1

(27)

DYNAMICAL ANALYSIS Our system of interest is not, strictly speaking, a gradient system; but we have seen that such a system can be set up with very similar properties. If we consider all exogenous variables to be fixed except a , (3, and (or analo­ gously for , k, since k = /a), then structural singularities are possible up to the swallowtail. However, we will simply use this as an argument to suggest that we should be on the alert for this result; we will not attempt either to translate our mathematical system into canonical form or to make any as­

7

7

7

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166

sumptions based on classification theorems. Instead, we will try to gain some insights into the mechanics of catastrophic change directly for this system and by this means we show the existence of folds. The detailed argument is presented in a paper by Harris and Wilson (1978) and only the main ideas and results will be given here. They will then be used as the basis for further explorations and extensions. The argument is presented here for the case a > 1. When a = 1, the results are qualitatively similar, and when a < the behavior is smoother. The argument proceeds in a number of stages. We identify first possible equilibrium points and second their stability. We then relax the assumptions on which this analysis was based and proceed towards a full dynamical analy­ sis which includes an account of the evolution of structure. We can get some insight into the nature of the equilibrium values of {W,} by a simple trick. The manipulation which generated this in terms of Wj can be used to show D, as a function of W, as

1

e,P,W,e XW'e"ec'*

= D/'XWj) ,

(28)

say, and we also require Dj = kWj = D f 2\ W ) .

(29)

We can thus plot Df X) against Wj9 and D/2) against Wj and the intersections are the solutions to equation (26) and the possible equilibrium points. The shape dD d2D of the D.(1) curve can be obtained from an analysis of — 1 and---- 1 . It can be dWj dW2 shown that it has essentially a logistic shape (though with the possibility of additional points of inflection, though we will neglect this complication to deal with only the essentials of the argument here— see Wilson, 1979, for further details). This generates such plots as those in fig. 8.1(a) and fig. 8.1(b). It can immediately be seen that there are two fundamentally different cases: when the line has the upper intersections Wf and Wf, and when it does not. There is a critical case when the line touches the curve, as shown in fig. 8.1(c). The slope of the line in this case depends on the curve, and is, hence, y-dependent (ie. location dependent) and is denoted by k f \ k is a system-wide parameter, which can be interpreted as the average cost of supplying facilities (and could itself be y-dependent, but again we will neglect this complication here to concentrate on the essentials of the argument). It can easily be seen by using the differential equations (23) that the ori­ gin is always a stable point j in case of fig. 1(b), Wf is stable and Wf is unsta-

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Fig. 8.1. Graphical solutions of equation (26): the intersections of D /0, equation (28), and Df2\ equation (29) as a function of Wj are shown.

ble. Hence, if k < k f \ Wf exists and there is the possibility of development at j\ if k > k]n\ there is no development at j. These ideas can be collected together in an alternative graphical form: consider k to be a variable parameter and plot the stable values of Wj (the origin or Wf when it exists) against k. This leads to fig. 8.2 which is imme­ diately recognizable as an example of the fold catastrophe, though with the stable state at Wj = 0 for all k added. The structural singularity occurs, as we have seen, at k = kfix and at point, the tangent to the curve is vertical, which can be expressed as:

We have so far said little about the practical problem of computing equi­ librium values numerically. The equations (26) can be solved numerically by various methods and these solutions turn out to be equivalent to the solution to the mathematical programming problem (16). The details of various methods are presented elsewhere (Harris and Wilson, 1978; Leonardi, 1978 and Wil­ son and Clarke, 1978). The analysis so far has been presented in terms of the parameter k. In fact, when k = k f \ the zone’s situation is also critical in all the other param­ eters. The focus on k provides a useful introduction as the effect of changes is so clearly visible with k being simply the slope of the D /2) line. However, if a and (3 change, the D / l) curve moves. The values of a and p at which this curve touches the line can be labelled a ;cnt and Pjm, and fold curves, analogous to that in fig. 8.2, can be constructed for these parameters. Development is

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Fig. 8.2. Graphical solutions of equation (26): the stable values of W, are shown as a function of a.

more likely in a zone if a is high and (3 is low and so the fold curves are as shown in figs. 8.3 and 8.4— in the former 1/a is used as the horizontal axis. This result can also be expressed algebraically. Denote by hj the follow­ ing functions:

hm =

¿vTO-0/w.

(3D

At the critical point, D / 2) touches the D / l) curve, and so dD,(2) dWj

dWj

(32)

Thus, differentiating equation (31) and using equation (32), we see that dhj — = 0 dWj

(33)

at the critical point. Now hj is a function of the parameters a , (3, k also; this is written explicitly as h^Wj,a ,(3,/:). The implicit function theorem shows that

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Fig. 8.3. Graphical solutions of equation (26): the stable values of Wj are shown as a function of a ' 1.

Fig. 8.4. Graphical solutions of equation (26): the stable values of Wj are shown versus the parameter p.

dk dhJdW, -----= ------- ------1 = 0 , dWj dhj/dk

(36)

where each right-hand side is zero because of equation (33). This is an al­ gebraic representation of the folds in figs. 8.3, 8.4, and 8.2, with the vertical tangent occurring at the critical points. For any given a and p, there exists, as we have seen, a critical value of k, k f l. These associated values of a and p are also critical. There is thus a whole surface traced out in (a, p, k) space of critical parameter values ( a f \ Pf \ k f l) for each zone j. We have demonstrated that the equilibrium surface in (Wj9 a, p, k) space has folds at the critical values of a, p, and k. Although there are hints from catastrophe theory that more complicated forms of struc­ tural singularity— such as cusps and swallowtails—may exist in that surface,

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it remains an interesting research task to see whether this is so in this particu­ lar case. So far, we have assumed that the other independent variables are fixed in our investigation of what can happen to Wj at zone j as a function of a, (3 and k. The argument can easily be extended to incorporate {e}, {P,} and {cj. If these are treated as parameters, then the argument of equations (31) to (36) can be extended in an obvious way and, thus, show that, in the critical state, they also have critical values. For modeling the evolution of urban develop­ ment, this result is important since changes in {P,}, say as a city grows, are likely to have greater effects than changes in a, (3 or k. However the formal argument remains the same, and we will pick up its implications in this re­ spect in the next section. We should note in this case, however, that by adding many new varying parameters, we go much beyond the limits of elementary catastrophe theory (for the associated gradient system) and the structural sin­ gularities may be much more complicated. A more difficult problem, for zone j, is presented by {W*}, ki= j. These also have been assumed to be fixed. They also, if and when they vary, are critical parameters for zone j when everything else is critical. The difficulty here is that a similar analysis to that used for zone j applies to each other zone. This added complication will also be taken up in the next section. There is also the problem that, although we have constructed the D/2) (W,) curve for {W*}, ki= j , fixed, this is unlikely to be a realistic assumption for the range of Wj considered. The basis of the analysis for zone j can probably be maintained in the following way: Near the value of W} at which parameters are critical, there will exist a set of {W*}, ki= j , which are “realistic”—perhaps existing values. It is likely that we only need to construct the curve “near” such a point, as in fig. 8.5, for the argument to hold. The further implications of this will also be considered in the next section. A range of alternative assumptions for {W*}, ki= j , are explored in another paper (Wilson and Clarke, 1978).

THE DYNAMICS AND EVOLUTION OF SPATIAL STRUCTURE We can now explore in outline the implications of this analysis for the model­ ing of the evolution of spatial structure. The growth of cities and development of economies can be represented by the growth of {e,}, {P,} and {c,y}. We can assume, for simplicity, that a , (3 and k either are given and do not change or also change in some specified way. Then, for each zone j, at each point in time, the analysis of the preceding section could be used to determine whether that zone was in a no-development-possible state (NDP) or in a developmentpossible state (DP). As the whole system evolves, zones will switch from

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Fig. 8.5. Graphical solutions of equation (26): the “near” regions of fig. 8.1(c) are emphasized.

NDP to DP and W, will become nonzero for such zones. In principle then, we can now see how to model the evolution of structure in the whole system. The reality is likely to be different at least in the following sense: what actual development occurs will be determined by entrepreneurial or govern­ mental agencies and they are unlikely to be doing these precise calculations. Such developments (or lack of them) can be viewed as fluctuations around the equilibrium state evolutionary path at each point in time. However, we have seen that a given zonal development affects the calculation of criticality for all other zones. In this sense, fluctuations will drive the whole system to new states. This is an analytical representation of common sense results: if a large shopping center is established at j , say, by entrepreneurial whim, then, even though that zone at that time may be in the NDP state and indeed nowhere near the top of the list for development, from the moment of its construction all the criticality conditions must be recalculated and future developments will be correspondingly affected. In spite of this randomness, there is likely to be a degree of order in the spatial structure which evolves. For example, the num­ ber of centers in particular size groups and their average spacing may still be determined by the mechanism of evolution proposed. In this sense, the model proposed here generates order from fluctuations in the same sense as that of Allen and his coworkers (1978), even though the mechanisms are somewhat different. A further complication arises because of the way in which all the Wks affect the criticality computation for each zone j. When there is a jump, say from NDP to DP for zone j , this will cause jumps in the critical parameter values for other zones—that is other zones k, ( a f , P f , &f) jump in value and this jump may take the current (a ,p ,k) value for the system from NDP to

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DP state, or vice versa, for some such zones. Although this is a complication, it does not present formidable new analytical difficulties. Finally, we should note a further necessary theoretical extension which is of great importance. So far, we have assumed the temporal variation of {P,} to be given. In reality, the evolution of {P,}, essentially a residential structural variable, will depend on mechanisms rather like those given for {W}. Further, development in such a residential model will depend on the location of jobs— which will be partly associated with {W}—and on access to shopping facili­ ties—the {W,} directly, so that the two models will be strongly coupled. Again, we can easily see the principles on which such a more general model can be based, but the detailed working out will be a complex task. In effect, such a model provides the basis of a dynamic central place theory (cf. Wilson, 1978).

CONCLUDING COMMENTS It is clear that the methods presented here offer insights but are nonetheless only the beginnings of a research program which is a major task. This pro­ gram will lead to new questions, both theoretical and empirical. For example, the model presented here is an aggregated one. Central place theory has con­ cerned itself more explicitly with hierarchical structure and the presence or absence of certain kinds of function. This style of disaggregation could be added to the models presented here. It is also likely that more complicated differential equation systems can be developed, though the mechanism pre­ sented here may offer some insight into the solutions of those. For example, the equations (27) are special cases of some which occur in ecology and which are analysed by Hirsch and Smale (1974). They note bifurcation prop­ erties involving a separtrix-crossing jump from a state with all WjS nonzero to another state with one or more zeros. The “all-nonzero” state is unique if it exists, and our analysis may contribute knowledge of the conditions in which it does exist. The empirical task is to match data against the predictions of the model. The biggest difficulty is in the poor availability of relevant data, and this case may be one in which the provision of a theory may aid the formulation of plans for data collection and organization; but that will be a mammoth task for a long time. This lack of data also leads us into a final comment on theoretical issues. Perhaps the main tasks involve the interpretation of jump behavior: identifying the conditions under which structures can suddenly emerge and finding examples (though the theory says much about smooth change also). We have seen that jumps can be triggered by changes in any of the param­ eters, and that the critical parameters are strongly coupled among all the

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zones. This fact alone makes the interpretation of jumps difficult. The situa­ tion is further complicated by the existence of other mechanisms for causing jump behavior. These include the separtrix-crossing jumps mentioned briefly earlier, so-called constraint catastrophes evident from mathematical program­ ing formulations, and alternative models of jump behavior such as that pre­ sented by Poston and Wilson (1977). It is hoped that the methods presented here offer substantial new in­ sights. It is equally clear that the research program which must follow is a very extensive one.

REFERENCES P. M. Allen, J. L. Deneubourg, M. Sanglier, F. Boon and A. de Palma (1978). The dynamics of urban evolution, Volume 1: interurban evolution. Volume 2, intraur­ ban evolution. Final Report to the U.S. Department of Transportation, Wash­ ington, D.C. J. D. Coelho, and A. G. Wilson (1976). The optimum size and location of shopping centres. Regional Studies, 10, pp. 413-21. B. Harris and A. G. Wilson (1978). Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environment and Planning A, 10, pp. 371-88. M. W. Hirsch and S. Smale (1974). Differential equations, dynamical systems and linear algebra. Academic Press, New York. G. Leonardi (1978). Optimum facility location by accessibility maximising. Environ­ ment and Planning A, 10, pp. 1287-305. T. Poston and I. N. Stewart (1978). Catastrophe theory and its applications. Pitman, London. ---------and A. G. Wilson (1977). Facility size vs. distance travelled: urban services and the fold catastrophe. Environment and Planning A, 9, pp. 681-6. R. Thom (1975). Structural stability and morphogenesis. Benjamin, Reading, Mass. A. G. Wilson (1970). Entropy in urban and regional modelling. Pion, London; dis­ tributed by Academic Press, London and New York. ---------(1974). Urban and regional models in geography and planning. John Wiley, Chichester. ---------(1978). Spatial interaction and settlement structure: towards an explicit central place theory. In A. Karlqvist, L. Lundqvist, F. Snickars, and J. Weibull (Eds.) Spatial interaction theory and planning models. North Holland, Amsterdam, pp. 137-56. ---------(1979). Catastrophe theory and bifurcation, with applications in urban geogra­ phy. Forthcoming. ---------and M. Clarke (1978). Some illustrations of catastrophe theory applied to ur­ ban retailing structure. Working Paper 228, School of Geography, University of Leeds. E. C. Zeeman (1978). Catastrophe theory. Addison Wesley, Reading, Mass.

9. Trip Making and Locational Choice

Martin J. Beckmann

1. The markets for urban housing and urban land are important cases of mar­ kets in temporary equilibrium. In the absence of rent controls and of alloca­ tion by central planning, supply and demand of existing housing space will be balanced, and this balance is achieved through housing rents, the competitive market price of housing of various types in various locations. The market for land moves more slowly. Land values are established only by transactions which occur infrequently. The estimated land values and the observed housing rents may be in disequilibrium as indicated by existing opportunities for profit making, construction, or other conversion of existing land use. If the city un­ der consideration is in a long-term process of population or income growth such disequilibria may exist for a long time. It is not our purpose here to study the dynamics of the economic ac­ tivities that are set in motion by such disequilibria. Rather we want to focus on the temporary equilibrium in the housing market. Specifically, we are in­ terested in this question: how does location affect housing rents? To what ex­ tent can these rents serve as a measure of attractiveness of a location? In the absence of topographic or cultural features, such attraction must be the result of proximity and distances to destinations for possible interactions in the widest sense. In other words, the attraction of a location as revealed by hous­ ing rents in the market must be an indicator of accessibility. Now, since dif­ ferent households are interested in different activities, accessibility so defined is a subjective magnitude. The actual price paid for housing is the bid of that household whose rent bid function has surpassed those of all others. Acces­ sibility, therefore, means access to the most interested party. Against this background we may ask the more specific question as to the functional form of the rent function. How do distances and attractions of po­ tential destinations combine to determine an accessibility measure? This is the object of study of this paper. Martin J. Beckmann is with the Institute for Statistics and Advanced Research, Tech­ nical University of Munich.

Trip Making and Locational Choice

175

We mention in passing that urban economics has dealt with the other physical attributes of housing by constructing an appropriate index and con­ verting all housing qualities other than location into equivalent space units. We shall follow this convention [Beckmann-Büttler]. Thus all demand and rent bid functions are for one standard unit of space in a particular location. 2. The standard model of the new urban economics considers a monocentric city where all shopping and work are concentrated in a central business dis­ trict whose extension can be neglected, so that location simply means distance from the center, the center being a well-defined point. It has long been felt that economic analysis should move away from this oversimplification and consider urban location patterns where work and shopping opportunities are more dispersed. Thus, let us turn to a “representative household” considering locating in a location i. This household is interested in interaction with vari­ ous destinations k located at distances rik. Some of these destinations may be perfect substitutes for each other, such as branches of the same department store chain. In this case only the nearest of the perfect substitutes should be considered and all others dropped. What remains are destinations which are imperfect substitutes. This means that interaction with a pair of these, ceteris paribus, is not equivalent to twice the interaction with any one of them. In view of the convexity of preferences usually assumed in consumption theory, interaction with a pair will be preferred to twice the amount of interaction with any single object. From now on we call interactions “trips.” In other words, we disregard that type of interaction that does not require physical presence, such as tele­ phone calls and letter writing, the assumption being that these have no signifi­ cant effect on the accessibility qualities of a residential location. 3. One way of modeling this interaction with destinations that are imperfect substitutes is to introduce a utility function whose arguments are, among other things, the amounts of interactions, that is, the number of trips with destina­ tions that are imperfect substitutes, u = u(xu . . . xk, . . . x„) , where xk is the number of trips to destination k. Another variable in the utility function must be the amount of housing 5. As remarked before, this quantity s is also a proxy for all physical attributes of housing other than location. We assume that at location i the household may choose freely the amount of space s. This assumption means that we treat supply as sufficiently flexible. A limitation of the available supply in one loca­ tion to just one type of housing with fixed space would entail a certain loss of

Martin J. Beckmann

176

utility. Even where it is observed that one location has only one type of hous­ ing of a certain size, this fact may be the result of demand rather than supply, a point to which we shall return. Let Pi be the price of one space unit of housing in location i. A meaning­ ful utility function must contain not only the number of trips to various loca­ tions, but also the amount of space consumed. Furthermore, it must include at least one variable for general consumption. If y is household income and r (x, < 4 X X —» 2X Ea+X —> E ]

(transformation of a normal cell into a cancer cell) (autocatalytic growth of cancer cells) (2) (recognition and destruction of cancer cells by cytotoxic cells; Michaelian kinetics)

E0+P

It should be remarked that eq. (1) describes quite general kinetics and could also find application in enzyme kinetics and even for some chemical reactions. Let us first analyze the deterministic steady state properties of eq. (1). This model displays bistability for a certain region in parameter space. The critical point is given by (1

-

0 )3

(1

+

20)3

1

-

0

a c = ------—— , £ r = ------------- , xc = ------- , for 2703 2702 30

1

0< 1

3

( )

1

and if a and 0 the interval for (3 in which two simultaneous stable steady states exist is given by

)2

(1 + Vot < P < ^ + ^ 40

.

(4)

The curves of the steady states for different values of the parameter a are represented in fig. 11.1 by the solid curves. We see that the situation is com­ pletely analogous to that of a van der Waals’ gas.

Noise-Induced Nonequilibrium

195

7.

2.

2

6

10

x

Fig. 11.1. Solid curves give steady-state solutions of eq. (1) as a function of (3. Broken curves give the extrema of the probability density (7) as a function of (3 for a = 4.5 and various values a2.

