TOWARDS A UNIFYING THEORY OF VEGETATION DYNAMICS

by Madhur Anand

Department of Plant Sciences [Environmental Science]

Subrnitted in partial filfillrnent of the requirements for the degree of

Doctor of Philosophy

Faculty of Graduate Studies The University of Western Ontario London, Ontario May 1997 National Libraiy Bibliothèque nationale !*m of Canada du Canada Acquisitions and Acquisitions et Bibliographie SeMces senrices bibliographiques 395 Wellington Street 395, rue Wellington OttawaON K1AON4 Ottawa ON KIA ON4 Canada Carlada

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The thesis to be argued is that vegetation dynamics is a unified, complex chaotic process, hierarchically unfolding over spatio-temporal scales, where determinism and randomness convolute, shifting in dominance through sequential phases. Conventional theory, known as the Clements-Gleason controversy, views determinism and randomness as mutually exclusive phenornena. My goal is to prove conventional theory wrong.

1 use the language of dynarnical systems theory to conceptualize vegetation dynamics as an analytical space-time trajectory. The phase space coordinates are derived from cornpositional data. Two case studies are presented. The first short-term dynamics in an Atlantic heathland recovering from fire and grazing. Dynamics is obsewed to be transient, beginning with linear determinism and then collapsing into a chaotic nonlinear phase. The stationary Markov chain serves as an excellent reference set for detecting determinisrn, and the model fits the observed process quite well. However, this simple linear model cannot account for the phase transition defining the natural process.

Interestingly, simulation experiments on the stationary Markov chah with various levels of random perturbation recover the obsewed two-phase structure surprisingly well. The simulated process is deterministically chaotic, possessing a positive Lyapunov exponent and high fractal dimension. This verifies that passage between dominant deterrninism and dominant randornness is a distinctly possible event and suggests that the heathland process is highly complex and unpredictable in the long-tem.

The second case study of long-term postglacial vegetation dynarnics revealed a surprising repetition of the pattern seen in the heathland. The two-phase complex chaotic structure re-emerged, revealing the hierarchical nature of vegetation dynamics.

Against better intuition, apparently random effects can enhance the appearance of determinism through a synergistic accumulation of small feedback effects. Determinism is thus not always dominant, but rather appears as an underlying wave which may at times be overwhelmed by randomness. The fact that determinism and randomness are not easily teased apart suggests that both Clements' and Gleason's views are necessary. "Chaos" provides the basis for a unified theory of vegetation dynamics. Classical theones are reconciled and in fact reside as special cases. keywords: chaos, coding, complexity, determinism, diversity, entropy, information theory, modelling, succession 1 thank my supervisor, Dr. L. Orloci, for his unfailing guidance, wisdom and inspiration, which were fundamental in defining the path which brought me here. 1 am eminently grateful to hirn for providing me with an open and fertile acadernic environment, which included access to facilities, opportunities to travel to conferences, participation in scientific research at an international level and, most importantly, a passion towards science and life in general. 1 thank the members of rny advisory cornmittee for much guidance and expert advice. Specifically 1 thank the late Dr. R. C. Jancey for instructive and clarifying discussions. I am most grateful to Dr. M. A. Maun for sage advice and contagious enthusiasm toward my research. 1 consider myself extremely lucky to have crossed paths with Dr. A. L. Szilird who provided invaluable insight and a truly fresh perspective to me on the subject. 1 thank the institutional and panel members of the Natural Science and Engineering Research Council of Canada for scholarships and the Chair and members of the Department of Plant Sciences at the University of Western Ontario for the quality intellectual and collegiate environment. Colleagues, fnends, and fellow graduate students. Dr. R. C. Bailey, Dr. J. Bowles, L. Coklin, Dr. G. Gray, Dr. R. H. Green, X. S. He, Dr. A. Heinicke, Dr. L. Kari, T. Kavanaugh, M. Kennedy, Dr. M. A. Lachance, L. R. Little, Dr. R. Martin, A. Raake, S. Tavares, Dr. C. G. Trick, A. Tsang, Dr. D. Urban, Dr. D. B. Walden: thank you for your confidence in me. To my family, I give rny love and warmest thanks for the encouragement, high hopes, and unconditional faith which kept me looking forward and up to new vistas. TABLE OF CONTENTS

Page . . CERTIFICATE OF EXAMINATION...... il .*. AB STRACT...... iii ACKNO WLEDGMENTS...... vi TABLE OF CONTENTS...... vii LIST OF TABLES ...... xii ... LIST OF FIGURES ...... x~ii LIST OF APPENDICES ...... xv

Chapter 1. ABOUT PERTINENT CONCEPTS IN GENERAL ...... 4 1 .1 Introduction...... 4 1.2 Pattern, process and mechanism: the distinction ...... 6 1.3 The scale paradigm...... 9 1.3.1 Definition ...... 9 1.3.2 Views of the medium...... 11 1.4 Unsolved problems in the pattern, process, mechanism complex ...... 15 1.4.1 Uniqueness ...... 19 1.4.2 Stability ...... 22

In terlude...... 24

vii Chapter 2 . THE MEDIUM ...... 25 2.1 Characteristics of the vegetation system...... 25 3.1.1 Definitions ...... 25 2.1.2 Scale...... 26 2.1.3 Views: Reductionist or Holistic, Static or Dynamic ...... 28 2.2 Pattern analysis in vegetation ...... 32 2.3 Process in the vegetatioa system ...... 38 2.4 Mechanisms in the vegetation system...... 42 2.5 Summary ...... 45

In terlude ...... 46

Chapter 3 . HOW COMPLEX IS COMPLEX? ...... 47 3.1 The vegetation system is complex ...... 47 3.2 Kolmogorov complexi ty ...... 48 3.2.1 Entropy ...... 49 3.2.2 Coding ...... 51 3.2.3 To ta1 complexity ...... 56 3.3 Test cases and results ...... 57 3.4 Complexity and hierarchy ...... 67 3.4.1 Models of partitioning ...... 70 3.4.2 A worked example...... 72 3.4.3 Conclusions from hierarchical ... partitioning ...... 78 3.5 Discussion ...... 79

Interlude ...... 84

... Vlll Chapter 4 . CONTROVERSY OF HISTORIC PROPORTIONS AND CALLS FOR ITS RESOLUTION...... 85 4.1 Foundation. edifice. superstructure ...... 85 4.2 Clements-Gleason ...... 88 4.3 Classics examined anew ...... 89

Interlude ...... 92

Chap ter 5 . MODELLING DETERMINISM ...... 93 5.1 The role of models ...... 93 5.2 Clernen tsian determini sm...... 96 5.3 Abstraction of Nature ...... 98 5.3.1 Reducing dimensionality ...... 98 5.3.2 Analogues ...... 98 5.4 Mathematical background ...... 99 Dynamical systems theory ...... 99 Where? .. the attractor ...... 101 5.4.2.1 Point attractors ...... 102 5.4.2.2 Limit cycles ...... 106 5.4.2.3 Strange attractor ...... 107 How? .-the path ...... 112 Constraints ...... 116 Generalizations...... 117 5 -5 Application ...... 117 5.5.1 Fast attempts ...... 117 5.5.2 1s chance a necessity? ...... 122

In terlude ...... 125 Chapter 6 . CASE STUDIES...... 126 6.1 Short-term dynamics: Atlantic heathland recovery ...... 126 . . 6.1.1 Data description ...... 126 6.1.2 Propositions ...... 128 6.1.2.1 A cornmunity level view ...... 128 6.1.2.2 Determinism in the phase structure ...... 133 6.1.2.3 Determinism cannot be simply decomposed ...... 135 6.1.2.4 Small chance effects can produce dramatic outcornes ...... 138 6.2. Simulated data, detenninis tic chaos ...... 143 6.2.1 Relevant chaos-theoretical concepts ...... 143 6.2.2 Testing for chaoticity ...... 145 6.2.2.1 Lyapunov exponents ...... 146 6.2.2.3 Fractal dimension ...... 147 6.2.3 Results ...... 149 6.3 Long-term dynamics: Postglacial Jackson Pond ...... 151 6.3.1 Data description ...... 151 6.3.2 Analyses ...... 152 6.4 Recapinilation of case studies ...... 161

In terlude ...... 163

Chapter 7 . A UNIFYING THEORY OF VEGETATION DYNAMICS ...... 164 7.1 Cornplex ecological systems are paradoxical ...... 164 7.2 The necessary ingredients of a uniQing theory ...... 166 7.3 What the simulated data show ...... 168 7.4 A Prigoginian interpretation ...... 169 7.5 Unified theory: a first approximation ...... 172

Coda ...... 176 Appendix A . DEFINITIONS ...... 177 Appendix B . ALGORITHMS ...... 181 Appendix C . CODING-THEORETICAL BACKGROUND...... 190 Appendix D . JACKSON POND DATA ...... 197 REFERENCES ...... 215 VITA ...... 240 LIST OF TABLES

Table Description Page

Typical relevé from recovering vegetation on abandoned agricultural land ...... 53 Binary coding of the relevé in Table 3.2.2.1 by Huffman's (1952) method ...... 54 Levels of evenness in a hypothetical cornmunity ...... 60 Coefficient of determination (R~)in simple linear regression of A on structural properties as identified ...... 65 A typical phytosociological relevé of a 10m xlOm plot showing strata levels ...... 74 Partitioning of quantities through directional . . decomposition ...... 76 Partitioning of quantities through analytical nested . . decomposition...... 78 Point cover estimates in Lippe et. al (1985) data ...... 125 Partition of sums of squared deviations from centroid within and between recovery phases ...... 132 Cornparison of observed and simulated distance configurations in Atlantic Heathland recovery by phase and by phases cornbined ...... 136 Fit of various Markov models to observed process ...... 140 Fractal dimension and Lyapunov exponent of reconstructed attractors in processes as shown ...... 151 Fit of stationary Markov chah to postglacial dynamics in Jackson pond, Kentucky ...... 159 LIST OF FIGURES

Figure Description Page

Formal mapping A from model space M to data space P ...... 17 Non-uniqueness of the Mechanism-Pattern relationship .... 20 The relationship between complexity and chaos...... 23 Five kinds of spatial pattern ...... 34 Vertical stratification in an idealized cornmunity ...... 35 The behaviour of total complexity L. Rényi's entropy H of order 12. and A in artificial data sets of 5 and 20 species...... 61 Relationships of A and H expressed as a deviation from its maximum in the artificial data set of 5-species ...... 62 Spatial dynarnics of A in real data ...... 63 Covariation of A and maxH .H in real data ...... 64 Temporal dynamics of L. H and A in real data ...... 66 Behaviour of A under different species response-types .... 67 Vertical stratification in a mixed conifer-deciduous stand in Boreal Ontario ...... 69 Vertical stratification in an idealized cornmunity ...... 70 Analytical nested hierarchy emerging from cluster analysis of association matrix ...... 77 An oscillating pendulum ...... 103 Convergence in the stationary Markov mode1...... 105 Population-level dynamics in Lippe et . al . (1985)...... 130 Two-dimensional Eigenmapping of the 19-step recovery trajectory (A) fiom Atlantic heathland (Lippe et. al. 1985)...... 13 1 Joint scaner of relevé distances based on analysis of the data in Table 6.1.1.1 ...... 135 Decomposition of determinism in sequentiai phases ...... -137 Eigen mappings of various Markov processes ...... 141 Euclidean distance profiles of the Markov process fitted to heathland data ...... 142 Phase space mappings of pure and perturbed Markov process and reconstnicted attractors ...... 150 Three-dimensional Eigenmappings of the 205-step postglacial vegetation trajectory at Jackson Pond, Kentucky (Wilkins et. al 199 1)...... 153 Euclidean distance profile of the observed postglacial vegetation dynamics in Jackson Pond, Kentucky...... 157 Reconstnicted attractor of postglacial vegetation dynamics in Jackson Pond, Kentucky ...... 160 LIST OF APPENDICES

APPENDIX A: Definitions ...... 177

APPENDIX B: Algorithms ...... 181

B . 1 General description ...... 181 B .2 Input and preliminary calculations...... 181 B.3 Chaos algorithms ...... 181 B .3.1 Lyapunov exponents ...... 181 B.3.2. Fractal dimension ...... 184 B.4 Relevant Output...... 189

APPENDIX C: Coding-theoretical background ...... 190 ... C .1 Basic definit~ons...... 190 C.2 Average Length L ...... 191 C.3 Relationship between H and L ...... 192 C.4 Huffman coding ...... 194

APPENDIX D: Jackson Pond Data ...... 197

D . 1 Reference information ...... 197 D.2 Variable list ...... 198 D.3 Raw core data ...... 200 PRELUDE

Here 1 trace the general structure of the thesis and comment upon my intentions to formulate a unifying theory of vegetatioo dynamics. I anchor my approach in the realization that inescapably al1 dynamical processes corne to a head in three general things: complexity, deterrninism and chaos. I see these as the three main ingredients that a unifying theory of vegetation dynamics should interlink. We find these main lines of inquiry braided throughout the chapters. Clearly, 1 must address the general aspects before 1 can discuss the specific. This requires an in-depth look at what others have done. 1 do not restrict myself to the field of ecology, for it is precisely what 1 find beyond that lays a foundation for unification. This cornes both directly through the provision of tools and indirectly through the ability to see links in the perspective of high level dynarnical systems. Inevitably, parts of this thesis will perplex some readers. In anticipation of this 1 apologize, in advance, to the ecologist reader for the technical jargon of mathematics, computer science and statistics that 1 must use by circumstance. To the mathematician, computer scientist, or statistician, 1 apologize for the vague, though standard ecological terminology, which we use. To aid the reader 1 have provided a brief Appendix of definitions. 1 begin in Chapter 1 with a rather philosophical discussion of rny topic. 1 do so because 1 believe, dong with others (e.g., Loehle 1988; Peters 1991), that eculogists have much to gain from the tools and docmnes of philosophy. 1 believe that current ecological thinking, and thus ecological theory, is stagnating within an application of outdated philosophical dogma. For my ultimate goal of offering a new and unifying theory of vegetation dynamics, I will substantiate this claim. In Chapter 2 1 focus upon the vegetation system at large and give an extensive review of the study of pattern, process and mechanisms that govern the vegetation system, covering the period over the past century and more. 1 develop the idea of 'complexity' in Chapter 3. This may appear a slight sideshow, despite its extreme relevance in building an ecological theory. I assert here that we have not embraced the notion of complexity in ecology and go so far as to present a new definition. I then show its application to test cases. I turn to dynamics in Chapter 4 and discuss the Clements-Gleason controversy. In doing so 1 atternpt to shed new light on ths controversy and offer interpretations. Chapter 5 details the logic and methodology for modelling vegetation dynamics as a forma1 dynamical system. This essay is quite technical. In Chapter 6 1 present the two major case studies. Using the first one as both a protocol and an example for short-term recovery, and the second one as an example of long-term dynamics, as seen in the fossil pollen and tissue record of paleontologists, I examine the reasoning behind the decisions taken to resolve the controversies outlined in Chapter 4. Chapter 7 is my attempt, through summarization, reflection and connections to show that this dissertation gives a basis for a unifying theory of vegetation dynamics. 1 have subrnitted parts of this dissertation for publication in various acadernic journals over the past year. My curriculum vitae includes the reference to these. Chapter 1

ABOUT PERTINENT CONCEPTS IN GENERAL

The sigruficant problems we face cannot be solved ai the sarne level of thinking we were at when we created them. Albert Einstein,