We will now analyze the steady state properties of our model if p, fluctu­ ates around the value (3 with a variance a 2. We assume that (3, is a stationary process, since we are not interested in the effects of a systematic variation of the environment, and furthermore that the correlation time of (3, is very small compared to the macroscopic time scale so that we can make the idealization of Gaussian white noise. This is necessary and sufficient for jc, to be Marko­ vian. Thus interpreting (3, as a Gaussian white noise parameter in eq. (1) we obtain the following Ito-stochastic differential equation (I-SDE): dxt = ( a + (1 \

-

6------\dt

+ a —— dW,

l+x)

\ + x

(5) = f(x,$)dt + G(x)dWt which is equivalent to the Fokker-Planck equation (FPE) for the probability density: dtp(x,t) = ~dx | | a + (1 -

- (3

j p(x,t) (6 )

Werner Horsthemke

196 The stationary solution of (6) is, p st(x) = N exp-

"(1 ~

20)x2 (7)

The extrema of p sf(x), which we identify as the macroscopic steady states— maxima as stable, minima as unstable—can be calculated from

The results of this analysis are displayed by broken curves in fig. 11.1 for the value a = 4.5, i.e. above the critical value a = 2.7. Under the influence of external noise the behavior of the system is funda­ mentally different from that predicted by the phenomenological description: Even though the system is above the critical point, the stationary solution of the FPE ( ) has three extrema if the variance is within a certain range of val­ ues. This behavior is shown in more detail in fig. 11.2 where psr(x) is plotted for three different values of ct2. If the variance is sufficiently small pst has one extremum, a maximum which corresponds to the deterministic steady state x5t. On increasing the variance above a certain value, pst acquires an additional maximum near x = 0. The height of this peak grows with a and finally the maximum near xst disappears. This example shows that by increasing the strength of the external noise, without changing its mean value, the system can be made to undergo a phase transition well above the deterministic critical point. For this effect to occur in one variable system it is essential that the fluctuating parameter be multiplied by a function of x. For an additive noise source the steady state behavior will be essentially the same as in the deter­ ministic case.

6

2

6

TRANSITIONS INDUCED SOLELY BY EXTERNAL NOISE [ ] In order to elucidate the mechanism of these phenomena and to establish a general criterion to determine when transitions can be induced solely by exter­ nal noise we will now consider the following phenomenological equation: (9)

Noise-Induced Nonequilibrium

197

Ps (x)

0.2

0.1 0

5

10

x

Fig. 11.2. Probability density (7) versus jc for three values of a 2 (a = 4.5, p = 7.5).

where x can take values in the interval [0,1]. A possible reaction scheme is Y 2Y+A fl'+ X + y -» 2X+B

( 10)

These reactions conserve the total number of X and Y so that x denotes the fraction of particles of X in the reactor. The stationary states of eq. (9) are given by the relation

( 11) which is plotted in fig. 11.3 (curve labeled 0). It is easily verified that xs( is stable for all P, that is, this model does not display any instability. We will now analyze the influence of fluctuations on the concentrations of A' and B f and make again the idealization of white noise. This procedure leads to the following SDE ( 12)

which will again be interpreted as an Ito equation. (The results of this section and the preceding do not change qualitatively if the SDEs are interpreted in the sense of Stratonovic.) The corresponding Fokker-Planck equation reads

198

Werner Horsthemke

Fig. 11.3. Steady states of (9) and extrema of the probability density (14) as a func­ tion p.

d,p(x,t) = - d ,

^

- * + px(l - x) j p(x,t) j (13)

2

+ —(T2d„ j x 2(l —x) p(x,f)J , and has the stationary density Ps, (x) = Jf

Using relation

1

2n exp - I x\\~x)2 cr2\

: ~ p/n —

2x(\ - x)

] . x /

(14)

8we obtain for the extrema

Xm + p jr„ (l-x J - (J2x j l - x j ( l - 2 x j = 0 .

(15)

Note that the degree of this polynomial is increased by one compared to that for the deterministic steady states allowing for qualitatively new features in the stochastic description. In the language of catastrophe theory the form of the potential has been modified leading to a different topological structure of the stationary manifold and, hence, to a different structure of the set of the instability points. Let us first discuss the case p = 0, that is the steady state of eq. (9) is Xx = Vi. For the extrema of pst we obtain from eq. (15)

199

Noise-Induced Nonequilibrium

2

We have the following situation: For or2 < 2 xml = Vi is a maximum. At cr xmX is a triple root and for a 2 > 2 xml becomes a minimum and two maxima appear at xm± which tend to respectively , that is, the asymptotes of as a —» oo. Approaching the critical point from above cr I a the distance be­ tween xm+ and xm_ tends to zero like Vcr - a . For p ^ the transition is a hard one as can be seen from fig. 11.3. For (3 > 0(3 < ) the peak corre­ sponding to the steady state of eq. (9) moves towards 1 (0), the asymptotes of xjr((3), with growing a and if a depasses o-,\(|P|) > = a a second peak appears at a finite distance from the original one, near the other boundary of phase space and corresponds to the asymptote for p — > - oo (p — > oc). These facts show that, as to the extrema of pst(x), in the (P, a 2) half plane we have a cusp catastrophe with critical point at ( , ). These results were derived making the idealization of white noise, which is useful since only in this case the process xt is Markovian. However, in some applications this idealization can be inadequate and it has to be verified that noise-induced transitions are not an artifact arising from white noise. In ref. we showed that they indeed occur also under the influence of real noise: If P, is modeled by a stationary ergodic Markov process—such a noise is called colored noise—qualitatively the same phenomena as those in the case of white noise are observed. Since the mathematics is, however, rather in­ volved I will not present the derivation of this result here. Our analysis leads to the conclusion that noise-induced transitions occur in general in systems in which the curve of the steady states as a function of the external parameter has one or two finite asymptotes. This can be under­ stood by the fact that the stationary probability density of xt is essentially the transformation of the stationary density qst of p, by way of the curve xif(P). If x.XP) has a finite asymptote then on increasing a a finite part of the proba­ bility mass of qst is mapped into a narrowing domain near the asymptote, ac­ counting for the growth of a peak in this neighborhood.

0

2

1

2 2 0 0

2 2

2

2

2

2

02

6

2

EXPERIMENTAL EVIDENCE FOR NOISE-INDUCED TRANSITIONS AND CONCLUSIONS To my knowledge the first experimental evidence for noise-induced transi­ tions has been obtained by Kabashima and his coworkers [7] on an electrical circuit system, namely a parametric oscillator that undergoes a second-order transition from nonoscillatory to oscillatory behavior. The evolution of its am­ plitude can be described by x = Ix - kx3 .

(17)

Werner Horsthemke

200

The electrical circuit was coupled to a wide-band noise generator making 1 a fluctuating quantity. This type of equation was the first example in which we had studied the influence of external white noise and found a noise in­ duced transition [ ]. Kabashima’s results confirm completely our theoretical predictions. More recently I studied experimentally in collaboration with Dr. De Kepper at the Centre de Recherche Paul Pascal, Bordeaux, the influence of a fluctuating light source on transition phenomena in the Briggs-Rauscher reac­ tion in a continuous-flow stirred tank reactor This reaction has been exten­ sively studied since, like the Belousov-Zhabotinskii reaction, it constitutes an example for dissipative structures. Although the reaction mechanism is not yet completely known, there is a good evidence that the step

8

.9

CH2( COOH

)2+ I2—» I~ + CHI( COOH )2+ H+

(18)

plays a major role in the Briggs-Rauscher reaction. Since it obeys a Michealian type of of kinetics and the incident light source acts on I by photolyzing it, one might expect that the result of the process discussed in the second part of this paper could have some bearing on the behavior of the system. The experimental results are represented in fig. 11.4 and fig. 11.5. In both figures the optical density of the reacting mixture at 460 nm (absorption of I2) is plotted versus the mean intensity of the incident light. The solid lines correspond to a nonfluctuating intensity, the broken lines to a noisy light in­ tensity. The noisy light intensity was obtained by sending the incident light beam through a small box containing little polystyrene balls agitated in a tur­ bulent air stream. This device generated a Gaussian noise with a relative vari­ ance of 13 percent and an exponentially decreasing correlation with a charac­ teristic time of 0.05 seconds. Fig. 11.4 depicts a region of bistability between an oscillating state A of low optical density and a stationary state B of high optical density for a constant light intensity. The constraints correspond to a situation near the critical point, the hysteresis being rather small. The use of the fluctuating light source leads to a broadening of the region of bistability and to such a shift to lower mean intensities that it is disjunct with that of the nonfluctuating case. This result implies that, for intensities between I'Aand IBf transitions can be induced by the external noise. Noise-induced transitions were easily observable experimentally for slightly different values of the con­ straints corresponding to a situation above the critical point as shown in fig. 11.5. For intensities between VB and IA we made the system undergo a transition by switching on the noise. These experimental results clearly confirm that external noise can deeply modify the macroscopic behavior of a system. Especially for applications to natural systems, this finding implies that ambient noise should not in every

2

Noise-Induced Nonequilibrium

201

[KIO 3] : 0.047 mole I ' 1 tHz02l =i . I mole I ’ 1 [HCI04> 0 .0 5 5 mole I *1 (MnS04] : 0.004 mole I ' 1 T -25°C Constant \ Incident ------ Fluctuating/Light Intensity

I (arbitrary units)

Fig. 11.4. Plot of the optical density of the Briggs-Rauscher reaction versus the mean incident light intensity for the constraints indicated.

O.D. 1.0

Fig. 11.5. Plot of the optical density of the Briggs-Rauscher reaction versus the mean incident light intensity for the constraints indicated.

202

Werner Horsthemke

case be considered as an effect that “unfortunately” perturbs an “ideal” sys­ tem” but should be considered, on the contrary, as one that could play an important role by allowing for phenomena which would be impossible in a constant environment. Furthermore, the good qualitative agreement between the theoretical predictions and the experimental results supports the idealiza­ tions made and shows that it is worthwhile to pursue the study of model sys­ tems to find general principles and to gain a qualitative insight into the influ­ ence of external noise on nonlinear systems.

REFERENCES 1. G. Nicolis, I. Prigogine: “Self-Organization in Nonequilibrium Systems,” Wiley, New York 1977. 2. H. Haken: “Synergetics. An Introduction,” Springer, Berlin 1977. 3. G. Nicolis, J. W. Turner, Physica 89 A (1977), 326; W. Horsthemke, M. MalekMansour, L. Brenig, Z. Physik B 28 (1977), 135. 4. W. Horsthemke, R. Lefever, Phys. Lett. 64 A (1977), 19; R. Lefever, W. Horst­ hemke, Bull. Math. Biol, (in press). 5. R. Lefever, R. Garay: “A mathematical model of the immune surveillance against cancer” in G. I. Bell, A. S. Perelson, G. H. Pimbley, eds., Theoretical Immunol­ ogy, Dekker, New York 1978. 6. L. Arnold, W. Horsthemke, R. Lefever, Z. Physik B 29 (1978), 367. 7. S. Kabashima, S. Kogure, T. Kawakubo, T. Okada, J. Phys. Soc. Japan (in press). S. Kabashima, Statphys 13, Ann. Israel Phys. Soc. 2(2) (1978), 710. 8. W. Horsthemke, M. Malek-Mansour, Z. Physik B 24 (1976), 307. 9. P. De Kepper, W. Horsthemke, C. R. Acad. Sc. C 287 (1978), 251.

12. Nucléation Paradigm: Survival Threshold in Population Dynamics W. H. Zurek and W. C. Schieve

INTRODUCTION The purpose of this paper is to review some aspects of the phenomenon of nucleation and then to suggest a specific application in the field of population dynamics. Nucleation can serve as an interesting paradigm in the description of the stability of a system, other than vapor-liquid phase transition for which the concept of nucleation was originally defined We begin below by giving what we consider to be a very general definition of nucleation and then illus­ trate, with the help of a molecular dynamics computer experiment, nucleation as an instability occurring in an imperfect gas. We will then discuss a general­ ization of nucleation based on the one-dimensional Fokker-Planck equation, with the calculation of the switching time replacing the usual nucleation rate as a main result. Detailed description of an example derived from the field of population dynamics will then be given. Let a system be approximately described by a set of differential equa­ tions involving only macroscopic parameters of physical or chemical origin. We define a solution of this set of equations to be a locally stable state if a system disturbed from it by any infinitesimal perturbation returns to the origi­ nal state. Assume that at least two such solutions do exist. In such a situation transitions between these solutions may occur. They will be caused either by external disturbances or by fluctuations intrinsic to the system but omitted in the deterministic macroscopic description. Nucleation can now be defined as a fluctuation-induced transition be­ tween two locally stable states. A macroscopic notion of local stability, central to the definition of nu­ cleation, can be based on the mathematical foundation of the stability the­ ory The notion of local stability already has been applied successfully to the description of thermodynamic systems both close and far from equilibrium

.123

.45

W. H. Zurek and W. C. Schieve are with the Center for Statistical Mechanics and Ther­ modynamics, University of Texas at Austin.

W. H. Zurek and W. C. Schieve

204

.6,78

by Prigogine and his coworkers The microscopic phenomenon of fluctua­ tions is the other cornerstone on which the theory of spontaneous nucleation is founded Such fluctuations are inherently present in all macroscopic systems. We now turn our attention to the classic example of droplet nucleation in the course of the vapor-liquid phase transition illustrated by a molecular dy­ namics computer experiment.

.6,7,8

DROPLET NUCLEATION A celebrated, though not totally understood, example of nucleation occurs during the vapor-liquid phase transition. Even in the supersaturated gas or su­ perheated liquid, the emergence of a sufficiently large cluster is a rare fluctua­ tion. The metastable phase proves to be surprisingly stable. For instance, water can be superheated under atmospheric pressure to 279.5°C before boil­ ing occurs, if the experiment is carefully performed, even though everyday experience teaches us that (when the nucleation embryos are present) boiling occurs at 100°C. Yet, after a sufficiently long time a new phase will form. A random walk model is proposed in nucleation theory in an attempt to understand such phenomena. It is assumed there that the cluster Ae containing € particles increases and decreases its size because of the absorption and evap­ oration of single molecules (monomers):

4 W(€->€+l)>A, e+\ , A, ^+1W(€+l->€)>Af,

A€

.

1

( )

Consequently, the problem of cluster formation becomes equivalent to a ran­ dom walk in the one-dimensional lattice where €—number of particles in the cluster— is the coordinate. When the initial metastable state is vapor, then the walk starts from small clusters, which may grow as they approach properties of the bulk liquid. The free energy of formation of an €-particle cluster, , is used to deter­ mine relative probabilities of the up and down jumps (W(€—>€+l)/W (€+l —»€) = exp (—(^ = 4 is always positive. As a result, jumps down (€ -> € - ) are more likely than the jumps up (€—»€ + ), as one can see from the formula for W(€— 1). This explains qualitatively the behavior of the metastable states. Quantitative predictions can be made when the values of surface-free energy s and chemical potential (jl are known, and the agreement between the experimental and theoretical results is encour­ aging, although not consistent. The discrepancies arise because of the use of macroscopic values of s and |x, a clear extrapolation of validity of thermody­ namics to clusters of microscopic size. The basis for calculation of the evolution of number densities of €-sized clusters ne(t) as a function of time is given by a set of coupled, linear firstorder differential equations:

>€+1

1

1

1

1

h({t) = W(€ — —>€)n€_j(i) + W (€+l-*€)/i,+ (f) (3)

The above set of equations gives deterministic predictions for the evolu­ tion of concentrations of various clusters. It can also be reinterpreted as a stochastic master equation for the evolution of probability (Pe( t ) ^ n e(t)) of finding that a certain cluster of size at time t = has reached size i at time t. It is, therefore, not surprising that equation (3) leads, under the usual ap­ proximations, to the Fokker-Planck type equation:

€0

0

(4) Here € is a continuous variable, he is the equilibrium distribution of clusters: h( = n x exp ( - Q j k T ) ,

(5)

and R is a rate at which clusters evaporate and absorb monomers. (R can also depend on €.) One can verify the qualitative character of assumed by the nucleation theory by performing molecular dynamics simulation of a system in a phase transition region (see fig. 12.1, also refs. 13 and 14). Results discussed below were obtained for a system of hard-core square well discs in periodic boundary conditions. All the particles are identical, with mass m = 6,628 x 10~ g, radius of the hard core a, = 2.98 x 10 cm, radius of the outer wall cr = 1.96 X a „ and the depth of the well e = 2.305 1 0 erg. All particles

100

23 2

'8

14

W. H. Zurek and W. C. Schieve

206

are enclosed in a square well of size L = 112.08a,. For convenience we have expressed time in reduced units of (e/ma,2)12, and we shall also use reduced temperature and energy units (T* = kT/e, £* = Eli). The evolution of the system is followed starting from artificially created initial conditions Bound states arise naturally in the course of many-body collisions. The resulting liberation of the binding energy into the kinetic en­ ergy of vibrations causes increase of the kinetic temperature. This heating up is relatively fast, and equilibrium temperature is usually reached after t* = 500-1000 or t = x 10 — x 10 second in the laboratory time. The evolution of each system was monitored for t* = 4000-6000 or up to 10 seconds in the laboratory time (two to three hours of CDC 64/6600 central processor time). This elapsed laboratory time corresponds roughly to full collisions—encounters starting from the collision of the two sepa­ rated particles and ending by the particles leaving their radius of interaction. The total number of binary encounters of different types was larger, typically by one order of magnitude. The energy of the systems is constant— all the investigated systems can be considered adiabatic. The formation of a chemical link between two particles is indicated by a collision in which their hard cores are reflected inward off their outer wall of the square well. A graph consisting of all the particles (vertices) and all the links (interparticle bonds) describes uniquely the topology of the system. Connected subgraphs of this graph are equivalent to physical clusters, and this definition enables one to construct an efficient pattern recognition algorithm in which the computer follows dynamically the formation and destruction of clusters. Vapor-liquid phase transition occurring in the system is evident in figs. . a and . b, where the dependence of the total energy £* on the tem­ perature T* and a “snapshot” of the system (E * = 0.2, T* = 0.5) are shown. The temperature reached in equilibrium by all systems with total energy £* < is almost the same, T* = T*c = 0 .5 , while for £* > 0.3 the temperature is roughly proportional to the total energy, as would be the case for a perfect gas. Mole fractions proportional to the concentrations of clusters of different sizes are shown in fig. 12.2a-d. For £* = 0.5 (fig. 12.2a) and £* = 0.3 (fig. . b), the distribution of clusters (as represented by mole fractions) rapidly decreases with the increasing cluster size €. In contrast to that monotonic decrease, the cluster distribution in the phase transition range becomes bimodal (fig. 12.2c, d). The second max­ imum corresponds to supercritical clusters. It is divided from the subcritical clusters by a pronounced minimum. Clearly the cluster distribution reflects expected character of the free energy of cluster formation qualitatively similar to the one predicted by equation ( ).

.1314 8

-10 1.6

~9

8

20,000

121

121

0.2

122

2

Survival Threshold in Population Dynamics

207

Fig. 12.1. (a) The £*(7*) dependence showing the characteristic shape for 7* = 0.5; £* < 0 was achieved by cooling down the system that has already evolved in the higher energy (E* > 0) for some time; the slope of the curve £*(T*) is a specific heat cv. (b) The spatial distribution of the particles in the system at a given instance shows a well-developed droplet containing nineteen particles; the size of the points in the pic­ ture corresponds to the size of the outer wall of the potential (a 2).

W. H. Zurek and W. C. Schieve

208

m, i i i | i i P i-p r i i i 1 i i i i 1 i a.) E*=0.5 , T*=0.66 01 \ * 0.01 4 0.001

m

i | i i i r |-

4

Oil

\ 0.01

E*= 0.3, T*s 0.56

i

»

0.1

►

—r

0001

■

■

OLOOl 0.1

■

0.001

■■

■

■

■ ■

★

d.) E*5 - 0.2 , T*'0.51

’ * aoi

T*=0.53

c.) E*s 0 ,

■

aoi

★

if

★

★ * ** ★ i i i

2 3 4 5 6 7 8910

1i 15

* 20

***** 25 30

Fig. 12.2. Mole fractions, m e for clusters formed in a system at four different values of energy £*: (a) £* = 0.5; (b) £* = 0.3; (c) £* = 0.0 (d) £* = -0 .2 .

The finite system effects are responsible for the falloff of the distribution function at higher values of €. In an infinite system, once the size of the clus­ ter exceeds the critical size €f, the cluster is forced to grow until it will coagu­ late forming a new liquid phase. The situation is different in the finite, adia­ batic system investigated here. New bonds formed within the system increase the kinetic energy and decrease the pressure of the monomer gas. Both effects act as a negative feedback, preventing formation of ever larger clusters. This effect takes place in the region well above the minimum. The described computer experiment gave first direct confirmation of the basic assumption of the nucleation theory—existence of the maximum in the free energy of cluster formation, 4>(. This shape of is reflected in the bimodal cluster distribution with a minimum corresponding to a maximum at € = ( c. It also can be shown that energetic properties of the clusters lead to a bimodal character of thermodynamic estimate for

14,17

Survival Threshold in Population Dynamics

209

NUCLEATION, FOKKER-PLANCK EQUATION AND BISTABLE SYSTEMS We now shall extend our attention to all these problems in which a competi­ tion between a systematic evolution, deterministic on a macroscopic level, and random diffusion on the microscopic level can be idealized by a FokkerPlanck equation for the probability density P(x, i

)-910

P(x,i) = - - [a(x)P(x,t)] + ~ dx

b(x)P(x,t) ,

2 dx2

6

( )

6

where x is the continuous random variable, and t is the time. In equation ( ) a(x) and b(x) can be interpreted as force and diffusion constants by analogy with the problem of the motion of a Brownian particle We shall restrict ourselves, for simplicity, to one-dimensional systems, although extension of our treatment to instances where x is a vector is conceptually straightfor­ ward whenever a{x) is a conservative force, that is, there exists x) such

.1516

that a(x) = ---- ^(x). dx

The equilibrium distribution is given by: P{x) = P(x,t=oo) = Lim P(x,t) .

(7)

f — >0C

When it exists it satisfies a time-independent relation derived from the Fokker-Planck equation:

1

a a(x)P(x) = ----- b(x)P(x) . 2 dx

8

( )

It is not difficult to convince oneself that P(x) = C exp l - £ n b(x) + 2 I dx a(x)/b(x) \ ,

(9)

C being the normalization constant. We shall consider instances when P(x) is bimodal, which is equivalent to the existence of the two minima of the function: U(x) = i n b(x) - 2 j o dx a(x)/b(x) .

(10)

W. H. Zurek and W. C. Schieve

210

+00

A situation in which t/(+°°) = U(-«>) = and for which the intermediate U(x) assumes two minima separated by a maximum was illustrated in fig. 12.3. A problem of considerable interest arising in such situations is to calcu­ late the evolution of the probability of finding a system in the vicinity of one of the locally stable states, providing it was distributed between them initially (t=0) in a known fashion. We shall solve this problem by approximating the original Fokker-Planck equation with a discrete, two-level master equation. Let us define xB PM =

P(x,t)dx = 1 - P2(t) ,

(11)

xB being the border between the two locally stable states. Now the goal we pursue can be redefined as finding, from the complete Fokker-Planck equation , a specific equation that P^t) and P2(t) must satisfy. We note that from equation ( ) it follows directly that x*

6

6

P,(0 = | P(x,t)dx ={ - a(x)P(x,t) - I

2dx b{x)P{x,t)

( 12)

Therefore, if we could find P(x = xB, t) exactly, we could calculate Px{t). There is one xB particularly suitable, the one for which a(x = xB) = 0, and da(x)/dxI _ < 0 . For, in the absence of fluctuations, this is the point of x

XB

unstable equilibrium, the “nucleation point,” located in the range of the max­ imum of U(x). Therefore, if the bottleneck around x = xB is sufficiently nar­ row, (P(xB) P(xA) and P(xB) < P(xc)), and sufficiently long, one can ex­ pect that, for all times beyond some initial and short period, local equilibrium will be established around xA and xc, the minima of U(x). In such a case one can consider evolution of P(x, t) in three different, coupled regions of jc. In the range around it is approximately true that P(x,t) ~ P{x)Px(t) ,

(13)

with a corresponding relation valid in the vicinity of xc. In the region x e (xB—h A' xb+Ac) a steady state is established with the current J, deter­ mined by the time-dependent boundary conditions, which in turn are deter­ mined by the local equilibrium equation (13). Current J can be calculated ap­ proximately by solving a time-independent equation:

Survival Threshold in Population Dynamics

211

Fig. 12.3. Force a, diffusion coefficient b, resulting U(x) and corresponding proba­ bility distribution in equilibrium.

1 d

J = a{x)P(x)-------b(x)P(x) . dx

(14)

2

Going through an analysis similar to the one used customarily in the case of nucleation, we can calculate P t(t) = \ ( p 2(t)IP2 - P M / P A ,

(15)

where P, = lim P,(f) and similar definition gives P2. An analogous equation holds for P2(t). Constant X can be expressed from equation (14): + A,c

XB

iI

(/V) = N | a + N(g+c) - N 2g2b/2

2

Allow us, furthermore, to approximate diffusion coefficient co (A0 by o = Na. Now the Fokker-Planck equation reads:

)2

P(N,t) =

[ N ( - a + N ( g - c ) - N 2g 2bl 2] dN

(30) 1

d2

+ ----- [Aia] \ P ( N , t ) .

2 diV2

The time-independent deterministic solutions are now given by: Ne =

0,

n , = f c £ l z V f c Z H 3 i!ii’ , g2b 1Vp

(g~c ) + V ( g - c

(31)

)2- 2abg2

g2b

2

The qualitative character of w,(A0, w (A0, U(N) derived from equation (10) as well as exp (-U(N)) is shown in fig. 12.6 The analogy between the bistable character of a general Fokker-Planck equation considered in the previous sec­ tion and a bistable behavior predicted by equation (30) with to, (AO and w (A given by equation (31) leaves little doubt. The only (technical) difference is caused by the nonnormalizable exp (—U(N)), caused by the so-called built-in

20

W H. Zurek and W. C. Schieve

220

Fig. 12.6. Force il, (AO, diffusion coefficient i l 2(N), U(N) and exponent (-U (N )) for the Fokker-Planck equation (30).

2

.22

boundary condition at /V = 0, where w (A0 = 0 Therefore, P(N) proper does not exist. This result has no significance for the calculation which we are about to perform. One can now pose the usual, nucleation-inspired question: What is the switching time (extinction time may be a better word in this context), that is, the decay time for a population that is initially located in the neighborhood of N = NP > Ns). This is the time associated with the random walk going over the U(N) barrier at N = Ns. The answer is provided by a method described in the previous section. As before, we have to consider the steady-state FokkerPlanck equation to evaluate the current J from N > Ns to N < Ns over the barrier of Ns: Ne + AjV

J" =

f

N r ~ AN

dNI

2

(o (A0 C exp ( - t w o )

(32a)

Survival Threshold in Population Dynamics

221

Here AN is sufficiently big to contain the peak of u>2(N)e+U(N) in the vicinity of N = Ns, and C is a normalization constant, which we shall fix by requiring that:

C

J

exp ^-U (N )JdN =

1,

(32b)

that is, that a system is initially of a supercritical size. Employing Gaussian approximations to equation (32) and disregarding second-order contributions of In (AO in the expansion of U(N), we arrive at the formula for the decay time t , or the inverse of the decay constant X: Nc + AN T

"1

= X =

I

dN!

20

o) (A exp

7V5-AV -(-

f

(33)

X

exp |- t/( A o ) dN

Throughout our discussion we have tacitly asserted that the basic as­ sumption justifying nucleation approximation is satisfied, that is, eU(Np) ~ U{Ns) < 1. This is not necessarily the case, and the validity of the above assumption depends on the values of a, b, c, and g of the original population dynamics equation (21). This technical detail will not be considered any further. Instead, let us discuss the meaning of equation (33). It indicates that even if the population of certain species is initially in the vicinity of a stable point Np, there is always a finite, even if very large, time t after which the entire population may disappear. This time t is a sensitive (exponential) function of the properties determined by the ecosystem and is likely to be extremely large (age of the universe or longer) for species living in an environment with ade­ quate food supply and a natural deathrate sufficiently small in comparison with the birthrate. The limit of equation (21), that is, the usual MalthusVerhulst equation 18, is an excellent approximation, and the problem of ex­ tinction threshold does not arise at all. When, however, the conditions in the ecosystem become less favorable (c and a increase), the stable population Np may approach the survival threshold Ns. In such a case, extinction time t may be of the order of years, and it is unlikely that such a species will survive.

W. H. Zurek and W. C. Schieve

222 DISCUSSION

The previous section showed that the Markovian character of the process of fertilization leads to a bistable equation for N, the average size of the popula­ tion. From a mathematical point of view, taking into account a nonlinear birth term in this simple model results in a change from the bifurcation observed in the original Malthus-Verhulst equation to the imperfect bifurcation occurring in its generalized version. The two stable states (N = Np > 0 and N = NE = 0) are now divided by an unstable state at N = Ns. Ns gives then the smallest size of the population that, according to the deterministic, macroscopic equations, could survive. In the description on the deterministic macroscopic level, the two stable states, once reached, would persist forever. To allow for the transitions be­ tween them, we have to take into account fluctuations inherent in the system. The postulated master equation, as well as its approximation by means of a Fokker-Planck equation, leads to the possibility of transitions between the for­ merly stable, and now metastable, states. The switching time can now be cal­ culated by means of an approximation familiar from the similar problems aris­ ing in the context of vapor-liquid nucleation. The extinction threshold Ns that appears in these considerations is strictly corresponding to the nucleation size. It is important to stress that this generalization of nucleation does not start from the concept of a droplet, which is anyway the most controversial part in the classical homogeneous nucleation theory Instead, it takes the stochastic bistable process as a fundamental object. Let us also note that a survival threshold obtained in our paper has a very different origin from the extinction threshold discussed in the context of population dynamics There the threshold appears only on the level of the stochastic equation and bears the qualitative character of “gambler’s ruin.” It would be interesting to consider an extension of the model leading to consideration of the generalized equation (21) to include diffusion. It would also be intriguing to consider predator-prey models based on the equations with nonlinear birth terms of the type appearing in Eq. (21). From the qualita­ tive considerations, one can expect that, for the appropriate conditions, wave­ like phenomena would occur, in analogy to the behavior of nonequilibrium chemical systems

.123

.21

.6,7’8’27

REFERENCES 1. 2.

A. C. Zettlemoyer (ed.), Nucleation (Dekker, New York, 1969). A. C. Zettlemoyer (ed.), Nucleation Phenomena (Wiley, New York, 1977).

Survival Threshold in Population Dynamics

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17.

18. 19. 20.

21.

22. 23.

223

F. F. Abraham, Homogeneous Nucleation Theory (Academic Press, New York, 1974). D. Sattinger, Topics in Stability and Bifurcation Theory (Springer-Verlag, Berlin, 1973). A Liapunoff, Ann. of Math. Studies 17; for a modern perspective see J. La Salle and S. Lefschetz, Stability by Liapunoff’s Direct Method (Academic Press, New York, 1961). I. Prigogine, Nobel Lecture, reprinted in Science, 201, 111 (1978). P. Glansdorff and I. Prigogine, Structure, Stability and Fluctuations (Wiley-Interscience, New York, 1977). G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (WileyInterscience, New York, 1977). S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). I. Oppenheim, K. E. Shuler and G. H. Weiss, Stochastic Processes in Chemical Physics: The Master Equation (MIT Press, Cambridge, 1977). R. E. Apfel, Nature 238, 63 (1972). J. Zeldovich, Zh. Exp. Teor. Phys. 72, 525 (1942); K. Binder, D. Stauffer, Adv. Phys. 25, 343 (1976). W. H. Zurek and W. C. Schieve, J. Chem. Phys. 68, 840 (1978); W. H. Zurek and W. C. Schieve, Physics Letters 67A, 42 (1978). W. H. Zurek and W. C. Schieve, Proceedings of the 11th International Sym­ posium on Rarefied Gas Dynamics, Cannes, July 1978, R. Campargue, ed. (Com­ missariat a L’Energie Atomique, Paris, 1979). A. Einstein, Investigations on the Theory of the Brownian Movement (Methuen, London, 1976). M. Smoluchowski, Ann. d. Phys. 21, 756 (1906). W. H. Zurek, Homogeneous Nucleation and Clustering in Vapor-Liquid Phase Transitions as an Example of Stochastic Bistability, Ph.D. Dissertation, Dept, of Physics, University of Texas, Austin, Texas, 1979. H. A. Kramers, Physics 7, 284 (1940). N. G. Van Kämpen, J. Stat. Phys. 17, 71 (1977). J. C. Englund, W. C. Schieve, W. H. Zurek, and R. F. Gragg, Proceedings of the International Conference on Optical Bistability, Asheville, N.C., June 1980 (Plenum, 1981). N. S. Goel, S. C. Maitra, and E. W. Montroll, Rev. Mod. Phys. 43, 231 (1971); N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic Press, New York, 1954); A. J. Lotka, Elements o f Mathematical Biology (Dover, New York, 1956); E. W. Montroll, in Quantum Theory and Statistical Physics, Boulder Lectures in Theoretical Physics 10A, A. O. Barut and W. E. Brittin eds. (Gordon and Breach, New York, 1968); E. Batschelet, Introduction to Mathematics for Life Scientists (Springer, Berlin, 1971). H. Haken, Synergetics (Springer-Verlag, New York, 1977); section 5.4 and chap­ ter 10 are of greatest significance for the present discussion. N. G. Van Kampen, in Advances of Chemical Physics, I. Prigogine and S. Rice, eds., 34, 245 (1976).

224

W. H. Zurek and W. C. Schieve

24. One can also take g = f • N /il, if fl is the actual physical volume of the habitat and not the symbolic measure of available food supply. This simple modification can be introduced into Eq. (19) and taken into account in other calculations. 25. E. L. Reiss, in Applications of Bifurcation Theory, P. H. Rabinovitz, ed. (Aca­ demic Press, New York, 1977). 26. W. Feller, Ann. of Math. 55, 468 (1952); also J. Keilson, J. Appl. Probability 2, 405 (1965). 27. J. S. Turner, this volume.

13. The Efficiency of Oxidative Phosphorylation and the Thermodynamic Buffer Enzymes J. W. Stucki

INTRODUCTION In living organisms, ATP is the most important form of chemical energy for the performance of work. In aerobic tissue the production of ATP is driven by the oxidation of substrates in a process called oxidative phosphorylation. This process is localized within subcellular organelles, the mitochondria, which are therefore regarded to be the powerhouse of the cell. Although several hy­ potheses concerning the mechanism of oxidative phosphorylation have been proposed during the last decades the available experimental evidence does not yet permit us to decide which one of these hypotheses is correct. In view of the great difficulties in elucidating the mechanism of oxidative phospho­ rylation, we decided to investigate the energetic aspects of this process within the framework of a purely phénoménologie theory by using the methods of linear nonequilibrium thermodynamics. This paper summarizes some of the first results of this investigation. More detailed accounts of this work can be found elsewhere

,1

.23

MAXIMAL, IDEAL, AND NATURAL DEGREE OF COUPLING OF OXIDATIVE PHOSPHORYLATION In the linear regime of nonequilibrium thermodynamics, oxidative phospho­ rylation can be described by the linear phenomenological relations between flows J, and forces X, J , = L uX x + L nX 2

(1)

J. W. Stucki is with the Service de Chimie-Physique II, Université Libre de Bruxelles, and Pharmakologisches Institut der Universität Bern.

J. W. Stucki

226

21

(2)

J 2 = L X, + L22X 2

where 7, is the net flow of ATP production, J2 the net flow of oxygen con­ sumption, X, the developed phosphate potential and X2 the redox potential of the oxidizable substrates applied to the respiratory chain of the mitochondria. The phosphate potential is a measure for the distance of phosphorylation from equilibrium, X, = - A G° -RTln ATP/(ADP.P,), that is, a generalized force or affinity. Note that since oxidation is the driving reaction X2 ^ 0 and since phosphorylation is the driven reaction X, < 0. The linearity of the above relations can be verified experimentally. Mitochondria from rat liver were in­ cubated with an excess of glutamate + malate as oxidizable substrates. The hexokinase reaction was used to vary the phosphate potential by the addition of limiting concentrations of hexokinase in the presence of a large excess of glucose. The metabolites and oxygen were measured by enzymatic, radioac­ tive, and polarographic techniques after the incubuations. These measure­ ments allowed the calculation of the two flows and forces of interest. The plots of /, and J2 against X, at constant X yielded two straight lines. In a typical experiment it was found that L u = 1.218, L = 0.358, L2] = 0.364 and L = 0.127 per mg. of mitochondrial protein with regression coefficients of 0.95 and 0.94 for the two linear plots. This result shows that in addition to linearity of the phenomenological laws the Onsager symmetry (L]2 = L2] is also fulfilled within experimental error. For such symmetric linear schemes the degree of coupling

2

22

q

L,2/V L uL 22

12

(3)

4

has been introduced by Kedem and Caplan as a dimensionless measure to characterize how tightly the driven process (phosphorylation) is coupled to the driving process (oxidation). Furthermore, it is convenient to define the phenomenological stoichiometry z = V LJLn .