1.1 Introduction

Science is concerned with a search for patterns, the discovery of processes giving rise to these patterns, and the uncovering of the mechanisms which ultimately drive such patterns and processes. The goal of this thesis is no exception, however, inspired by the words of Einstein, the approach 1 take is unconventional. To appreciate this, I begin with a discussion of these concepts in general. I start out with basic definitions and then identify the problems with the conventional approach. I suggest ways in which these problerns may be overcome. The distinction between pattern, process and mechanism, albeit always implied, is rarely explicit. Typically, studies focus on duplexes like pattem/process or process/mechanism and rely on inference to link them together. It is often the case, however, that each of the pair is considered independently and thus approached from different directions. This can make establishment of connections between any two difficult. It is even mer to find links between triplets such as pattem, process and mechanism, simultaneously. When such is attempted, it usually includes ambitious assurnptions and far-fetched simplifications. As a mathematicdly-trained ecologist, 1 am fascinated by the notion of relationship involving pattem and process in ecological systems and intrigued by the reality that this fascination is both a product and a catalyst of reasoning. But there is another level of investigation, narnely that involving pattem recognition, which the practitioner may cal1 the-cri tical phase of inquiry. This phase is at the mercy of the analytical tools and, inherently, of deceptive perception. Once a pattern has been noticed, process and/or rnechanism are then inferred. The reliability of this inference is, of course, limited by the arnount of observation available on the system, the method of inference and, in general, scale effects. Stability and uniqueness, properties characteristic of simple systems, make the resolution of pattern, process and mechanism less sensitive to the complications of perception and scale. When the medium itself is complex, it is typically unstable and unpredictable (small inputs lead to large andor divergent outcornes), or non-unique (plurality of causes and effects) in its behaviour. It is likely bat, as a consequence, the relationships between causes and effects (pattern, process and mechanism) are also complex. In essence, the problems identified rnay be ill-posed, causing implicit rather than explicit difficulties. Although establishing the relationships between pattem, process and rnechanism through inference thus entails several practical diffïculties, it may be that the fundamental chzllenge lies in the theoretical complexities. Being conscious of the theoretical complexities is critical, but requires at the outset an a priori consensus on what is meant in the distinction between pattern, process and mechanism.

1 begin with a philosophical discussion of this within the broad domain of my topic. In Chapter 2 1 examine how the concepts apply to vegetation systems which are inherently cornplex.

1.2 Pattern, process and mechanism: the distinction

Pattern is defined by Webster's dictionary as "a discernible coherent system based on the intended interrelationship of component parts." While cornprehensive, this definition is unsatisfactory due to the infini te ways in which 'discernible' and 'coherent' may be interpreted and themselves defined. Furthemore, patterns may exist regardless of the 'intention' of interrelationships. Clearly, if a useful definition of pattern is desired, it cannot be such a general one. In application, pattern must be defined specifically, that is, with a particular purpose in rnind. Pattem defined as such, process can then be defined following Webster as "a series of operations or actions conducting to an end or pattem". This is different from mechanism which is the "fundamental physical or chemical process involved in or responsible for action, reaction, or other phenornenon." While there are obvious connections, it is very important to dis tinguish between the concepts of pattern, process and mechanism. Generally, and most simply put, in a given system, patterns are what we perceive, processes describe how these patterns corne about, and the mechanisms provide explanations or causes for why these patterns occur (Wiegert 1988). When considered in this manner, the concepts of pattern, process, and mechanism appear to be universal in Nature. In fact, van Leeuwen (1966) clairns that these are what constitute life itself. But an investigation of these concepts, fiee of metaphysical entanglement, requires the specification of an application. The concepts of pattern, process and mechanism are naturally alluded to in any scientific investigation. A classic illustration of this framework was Darwin's (1859) recognition of a pattern in the distribution of species which led to the conception of the process of evolution and the mechanism of natural selection. In a syntactic context, Dale (1980) and Orloci (1988) suggest that pattern is the "primitive", while function or process is the "gramrnar", and structure is the "message" or information content. Sornetimes the confusion between terms is merely semantic in ongin. For example, process is often called pattern in time. The terms 'process' and 'mechanism' are often used interchangeably. McCook (1 994) States that "the goals of ... ecology are to describe the similarities and differences in both the pattem and the process: to extract generalizations, and to identify the bases for differences in process." Clearly, while only two of the terms are used, narnely pnfiem and process, the third, mechnnism, is implied and distinguished as the "bases". Of course, it would be ideal to be able to study al1 of pattern, process and mechanism in a system, however this is not practical. Instead, usually the simplest and most apparent of these, namely pattern is studied and then an attempt is made to infer one or both of the other two. A set of non- experimental observations is made, and pattems (or the lack of them) are discovered. From these patterns, theories and generalizations are derived to explain processes. tlCustomarily, work begins with the contemplation and collection of data from some part of the biosphere. The ecologist then mes to argue back, inductively, from obsewed effects to hidden processes" (Scheiner and Gureritch 1993). McAuliffe (1988) makes the point that "the first descriptions of a heliocentric solar system, atomic structure, continental drift, and plant succession in forests were mode1 representations of the dynamics of various systems, al1 initially formulated without knowledge of the mechanisms responsible. In each of these cases, the accurate descriptions of the movements of objects (planets, subatomic particles. continents, or tree species) through time ... were necessary preliminaries to understanding the natural forces that underlie such movements" . Inductive inferences are by nature probabilistic. Pielou (1 977) realizes this when she States that "the mathematics used is usually statistical". What is actually desired, in the end, is the proximate or ultimate causes or mechanisms. Thus, of the three, it is pattern recognition which can be regarded as a precursor or definitive step in the pursuit of understanding. May (1986) suggests that the search to understand any complex system is primarily the search for pattern. Southwood (1980) sees the function of pattern recognition as a fundamental prerequisite to the development of theory. In particular, pattern recognition often leads to the formulation of specific, testable hypotheses, which attempt to explain the observed pattems. 1 end this section with an except frorn Frank Herbert's famous novel 'Heretics of Dunett. He perfectly illustrates the danger of prernature induction between pattern, process and mechanism.

There was a man who sat each day looking out through a narrow vertical opening where a single board had been removed from a tall wooden knce. Each day a wild ass of the desen passed outside the fence and across the narrow opening-first the nose. then the head. the forelegs. the long brown back. the hindlegs, and lady the tail. One day. the man leaped to his feet with the light of discovery in his eyes and he shouted for dl who could hear him: "It is obvious! The nose causes the tail!"

Stones of the Hidden Wisdom, from the oral history of Rakis Frank Herbert 1984. Heretics of Dune.

1-3.1 Definition

Pattern, process, and mechanisms cm only be measured within specified lirnits in space and time, which may or may not be conscious to the observer. These lirnits are commonly referred to as scale. Gleick (1987) describes the concept of scale and perception well:

How big it is? How long does it last? These are the most basic questions a scientist cmask about a thinp. They are so basic to the way people conceptualize the world that is not easy to see that they imply a certain bias. They suggest that size and duration. qualities that depend on scale. are qualities with meaning, qualities that cm help descnbe an object or classify it. Scale becomes important both in the selection of points or units in pattern identification (or "grain" Wiens (1989)), and in the overall space or time span allowed in pattern or process recognition (or "extent" Wiens (1989)). Nietsche teaches us that "There are no facts, only interpretations." But interpretations are always abstractions. We well know that the process of abstraction need not be unique, mainly because description of the natural system is not objective but scale-dependent. Scale permeates al1 aspects and brings into focus the notion of resolution: how you measure your position in time and space. The spatial and temporal components of scale may not be mutually exclusive, and both are dynamic properties. In addition, scde is defined by perception, but at the sarne time it may influence perception. This is made clear when trying to understand Nature from research results of ecological studies based on a single technique or a single scale. But dynamics cm only be defined within specified lirnits. Allen and Starr (1982) define scale as "the period of time or space over which signals are integrated or smoothed to give the message." They break scale of the 'message1down into three components: the scale of transmission, the scde of acquisition, and the scale assumed in observation (perception). The 'message' in our case is 'vegetation', specifically, the compositional and performance patterns (Orloci 1988) therein. Pattern recognition in dynamics may only be possible in the first place at a particular scale. That is, given more or less area or time, patterns may appear or be missed. When the lower or upper limits of the quantities of area and tirne are undefined, it becomes extremely dificult to decide on the appropriate scale. It also becomes difficult to interpret the relative importance of large scale patterns and their underlying component patterns. Thus, generalizations of the view are difficult, unless pattern is defined as a scale independent (e.g., time-static) process. This topic is well discussed by Podani (1984) . Deciding on the sufficiency or approptiateness of scale is important in delineating pattern, process and mechanism and thus is a pre-requisite to making the connections.