(4)

The efficiency of the linear energy converter (eqns 1-2 ) is then given by output power _

J ]X ]

x + q

input power

J 2X 2

q + l/x

4

(5)

Here x is the so-called force ratio zXJX2. A graph of efficiency versus force ratio for several degrees of coupling is given in fig. 13.1. As is apparent from

The Efficiency of Oxidative Phosphorylation

Force

227

ratio

Fig. 13.1. Dependence of efficiency on force ratio: plot of eq. (5) for the values of q indicated in the figure, z = 1.

1

this graph, two steady states are characterized by a zero efficiency: ( ) the static head state near minimal values of the force ratio where 7, vanishes and (2) the level flow state at a zero force ratio where X, vanishes. Static head corresponds to an open-circuited and level flow to a short-circuited situation. In between these states efficiency passes through an optimum. The optimal efficiency is given by

^

= ( i + V 7 ^ 2)2

(6)

and turns out to be a function of q only. Since efficiency is zero at both the open-circuited and the short-circuited situations, there must be a load resis­ tance attached to the energy converter that permits optimal efficiency. This resistance, or alternatively the load conductance, can be calculated from the entropy production (7)

J. W. Stucki

228

where positive definiteness is imposed by the second law of thermodynamics. The term Av introduced above is the load flow of ATP. Physically this term summarizes all ATP-using reactions which occur in the cell. A is the driving force of these processes and v their velocity. By assuming again linear rela­ tions between flows and forces and by further making the natural assumption that the ATP-using reactions are driven by X, we can put

(8)

Av = X]L33

33

where L is the phenomenological conductance of the work load imposed on the mitochondria. By keeping the applied redox potential X2constant, the en­ tropy production can now be expressed in terms of the force ratio

22 2

S = |jc2(1 + L J L U) + 2qx + 1j L X .

(9)

The minimum of S is at the state of optimal efficiency iff (10) The important relation ( 10) is the impedance matching of phosphorylation and ATP-use that, by virtue of the theorem of minimal entropy production at a steady state, guarantees that the state of optimal efficiency is the natural steady state of the mitochondria in the cellular environment. Later on an ex­ perimental proof for impedance matching in liver cells will be presented. For the moment, however, let us assume that impedance is matched and calculate some functions of interest at the state of optimal efficiency of oxida­ tive phosphorylation. The net flow of ATP synthesis at optimal efficiency is

(ii)

where the parameter angle a = sin xq has been introduced. The maximum of this function at constant X 2 is at

( 12)

(see fig. 13.2) thus defining the maximal degree of coupling where the net rate of ATP synthesis at optimal efficiency is maximal. Another function of inter­ est is the output at optimal efficiency

The Efficiency of Oxidative Phosphorylation

229

q Fig. 13.2. Optimal functions: plot of eqs. (11, 13, and 15) versus q. All functions were normalized by z = L22 = X2 = 1.

(J.XOop, = tg2 - cosa L12X\ -* ■ (!)

(13)

(see Fig. 13.2), which is maximal at q,d = J n y / ï - l ) = 0.910 ,

(14)

thus defining the ideal degree of coupling where output, that is, the product of net ATP production and developed phosphate potential at optimal efficiency is a maximum. The third function which we must consider is the output times the efficiency

4

22

(y,X,)0Pt iriop, = ig ^ j c o s a L X^ .

(15)

J. W. Stucki

230 The maximum of this function is at qM =

V 5 - 2 = 0.972 ,

(16)

thus defining the natural degree of coupling. A question arises now: Which of these three distinguished degrees of coupling might nature have selected? Later on it will be demonstrated on the basis of experimental data that in the case of liver the answer is the last mentioned natural degree of coupling.

THERMODYNAMIC BUFFER ENZYMES The attempt to verify experimentally the theoretical dependence of the effi­ ciency on the force ratio given in equation (5) led to the discovery of a new functional class of enzymes, which we call thermodynamic buffer enzymes for reasons which will be presented in this section. As described above, mito­ chondria from rat liver were incubated in the presence of different hexokinase concentrations at constant X2. The results of eight experiments are sum­ marized in fig. 13.3, which shows plots of experimentally determined effi­ ciencies versus the force ratio. In the control experiment (fig. 13.3a, no DAPP added) it was not possible to increase the force ratio beyond the value corresponding to optimal efficiency even though very high concentrations of hexokinase were added. At high hexokinase concentrations a considerable net accumulation of AMP was observed. AMP is produced in the adenylate ki­ nase reaction 2ADP

ATP + AMP

The enzyme adenylate kinase is localized within the intermembrane space of mitochondria. This enzyme has been known and characterized for a long time, although its physiological significance remained essentially not under­ stood. Nonetheless, the ubiquitous distribution of this enzyme in cells and mi­ tochondria from different tissues points to an important physiological role. There is a specific inhibitor known for adenylate kinase, the Di-adenosine-pentaphosphate (DAPP The experiment shown in fig. 13.3b reveals that after inhibition of the adenylate kinase by DAPP it was easier to establish force ratios closer to by adding hexokinase than it had been without inhibi­ tion of adenylate kinase. In the presence of the inhibitor, there was no longer a net accumulation of AMP to be observed. From this experiment it appears that the adenylate kinase reaction is able to buffer the phosphate potential near (X,)opt even if the load imposed on the system is too big with respect to the matched value given in equation ( ).

).5

0

10

The Efficiency of Oxidative Phosphorylation

no DAPP

231

2 0 f i M DAPP

1.0 r

0.0

-

1.0

Force r at i o

Fig. 13.3. Experimental verification of efficiency-force ratio relation: left panel de­ picts a control experiment without DAPP and right panel an experiment with DAPP. Theoretical curves are shown for q = 0.91, z = 3. Experimental points were pooled from four pairs of experiments with mitochondria isolated from different rat livers.

The experiment just described contained no AMP initially. If, in contrast, AMP were added at the beginning of the incubation, then the adenylate kinase reaction could also buffer loads which were lower than the matched value in equation (10) (results not shown). Thus the buffering action of the adenylate kinase works for both cases, that is for loads higher and loads lower than the value corresponding to impedance matching. This result arises from the ready reversibility of the reaction catalyzed by adenylate kinase. These experimental findings suggest the following possibilities: (1) In order to have optimal efficiency of oxidative phosphorylation in the cell, the condition of impedance matching (equation ) must be fulfilled on the aver­ age. (2) Temporary mismatching of the load conductance L33, which would lead to a decreased efficiency of oxidative phosphorylation, can be compen­ sated for through the adenylate kinase reaction. (3) Since this compensation is essentially a buffering of a thermodynamic potential, we propose that the en­ zymes which display such effects be called thermodynamic buffer enzymes. This buffering effect can intuitively best be understood on the basis of the entropy production. Consider the previously defined phenomenological load conductance L in equation ( ) to consist now of two terms: the intrinsic load conductance L of the ATP-using reactions plus an additional term corre­

10

33 33

8

J. W. Stucki

232

sponding to the adenylate kinase reaction. By convenience we express the adenylate kinase potential in terms of X, and obtain after a straightforward calculation the overall load conductance: 8

where

+ RTlnjl+exyi-iCi+XÙIRI}) X,

(17)

M 1 C, and C, = AG°phos+RTln- . 8= AG°AK + RTln-1-------—M P P,

M is the

normalized AMP concentration and Pt the concentration of inorganic phos­ phate. L is the phenomenological conductance of the adenylate kinase reac­ tion. The squared expression in equation (17) can be regarded as a weight of this conductance. Introducing (17) into equation (9) gives now the entropy production for the buffered case:

44

SB =

1

|'Ô+/?77n[l+exp{-(C,+X )/Æ7’}] x 4 \+ ^ . . \ Lu x. + #x+l L X

1 22 22

2

1 ¿33\ J

Lu) (18)

Fig. 13.4 shows a plot of the unbuffered entropy production, Su, (Lu = 0) and the buffered production SB (Lu =£ 0) for different values of the load con­ ductance L versus the force ratio. From this figure it becomes evident that in the buffered case variations of the load about the matched value lead to smaller deviations about (X,)opt than they do in the unbuffered case. At the present time it seems very difficult to design an experiment which would per­ mit one to see the buffer action of the adenylate kinase at work in the intact cell. We can, however, give such a demonstration by a computer experiment. Certainly the load imposed on the mitochondria by the cell is not con­ stant. It is, therefore, realistic to consider fluctuations of the load conductance about a mean value. These fluctuations stem from the different states of ac­ tivity of the cell. Thus by considering a fluctuating L we can mimic the real situation in the cell by a computer simulation. The rate laws for the adeninenucleotide concentrations are:

33

33

12 2 33

44 4

ATP —L nX, + L X + £ X, + L X ADP

12 2

(19)

-LnX, - L X - 2L uX4 - L 33X,

(20 )

44 4

(21)

AMP = ¿ X ,

The Efficiency of Oxidative Phosphorylation

233

F o r ce r a t i o

Force

ratio

Fig. 13.4. Influence of buffer enzyme on initial entropy production: plot of Su(eq. 9,

44

a) and Sb (eq. 18, b) versus force ratio. The parameters are L = L22 = X2 = 1, # = 0.91,M = 0.1. Each curve depicts the entropy production for a different value of L33: 0.1 -1.0 in intervals of 0.1. Dashed lines represent the loci of the steady states at initial time. For t —> =

-AG° z

1

( + V l - q 2)RT

and - ( 1 + V l - ? 2)C, + - A G ° oxid Z

(1+ V l - q 2)RT

C, has been defined above, 2 is the sum of the adeninenucleotides in the cell. These steady-state equations, thus, turn out to be functions of the degree of coupling only. Fig. 13.5 shows a plot of equations (22-23) versus q in the relevant range q = 0.9 - 1.0. The cellular concentrations of extramitochondrial adeninenucleotides in perfused livers have been measured by Elbers, Heldt, and their coworkers with sophisticated techniques. As is evident from the figure, all three experimentally determined values coincide with the corre­ sponding steady-state curves at the same degree of coupling qnax. This pro­ cedure proves experimentally ( ) that the assumption of impedance matching is indeed fulfilled since this assumption was introduced in the derivation of the steady-state equations and ( ) that mitochondria in liver cells have evolved to the natural degree of coupling. This finding is a very convincing demonstra­ tion of the remarkable economic beauty of nature. Let us now consider the deterministic system (equations 19-21) with a fluctuating load

6

1

2

The Efficiency of Oxidative Phosphorylation

235

Fig. 13.5. Steady states of extramitochondrial adeninenucleotides: plot of eqs. (19-21) versus q in the interval 0.9-1.0. The experimental data (circles) were taken from ref­ erence 6.

L 33

=

(25)

¿33 + P,

33

where L is the temporal average of the load conductance and p, is a real noise generated by an Ornstein-Uhlenbeck process pf = —ap,+o-£, ; a > 0

7

(26)

with the inverse correlation time a and the white noise with variance a 2. Equations (19-21, 24-25) were integrated numerically with a computer. A typical result of such a simulation is shown in fig. 13.6 where the phosphate potential and the efficiency are plotted versus time for the buffered (L ^ ) and the unbuffered case {Lu — 0). As expected the adenylate kinase reaction can effectively damp out the fluctuations of the phosphate potential and in­ crease the efficiency of oxidative phosphorylation. Further numerical analy­ ses, considering power spectra and so forth, are currently under investigation

0

1.0

unbuffered

Time

Tim e

Fig. 13.6. Simulation of thermodynamic buffering with a fluctuating work load: eqs. (19-21, 24-25) were numerically integrated as described in the text for the un­ buffered case (a) and the buffered case (b). The parameters are: a = 0.035, a = 3.5, L n = 0.247, L n = 0.075, L33 = 0.1025, z = 3, L44 = 5.0 (buffered), L44 = 0.0 (un­ buffered). The initial conditions are ATP0 = .1985, ADP0 = 1.26, AMPn = 3.5. From the results, the force ratio x and the efficiency r] were calculated and plotted versus time.

The Efficiency of Oxidative Phosphorylation

237

in our laboratory. Preliminary studies showed that adenylate kinase is not the only thermodynamic buffer enzyme. Another important reaction capable of thermodynamic buffering of the phosphate potential is catalyzed by creatine kinase ATP + creatine ^ ADP + creatinephosphate The creatine kinase system shows a thermodynamic buffering behavior very similar to the one observed with adenylate kinase. The buffering capacity of the creatine kinase system, however, is considerably higher than that of the adenylate kinase system because of a large pool size of creatine and creatine­ phosphate in muscle where the activity of creatine kinase is prevalent. Since fluctuations of the load in muscle are bigger than those in liver, it is under­ standable that nature has selected an additional thermodynamic buffer enzyme with a high capacity This tissue-specific localization of creatine kinase under­ lines once more the important physiological role of the thermodynamic buffer enzymes.

CONCLUDING REMARKS The most obvious role of ATP in biological organisms consists in furnishing the chemical energy to perform such work as muscular contraction, biosyn­ thesis of the building blocks of the organism, active transport across mem­ branes and so forth. Apart from that, ATP has an additional and perhaps deeper significance than just to provide energy for work. Because of ATP pro­ duction the phosphate potential is shifted far from equilibrium. Since ATP drives many processes in the cell, the potentials of these processes depend directly on the phosphate potential. Hence, the primary shift of the phosphate potential far from equilibrium results in a shift of all the other potentials, de­ pending on the phosphate potential, into the far-from-equilibrium regime. For example, the phosphate potential gives rise to electrochemical gradients of Na+ and K+ across neural membranes through active transport processes. These gradients are necessary for the brain to work. From the work of Prigogine and his school it is well known that a farfrom-equilibrium regime is a necessary condition for the establishment of the dissipative structures These structures show coherent and ordered behavior on a macroscopic level and thus share the main characteristics of living orga­ nisms. Therefore, living systems can be regarded as nature’s most striking examples of dissipative structures. Within the framework of the theory of dis­ sipative structures, the thermodynamic buffer enzymes, thus, acquire a very important meaning: They represent a new bioenergetic regulatory principle

.8

238

J . W. Stucki

for the maintenance of a far-from-equilibrium regime and, with this, for the maintenance of the very essence of life.

ACKNOWLEDGMENTS The author wishes to thank Professors I. Prigogine, G. Nicolis, and L. Ernster for stimulating discussions. This work has been supported by grants from the Swiss National Science Foundation. The expert technical assistance of Miss M. Over and L. Lehmann is gratefully acknowledged.

REFERENCES 1. P. D. Boyer, B. Chance, L. Ernster, P. Mitchell, E. Racker and E. C. Slater, Ann Rev Biochem, 46 (1977) 955. 2. J. W. Stucki, in “Energy Conservation in Membranes,” 29th Mosbacher Collo­ quium (edited by G. Schäfer and M. Klingenberg), Springer Verlag, Berlin, 1978, p. 265. 3. J. W. Stucki, Eur. J. Biochem. 109 (1981) 257-267; 269-283. 4. O. Kedem and S. R. Caplan, Trans. Faraday Soc., 61 (1965) 1867. 5. G. E. Lienhard and I. I. Secemski, J. Biol. Chem., 248 (1973) 1121. 6. R. Elbers, H. W. Heidt, P. Schmucker, S. Soboll and H. Wiese, Hoppe-Seyler’s Z. Physiol. Chemie, 355 (1974) 378. 7. Ch. Wang and G. E. Uhlenbeck, Rev. Modern Physics, 17 (1945) 113. 8. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, John Wiley, New York, 1977.

14. Patterns of Nonequilibrium Organization in a Marine Bacterial Population J . Wagensberg and J. Rodellar

EXPERIMENTAL FINDINGS

1

In a recent work it has been shown that a specially sensitive flow micro­ calorimeter provides a powerful technique for detecting microbiological be­ haviors. We have obtained thermograms (time evolution of the heat dissi­ pation rate) of the growing histories of several wild marine bacteria that were isolated in the sea (oceanographic expedition ATLOR VII, N.W. Africa, 1975, Instituto de Investigaciones Pesqueras) in different environmental con­ ditions (differing depth, nutrient concentration, oxygen concentration, and so forth). The culture develops at 21°C in the hermetically closed laboratory ves­ sel of the microcalorimeter, a fact that shows a certain “memory” in its metab­ olism related to its original ambient. Fig. 14. la shows the thermogram of one of these strains, a Flavobacterium isolated in the following conditions: depth, 20 m.; salinity, 36.15 g./l.; temperature, 22.3°C; and population density, 185 cells/ml. This thermogram has the typical profile of a bacterium coming from an ambient rich in oxygen. It starts with a first phase (A) corresponding to the aerobic growth. An adaptation or diauxia phase corresponding to the exhaustion of the oxygen within the vessel follows. The cells then look for a new metabolic trajectory compatible with the new, unpleasant external con­ straints. A second and anaerobic phase (B) is reached when the contamination and exhaustion of nutrients in the medium become dramatic. The population is now in the death stage (C), which can be very long. This operating method that shows how a population defends itself in a limited ambient was applied to more than twenty different strains. One of them—the Flavobacterium of fig. 14.1—displayed a very special behavior: 1. The series of thermograms recorded with this strain exhibited an evo­ lution tending to show an increase in the ability to carry out the anaerobic phase (figures 14.1a, 14.1b, and 14.1c). J. Wagensberg and J. Rodellar are with the Facultad de Física, Universidad de Barcelona.

./. Wagensberg and J. Rodellar

240 Q Ou,w)

(a)

(b)

(c) Time (days) Fig. 14.1. Three consecutive thermograms of the growing of the marine strain; A, B, and C denote the aerobic, anaerobic, and death phases.

2. A temporally energetic rhythm was observed with this phenomenon. If we look more closely at the thermograms (fig. 14.2a, 14.2b, and 14.2c corresponding to thermograms 14.1a, 14.1b, and 14.1c we can see that the energetic peaks appearing in phase A tend to reach a temporal organization of constant period at the culmination of phase B. In the last phase C the system approaches a thermodynamically closed system, and the temporal structure degenerates by increasing progressively the distance between consecutive peaks. These oscillations detected by the microcalorimeter were identified as glycolitic oscillations by means of the fluorimetrical recording of NADH con­ centration, which is an indicator metabolic parameter (fig. 14.3).