1.3.2 Views of the mediztm

The specification of scale as discussed above helps to define a view. A view depends not only on how you choose to look (scale) but inherently on what you are looking at (the medium) and where you are (historical context, consciousness, inherent bias). Generalizations of the effect of view are difficult. If the medium is complex (see subsequent chapters) generalization of the view becomes even more difficult. The easiest, and most common route to obtaining generalizations in this case bases itself on the justification of feasibility. But in reality, I believe there are three broad constraints: visualization, feasibility, and imagination, the first being the most stringent. If the number of state variables in the system is greater than three, which is the rule rather than the exception in ecology, then visualization is not possible. If however, the information contained in al1 state variables rnay be parsimoniously surnrnarized by three indirect state variables then visualization may still be realistic, but not exactly precise. If visualization is not essential, then one is only resmcted by feasibility which includes such things as computational constraints. 1 include imagination as a constraint because, while to me obvious, it is often overlooked. I will address some of the usual approaches to reducing complexity of the natural system for the purposes of abstraction. This results in working with an over-simplified view of the medium. 1 put several kinds of approaches into this category. The first has to do with focusing on some part of the medium rather than the whole. In the extreme, this approach is called "reductionistic". This is often the case in modelling where insufficient knowledge of the medium is typical. A critical choice is of the coordinate system into which we "map" the problem, as it were. Although solution of the problem should not depend upon the choice of coordinate system, an appropriate choice often simplifies the problem, perhaps to the point of delineating some general principles. Choice includes selection of variables to be considered, and in delineation of scale. Studies range fkom examining the single variates over space or tirne, to 'multiscale', 'multivariate' studies, which attempt to include the dynamics of many variables, over several spatio-temporal scales. Of course, the more complex the medium, the more variates and scales will be needed to comprehend pattern, process and mechanism. Selecting a subset of variables or sample space obviously carries the problem of loss of information which may or may not easily be inferred back from the subset. But how do we choose variables to describe the system when faced with seemingly infinite candidates? And how do we handle the problems of many variables without "over-inflating" the problem? Then, a consideration of variates independently may reveal nothing about how they behave within the phase space of the whole system (i.e., in the presence of al1 other variates). The importance of these questions is what defines the field of "systems science" which, as Wiegert(1988) points out, considers the interaction between die structure and function of a system. The view is "holistic" maintaining that a decomposition of a system into parts is impossible because the whole is greater than the sum of its parts. Lf interaction effects are important, then patterns, processes and mechanisms at the higher levels are not predictable from those at lower levels. What could be wrong with this view? hcluding many variables or rnanyflong scales in the description cm not only be impracticable, but redundant to the problem at hand. Often we are not interested in everything about the system; we cannot ever know everything about the system. Focus tums to the so-called 'emergent properties', which corne from a top-down view. These include properties such as diversity, stability, cornplexity, and others which are connected to dynamics. These properties are not only a function of the components of the system, but also of the interactions between the components. They essentially sumrnarize al1 these things, but we do not end up with the functional relationship between the global properties, and the local, component effects. If emergent properties are of interest then the reductionistic approach is not applicable. Odum (1983) puts it nicely: "In choosing the boundaries for an analysis of a system of interest, one may focus on the unit of interest by making al1 its influences outside forcing functions. However, ths procedure is reductionistic and concentrates on details of a single unit and limits one's ability to understand the interplay of pathways to forcing functions. We sometimes Say that one must mode1 and simulate a system that is one size larger than the one of interest". Focusing on preconceived ideas about trends to be expected may also distort the view. This problem is best represented by the historie fixation on the application of simple models to describe Nature. The reason for this probably lies in the fact that these models are simple to understand, and the idealistic view that Nature can be tarned (Berry 1988; Huxley 1902; Freud 1961). The latter can be traced back to Laplacian-type determinism, and the natural philosophy that there is an underlying simple order in Nature (Lewontin 1966). Ian Stewart (1989) makes a good argument for the limitations of relying on preconceived notions:

"And if you think the name of the game in dynamics is finding formulas for the solution of differential equations, your mathematics will only be able to study regular behaviour. You will actively seek out problems to which your methods apply, and ignore the rat. Not even sweep them under the carpet: to do that, you must at least acknowledge their existence. You're living in a fool's paradise ..."

In reality. we know that, while they indeed may display a high degree of order, natural systems do not slavishly follow Our simple models, vis-a-vis the factor influences or the interactions (Orloci 1978). While there is only one kind of linearity (straight line, plane, hyperplane), nonlinearity rnay exist in many foms. The question, however, is how much of it exists, and then how much cmwe afTord to ignore as simply "noise". We are dealing with a typical paradox of modelling. How much reality are we willing to give up for the sake of a workable model? Over- simpliQing assumptions on variation and heterogenity permeate the medium and may escape even the most reaiistic models. The real question relates to understanding how much exists, and then how much can we afford to ignore in order to make some sense of pattem, process and mechanism. 1 believe we cannot focus on preconceived linear trends and ignore al1 deviations from them as noise. In throwing away nonlinewity we are throwing away a definitive, and probably most defining aspect of the system, the 'baby with the bath water'. 1 advocate new reference systems for Nature which are more redistic, perhaps ones which focus not only upon trends of the linear nature, but those of the nonlinear nature as well, and of course on random variation.

1.4 UnsoZved problems in the pattern, process, rnechanism cornplex

Making deductive statements about of pattem, process and mechanisrn implies that the relationships between hem are simple and predictable. But as Cale et al. (1989) make it clear often processes are not deducible from pattern and rvely are mechanisms deducible from pattern (Kershaw 1963) or process. In this section, 1 describe a few main unresolved problems and the challenges which must be faced. Complications becorne apparent when the direction of the relationships between pattern, process and mechanism is considered. There are two directions of interest. On one hand, knowledge of relationships rnay allow the prediction of observed data (usually pattem or process) fiom a model (usually process or mechanism). This is the direct problem. Conversely, a model may be inferred from obsewed data. This is the inverse problem. Since one cm only record limited amounts of data, and since one can only fit a lirnited number of parameters into a model. it suffices, in practice. to consider model space and data space to be finite dimensional Cartesian spaces (Scales and Gersztenkom 1988). Consider the following simplistic formalization of the mechanism-pattern problem (in which the relationship between mechanism and pattern is chosen as an example):

where rn is a given mechanism in M (Mechanism space), p is a member of P (Pattern space) and A is some (linear or non-linear) operator representing the mapping of relations from model space (M) to data space (P). Perhaps the mapping could represent a process (see Fig. 1.1). Then, given a mechanism (m),the pattern @) could be predicted. For example, if we know how the mechanism of competition (m)works (A) then the composition or distribution of vegetation (p)will always be determined through (1.1). The calculation of Am with given m is the direct problem. There is no question that in practical applications a critical step is modelling the process, resulting in the symbolic description in A. Now, suppose we were interested in just the opposite. Given a pattern, we would want to infer the mechanism. This would require the definition of the inverse of A, (A-l), which could be represented by: This is the inverse problem (the determination of rn givenp and the relation in ( 1.1)). B y definition two problems are inverses of one another if the formulation of each involves al1 or part of the solution of the other (Keller 1976; Cannon and Hornung 1986). Inverse problerns concem the determination of properties of some inaccessible regions frorn observations on the boundary or outside the boundary of that region (Anger 1993). Clearly this is applicable to the problem of pattern and mechanism in ecology. Knowledge about mechanisrns is not always accessible. Al1 we have to work with are patterns. These patterns are then used to infer mechanism.

Mec hanisms (M) Patterns (P)

Fig. 1.1. Forma1 mapping A from mode1 space M to data space P.

The establishment of the nature of A-' is critical because it will determine the nature of the inverse problem. In particular, depending upon the nature of A-', the inverse problem may be properly-posed, or ill- posed. Moritz (1993) points out the conditions (originally introduced by Hadamard (1902)) which must be satisfied for a rnathematically properly- posed problem to correspond to reality: 1. The solution must exist (existence). 2. The solution must be uniquely determined by the data (uniqueness). 3. The solution must depend continuously on the data (stability). In the case of Our application, the first requirement implies that for every rn in M, there exists a solutionp in P. The second requirement implies that A be a one-to-one mapping, in which case A-1 is also one-to-one. The third requirement implies stability of the problem on the spaces (PTM). If one or more of these conditions is violated, then the problem is ill- posed. Most direct problems are properly posed. however it turns out that most inverse problems are ill-posed (Moritz 1993). To concern ourselves with the violation of the existence requirement at first seems rather absurd. It seems intuitive that every cause will have an effect and thus a solution will always exist. But it is entirely possible that a mechanisrn has no apparent effect (more familiarly, a rejection of the nul1 hypothesis). In this case rn would not be considered a member of M. This would then lead to the reformulating of the problem. It is much more interesting to examine violations of uniqueness and stability. It turns out that these violations lead to some very interesting questions about the relationship berneen pattern, process and mechanism. While many daim that such ill-posed problems are difficult or nearly impossible to solve (e.g. Anger 1985) others (e.g. Tikhonov and Arsenin 1977) are optirnistic. 1.4.1 Uniqueness

If uniqueness is the case, that is, for every perceived pattern it is known a priori that there exists a unique mechanism proceeding to its formation, then inference is valid. Given a mechanism. pattern could be predicted with certainty. Given a pattern, the mechanism could be inferred. However, this power relies on the presupposition that, not only is it known that there exists a unique relationship between mechanism and pattern, but also that the relationship is known. It may very well be that the problem of finding these relationships is not simple. But is anticipation of unique relationships realistic? 1s it not reasonable to believe that certain patterns could be achieved through several different mechanisms? Similarly, could one mechanism not lead to several different patterns? These relationships are sketched in Figure 1.2. Copi's (1978) logic may be useful in evaluating these situations, especially in the case of different mechanisms leading to the sarne pattem, what he terrns 'plurality of causes' (Fig. 1.2 a). He makes use of the idea that evey difference must make a difference. That is to Say, if the effects of two mechanisms are not different, then the rnechanisms themselves must or should not be different. In other words, differences in effects are really the basis upon which mechanisms are distinguished. But to accept that every difference makes a difference may not always be prudent in practice. Mec hmism Palt ern

Fig . 1.2. Non-uniqueness of the Mechanism-Pattern relationship. (a) plurality of causes (b) plurality of effects.

Copi (1978) argues against the docmne of 'plurality of causes'. He suggests that this predicarnent is often the result of the insufficient resolution of the effect. Hidden in this is the idea that changing scale changes Our ability to perceive patterns and understand their causes. Thus, as an effect is more and more precisely specified, the number of possible causes is reduced. Eventually, the effect is defined such that only one possible cause remains. Hence the doctrine of uniqueness of cause becomes the product of inductive reasoning. These arguments, while rather convincing, do not solve the problem at hand. Namely, it may be impossible to reduce a given pattern to the desired level. The limits of scale are a reality. Furthemore, how should one assess relative success? Are relationships discovered at those levels even desired? Does this mean that generalization becomes impossible? Surely this is not in line with general experience. Finally, it is still conceivable that two or more very different mechanisms could result in exactZy the same pattern, which tempts one to believe that every difference does not need to make a difference. The observation of evolutionary or ecological convergence actuates such a suspicion. Gilpin and Case (1976) and May (1977) have shown, for example that a number of alternative stable equilibrium cornmunities (pattern) may result from definably equivalent processes. Plurality of causes clearly leads to questions about the ability to infer mechanisms from a given pattern. What about predicting pattern from a given mechanism? In particular, how is this possible if a given mechanism can result in several different patterns (Fig. 1.2 b)? Of course if one is based on the "every difference ..." doctrine, then this is not an issue. However, plurality of effects may exist, especially if the impact of stochastic inputs is considered. It is also possible that while effects may be apparent- different, they may dl in fact be part of the same process or causal link. This is typically the case when effects are perceived at different (temporal or spatial) scales. It remains a challenge to incorporate scale effects into the problem of pattern, process and mechanism.

If the third condition of properly-posed problems is not met, then the problem is unstahle. This implies that small fluctuations in the initial specification of the initial conditions (data) may have a large effect on the outcome (model). In essence, for a problem to be stable, the solution must depend continuously on the data. Deterministic processes, as considered in classical mechanics, depend uniquely and continuously on the initial data, and, in fact, this is the essence of causality. Modem theory of nonlinear dynamical systems, however, has revealed that even classical systems can show the property of instability. This has also been referred to as 'sensitive dependence on initial conditions' or The Butterfiy Effect" (Lorenz 1963) in the terms of the newly developed chaos theory. Sensitive dependence on initial conditions has some interesting consequences with respect to the classical views of pattern, process and mechanism. For example, it seems intuitive that complex patterns are the result of equally complex processes or mechanisms. "This is true in rnany cases, but at the same tirne the long starxiing paradigm is far from being tnie in general. Rather, it seems, and this is one of the major surprising impacts of fractal geornetry and chaos theory, that a complex pattern is generated by a very simple process. In other words, the simplicity of a process should not mislead us into concluding that it will be easy to understand its consequences" (Peitgen et al. 1992). Chaos theory yields interesting relationships where the nature of the effects cannot be predicted by knowing the nature of the causes. But it must be remembered that chaos is only one subset of complexity. The reiationship between complexity (see Lewin 1992) and chaos theory is presented in Fig. 1.3. Clearly, in attempting to predict or infer in a system of patterns, processes and mechanisms, one would be lirnited by conventional logic. simple cauaes cornplex cause#

simple effects complex eff ects

Fig. 1.3. The relationship between complexity and chaos. Chaos manifests itself through the unconventional paths a. and b.

Some would argue that the implications of chaos theory render complex systerns unpredictable and thus it is futile to attempt to study them (Mullin 1993). But while chaos theory implies unpredictability, chaotic systems are deterministic (Kellert 1993). Chaos theory could still help to provide explanations for apparent chaotic patterns, that is, suggest some order in chaos. This idea will be explored in later chapters.

Having set the stage with critical philosophical considerations, I now begin to define the characters. This begins with a definition of the medium. In Chapter 2 1 examine the concepts of pattern, process and mechanism as they apply to the vegetation system. the skeptic

I saui Uiolet's extent. She must haue been blinking.

I srnelled the Oxygens spent. She must haue been breathing.

I felt the Leaf's torque. She must haue been spinning.

I heard the Millipede's fork. She must haue been screaming.

I tasted hue. I tasted Mue. I tasted few. 1 tasted two. I tasted skew. I tasted true. I tasted you. Chapter 2

THE MEDIUM

2.1 Characteristics of the vegetation system.

2.1.1 Definitions

The study of pattern, process and mechanism in vegetation has more than conceptual limitations. How are the three concepts defined and what arc the arising limitations when the medium is vegetation? To answer this I begin with a discussion of how the vegetation is viewed by others. Vegetation is a system in theoretical terms. It is composed of many different parts. When we attach quality and quantity to these parts, we are influenced by how exactly we wish to define vegetation, and also by how we view it. Let us consider the quality: If we define it as the "plant cover", using the primary dictionary definition, then its parts may be units of area deriving from a classification of space. If we define it as "plant Me" the secondary dictionary definition, then its parts are logically different species as identified in some systematic classification of life. In either case, subjectivity is involved. No one would argue against the idea that collectively individu& as populations and communities constitute vegetation. But these are fuzzy entities, a point Orloci (1993) has made with clarity. It follows then that the vegetation system in itself is a fuzzy and ill-defined medium. In its multiplicity and fuzziness, the vegetation system displays universal characteristics. For one thing, it varies non-randomly over space revealing heterogeneity, diversity, and structure. The vegetation system is also dynamic. It varies with time. This variability is highlighted by interactions. chance and stabilityhnstability. This is yet another dimension in the study of the vegetation system, narnely, process. But perceiving process is particularly difficult owing to the usually large time scales and that the system is not easily manipulated experimentally. Also, there are many underlying mechanisrns at work which produce the patterns and processes perceived. Simply stated, the vegetation system is complex (Greig-Smith 1986; Orloci 1988). I fully realize that a discussion of complexity is not a trivial matter and explore the topic further in Chapter 3. Generally speaking, the ecologist's way of reducing the complexity to facilitate its study is to limit the scale of study. This can be done in many ways. But the first consideration is the definition of the vegetation unit and the scale at which pattem is sought.