Marine Bacterial Population

241

(a)

(b)

Fig. 14.2. Temporal organizations (energetic peaks) appearing in the corresponding cases of fig. 14.1.

3. The populations that displayed rhythmical behavior (figs. 14.2b and 14.2c) exhibited a visually observable strong aggregation of cells (fig. 14.4) in the surface of the culture. The size of the aggregation is related to the pro­ file of the thermogram and to the period and amplitude of the peaks (i.e., the aggregation of case 14.2c is larger than that of case 14.2b). There are probably oscillations perpetually occurring within individual cells, but this can only be observed in the thermogram (> 5fxw) when the culture, or a sufficiently large part of it, is brought into synchronization. It is, however, difficult to conceive of a mechanism which would give rise to such a sustained synchronization between such a large number of cells. A charac-

J. Wagensberg and J . Rodellar

242

Fig. 14.3. The spontaneous temporal organizations of the bacterial population in the same phases of fig. 14.2c recorded by measuring the NADH-fluorescence. Exitation light was 350 nm and the emission fluorescence was isolated at 420 nm. Ordinate is in arbitrary units.

teristic situation of Goldbeter’s allosteric enzyme model for glycolitic sys­ tems is suggested in the next section as a description of this cooperative effect.

2

DISCUSSION The final kinetic equations of this model are: da - =

d2a * + o . -

dy — = dt

d2y k.-y + D — dr2

( 1)

where a and y denote the concentrations of the substrate and the product; cr] is the injection rate of substrate; ks is related to the outflow of product; Da and Dy are the diffusion coefficients of the substrate and the product along the

Marine Bacterial Population

243

Fig. 14.4. A transversal cut of the aggregation film observed through the optical microscope.

7

single coordinate r. O is a nonlinear function of a and which depends on the structural hypothesis of the model. There are good theoretic-experimental convergences between this model, without considering diffusion, and the ex­ perimental conditions under which glycolitic systems isolated from yeast or muscle cells are continuously stirred. In our—in vivo—case, neither diffu­ sion nor the geometry of the system and the boundary conditions can be ne­ glected. The stability analysis of equation ( l shows that the resulting organi­ zation is very sensitive to the length L of the system. With enough value of L (supracellular distances) propagating concentrations waves appear in the sys­ tem. If the steady-state value of a and are imposed symmetrically at the boundary, two sharp wavefronts of the reaction product are formed near the boundaries. These wave fronts move to the center where they collide, then the resulting peak decreases until new wave fronts build up near the boundaries. This periodic effect is sustained in time. It is not difficult to realize that the central point of the system will undergo relaxation oscillations as observed in the calorimeter.

)2

7

J. Wagensberg and J. Rodellar

244

Some evidence regarding the establishment of propagating waves be­ tween cells of Flavobacterium (on which experimental analysis is presently being undertaken) suggests that in this regime the system is given the capacity to exceed, in a very short time, some threshold of chemical substances that are then released into the extracellular medium as a pulse. The heat dilution of this mixture would then translate into the peaks recorded by the thermogram. In this image the unit of the supracellular level does not conserve the charac­ teristics of the cell. The new unit— a diffusion-reactive unit— for the mac­ roscopic structure is a set of cells that oscillate in phase. Each unit has a dif­ ferent but cooperative behavior in the whole system which can be considered as a new hierarchy of biological order, that is, that of aggregation. The forma­ tion of spatial patterns in Flavobacterium aggregations is now being explored by means of a digital computer analysis of micrographs. There are some analogous cases in the literature, for example, the peri­ odic aggregation process of the cells of Dyctiostelium discoideum, a slime mold which exhibits nonequilibrium organization processes in order to reach the aggregation state of the population The relevant aspect in the Flavobac­ terium case is that the evolution from a “wild” to an “educated” state of the strain and the success of the cooperative behavior was recorded in the labora­ tory; it seems that nonequilibrium organizations can actually be a mechanism that increases the interaction between initially independent cells and provides a nondeterministic source of new structures whose survival is determined by a subsequent Darwinian selection. In our case it has been proved that the cells coming from the aggregation are clearly more viable than those coming from the liquid culture. The formation of the new biological hierarchy occurs when abrupt changes arise in the surrounding conditions; this new hierarchy appears to be an intermediate situation in the biological evolution between a popula­ tion of free cells and a state of strong and harmonious interactions, that is, the tissue.

.3

4

REFERENCES 1. Wagensberg, J.; Castel, C.; Torra, V.; Rodellar, J.; and Vallespinós, F. “Estudio microcalorimétrico del metabolismo de bacterias marinas: detección de procesos rítmicos.” Inv. Pesqu. 42 (179), 1978. 2. Goldbeter, A. “Patterns of spaciotemporal organization in an allosteric enzyme model.” Proc. Nat. Acad. Sci. USA. 70 (3255), 1973. 3. Gerish, G., and Hess, B. “Cyclic-AMP-controlled oscillations in suspended Dyc­ tiostelium cells.” Proc. Nat. Acad. Sci. USA 72 (2118), 1974. 4. Lurié, D., and Wagensberg, J. “Entropy Balance in Biological Development and Heat Dissipation in Embryogenésis.” J. Non Equilib. Thermodyn. 4 (127), 1979.

15. Boolean Equations with Temporal Delays André de Palma, Isabelle Stengers, and Serge Pahaut

INTRODUCTION Experience teaches us that factors influencing our decisions are multiple and depend very often upon previous decisions. Decision models could in fact be considered more appropriate for describing a possible social logic to be set up than for decoding a secretly rational human behavior. The fact that many indi­ viduals try to find a rational reason after having made their choice (mecha­ nism of “cognitive dissonance”) should also be taken into account [1]. A model of decision processes must explain the feedback loops which connect our choices. We believe that in many cases time should be introduced explic­ itly in the description of the decision process. When a family for instance de­ cides to have a child, we are clearly confronted with a temporal delay (rfl). A delay exists between the moment the decision is taken and the moment it is carried out; we shall assume that it is only when the decision is carried out that its issue can affect other decision processes. (In our formalism the time delay can also be negative: ta < , everyone knows that this is not an impossi­ ble situation.) Let A be a variable quantifying the decision of the family to have a child (A = 1 when the couple decides to have a child). Let a be the variable denot­ ing that the desired event has taken place. When a = 0 and A = 1 the decision to have a child has been taken but not yet acted upon. A modification of the decision to have a child can, in this case, have two types of consequences described in fig. 15.2. It seems that we can describe the decision by means of discontinuous variables (binary), however the temporal delay, ta, (which changes from one person to another), should be considered as being continu­ ous. In our example the value of variable A (A - 1 or A = 0) depends on a series of factors resulting from the consequences of previous decisions. From the moment at which the effects of the decision are irreversible, which we

0

André de Palma, Isabelle Stengers, and Serge Pahaut are with the Faculté des Sci­ ences, Service de Chimie-Physique II, Université Libre de Bruxelles.

André de Palma, Isabelle Stengers, and Serge Pahaut

246

A=1 A=0

\ y

Q-]

a

=0

Fig. 15.1

2

•>

--------- first case ---------second case

Fig. 15.2

assume is the moment at which its consequences have reached a given thresh­ old (A goes from to ), the decision will influence other decisions: to live in the town or in the country, to have or not to have a car, and so forth and it will influence the time delay between a decision and its irrevocable execution. The aim of the article is to demonstrate the use of a mathematical tool which can satisfy the requirements described above. A large number of math­ ematical tools are possible candidates, but one of these tools seems to de­ scribe the mechanisms very well and, moreover, is very simple: Boolean al­ gebra developed by R. Thomas [3] who generalized models used by S. A. Kaufmann [3]. The description of complex systems having a large number of elements in interaction, such as chemical systems, is made in terms of nonlinear dif­ ferential equations. Chemical kinetics [4] describes the evolution of systems as a function of the evolution of concentrations x( of the products present in the system:

0 1

x ,= m

i = 1,2, . . . N.

( 1)

Nevertheless, for a number of variables N > 2, the complete analytical study of such systems becomes very complicated or even impossible. That is why

Boolean Equations with Temporal Delays

247

other mathematical techniques [5], such as bifurcation theory for example, are used in order to provide information concerning the stability and existence of stationary states, (solutions of (1) f (x) = 0; i = 1, 2, . . . N). In some cases one only obtains information on the system by means of computer sim­ ulations. The limitation of these techniques, indeed, induced several authors [2, 3, ] to apply Boolean theory to the study of such complex systems. Be­ cause of the very small number of molecules of each type, the system of ge­ netic regulation seems suited to this approach. In the first part of our text we will describe briefly the Boolean formulation as applied in genetics. In the second part we will try to explain the same kind of approach, but applied to individual decisions. In another paper [7] we have shown that the methods developed by R. Thomas for genetic regulation can be applied to problems involving a large number of individuals, such as the problem of residential location. In that case, however, we assumed a strong nonlinearity hypothesis.

6

PRESENTATION OF BOOLEAN FORMALISM The Boolean formulation, as we will describe it, is a new type of approach of sequential circuits [ ]. The Boolean variables have two values of 0 and 1; the mathematical operations are governed by rules presented in fig. 15.3. One can distinguish two different types of systems. In a combinatory system the input variables (X) determine immediately the values of the output variables (Z). The Z, are logical functions of the variables (X). For instance,

8

Z = X x-X2+X3 .

Fig. 15.3

(2)

André de Palma, Isabelle Stengers, and Serge Pahaut

248

X1 X2 X3

z

0

0

0

0

o

o

1

0

1

1

1

0

1

0

0

1 0

0

0

1 0

1

1

1

1

1

0

1

1 0

0

Fig. 15.5

Equation (2) is equivalent to fig. 15.5. Thus we can also write

3

2

Z = X • x x• X .

(3)

In a sequential system the values of variable (Z) at time t depend, on the one hand, on the values of variable (X) at time t, and, on the other hand, on the values (X) at previous moments. In order to store information concerning the state of the system at previous times, one has to introduce memory vari­ ables y and memory functions F. Let us consider the case of a sequential sys­ tem having only one memory variable. The relation between the memory variables and the memory functions is (4)

Boolean Equations with Temporal Delays

249

A. Fig. 15.7

A is a temporal delay between the input and the output of the signal in the combinatory block C. Note that A, is the “on” delay and Ar is the “off” delay. The equations of a sequential system generally will be Yj = gj (x]9 x2, . . . xn9 y l9 y 29 . . . yr)

j = 1, 2, . . . r.

(5)

Zk =fk (*i, x 29 . . . xn9 y l9 y 29 . . . yr)

k=

1, 2, . . . p

( )

yi(t) = Y i ( t - A,)

/=1,2, . . . r

6

(7)

Equations (5) and (7) are an implicit system in y, and the trajectory can be studied since we know the initial state and the time delay. A stable state will be defined as follows: y, (t) = 7, (t) for every i =

1, . . . n .

8

( )

When we have at least one variable k such that y k (0 * Yk ( t ) ,

(9)

the corresponding state is unstable. When a system needs an infinite amount of time to adjust the values of y and Y (the memory variables and the memory

André de Palma, Isabelle Stengers, and Serge Pahaut

250

functions respectively), it forms a cyclic trajectory. If this time is finite, it evolves to a stable state. Let us consider a system having two memory functions A and B and two memory variables a and b. We suppose that the variables X (input) remain constant. Let us suppose that the equations of the system are A =0

( 10)

B = a-b + a‘b ,

(11)

then fig. 15.8 can be established. The circled states are stable; the bars on the memory functions mean that these may change. Starting from the initial state a - , b - , the system may evolve to the stable state or ; the evolution depends on the different values of the “off” and “on” delays tj, tA and t-b and tb. In this case we have a phenomenon of “critical course” because there is competition between two transitions. The transition in which the time delay is the smallest will prevail. We consider that the probability of the two delays being exactly equal is zero and has no physical significance. The different evolutions of the system are represented in fig. 15.9 (initial state a - 1, b = 0). The system will traverse the loops (w) a certain number of times before reaching one of the two stable states (see fig. 15.10). This phenomenon can only occur when the sequential machines simulating the equations of the model have a memory from one state to another (during which the process of off and on delay take place). The system goes round the loop (w) a certain number of times, because at each passage from state to state , we have the requirement of off delay for variable a. If the system did not have this memory, once it forms a loop this loop would become a cycle. A further ex­ ample, described below, will show the case of a system having two different types of loops. The first one cannot be traversed more than a determined num­ ber of times if the system keeps memory of an order, until a counterorder is

1

0

00 01

10

a

b

A

B

o o 0 i 1 Ï

(]O D fô ~ T ) 0 0

T o

o i

Fig. 15.8

11

11 Fig. 15.9

Boolean Equations with Temporal Delays

251

*

'b ‘b1*‘5 w

H-------- -------------

w

lc

i

B =1 -

8=0

b=1 b=0A=0-

a= l • o=0 Fig. 15.10

given. For the same conditions, the second loop will form a cycle. If the sys­ tem would keep the order in memory even when interrupted by a counterorder this cycle could become unstable. By means of some rapid algorithms, when giving the equations of the system [equations ( ) and ( )], it is possible to find all the stable states of the system and its possible cycles as a function of the different data of the problem (initial state, the system’s delays of the various memory variables). The system of genetic regulation could be described by this formalism [3]. Let S be a gene, A its state of expression (A = 1 when the gene expresses itself; otherwise A = 0). When the gene expresses itself, a time delay ta is required before the concentration of the synthesized product reaches a given threshold (a = ); when the concentration does not reach this threshold one may consider that the product does not exist = 0). When the synthesis of the product stops, the mechanisms of dilution or degradation will lower the concentration of the product after a time ta below the threshold of efficiency. The advantage of the Boolean description is its simple way of describing the characteristics of genetic regulation. If a product synthesized from a gene

10

1

11

(¿7

André de Palma, Isabelle Stengers, and Serge Pahaut

252

can inhibit its own synthesis after having reached a given threshold concentra­ tion, we can write A =a

(12)

Such an equation describes a typical negative loop. Its solution is as repre­ sented in fig. 15.11. More generally, it is possible to establish behavioral laws for systems having a large number of negative and positive loops [9, 10]. The formula­ tion, below, seems to satisfy the requirements of decision mechanisms as de­ scribed in our introduction. We will try now to formalize a fictitious case of decision.

AN EXAMPLE OF A DECISION MODEL The trees of decision associated with a variety of choices between which indi­ viduals can choose are often very complex because of the large number of variables considered. Most of the time the modeler has to isolate a specific problem (corresponding to a subclass of choices) and to study independently the other choices made by the individuals. If one considers for example the problem of residential location, it is known that a variety of other decisions may influence or be influenced by the locational decision of an individual: the number of cars, the means of transport used to go to work, the type of job, and other nonprofessional activities [11]. These different choices, though in­ tercorrelated, are determined by certain restraints, priorities, and exclusions which should allow the modeler to isolate certain problems. This simplifica­ tion can also be based on the fact that scales specific to different types of choices may be different. One may separate choices into independent classes that can be described by mathematical techniques and specific approaches. The hierarchy of decisions is an example of structuration and classification of choices into different classes. The aim of the example presented here will be

Q

l5 Fig. 15.11

=1

0 =0

Boolean Equations with Temporal Delays

253

to show how it is possible to represent, on the level of decision trees, the pri­ orities and the constraints to which individual decisions are subject. Employment Location Mobility Bundle + residential location and housing + automobile ownership + mode to work Travel Choices—Nonwork Trips + frequency + destination + mode + time of day + route

Let si denote a given choice. Memory function A is associated with choice si and will take value 1 when an individual intends to adopt si (A = 0 when the individual does not have the intention of choosing ¿4). Let us sup­ pose that at t = the individual has decided to choose si but has not yet acted upon the decision. After a certain time ta the individual will have carried out the decision to choose si and variable a, which measures the actual choice of the individual, will take value 1. In general, the on and off delays of variable a will be different. The processes described here may be described mathematically in the following way:

0

a (t) = A (t - AJ with

(13)

Aa = ta

if

Aa = ta

if a=

O

.• u :

a= 0

(14)

1.

(15)

—r

-i *--—> 1 a

Fig. 15.12

—

A =1 nA --0w a =1 o=0

André de Palma, Isabelle Stengers, and Serge Pahaut

254

One can really talk about a system when the input and output are connected. We suppose that in our model the command of A is determined as a function of the state of the system. If we consider a system of three variables corresponding to decisions d SS and % and if A, B, C are memory functions associated with each of the decisions, we will have A = A(a,b,c)

(16)

B = B(a,b,c)

(17)

C = C(a,b,c) .

( 18)

We want to discuss the following decision example. We suppose that in the given system an individual may choose between three nonexclusive and inter­ acting decisions connected by the following constraints: si is chosen in the case in which the individual is in a position which results from the choice SS or from the refusal of %. For the choices 95 and si plays a role of mediator; if the choice M has been adopted, the decisions Sft and c€, once adopted, keep on being adopted if nothing changes. If the choice si has not been adopted, the individual cannot choose between 2S and % separately. The equations of the system are, then, A =b+c

(19)

B = d'd + a*b

(20)

C = d'd + a-c ,

(21)

d = b-c + b-c .