Whittaker ( 1970) defines some cornrnon scales in pattem analysis of vegetation. He does this by posing three questions which, in essence, he sees as three scales. He thus makes the point clear that the question implies the scale and vice-versa. He asks, firstly, how are species populations distributed in relation to one another and communities dong an environmental gradient? Secondly, he asks, how are kinds of communities in an area related to the patterns of more than one environmental gradient (ecocline)? Thirdly, how should world-wide relations of communities to climatic gradients be interpreted (biome)? Palmer (1988) suggests that vegetation displays detail at al1 spatial scales of interest and is thus a "fractal" (Mandelbrot 1982). If so, considerations of space are not limited by the Euclidean spatial parameters such as distance (length, depth, thickness, etc.). He gives the example that, on a small scale, pattems are defined by the spacings and interactions of individual plants. On a larger scale, vegetation consists of patches, and on a yet larger scale, vegetation patterns are influenced by geomorphological features of the environment. He concludes that the fractal dimension (see Barnsley 1988) of vegetation varies as a function of scale, and thus al1 scales are 'important' and 'natural'. The universal and also critical importance of scale of observation has been recognized by many (e.;. Kershaw 1960; Levin 1988; Allen and Hoekstra 1990). Kershaw (1963) clairns that scale is important because it aids in the understanding of causal mechanisms. For exarnple, he suggests that if some pattem is observed at a scale of 10 cm, an explanation can be sought which is based on the morphology of the plant. Conversely, if the scale of pattem is of the order of several meters, the causal factors are more likely to be related to topography, drainage or soi1 composition. Delcourt et. al. (1983) employ a hierarchical mode1 to suggest methods of spatial and temporal scale resolution which is promising. Greig-Smith ( 1979) suggests that obvious large-scale pattern is well-documented and can be generally attributed to environmental or biotic factors, however less obvious large and intermediate scdes of pattern must be correlated with less obvious environmental differences. As the scales become smaller, it becomes more difficult to identify environmental correlations.

2.1.3 Vieivs: Reductionist or Holistic. Stark or Dynarnic

While scale can be critical in pattern resolution, it mut be recognized that the view at which patterns are sought is equally important. The reader is asked to recall the discussion in section 1.3.2. Many levels have been suggested (e.g. Odum 1977; Levins and Lewontin 1980). Wiegleb's (1989) six levels are typical: individual, population, species, comrnunity, vegetation unit and vegetation cornplex. These levels cncompass those most commonly used in population and community ecology. Wiegleb ( 1989) further suggests that some levels are 'real' (individuals, species, vegetation), and others are 'abstract' (community type, ecosystem type, flora). One of the controversies is pivoted on the question : what constitutes a commlmi~?Anderson and Kikkawa ( 1986) give a good review of the development of the concept. They trace the origins back to the primitive classification systems of Arktotle and Theophrastus (372-288 BC) and discuss the contributions in work on vegetation formations in relation to broad climatic belts (von Humbolt 1850; Warming 1909; Raunkiaer 1934; Kemer 1863). They also point out the classical phytosociological work in this regard led by Braun-Blanquet (1965). Mueller-Dombois and Eilenberg (1974) suggest that the community concept arises through the need to classify the vegetation. To them the community is a "subdivision" of vegetation cover such that "wherever the cover shows more or less obvious spatial changes, one may distinguish a different cornmunity". The concept of the cornrnunity was based purely on spatial classification and may not be interpreted in the reductionistic/holistic debate which relates to functional aspects. Community and individual or population level views have been heavily debated. McIntosh (1980) reviews this debate. He States that the distinction between the population-centered, individualistic position, credited to Gleason (19 17,1926) and the community level view credited to Clements ( 19 16) is commonly seen as a dichotomy between reductionistic and holistic approaches. Gleason ( 19 17) believed that a reductionistic view was the only realistic view since "the development and maintenance of vepetation is ...merely the resultant of the development and maintenance of the component individuals". He further stated that "according to this view, the phenornena of vegetation depends completely upon the phenomena of the individual. It is in sharp contrast with the view of Clements that the unit of vegetation is an organism, which exhibits a series of functions distinct from those of the individual...". Clements' views cm be considered 'holistic' because he considered vegetation to be an organism with properties and behaviours unique from those of individual populations. Clements ( 19 16) stated his argument as such: "a complete understanding of succession is possible only from the consideration of various viewpoints. Its most striking feature lies in the movement of populations, the waves of invasion which rise and fa11 through the habitat from initiation to climax. On the physical side, the fundamental view is that which deals with the forces which initiate succession and the reactions which rnaintain it. This leads to the consideration of the responsive processes or functions which characterize the development, and the resulting structures, communities, zones, alternes, and layers. Finally, al1 these viewpoints are summed up in an idea which regards succession as the growth or development and the reproduction of a compiex organism." Cain ( 1939) quotes Alechin ( 1925) in defining a community as "a complex of plants with mutual adaptations." In this definition the fundamental charxtenstics of a plant comrnunity are : layering in space, layering in time, variability, stability. He also finds in Alechin (1926) the staternent that "under community must be placed only such a plant complex as consists of elements of unlike value lawfully bound together, which build stable combinations (in variable equilibrium) and which can corne to exist only in the course of a long time." Whittaker (1975) defines the community as "an assemblage of species or populations that live in a defined environment at a defined spatial-temporal scale, and interact with one another forming together a distinctive living system with its own composition, structure, environmental relations, development and function". It is now well-established that communities represent "an ensemble of individuals representing numerous species which coexist and interact in an area or habitat. Accordingly, communities represent a prominent level of organization where individual survivorship, species persistence and community properties blend to forrn a dynamic assemblage" (Drake 1990). According to Tallis ( 199 1) plant cornmunities are particular assemblages of species, which in the history of global plant cover have corne together opportunistically and have then been re-ordered as climate conditions continued to change. Clearly, cornmunities maintain themselves through interactions and thus cannot be considered static. It seems clear that fundamental to the question of 'what is a community?' are the questions: 1. how are they maintained? 2. how do they corne into existence? The answers to these obviously require a dynamic view of the community. Once a spatial scale, both unit and view has been defined the temporal scale must be considered. This also has units and extent and can very much influence perception. For example, Orloci (1981), Hiusman et al. (1993) and Van Andel et al. (1993) suggest that it is obvious that there is a time constraint on interpretation since changes which are actually cyclic in the long run may appear linear in the short mn. Furthemore, 1 believe that there is more than enough evidence to conclude that the community is more than a simple sum of the parts, and that while a reductionistic view is sometimes the only choice, the utility of a holistic view on things should not be underrated. It has been suggested that as the units increase in scope (as the view broadens from reductionistic to holistic), information is lost through the accumulation of 'aggregation error' (Cale and Odell 1980; Gardner et al. 1982; Cale et al. 1983). This is based on the idea that ecological studies often deal with aggregate variables which attempt to summarize the seemingly infinite components of ecosystems. In general, it has been found that aggregate behaviour is different from the surn of the behaviours of the components (Cale and Odell 1980), which suggests that the reductionistic view may not be sufficient. Stone and Ezrati (1996) suggest that focusing on fluctuations at the level of populations reveals "only a fraction of the whole ecological picture".

2.2 Pattern anabis in vegetation

Vegetation science is defined by Austin and Smith (1989) as the study of those processes which determine the patterns of composition and emergent propertirs observed in vegetation. When we speak of pattern in vegetation dynamics, we may be referring to either pattern in space, or pattern in time. I distinguish these by calling the latter, "process". Hutchinson ( 1953) defined spatial pattern as structure which includes both order and arrangement. Whittaker (1970) suggested that community structure and composition involved vertical structure, horizontal pattern, tirne relations and niche differentiation. Spatial patterns are detected on the basis of variation in the texture, diversity, abundance or, most commonly, the distribution of species or plant cornmunities. Greig-Smith ( 196 1 ) defined pattern as departure from randomness of arrangement in relation to a defined area. He called this 'spatial heterogeneity'. Sirnilarly, ver Hoef et al. (1993) define pattem as the non-random horizontal spatial abundance of organisms. This approach of definition by exclusion, however, does not provide any insight into the meaning of pattern itself. More specifically, pattem is "the zero-dimensional characteristic of a set of points which describes the location of these points in terms of the relative distances of one point to another" (Upton and Fingleton 1985), where points are individual plants, patches. or comrnunities. To attempt to give meaning to many of the definitions of pattem, it appears that the concept of randomness must be first defined. Although the preceding definitions suggest that randomness is some type of anti-pattern, Pielou (1977) allows for a 'random pattern'. Zar (1974) defines a random pattern (or distribution of objects in space) to be one in which each portion of the space has the same probability of containing an object and the occurrence of an object in any portion of the space in no way influences the occurrence of any other object in any portion of space (independence). The idea of random numbers, random samples, random patterns, etc. is fuzzy because, in theory, the notion of randornness does not apply to finite things, but is rather a property of an infinite sequence, or a potentially infinite process (A. Sziliird, pers. comm). Ecologists, however, have been forced to put the idea of randomness into practice and thus the precision of the usage of the term has suffered. Pielou (1977) defines random pattern as one which results from individuals having been assigned independently and at random to the available units. The Poisson distribution is used to describe randomness, in the sense that it would be expected to find that the number of individuals per unit is a Poisson variate (Upton and Fingleton 1985). Unit size is of course a problem in that an arrangement may appear random at one size and something else at another. Others (R. H. Green, pers. comm) define randornness as a 'lack of pattern'. Ver Hoef et al. (1993) point out that very few organisms exhibit complete spatial randomness at al1 scales, and thus attention must naturally be focused toward the description of non-random patterns. Though random changes undeniably occur in ecosystems, they are of much less interest than those changes which are not random (Goodall 1977). Departures from randomness can be analyzed using several classical distributions (eg generalized Poisson, cornpound Poisson), but they rely upon the assumption of independence of events, which in Nature is highly unrealistic. Some examples of spatial patterns are presented in Fig. 2.1.

Fig. 2.1. Five kinds of spatial pattern: a. randorn or Poisson. b. clumped. c. regularly clumped. d. clumps of regular points. e. regular. (rnodified from Whittaker (1970) and Upton and Fingleton ( 1985). Clearly pattern may not be simply spatial heterogeneity in two dimensions as pictured above, but typicaliy is also manifested in three or more dimensions. The latter is still easily visualized by superimposing a vertical stratification of vegetation idealized in Fig. 2.2. Higher dimensional pattern rnay also be defined Orloci (1988), but of course in analytical spaces, and is not easily visualized.

Fig. 2.2. Vertical stratification in an idealized community (modelled after Dansereau 1957).

Some types of pattern are quite visible, while others rnay require unmasking through more elaborate observation. "If a large area, including widely differing habitats, is examined, pattern is evident on casual inspection, and, if the range of habitats and corresponding vegetation are great enough, different plant communities will be recognized in different parts of the area" (Greig-Smith 1961). In such a case, pattern detection is simple. However, even if vegetation is visually without a pattem, one may still exist. Detection of pattem in this case rnay however require methods of analysis more complicated than simple visual inspection. Probing analytical (geometric or theoretical) spaces for patterns may be useful (Whittaker 1970; Orloci 1988). Orloci (1988) quotes Benzecri (1969) who daims that such an approach is necessary, as he sees pattern as a property which should emerge "from a sea of data, not through a priori nominalistic postulates or axioms or by unduly fragmentary measurements of isolated facts ...but through the sirnultaneous synthesis ... of putting together a number of elementary facts ... ". Hill ( 1973) puts it another way in stating that the function of pattern analysis is to summarize data in a way such that those features which are not apparent are appreciated . Pattern recognition in itself has become an extensive field of research (e.g.,Legendre and Fortin 1989; Bouxin 199 1; Kissov et al. 1990; Kuncheva 1993). The detection and quantification of pattern in vegetation using classical anal ysis of variance approaches have been studied by many (e.g., Greig-Smith 1952, 196 1, 1970; Kershaw 1973; Hill 1973; Usher 1975). Greig-Smith ( 1952) invented an analytical technique which essentially compared observed variances in data grouped into larger and larger size units, and facilitated pattem detection at various scdes. His method has been widely used and has sparked new and (arguably) improved methods of pattem detection (Kershaw 1957; Williams et. al. 1969; Usher 1975; Carpenter and Chaney 1983; Ford and Renshaw 1984; Dale and MacIsaac 1989; Dale and Blundon 1990). As multivariate methods became more accessible, they became an essential tool for pattem detection in vegetation ecology (Willams and Lambert 1959, Orloci 1978, 1988; Digby and Kempton 1987). Detection of pattern in vegetation ecology is often done through the multivariate methods of ordination (e.g Ludwig and Reynolds 1988), classification (e.g., Gauch and Whittaker 198 1) and multiple regression (e.p. Scheiner and Gureritch 1993). Often this is done to reduce complexity, by eliminating unnecessary 'noise', without losing any critical information. Olsvig- Whittaker ( 1988) sees the primary conceptual problem as that of reducing high-dimensional pattern to a more recognizable (i.e. 2 or 3) dimension. A criticism of complex analytical techniques is that the perceived patterns are far-removed from immediate applicability . It may become difficult to relate both species and community pattern to the pattern of environmental variation in order to study plant-environment relationships. However work like that of Green (1993) clearly demonstrates that analytical techniques can be important in application. With so many different tools and methods of pattern detection, it is reasonable to ask, how is perception of pattern affected by the method of analysis? Usher (1975) found that no single analysis yielded complete information on pattern. That is, it appears that each method provides some unique insight into pattem detection. He suggests that several different analyses should be perforrned and some consensus of results accepted. Podani ( 1989) provides a cornparison of ordination and classification methods and attempts to provide a decision framework to help determine which methods are most appropriate for the types of patterns requiring detection. Greig-Smith's methods have been expanded upon using geostatistics (Robertson 1987; Halvorson 1994) and fractal geometry (Palmer 1988) to consider pattem analysis at several spatial scales. 2.3 Process in the vegetation systern

Temporal patterns (or processes) are revealed when areas are observed repeatedly over tirne. A relevant question is: How do spatial patterns in the vegetation system emerge? Dificulty may be expected in trying to distinguish between pattern and process: "Nature forms patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space" (Gleick 1987). Watt (1947) insists that the static descriptions of plant communities (pattern) provide information of some, but not criticai value to an understanding of them. It is the dynamic behaviour (processes) of how the individuals and the species are put together, and what determines their relative proportions and their spatial and temporal relations to each other which requires much more study. Theory in vegetation ecology, has clearly evolved through the recognition of patterns and processes and the subsequent suggestion of mechanisms governing them. The leap from the static view to the dynamic view of vegetation was indeed revolutionary (Pickett and McDonnell 1989). One way of distinguishing (spatial) pattern from process is to once again consider randomness. What some recognize as a random pattern can be generated by the Poisson process. The Poisson process is a function given by two parameters (population mean, population variance), which, when iterated (in theory, infinitely), generate the random pattern. In these terms, the pattem is called the descriptor and the process is the function. Temporal patterns have been given the general term 'successions' in ecology. So, how do we measure or identify processes? In other words, how do we study dynamics? Attention becomes focused on the dynarnic concepts of interactions, transience, stability and chance. 1 discuss how dynamics has been studied in the past and elaborate on a new approach to the study of dynarnics of the vegetation system in later chapters. Although the study of vegetation dynamics began earlier (e.g. Kerner Von Madaun 1863; Warming 1895), one of the first to formalize the dynarnical nature of vegetation was Cowles (1899). He stressed the existence of a constant dynarnic interaction between plant formations and the underlying geological formations. This view considered vegetation to be continuously-chmging. Through the recognition of spatial patterns related to geological and other environmental factors, he realized that it was important to "endeavor to discover the laws which govem the panoramic changes" (mechanisms) and that "ecology, therefore, is a study of dynamics" (processes). E. C. Clements viewed vegetation as an organisrn which "arises,