( )

with

22

This system of equations allows the calculation of the trajectory adopted by the system, if the time delay and the initial state are known. A good deal of information related to the time evolution of the system is contained in fig. 15.13. A state of the system is stable when the value of each of its mem­ ory variables is equal to the values of the corresponding memory functions. Equations (19), (20), and (21) admit, therefore, of three stable states: 110, , and . For the stable state 100, for example, a - 1, b = 0 and c = 0 says that the given individual has already chosen si. For this particular state A = I, B

100

111

Boolean Equations with Temporal Delays

a

b

c

255

IO

IO

IO

ABC 1

1

1

0

1 0

1 0

0

0

1

1

1

1

0 0

0

1

o o T 1

0

0

1

1 0

0 1 oj

1

1

1

Q_ i "D

1

0

1

O

0

0

0

0)

1

Fig. 15.13

Fig. 15.14

0

0

= and C = , which means that the individual has no desire other than to choose ¿4. It appears then that one is justified in considering the state 100 to be stable. From fig. 15.14 we can conclude the possible temporal evolutions. In our first graph, two cycles and two stable states seem possible (we will describe the type of cycle below). A second sequence is possible and is shown in fig. 15.15. This sequence is disconnected from the preceding. We may summarize this discussion by a graph, fig. 15.16, the top of which corre­ sponds to a stable or unstable state and each branch of which corresponds to a possible trajectory. It is necessary to define the problem more precisely in order to determine which are the possible sequences of the system. If we start with 001 as the initial state, the only possible next step must be 000. From 000, three se­ quences are possible (see fig. 15.17). If ta < tb and ta < tc, the following state will be 100 which is stable. If tc < ta and tc < tb, the system evolves towards

André de Palma, Isabelle Stengers, and Serge Pahaut

256

011-------- ( n T )

o o T * -^ -ô ô ô — ^ 1 0 0 t. | b ôïo

Fig. 15.15

Fig. 15.17

ni

Toi

000

*b v t -t. __ 5To -g - fe>

000 Fig. 15.18

Fig. 15.16

state 001. When it is in this state, the on order of the variables a and b van­ ishes and only an off order of variable c exists so that the system returns to the state 000. It will be in exactly the same condition as at the preceding time and the system will indefinitely oscillate between 000 and 001. If tb < ta and tb < tc9 the system will move towards state , which is unstable, and can end up in either 000 or the stable state 110. Since the on order of the variable a goes without a counterorder during the evolution of the system from state we will have

010

000

if ta 000

(24)

if ta>tb + tb the sequence

In 000 we will not be in the same state as before because the on order of the variable a has existed for a time t equal to tb + t-b. if ta 100

(25)

Boolean Equations with Temporal Delays

257

000—» 010

(26)

if ta>tb + (tb + t-b) then More generally,

if k(tb 4- tb)*, used by all vehicles on major roads per unit area of land per unit time c(>* = yl$(r) = AB exp ( - V r / a - V rib) ( k x +

(50)

While Eqs. (46)-(49) are rather simplified, the result stated in Eq. (50) is certainly of interest in that it provides a map of the fuel consumed over a city or town; this matter is being examined by Chang and the author. More realis­ tic equations would express some of the detailed structure, such as perhaps the average speed remaining fairly constant over a central business district and then increasing with r to reach another approximately constant value in the outlying suburbs. It may not be too much to expect some indications of angular and local variations in all the traffic functions noted above. The effect of stops in traffic has also been explored. Recently, we investi­ gated the potential fuel economy and exhaust emission benefits that might be obtained by smoothing the flow of traffic (36). Substantial improvements in fuel economy and reductions in exhaust emissions are possible if the flow of traffic is smoothed, that is, if there are a minimum of stops and an associated smooth speed-time history. We found that traveling during the smooth flow conditions of early morning (4 a .m .) as compared to travel on the same urban route during highly congested flow (5 p.m . rush hour) resulted in, for exam­ ple, a fuel economy improvement of over 30 percent. The importance of the distribution of stopped cars on the routes of a city network over the entire day-night period cannot be overemphasized. Another study concerning the smoothness of traffic was carried out by us in connection with car following. The dynamic characteristics of a driver-vehicle system in the task of following another car was analyzed using spectral analysis techniques on data obtained with eighteen driver-subjects (37). Results obtained in both freeway and test track environments indicate a strong dependence of driver behavior on intervehicular spacing. In turn the driver’s response affects the speed-time history and the stability of the system (38-40). A knowledge of the averages of flow, concentration, speed, fraction of vehicles moving, etc., over a large network area most certainly would be of great utility with regard to many questions concerning town planning, which is itself intimately involved with the spatial distribution of many such vari­ ables. This would also require examining the effect of inhomogeneities in the distribution of housing, industry, and services, as well as the effect of inho­ mogeneities in the traffic system on the character of the vehicular traffic.

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Robert Herman

Models involving such questions are being pursued by Allen et al. (3). Ques­ tions of the kind explored in this paper are clearly interrelated strongly with various types of economic considerations. While knowledge concerning specific and local traffic situations is im­ portant, we wish to emphasize that an overall view of traffic in a large city area as discussed here is imperative. Over the years we have taken advantage of the strength of combining theory with observation and experimentation in order to view the traffic system on all hierarchical levels. Experience has shown that for such large and complex systems we do not understand the vari­ ous nonlinear interactions between the parts of the system sufficiently well to construct a consistent overall picture. Therefore, we believe that it is impor­ tant to mount a serious effort to attack this complex traffic and urban problem, which is one of the major problems of our time because it deeply affects the way we live and the energy resources of humanity.

REFERENCES 1.

Chang, M.-F., and Herman, R., “An Attempt to Characterize Traffic in Metro­ politan Areas,” Transportation Science, 72, 58, 1978. 2. Prigogine, I., Allen, P. M., and Herman, R., “The Evolution of Complexity and the Laws of Nature,” in Goals In a Global Community. The Original Background Papers for Goals for Mankind, A Report to the Club of Rome, Ed. E. Laszlo and J. Bierman, Pergamon Press, New York, 1977, pp. 1-63. 3. Allen, P. M., Deneubourg, J. L., Sanglier, M., Boon, F., and DePalma, A., “Dy­ namic Models of Urban Systems,” Final Report to the Department of Transporta­ tion under contract TSC-1185, July 1977; Allen, P. M., and Sanglier, M., “Dy­ namic Models of Urban Growth,” Journal of Social and Biological Structures, 1, 265, 1978. 4. Lighthill, M. J., and Whitham, G. B., “Kinematic Waves, II. A Theory of Traffic Flow on Long Crowded Roads,” Proc. Roy. Soc. (London) 229A, 317, 1955. 5. Richards, P. I., “Shock Waves on a Highway,” Operations Research, 4, 42, 1956. 6 . Leutzbach, W., “Testing the Applicability of the Theory of Continuity on Traffic Flow at Bottlenecks,” in Vehicular Traffic Science, Ed. L. C. Edie, R. Herman and R. Rothery, American Elsevier, New York, 1967, pp. 1-13. 7. Edie, L. C., “Flow Theories,” in Traffic Science, Ed. D. C. Gazis, John Wiley & Sons, New York, 1974, pp. 1-108. 8 . Prigogine, I., Herman, R. and Anderson, R., “Local Steady State Theory and Macroscopic Hydrodynamics of Traffic Flow,” in Vehicular Traffic Science, Ed. L. C. Edie, R. Herman and R. Rothery, American Elsevier, New York, 1967, pp. 62. 9. Reuschel, R. “Fahrzeugbewegungen in der Kolonne bei gleichförmig beschleu­ nigtem oder verzögertem Leitfarzeug,” Zeit. Osterr, Ing. und Architekt. Vereines, 95, 59-62, 73-77, 1950; Oesterreich. Ing. Archiv. 4 , 193, 1950.

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10. Pipes, L. A., “An Operational Analysis of Traffic Dynamics,” J. Appi. Phys., 24, 274, 1953. 11. Chandler, R. E., Herman, R., and Montroll, E. W. “Traffic Dynamics: Study in Car Following,” Operations Research, 6, 165, 1958. 12. Herman, R., “Theoretical Research and Experimental Studies In Vehicular Traf­ fic,” Proceedings of the Third Conference of the Australian Road Research Board, 5, 25, 1966. 13. Lee, G., “A Generalization of Linear Car Following Theory,” Operations Re­ search, 14, 595, 1966. 14. Gazis, D. C., Herman, R., and Rothery, R., “Nonlinear Follow-the-Leader Mod­ els of Traffic Flow,” Operations Research, 9, 545, 1961. 15. Herman, R., Montroll, E. W., Potts, R. B., and Rothery, R. W., “Traffic Dynam­ ics: Analysis of Stability in Car Following,” Operations Research, 7, 8 6 , 1959. 16. Gazis, D. C., Herman, R., and Rothery, R. W., “Analytical Methods in Transpor­ tation: Mathematical Car-Following Theory of Traffic Flow,” Proceedings Ameri­ can Society Civil Engineers, Eng. Mech. Div., 6 , 29, 1963. 17. Gazis, D. C., Herman, R., and Potts, R. B., “Car Following Theory of Steady State Traffic Flow,” Operations Research, 7, 499, 1959. 18. Prigogine, I. and Herman, R., Kinetic Theory of Vehicular Traffic, American Elsevier, New York, 1971. 19. Herman, R. and Lam, T. N., “On the Mean Speed in the ‘Boltzmann-Like’ Traffic Theory: Analytical Derivation; A Numerical Method,” Transportation Science, 5, 314, 418, 1971; Herman, R., Lam, T. N. and Prigogine, I., “Kinetic Theory of Vehicular Traffic—Comparison with Data,” Transportation Science, 6 , 440, 1972; Herman, R., Lam, T. N., and Prigogine, I., “Multilane Vehicular Traffic and Adaptive Human Behavior,” Science, 779, 918, 1973. 20. Greenberg, H., “An Analysis of Traffic Flow,” Operations Research, 7, 79, 1959. 21. Evans, L., Herman, R., and Lam, T. N., “Gasoline Consumption In Urban Traf­ fic,” Society of Automotive Engineers, SAE Paper No. 760048, February 23, 1976. 22. Chang, M.-F., Evans, L., Herman, R., and Wasielewski, P., “Gasoline Con­ sumption In Urban Traffic,” Transportation Research Board Record 599, 25-30, 1976. 23. Evans, L., Herman, R., and Lam, T. N., “Multivariate Analysis of Traffic Factors Related to Fuel Consumption In Urban Driving,” Transportation Science, 10, 205, 1976. 24. Evans, L ., and Herman, R ., “A Simplified Approach to Calculations of Fuel Con­ sumption in Urban Traffic Systems,” Traffic Engineering and Control, 77, 352, 1976. 25. Johnson, T. M., Formenti, D. L., Gray, R. F., and Peterson, W. C., “Measure­ ment of Motor Vehicle Operation Pertinent to Fuel Economy,” SAE Paper No. 750003, February 1975. 26. Herman, R ., and Lam, T. N ., “Trip Time Characteristics of Journeys To and From Work,” in Proceedings of the Sixth International Symposium on Transportation and Traffic Theory,” D. J. Buckley, Ed., A. H. and A. W. Reed, Sydney, Aus­ tralia, 1974, pp. 57-85.

284

Robert Herman

27. Herman, R. and Prigogine, I., “A Two-Fluid Approach to Town Traffic,” Science, 204, 148 (1979). 28. Roth, G., “The Economic Benefit to be Obtained by Road Improvements, with Special Reference to Vehicle Operating Costs,” Department of Scientific and In­ dustrial Research, Road Research Laboratory, Research Note No. RN/3426/GJR, Harmondsworth, 1959 (unpublished). 29. Wardrop, J. G., “Journey Speed and Flow in Central Urban Areas,” Traffic Engi­ neering & Control, 9, 528 (1968). 30. Johnston, R. R. M., Trayford, R. S. and Wooldridge, “Fuel Economy in Peak Hour Travel,” the SAE—Australasia, 37, 53, 1977. 31. London, F., Superfluids, John Wiley & Sons, London, 1954, Vol. 2, p. 40. 32. Sharpe, R., “The Effect of Urban Form on Transport Energy Patterns,” Urban Ecology, 3, 125, 1978 (See Figure 3). 33. Hutchinson, T. P., “Urban Traffic Speeds—II: Relation of the Parameters of Two Simpler Models to Size of the City and Time of Day,” Transportation Science, 8, 50, 1974. 34. Branston, D. M., “Urban Traffic Speeds—I: A Comparison of Proposed Expres­ sions Relating Journey Speed to Distance from a Town Center,” Transportation Science, 8 , 35, 1974. 35. Vaughan, R. J., Loannou, A., and Phylactou, R., “Traffic Characteristics as a Function of Distance to Town Center,” Traffic Engineering & Control, 14, 224, 1972. 36. Herman, R., Rule, R. G., and Jackson, M. W., “Fuel Economy and Exhaust Emissions Under Two Conditions of Traffic Smoothness,” SAE Paper No. 780614, June 1978. 37. Herman, R., Rothery, R., and Rule, R. G., “Analysis of Car-Following Experi­ ments Using Spectral Analysis Techniques,” Proceedings of the Seventh Interna­ tional Symposium on Transportation and Traffic Theory, Ed. T. Sasaki and T. Yamaoka, The Institute of Systems Science Research, Kyoto, 1967, pp. 37. 38. Darroch, J. N., and Rothery, R., “Car Following and Spectral Analysis,” in Traf­ fic Flow and Transportation, Ed. G. F. Newell, American Elsevier, New York, 1972, pp. 47. 39. Herman, R. and Rothery, R., “Propagation of Disturbances in Vehicular Pla­ toons,” in Vehicular Traffic Science, Ed. L. C. Edie, R. Herman, and R. Rothery, American Elsevier, New York, 1967, pp. 14; Herman, R., and Rothery, R., “Fre­ quency and Amplitude Dependence of Disturbances in a Traffic Stream,” in Beiträge zur Theorie des Verkehrsflusses, Ed. W. Leutzbach and P. Baron, He­ rausgegeben vom Bundesminister für Verkehr, Abt. Straszenbau, Bonn, Germany, 1969, pp. 14; Lam, T. N., and Rothery, R., “The Spectral Analysis of Speed Fluctuations on a Freeway,” Transportation Science, 4, 293, 1970. 40. Rothery, R ., Silver, R:, Herman, R ., and Torner, C ., “Analysis of Experiments on Single-Lane Bus Flow,” Operations Research, 12, 913, 1964; Herman, R., Lam, T. N., and Rothery, R., “Further Studies on Single-Lane Bus Flow: Transient Characteristics,” Transportation Science, 4, 187, 1970.

17. Fluctuations in Demand and Transportation Mode Choice J. L. Deneubourg, A. de Palma, and D. Kahn

INTRODUCTION The problem this article addresses is the structuring of a system of transpor­ tation which is subject to fluctuations in human behavior. It is related to the article in this collection, by P. Allen, on dynamic urban growth models and employs the concept of order through fluctuation in nonlinear far-fromequilibrium systems and may be viewed as focusing on the transportation as­ pects of those models. In order to treat the transportation aspects explicity, several models covering different aspects of transportation were developed. These aspects in­ cluded the spatial structuring of two similar transportation networks in com­ petition for riders, the physical growth of a transportation system as a function of urban population density, and a model of choice between competing modes of transportation. This article will develop only the last in detail and summa­ rize one of the others in an appendix. These models have the characteristics both of being dynamic and of allowing fluctuations in individual behavior to play a role in the system’s evolution. The models, therefore, consider that a deterministic dynamic evolution of a system is always subject to change be­ cause of fluctuations. The method used was to find solutions of the deterministic equations of the model and then to examine the stability of these solutions to fluctuations. For example, if a stationary state solution in one regime of demand for trans­ portation is found to exist, the stability of this solutiuon to fluctuations in de­ mand must also be examined. Some stationary states may be unstable even to small fluctuations in de­ mand and others, though locally stable, may become unstable if a sufficiently large fluctuation were to occur. The system may then adopt a new solution J. D. Deneubourg and A. de Palma are with the Service de Chimie-Physique II, Uni­ versité Libre de Bruxelles. D. Kahn is with the Transportation Systems Center, Cambridge, Massachusetts.

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286

which is stable to the fluctuation. In general, there exist threshold values of demand at which the system’s evolutionary path is altered. These changes in a system’s evolutionary path are an essential part of the process, and both the time evolution within regimes and the evolution that occurs as thresholds are exceeded and bifurcations occur must be considered.

TRANSPORTATION MODE CHOICE MODEL Equations of the Model In developing the equations of the model of transportation mode choice, we note that a number of kinetic equations are possible. We choose one that cor­ responds to the logistic equation, but one could quite plausibly arrive at this type of equation from a consideration of generalized transition probabilities, as will be shown in an appendix. We suppose that the total number of transportation users evolves accord­ ing to the logistic law:

1

i + j = D - (x + y) ,

( )

where * represents the number of users of one mode of transportation, which we shall take to be the car, and y represents the number of users of a second mode of transportation, which we shall take to be the bus. D is the total num­ ber of transportation users. We further suppose that a function Fx(x,y) measures the attractivity for the car mode; and that Fv(jt,;y) measures the attractivity for the bus mode. The relative attractivities are then Fx(x,y)/(Fx+F,) and Fv(jt,;y)/(Fr+ F v). The equations of evolution for x and for y are then given by: .

D F x(x,y)

x

Fx(x,y) + F£x,y)

(2) .

D Fy(x,y)

J

F£x,y) + F,(x,y)

_ y ’

so that DFx(x,y)/(Fx(x,y) + Fy(x,y)), for example, represents potential demand for the car mode. Attractivity Functions In order to illustrate the kinds of results this model will yield, we shall take some very simple forms for the attractivity functions. In a real situation, the

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287

attractivities for particular modes will, of course, depend upon a number of factors, including, for example, perceived sociological status associated with a mode and the number of crimes occurring in a particular mode of transpor­ tation. Conceptually, it is not difficult to include such factors in the attractivity functions, rather the difficulties are empirical. If empirical data were avail­ able on the factors affecting mode choice, then we could, in principal, write attractivity functions which reflected these factors of mode choice. In the first model to be presented here, we shall assume that the most important characteristic of a transportation mode that affects mode choice is its speed. For this first model we neglect all other factors determining mode choice. In a second model, to be presented in a later section, we shall consider some psychological factors as well. Attractivity, a Function of Mode Speed Alone The attractivity functions will be assumed to depend only on the mode speeds as follows: Fx(*>y) ~ v/

Fv(x,y) = v/ ,

(3)

where vx and vvare the car and bus speeds and p and q are powers. We further assume that the car and bus speeds can be represented as in fig. 17.1 and fig. 17.2. The car-speed relation comes from empirical data that car speeds de­ crease with increasing congestion. The bus-speed curve assumes that, up to a certain density, the bus speed will increase with the number of users. This assumes that more buses will be put into service as the demand for buses in­ creases, thus reducing the overall time of bus transit. The curve also shows the possibility for bus congestion. The curves for the car and bus speeds can be represented analytically by v —---------------- — ? n>0 (3 + ji (x+vy)*

(4)

vv = --------- ----------- , rn^o P' + p (v'x+y)m r^ Q ?

(5)

m>r where the constants are attributes of the roadway and give the slope of the curves. In subsequent work, we shall be assuming special bus lanes so that there will be no car-bus interactions, (v = v' = ). We shall now illustrate the type of results that are obtained with this

0

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288

Fig. 17.2. Bus speed, bus user density relation.

model by taking some simple special cases. The first case assumes p = q — 1 and vv = l/x

Vv = y .

6

( )

With this assumption the equations of evolution of car and bus users become: D/x X

=

--------------

l/x + J

—

X

(7) y =

Dy ------------- y • Ux + y

The stationary states are defined as those for which x = 0, y = 0. We find the following stationary states for this system

Demand and Transportation Mode Choice

289

x = D, y = 0

( 8) x = 1, y = D - 1 . A stability analysis of equations (7) shows that the state for which y = 0 (no bus riders) becomes unstable if the demand for transportation D becomes suf­ ficiently large D > 1. Taking D as a bifurcation parameter, the bifurcation dia­ gram for this is shown in fig. 17.3. For D< l a stable stationary state exists, namely, (x = D, y = 0). The other stationary state (x = 1, y = D - 1) is unstable in this regime. For higher values of D, such that D > 1, the state (x = D, y = 0) is unstable and the other state is stable. There is thus a minimum value of transit demand before the bus mode becomes a viable possibility. In jc, y space, the situation is depicted in fig. 17.4. The car and bus speeds in these two regimes of transportation demand are shown in fig. 17.5 and fig. 17.6. The figures show that once bus service becomes an alternative, its speed will increase with increasing demand. The car speed will remain constant in this model. Finite car speeds. If we had made the car speeds somewhat more realistic by limiting them at the low end, the above results would not change in any qualitative way. When a constant factor, a, is introduced in the first equation of ( ), equations ( ) become

6

v, = — a + x

6

vv = y .

Fig. 17.3. Bifurcation diagram corresponding to equation (7).

(9)

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290

y

(a)

(b)

Fig. 17.4. Car user density versus bus user density; the stable state is represented by S,

the unstable state by U.

This form for vr limits the car speeds at low densities. The stationary states are now given by x = D, y =

0 ( 10)

2

x ± = ( - a ± V a + 4)/2 . The state (x = D, y = 0) is unstable and (jc+, j +) is stable for Z)>D* where

2

D* = ( - a + V a + X)!2 .

( 11)

The results are qualitatively the same as in the first case, here the critical den­ sity for stability is D * where before it had the numerical value of 1. We note that this stability condition is identical to the condition for existence of the solution x. Finite car speeds and bus congestion effects. The purpose of this section is to show that, if we also make the bus-speed term somewhat more realistic, the previous results do not change qualitatively. Putting congestion effects into the bus-speed expression we obtain in place of equation (9)

x

where

a + jc

v

0and 7 are constants.

7+ y

^

Demand and Transportation Mode Choice

291

Fig. 17.5. Car speed as a function of demand.

Again, one stationary state of the system is given by (y = 0, jc = D), which becomes unstable when the demand exceeds a critical value D c, where

2 47 /0)/2 .

Dc = ( - a + V a +

The other stationary state is + _ -a 9

-1 + V(a9

(13)

given by: + l

)2+ 4 6(D + 7)

The bifurcation diagrams for this case are shown in fig. 17.7 and fig. 17.8 and are seen to be qualitatively similar to the previous, using simpler forms for the car- and bus-speed relations. Internal perturbations near a stationary state are such that y < yQand since y0 = , the perturbations y must be introduced as an external factor corresponding to the introduction of a new mode of transportation. Thus the examples here give the conditions under which the system becomes unstable with regard to the introduction of a new mode of transportation. The condi-

0

8

8

292

J. L. Deneubourg, A. de Palma, and D. Kahn

Demand and Transportation Mode Choice

293

tion that the system accept this new mode, D > Dc, depends upon the charac­ teristics of the old, a, and those of the new, and .

7

0

Non-linear dependence upon speed. The previous attractivity functions were linearly dependent upon the mode speeds. It is interesting to ask how the solu­ tions might change if a stronger-than-linear dependence of mode choice on mode speed were assumed. For example, let us take Fx = v

x2

Fy = v 2

(15)

with the simplest form for the speed-density relation vx = l / x

vy = y .

(16)

We then obtain for the equations of evolution the following: D/x2 X

=

y=

X

2 2- y l/ x2 + y 2

------------------------

\/ x2 + y Dy2

,, 2;

(17)

The stationary states in this case are given by x = D, y = 0 (18) (D - yy = l/y . Taking the density D as the bifurcation parameter, we obtain a bifurcation dia­ gram for the second stationary state as shown in fig. 17.9. The figure shows the existence of a minimum value of D, namely, Dcbefore bus transit is possi­ ble; however, here two solutions are possible. One solution gives the number of bus users increasing with increasing demand and the other, decreasing with increasing demand. Examining the stability of these two solutions, we find the upper branch (see fig. 17.9) to be stable and the lower one to be unstable to fluctuations. Thus, once a critical density for bus ridership is reached, an in­ creasing number of bus riders will be attracted to the bus mode, while a de­ creasing number of car users will be attracted to the car mode. Beyond the critical density Dc any further increase in the number of cars leads to an unsta­ ble situation. This model thus predicts that fluctuations will drive the system to the stable branch in which the number of car users decrease with a further increase in demand. Fig. 17.10 shows the situation in (x, y) space. For a num-

J. L. Deneubourg, A. de Palma, and D. Kahn

294

Fig. 17.9. Bifurcation diagram for the system of equation (17), y versus D.

1/4

ber of cars greater than a critical value numerically equal to 3 and a number of bus users less than 9 there is a solution U which is unstable. For a num­ ber of car users less than the critical value and a number of bus users greater than 9 there is a stable stationary state labelled S in the figure. The figure also shows the curve which separates space into two regions, each region being under the influence of a stable stationary state.

“1/4

1/4

General discussion. It is worthwhile to point out that even with the very sim­ ple forms used for the attractivity functions, bifurcations appear and thresh­ olds of transportation demand exist before bus service becomes viable. If we were to develop more realistic forms for the attractivity functions, which like human behavior can be expected to be quite nonlinear, we could also expect an even greater richness in the evolutionary branchings of the sys­ tem. As each solution is subject to fluctuations the evolutionary path is also subject to new branchings and hence new paths of evolution will emerge. Fig. 17.11 schematically illustrates a general bifurcation diagram in which we use X as a bifurcation parameter. The stationary state is represented by the variable jc while X is a parameter which measures the feedback effects in the system. We see from the figure that if the feedback parameter is sufficiently small, X < X0, the system will have only one stationary state, x°. If, however, X is sufficiently large, X > X0, new stationary states may appear in the system. As X increases, the number of possible stationary states of the system in­ creases. If the system accepts the branch x +, then, as time increases, the prob­ ability that the system will jump to the branch decreases. Thus, the choice

Demand and Transportation Mode Choice

295

Fig. 17.10. Car user density versus bus user density for the system of equations (17); the critical numerical values are xc = 3 1/4 and y < y c; stable (»S') for x ^ xc and y ^ yc.

= 9 l/4; unstable (U) for x > xc and

Fig. 17.11. General bifurcation diagram with X as the bifurcation parameter.

J. L. Deneubourg, A. de Palma, andD. Kahn

296

of branches, or x~ made by the system near a bifurcation point deter­ mines the system’s further evolution. This phenomenon underscores the im­ portance of history in the evolution of systems. Attractivity, a Function of Mode Speed and Psychological Factors We now wish to look at the transportation mode choice problem with an added psychological factor. We shall assume that mode choice depends not only on the mode speed, but also on the publicity given to that mode and on imitation. By imitation we mean that people tend to copy the behavior of oth­ ers that gives rise to a positive reinforcement of mode choice. We take for the attractivity functions Fx (x,y) = v / F , (19) Fy (x,y) = v / F2 , where F, and F 2 are given by Fx= 0 + a x (20) F2 = 0' + a ' y . The constant terms 0 and 0' are publicity terms for the modes, and the terms depending upon the density of users are the imitative terms proportional to the number of people already using these modes. When equation (19) and equation (20) are used, the equations of evolu­ tion become . _

D(6/x + a)

_

0/x + a + 0'y + a y 2 (21)

. _

D(0> + a ’y 2)

_

d/x + a + 0'y + a ’y2 ^ ’

6

where we have used equation ( ) for the part of the attractivity function that depends upon mode speed. A stationary state of the system is (x = D, y = 0), while the other sta­ tionary states are given by the roots of the cubic equation:

Demand and Transportation Mode Choice

297

We shall discuss the case for which there is no publicity for car use (0 = 0). In this case the other stationary states of the system given by equations ( ) are obtained as the roots of the quadratic equation:

21

2

a ' y + (0' - a ' D)y + (a - D0') = 0 .

(23)

Two critical densities of demand emerge from the analysis: Dc2 = a/0'

(24)

Dc = ( V 4 a a ' - 0 ')/a ' .

(25)

The demand D\ is the critical density for the (x = D, y = 0) stationary state. For sufficiently high demand, D > D\ the y - 0 state is unstable to fluctua­ tions. The other critical density D c given by equation (25) gives the condition for the other stationary states to exist. There are two interesting situations, one for which there is great pub­ licity for bus use, ' > a a ', depicted in fig. 17.12, and one for which there is little publicity, 0' < a a ' shown in fig. 17.13. Fig. 17.12 shows that for D < D there is only one stable stationary state, (y = 0, x = D), labeled / in the figure. When the density becomes high enough, D > D\, this state / becomes unstable, but another one ap­ pears, that is labeled y+ and is stable and increases with increasing demand for transportation. This result is similar to the results of the previous models. For the case of little publicity for the bus, fig. 17.13 shows that there is a region of intermediate transportation demand in which the system will accept the stationary states y +, y~, and / . The y~ state is unstable and, hence, any perturbation will immediately move the system away from this state. In this region of transportation demand, a given size perturbation Ay,- is needed to bring the system from the state y (where y = ) to the state y+ (which in­ creases with increasing demand). The value of the perturbation Ay, needed for this transformation decreases as the transportation demand Df increases. That is, the size of the fluctuation necessary to cause the system to jump from the state y l to the state y+ decreases as the demand for transportation increases, as one would intuitively expect. For larger densities, D > D \ , whatever the value of the perturbation, Ay, the system will spontaneously go over to the stationary state (y+, x +). Fig. 17.14 shows how publicity for the bus is related to the total demand for transportation. The figure also shows the conditions under which coexis­ tence between the bus and car modes is possible. The figure illustrates that coexistence is more easily accomplished if the publicity for bus use is high, (in order for there to be both car and bus users the demand for transportation

0

5

1

0

298

J. L. Deneubourg, A . de Palma, andD . Kahn

Fig. 17.12. Bifurcation diagram for the system (21) with no publicity for the car and extensive publicity for the bus.

must exceed D c, and the demand will exceed this value more easily if the bus publicity term is large). The figure also illustrates that the condition for there always to be bus riders, (namely that the demand be greater than D\), happens more easily when there is much bus publicity.

CONCLUSIONS The models of transportation mode choice introduced in this article have il­ lustrated the importance of fluctuations in the demand for transportation on the viability of competing modes of transportation. Some stationary states of the system were unstable to even small fluctuations in demand or to the intro­ duction of a new transportation mode, while others, though locally stable, would become unstable if a sufficiently large fluctuation in demand occurred. The system in this case would adopt a new solution which was stable to the fluctuation. This adaptive emergence is one example of the concept of order through fluctuation whereby a system self-organizes to a new mode of behav­ ior when critical size thresholds for stability are exceeded.

Demand and Transportation Mode Choice

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Fig. 17.13. Bifurcation diagram for the system (21) with no publicity for the car and little publicity for the bus.

Fig. 17.14. Relation between demand and the bus publicity factor; the parameter D\ = 0 7 a \

J. L. Deneubourg, A. de Palma, and D. Kahn

300 APPENDIX A

Transition Probabilities and the Logistic Equation

In this appendix we show that the equations of evolution used in this paper follow quite plausibly from some general considerations of transition proba­ bilities and utility functions. We consider a system having N different possible states, N 2 possible transitions. We define g(i,j) as the utility function of the transition i to j. This utility function will be a function of g(i), g(j) (utility functions associated with each state) and the distance (or a generalization of the distance) between the state i and the state j. In general, individuals will try to make transitions which increase their utility function. The probability per unit time to make a transition from the state i to the state j will be / N P(Uj) = g(i,j) / 2 g(i,k) . / k=

(A-l)

1

Assuming that the transitional probabilities P(iJ) exist, we can describe the evolution of the population x(i) as the probability of a transition from state j to state i multiplied by the number of people at y, minus the opposite transition: N

N

x(i) = ^

P(j,i) x(j) - 2

j= 1

P(iJ)

X(i)

.

(A-2)

7=1

If we replace the transition probabilities P(iJ) with the generalized utility functions g(i,j) we obtain: •^

v» g 0 \0 *0 )

40 = y

i

v ;

Zg( j , k)

-----x(i) ,

(A-3)

which has the same form as the equations of evolution that we have been us­ ing. For example, consider the problem of transportation mode choice and assume that the utility function is given by g(ij) = g(i) g(j) •

(A-4)

This assumption leads to an equation of evolution of the form: x(i) = x TP(i) - x(i) ,

(A-5)

Demand and Transportation Mode Choice

301

where xT is the total population and N

P(i) = 8(0

^ g(k) • k=

1

(A-6)

If x is the number of car users the equations of evolution (A-5) become x = D P(x) —x (A-7) y — D P(y) - y , where D has replaced xT. The probabilities P(x) and P(y) are given by: P(x) = g(x)/(g(x) + g(y)

8

(A- ) ^00 = g O )/(£(•*) + g(y) ) •

With these definitions of the transitional probabilities and an identification of the utility functions with the attractivity functions used in the paper, the evolu­ tion equations (A-7) are of the same form as the equations (2) of this article. This analysis, however, proceeded from a probabilistic model to a determinis­ tic one involving average values for the variables. There is thus an implicit assumption that we are dealing with a sufficiently large number of individu­ als. The model may thus be said to be a phenomenological one for the fre­ quency of transition.

APPENDIX B Transportation System Growth and Density of Users In this appendix we briefly indicate how one may treat the interrelation between the growth of a transportation mode and the density of users of transportation. For one mode of transportation L, we have the evolution equations

J. L. Deneubourg, A. de Palma, andD. Kahn

302

where y is the number of users of the transportation mode L, D is the total demand for the mode, p is the attractivity of other transportation modes, and L is the number of miles of highway or railroad track or the like. As an example, if we assume that the attractivity of a transportation mode is only a function of the infrastructure L that it offers, then, DL

(B-2)

p+ L

where p represents competing modes. If we assume that the physical growth of the transportation mode depends only on the influx of money and the cost of running the system, K, we have L = {§ - K) L

(B-3)

If we represent the money flux $ as a(t) + vy where a(t) are state subsidies or capital grant monies, and vy is the revenue received from the users of the mode, then the equations of evolution become, assuming a = ,

0

DL J =' p+ L

y

(B-4) L = vy — K L When we study the stationary states of this system we find that one stationary state is (L = 0, y = 0). This is stable when maintenance costs are too high, K >Dv/ p. The other stationary state (L +,y+) becomes viable when the influx of money exceeds the maintenance costs, K P, then conventionally adopted functions such as the CES (constant elasticity of substitution) would yield an isoquant Q'\ with the thermodynamic constraint, one fits the points to the curve Q. It is quite apparent that Q falls more steeply than Qf in the region shown. (What happens at larger E depends on the form chosen for Q and Q'.) This fact implies that estimates of the optimum operating point at E prices higher than those yielding points a, and will be further to the left if one uses the family of curves of the form Q' than if one uses curves of the form Q. In other words, if the production system is operating in a region near enough to the thermodynamic limit to make the function Q' flatter than the function Q, the producer will underestimate the economically efficient amount of E (read energy or fuel for E , of course) if he neglects the effects of the natural thermodynamic limit. The reason underlying this apparent para­

.5

0 7

Thermodynamic Constraints in Economic Analysis

P

329

E

Fig. 2 0.4. Illustration of the fitting of “observed” points a, (3, 7 to formal representa­ tions of production functions without consideration of thermodynamic limits (Q') and with these limits (Q).

dox is that the substitutability of L by E, —(dL/dE)Q, is greater with Q than is implied by Q' and {hat, if the productivity of L is constant, the real productiv­ ity» (dQ/dE)L, of E is less than is implied by the family of curves with the shape of Q'.

THE NATURE OF THERMODYNAMIC CONSTRAINTS We have seen how the existence of a thermodynamic limit affects the behavior of a production system and the producer who uses an economic analysis to optimize his operating point. How does one locate the asymptote? The natural starting point is to choose the traditional thermodynamic limit. This is certainly a lower bound, but it is not necessarily the most useful lower bound. If one underestimates the value of the asymptote P, then one is liable to make the same errors, admittedly to a lesser degree, that one makes by forgetting that P is not zero. Because we carry out production processes at finite rates, we can be quite sure that P(r), the lower bound on E to produce Q at a rate r = Q, is greater than P(0), the bound on E for a reversible process. The relevant questions at this point are “How do we find P{r)l What is the difference between P(0) and P{r) and when is that difference significant?

330

R. Stephen Berry and Bjarne Andresen

How does P(r) depend on r?” Then we can go on to ask what path the process must take to yield or approach P(r). These questions bring us directly to a substantive problem of thermody­ namics. A meaningful bound P(r) must be the equivalent of a thermodynamic potential, in the sense that if the constraints are specified defining the range of allowable processes, then P(r) is the extremal value of the work (maximum if it is work extracted to a work reservoir, minimum if it is work done on the system) in the allowable space, and within that space, P(r) must depend only on the initial and final points and of course on the rate r. The one complica­ tion beyond the conventional definition of a thermodynamic potential is the inclusion of a constraint on the rate or time for the process. It was shown that such functions P(r) exist and that they can be constructed by a fairly sim­ ple transformation, a natural generalization of a Legendre transformation7; strictly, one generates integrals of motion in the sense of Cartan Naturally if one introduces time constraints in addition to the traditional constraints on temperature, pressure, entropy, and so on, the parameters of the thermody­ namic system—its heat capacities and the other relevant equilibrium param­ eters— must be augmented by parameters giving its relevant time responses (heat conductance, frictional coefficient, chemical rate constraints, and so forth). To get more physics out, we must put more physics in. It has been possible, in the cases studied until now, to choose a time scale so that only one or, at most, two or three relaxation processes are relevant. For example heat conduction between the system and its reservoirs may be the one significant rate process, whenever internal relaxation processes of the working fluid are rapid enough to establish internal thermal equilibrium in the fluid. If tech­ nological and time constraints do not fully determine the potential P(r), one may include others, for example the constraint of fixed budget. The difference between two points of a potential function chosen this way will be the energy required to carry out the process at the specified (average) rate, r, at the lowest possible cost. To illustrate the significance of letting the process proceed at nonvanish­ ing rates, let us look at the performance of an idealized auto engine operating between a high combustion temperature and the ambient temperature. This example becomes a rich nonlinear system when it is transformed into an ex­ tremal problem for operating an engine at a finite rate. Suppose we attribute three loss processes to the engine: friction, a finite heat conductance between reservoirs and system, and a heat leak into the low-temperature reservoir. Suppose the friction is taken to be linear in velocity or quadratic in the power (corresponding to a thick lubricant layer), and heat transfer between system and reservoirs follows Newton’s law: the rate of heat flow is directly propor­ tional to the temperature difference. Then the power, w, continuous or aver­ age, delivered by such an engine, is expressible in terms of the reservoir tem­

6

.8

Thermodynamic Constraints in Economic Analysis

331

peratures THand Tu the rate of heat flow —q from the hot reservoir, the heat resistance p/ , the friction coefficient a, and the rate of heat leak q0:

2

xv = - a

( Th Tl qp \ 2 r v ^ r )

( TH TL \ ■(—

)’ (4)

Let us choose to make power an extremum. We obtain a quartic equation whose four solutions are most naturally expressed in terms of the temperature difference AT^qp

(5)

and a ratio of heat resistance to friction

2

v = p/ a ,

(6)

as at

=

t h±

V

t j l

(V)

and

(8 ) The general form of the solution can be shown conveniently in a contour dia­ gram of pw, essentially the power on a plane with axes AT that we can inter­ pret as a scaled axis for q, and v, that we can interpret as the inverse of friction. Such a contour diagram is shown in fig. 20.5. The figure is drawn for Th/Tl = 9. The light lines are contours of equal power for fixed p. The heavy solid lines give the maximal power solutions; the heavy dotted lines give min­ ima. “Solutions” for A7 > 9 correspond to physically inadmissible regions, because they would violate the Second Law. Solutions with AT < 0 corre­ spond to the engine acting as a heat pump. Note that in the admissible heat engine region, there are two regimes. Over most of this region shown, to the left of v ~ - , there is only a single maximum to pw; this is the region in which the power is limited by the rate of heat transfer. To the right, there are two (equal) maxima and one minimum of pw, corresponding to each value of

4

332

R. Stephen Berry and Bjarne Andresen

Fig. 20.5. Contours of power production for a two-reservoir heat engine with finite heat conductance to its reservoirs and friction. Heavy solid lines are maxima of power; heavy dotted lines are minima. The ordinate AT is pq, and the abscissa v is p/2a, essentially the inverse of the friction, as explained in the text.

1

v, but to different values of A7 or q. The system exhibits a bifurcation point at (A = , v — 4); as friction increases, the system enters sharply into the friction-dominated region. In this region the maxima of pw correspond to two quite different rates of heat input and, therefore, to two quite different efficiencies. We can translate this model into real-world numbers using values ap­ propriate for an automobile engine Suppose that the combustion tempera­ ture Th = 2700 K and the ambient TL = 300 K; we take realistic values for p = —1KlkW and a = 4.167 X 10 kW~l, which correspond to q = 1800 kW and w = 600 kW at the bifurcation point that separates the regions domi­ nated by friction and by heat resistance. The efficiency of this system at the bifurcation point is only 33 percent, compared with a Carnot efficiency (i.e., reversible operation) for these two reservoirs of 89 percent. If the engine is improved so that the friction coefficient is halved, then w goes up to 1,200 kW with the same heat consumption as before, to give an efficiency of 67 percent;

71 6

.9 “4

Thermodynamic Constraints in Economic Analysis

333

if on the other hand a is doubled, then a power output of only 300 kW is obtained with q = 706 kW or 2,294 kW, corresponding to efficiencies of 42 percent and 13 percent. In short, the penalty in the thermodynamic efficiency for operating the engine at a finite rate so as to maximize power is, for the most favorable case described here, 89 percent minus 67 percent or 22 per­ cent. If one maximizes efficiency rather than power, one achieves a different set of optimal operating conditions. These conditions follow approximately the same qualitative behavior as those that maximize the power. Another illustration of rate dependence is the operation of a heat engine between two thermal reservoirs with finite heat conductance to each of them. The maximum power is extracted from this system at an intermediate rate of operation where the increase in power with rate is balanced against the in­ creased temperature drops across the thermal resistances. The efficiency at this point is 1 - V r high/ r iowas opposed to the traditional Carnot efficiency l-^gh/riow. Note that this result is independent of the thermal resistance, which is not true for the power produced. So far we have concentrated on the optimization of power produced for the heat cycles, but that is but one of an infinite number of possible objective functions. Other desires might be to maximize the efficiency or minimize the entropy production. Even more likely, the goal might be nontechnical, in par­ ticular the minimizing of unit cost or total cost of operation or a weighted combination of all these possibilities. The choice is completely open and will generally lead to different optimal behavior for different objective functions An example is given in fig. 20.6 where we again consider a heat engine with thermal resistances to its reservoirs. The figure shows the optimal allocation of time between the hot ( t , ) and cold ( t 2) isothermal branches of the cycle when (a) the total entropy production is minimized, (b) the efficiency is a maximum, and (c) the power produced is maximized. The constraint of fixed cycle time t , + t 2 forces the operating point to lie on the diagonal indicated. Positive work is only produced for times above the hyperbola. Clearly, the different objective functions require different operating conditions (allocation of time), and, under conditions of very short time, it may even be possible to optimize one function but not another. The preceding thermodynamic analysis has produced limits on various quantities of choice for the process considered, either directly or as the dif­ ference between values of a potential function in the initial and final states. However, no indication is given of how this optimal performance is achieved. Finding the time path of that process is an entirely different, and much more difficult, project usually solved by optimal control theory. Although that in­ formation is essential for the actual operation of the process, we only need the limiting values of energy consumption and similar values for the economic analysis and will bypass further discussion of the optimal path.

.10

334

R . Stephen Berry and Bjarne Andresen

Fig. 20.6. General form of the extremals for a Carnot-like heat engine with finite heat conductance that may spend time t, in contact with THand t 2 in contact with TL. The parallel lines marked 5 and r\ correspond to minimal entropy production and maximum efficiency; the line marked w corresponds to maximum power. The hyperbola divides the region of positive work (above) from a region in which the work would be nega­ tive. Any engine with a fixed period lies on a line t , + t 2 = t , a constant.

In this section we have looked at the thermodynamic behavior of pro­ cesses proceeding at nonvanishing rate and found the energy requirements to depend strongly on the time period allocated for the process. In the terms of the previous section, the proper asymptote to use in the economic analysis of a technology is often a sensitive function of the rate at which we want the prod­ uct delivered.

APPLICATION OF CONSTRAINTS The thermodynamic constraints for finite time operation described in the pre­ vious section can be used in the economic analysis in two different ways: 1. One can calculate the total cost C of producing the quantity Q of prod­ uct in a given time, consisting of the capital cost for a plant of a certain size plus the operating cost for optimal operation at a rate r necessary to produce Q. Obviously, there is a tradeoff between a larger plant operating at a lower rate (and therefore usually at a higher efficiency) and the capital cost of that

Thermodynamic Constraints in Economic Analysis

335

plant, so one can find the economic optimum by either minimizing C for fixed Q or maximizing Q for fixed C. 2. One can add information about the operating and capital costs as func­ tions of rate to the definition of the extremal equations and solve them with constrained product quality Q. Both procedures will arrive at the same optimum, so it is a matter of taste whether one prefers the black-box approach of or is interested in the inter­ mediate physical information of . This problem can be made more precise and realistic by taking into ac­ count discounting, changing productivity of capital and the dependence of productivities on the rate of production. Here, we merely give a formal state­ ment of the problem, not a solution. Let the objective function it be the net profit, the difference between revenue R and cost C. The discount rate is r; PQ(t) is the unit price of product Q at time t\ Q and C are the time rates of change of physical output Q and cost C. Thus

1

n =

r

-

2

c

= I Q(t)pQ(t)e~rtdt - I C(t)e~ndt J 0

0

where the total cost per unit time is the sum of the input costs c -

sz / op / o

.

j

For example we may take a three-input production system with capital K , ma­ terials M, and energy Ex. Let us suppose that the cost of capital bought at time t is fully paid at that time; the forward costs are included in the cost. Then C (t )

K(t)pK(t) + M(t)p 2(t) + E(t)pE(t) e rldt ,

=

although one might choose to use different exogenous discount rates for the different input factors in special cases. The output rate Q(t) is a function of the amount of capital K(t) and of its distribution of vintage and productivity. We denote the output rate at t' per unit of capital purchased at f . Thus

0

Our finite-time thermodynamics enters in the physical basis of the func­ tion (f'-i"). To be as precise as possible, we designate c(> as a rate-ofproduction function, a relation between rate of production and rate of use of

336

R. Stephen Berry and Bjarne Andresen

input factors (and possibly of amounts of inputs as well in some cases) to dis­ tinguish it from a relation between amounts of output and amounts of input factors. Note that $ is presumed here to relate physical variables. Then for our three-factor system, we require that c\>(t'-t") be a function of E(t'), of M(i') and of a set of parameters K i t '- t ”) that characterize the rate-ofproduction function. Strictly, then is a functional, rather than a simple function. The problem is then one of finding the maximum of tt, the objective function, by suitable adjustment of the control functions K , M and E. We may suppose Q{t) is given exogenously; the functional [M, E\ X7(f'-/")] con­ nects M and E and, thus, can be treated as a constraint. The usual constraints on positivity of M and E apply. At this point, one can begin making approx­ imations and introduce empirical or theoretical c}>s and reduce the problem to a simple enough form to be soluble. As an example, one might use for one’s cj> a function like that of a Carnot-like heat engine operating between two reservoirs at temperatures TH and Tl , at a finite rate t _1 per cycle but with finite, nonzero heat conductance k connecting the engine to its reservoirs and with minimum loss of avail­ ability. The heat conductance can be interpreted as K. Such an engine has an extremum of useful power— a maximum of power if it produces work or a minimum of power required if it acts as a consumer of work—equal to

10

w = w rcvcrslblc - Tl(

^

'branches

|< t ,|W t .

’

Here |cr,| is the absolute value of the entropy carried in branch i of the process and is characteristic of the particular machine. The amount of materials and fuel or energy consumed of course enter in the as. In the simplest case, we neglect M altogether and E is simply the fuel consumed per cycle, expressed in units of heat per cycle, which is, in turn, THu ], where a, is the entropy transported in the hot isothermal branch of the cycle. The reversible work in this system is the Carnot work [1 - (TL/TH)]THv l. Entropy is only transported in two branches of the Carnot-like cycle and the amounts in the two branches are the same, so W =

1 - (Tl/Th) 7 > , - Tl(2ct, ) 2/ k t 1 - (Tl/Th) E - [ATJTl k t ] E2

Thermodynamic Constraints in Economic Analysis

337

If the output Q is taken to be power, then this expression is precisely a physi­ cal rate-of-production function connecting the physical inputs of rate of fuel consumption E and capital K , in the form of heat conductance, with the sal­ able output W. Formal application of an economic constraint to the problem of finding a thermodynamic extremum leads to a simple corollary to a general and ele­ mentary theorem of economics Suppose we want to find the extremum of the change of a generalized potential AP, with P a function of the parameters X lf . . . ,X„ which may include rate or time per unit as well as the usual pa­ rameters of a thermodynamic system. Additionally, suppose one constraint is a fixed budget, C(X}, . . . ,X„) = constant. Then the fundamental variational equation is

.11

d (A P )-\d C = 0 ,

(9)

or, for each parameter X i9 dAP _

dC

dXt

dX{ ’

( 10)

This equality in turn implies that the constant X satisfies X = dC/dAP =

(11)

that is that the ratio of marginal cost of each factor to its marginal thermody­ namic (work) productivity is a constant, the same for all factors, and equal to the marginal cost of optimally produced work. Finally, we can look toward using thermodynamic optima, especially those based on finite time constraints, as guides for the allocation of research and development effort. Even without carrying out optimal control analyses of the paths that would yield the extremal values of work, we can determine the minimum fuel requirements of processes, per unit of product, or the max­ imum work delivered per unit of fuel for engines for operations at nonzero rates and compare them with the fuel per unit output of actual operations. The analyses tell us immediately whether the operating rates are optimal and, as in the example of the engine given in the previous section, which physical pa­ rameters should be changed to bring the system to a more nearly optimal con­ dition. At a more fundamental level, we can see from the comparison of real and ideal requirements of fuel per unit output which processes have the poten­ tial for large improvements and which are already operating so near their ther­

338

R. Stephen Berry and Bjarne Andresen

modynamic limits that new technology based on the same ideal model can only yield a very small increment of fuel saving, for example. The generation of electricity from steam is in the latter category; modern generating systems operate so near to their Carnot efficiencies that we cannot possibly achieve large reductions in fuel use (so long as we use available materials) by tech­ nological improvement. Such processes as the manufacture of ammonia oper­ ate at thermodynamic levels only about one-third as efficient as their ideal reversible limits. We may be able to make technological changes to reduce the apparent inefficiencies of these systems. Their minimum energy require­ ments, however, are not yet known when they are constrained to produce products at their present rates. We do not really know yet how much we could hypothetically improve the technology of manufacturing ammonia, if we con­ strain the process to produce at the rate it now does. Finding the answers to such questions will be a next step toward using thermodynamics to guide prac­ tical policy choices.

REFERENCES 1. R. S. Berry, P. Salamon, and G. Heal, Resources and Energy 1, 125 (1978). 2. H. T. Odum, Ambio 2, 220 (1973). 3. M. W. Gilliland, Science 189, 1051 (1975). 4. B. M. Hannon, “Energy, Growth and Altruism,” Proceedings of the Limits to Growth Conference, Woodlands, Texas, 1975 (The Club of Rome, 1975). 5. N. Meshkov and R. S. Berry, Proc. Int’l. Conf. Energy Use Management, Los Angeles, Calif., October, 1979 (Pergamon Press, 1979), p. 374. 6 . P. Salamon, B. Andresen, and R. S. Berry, Phys. Rev. A 75, 2094 (1977). 7. H. Callen, Thermodynamics (Wiley, New York, 1960). 8 . E. Cartan, Legons sur les invariants integraux (Hermann, Paris, 1922); Les Systems differential exterieurs et leur applications geometrique (Hermann, Paris, 1945). 9. B. Andresen, P. Salamon, and R. S. Berry, J. Chem. Phys. 66, 1571 (1977). 10. P. Salamon and A. Nitzan, J. Chem. Phys. 74, 3546 (1981). 11. B. Andresen, R. S. Berry, and P. Salamon, Proc. Int’l. Conf. Energy Use Man­ agement, Tucson, Ariz., October, 1977 (Pergamon Press, New York, 1977), vol. II, p. 1.

21. Economic Dynamics

Russell Davidson

The aim of this contribution is to trace links between the theory of dissipative structures and evolution in the natural sciences and some of the notions of economic structure, stability, and equilibrium used by economists in their analyses of human economic interactions. A specific model of urban housing and decay will be used as an illustrative example of such an analysis, and the similarity of the concepts used to those used in physical theory will be pointed out. It is convenient to begin, as is usual in attempts at communication be­ tween disciplines, with some clarification of terminology. The word equi­ librium is used rather differently by natural scientists and economists. An economic equilibrium is defined simply as a state of affairs where demand is equal to supply. No implication is intended that this state of affairs is unchang­ ing over time: in that event economists speak, like everyone else, of a steady state. Again, a steady state need not be an equilibrium, in anyone’s language. The economist’s long-run equilibrium is more like that of the physicist, es­ pecially in the sense of John Maynard Keynes’ well-known remark, “In the long run, we’re all dead.” Of much more interest to economists is the idea of a steady state in which some kind of structure is maintained not, as in a chemi­ cal dissipative structure, by entropy production and dissipation, but by the more general phenomenon of a flow. The distinction between stocks and flows is vital to economics: one may think of our stock of productive capital and other resources as sustaining the flows of goods and services which it is the aim of an economy to provide for its participants. These flows create eco­ nomic structures, some stable, some not, some subject to more or less rapid continuous or discontinuous evolution. The driving force behind an economic structure is the sum of the motives or incentives of economic agents, the individual households and firms of the structure. These motives are usually taken to be those of profit and well­ being. Economic agents interact by exchanging commodities or claims to Russell Davidson is with the Department of Economics, Queen’s University, Kingston.

Russell Davidson

340

them in markets, usually, in the western world, in markets where to each com­ modity there corresponds a price. It is by the price mechanism that economic equilibrium is achieved, when, that is, it ever is achieved. To describe the dynamics of an economy, then, it is necessary to have equations giving the motion through time not only of the economic quantities, be they stocks or flows, but also of the market prices. Since everything depends, in general, on everything else, the equations will be coupled and nonlinear. Perhaps I should remark that not all economic models explicitly exhibit price dynamics. The celebrated Solow model (Solow 1956) omits all mention of economizing behavior. It is only when the missing ingredient is restored, as in Phelps (1961), however, that any conclusions may be drawn about the wel­ fare of economic agents. The equations of motion of an economy are derived, not (or not only) from physical laws, but from the behavioral assumptions made about eco­ nomic activity. To illustrate these, I shall now describe a model of urban hous­ ing I developed in Davidson (1977). The agents are the tenants, who are as­ sumed to rent their housing in a competitive market where units of varying standards of comfort or upkeep are offered at a range of prices (rents), and the landlords, who construct the housing units and maintain them at varying lev­ els of upkeep over their useful life. The landlords, who for simplicity are re­ garded as absentees, are assumed to maximize the stream of profits that they derive from their property. The tenants’ behavior is assumed to follow from their exercise of rational choice over the options open to them given their in­ comes. It is well known (see for example Arrow and Hahn 1971) that rational preferences may be represented by continuous utility indicators, defined on the set of possible flows of consumption of commodities, and that rational choice corresponds to maximization of such utility indicators. Let us imagine now that the competitive market rent for use of a housing unit of upkeep level k for unit time is R(k,t). Further, let upkeep depreciate exponentially at rate unless maintained by investment. We may say, then, that upkeep (a stock) changes over time according to the equation k = I - §k, where I is the level of investment (a flow). If the cost per unit of time of an investment rate I is C(/), each landlord will seek to maximize

8

T-

0 subject to the constraints k = I - hk, K > 0, / > 0. Here, T is the life of a unit, v its age, and P the discount rate or market interest rate. This is now a problem in optimal control theory (see for example Pontryagin 1962, Bellman 1967, Bryson and Ho 1969). If a landlord knows or thinks he knows the form

Economic Dynamics

341

of the functions R and C, he can solve the problem. It can be shown that, if the costate variable p is chosen to satisfy the differential equation

8

p = (P + )/? - - (k,t) dk with boundary condition p(T—v) = 0, then the optimal investment rate /* at time t is given by the equation C'(/*) = p . The dynamical system is com­ pleted by the consistent equation k = I* - bk. A tenant faced with market rents R(k,t) will maximize his utility subject to the constraint of his income. It is easily seen that, if for simplicity we as­ sume that intertemporal effects do not matter in the tenant’s welfare so that he simply maximizes his utility flow at each moment, then the condition for a utility maximum can be written (

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