Eorows, matures, and dies." The mature state, the climatic "climax" formation is, according to him, "able to reproduce itself with essential fidelity". Significantly, Clements believed that dynamics in vegetation is a "cornplex but definite" process. It is a logical extension of these that vegetation dynamics, to Clements, was universal, progressive, and orderly. If Clements was in fact correct then, with sufficient knowledge of the arnbient conditions, the process is predictable. The Clernentsian view is of course predicated on his assumption that diverse pioneer communities eventually converge to a self-perpetuating monoclimax state under the control of the regional climate. While Clements realized some limits to predictability, he still believed that there is design, purpose and unity in Nature, and that the plant association is unique and, in its totality, more than the sum of its individual components. His work incited immense controversy and he was repeatedly criticized for his teleological implications that design, purpose and unity were functioning in vegetation processes (Gleason 19 17, 1926; Colinvaux 1973; Dmry and Nisbet 1973). Miles (1980) displays the sharp criticism when he quotes Egler (195 1): " We have Clements the uncompromising idealist, the speculative philosopher, driver by some demon to set up a meticulously orderly system of nature. as neatly oganized and arranged as the components of Dante's Inferno". On the other extreme lies the Gleasonian diction ( 1926) that "succession is an extraordinarily mobile phenornenon", and "not to be stated as fixed laws, but only as generd pnnciples of exceedingly broad nature". He defined succession in 1927 as "al1 types of change in tirne, whether they are rnerely fluctuating or produce a fundamental change in the association". Gleason maintained that vegetation dynamics "need not ensue in any definitely predictable way" based on the belief that "every species is a law unto itself'. To Gleason vegetation dynarnics was individualistic, unpredictable, and under the strong influence of chance effects. Gleason asserted, for example that dispersal was the governing process in distribution of vegetation, which Roberts (1987) identifies as a purely stochastic process. Gleason's individudistic hypothesis of succession was in complete contrast to Clements' holistic view, emphasizing the importance of adaptations of individual species as a functioning mechnnism, independent of any transcendental properties of the whole community. Gleason thus provided a slightly more complicated picture of the processes involved in vegetation dynarnics insisting that they could not be reduced in complexity by the attribution of rules. In Gleason's 'Inferno' neither sinner nor saint would be given preference. Subsequent work (Cooper 1926; Tansley 1935; Watt 1947) built upon the basic ideas put forth by Clements and Gleason. The importance of scale in the perception of vegetation dynamics, and the possible existence of several climaxes (polyclimax) was introduced. This suggested that vegetation dynamics could follow more than one pathway or process, leading to possibly different patterns. Whittaker (1953) suggested the climax pattern view where the climax is not a fixed point, nor a fixed set of points, but rather a dynarnic continuum with a defined stochastic range. Van der Maarel (1988) provides a classification of the processes identified to date. He suggests eight kinds of changes in vegetation: fluctuations, gap dynamics, patch dynamics, cyclic succession, regeneration succession, secondary succession, prirnary succession, and secular succession. His classification is based on temporal scale, the degree of isolation of the vegetation (spatial pattern) and the 'level-of-integration' (spatial scale) at which vegetation is actually studied. These basic ideas of the pioneers remain of conternporary importance (Burrows 1990). Miles (1980) suggests that the 'organisrnic' concept of Clements is now the 'systems approach' of the cornputer age (Margalef 1958, 1968, Odum, 1969, Patten, 1975). However, much contention still exists over the fundamental nature of the process. Despite the emergence of succession as a 'theory' found described in most ecological textbooks, according to several authors (McIntosh 1980; Roberts 1987; Niering 1987; McCook 1994) there is yet no generally accepted theory of vegetation dynamics.

2.4 Mechanisms in the vegetation system

Much of the work in vegetation ecology still focuses on the recognition of pattems and processes (e-g., Silvertown and Wilson 1994). While processes are the agents of change, mechanisrns are the causes of change in vegetation pattern. While these have been essential towards theory formulation, perhaps more importantly, several mechanisms have been identified. I briefly review these because they are inherently related to pattem and process and provide a link to application. Recognition of some basic mechanisms responsible for pattern dates at least as far back to Cowles (1899) who proposed such principle factors as light and heat, wind, soil, and water. With respect to the distribution of organisms, Hutchinson (1953) suggested five types of pattem, which were defined by their underlying mechanisms: 1) Vectorial pattems are those which are determined by environmental factors. 2) Reproductive pattern is the product of processes such as dispersal. 3) Social pattern results from signaling of various kinds leading to spacing or aggregation. 4) Coactive pattem, he defined as the product of interspecific cornpetition. 5) Finally, there is also stochastic pattem, which depends solely on random forces. These broad categories of mechanisms have more or less served as a basis for explanation of pattern in vegetation ecology. For example, Kershaw ( 1963) names the three major causes of pattern as morphological, environmental, and sociological. These causes of pattem resemble very much Whittaker's (1970) reasons for the occurrence of patches (pattern) in vegetation. He attributes pattern to dispersal, differences in environment and species interrelations. More importantly, he recognizes that these three causes must interact. He also suggests that differentiation in time could be related to rhythms of environment caused by temporal cycles. Similarly, Greig-Smith (1979) outlined some of the major causes of vegetation spatial heterogeneity (pattem). These were: environment, animals, interrelations between plants, disturbance and fire, reaction of the whole vegetation, inefficiency of dispersal, historie causes, and chance. Mueller-Dombois and Eilenberg (1974) explain that these mechanisms can occur in space, or they cmbe tirne-dependent as in the cases of successions. Spatial mechanisms can further be divided into what have been termed endogenous and exogenous causal factors (Webb et al. 1972), or autogenic and allogenic concepts of vegetation change (see Odum 197 1). The former involve characteristics within the vegetation itself, whereas the latter are environment-related. Time-dependent mechanisms must be considered when process is of interest. Clements ( 19 16) for example distinguished between "proximate" and "ultimate" causes of vegetation processes which differed mainly in temporal scale. He stated that "since succession is a senes of complex process, it follows that there can be no single cause for a particular sere. One cause initiates succession by producing a bare area, another selects the population, a third determines the sequence of stages, and a fourth terminates the development...As a consequence, it is difficult to regard any one as paramount. Furthemore, it is hard to determine their relative importance, though their difference in role is obvious. It is especially necessary to recognize that the most evident or striking cause may not be the most important. " Egler (1954) provided two possible pathways "relay floristics" or "initial floristic composition sequence". The former is actually anaiogous to Clements' "reaction" model. Hom (1976) and Comell and Slatyer ( 1977) independently proposed three mechanistic models to help idealize species replacements over time. The first describes the case where there is chronic, patchy disturbance. In this scenario, interactions between species take the form of a race for unchallenged dominance in a recent opening, rather than direct competitive interference. In this scenario, succession is expected to be rapid, convergent to a stable state, and relatively independent of the characteristics of the disturbance itself. In the second model "obligatory succession" (Hom), "facilitation" (Connell and Slatyer) or "relay floristics (Egler), later species must wait for earlier species to die out. The third model "competitive hierarchy" (Hom) or "Tolerance" (Connell and Slatyer) suggests that late successional plants ue increasingly capable of dominating in crowded competition with early successional species and are also capable of invading the earliest States of succession. Hom (1976) recognizes that these models need not be mutually exclusive and that several mechanisms could be simultaneously driving succession. The relative importance of causal factors remains debatable. It still remains a very difficult task to determine which mechanisms are dominant behind patterns and processes. Van Andel et al (1993) give an up-to-date review of the mechanisms of succession and emphasize that 'the study of processes does enable at least as much understanding of vegetation succession as does the mechanistic approach'. It is critical to realize that it is extremely difficult to isolate mechanisms in vegetation dynamics, since they occur simultaneously and will be CO-dependent,and Iikely hierarchically related (Allen and Wyleto 1983; Pickett et al. 1987).

Several aspects of the vegetation system were discussed, which clearly suggest that Our perception forces us to conclude that it is extrernely complex. This complexity occurs because of and despite the fact that there are many ways to view the system and that it is dynamic. The task of determining the fundamental nature of the dynamics of vegetation system requires first that we corne to terms with this complexity. Mile this may be the case intuitively, there is not agreed upon definition of complexity. In Chapter 3 1 present my investigation into the question "How complex is complex"? 1 found that problem had not been satisfactorily resolved and 1 took upon the task of quantification of vegetation complexity. It's hard not to want to sharpen these blades. They have served me well. Rercing Mt. Fine precision. Pointed nonchalance. Dagger for rny dreams. Sword for rny honour. Unalloyed allegiance. Deathly vanity. Bravo! Beware . There were so many grindstones on my path. Mostly. 1 just kicked them around. or made a game of stepping fiom one to the next. But I put a few pretty ones in my pocket and forgot about them. Didn't how they would be so hard to throw away.

1 find one when I am cleaning my desk. 1 put my nose to it. It looks so dull, yet 1 still rememeber sharp. 1 decide it is useless and try to throw it away. But it the sound it makes As it hits rock bottom Startles me. A most absurd destiny For this ancient density. So 1 dig it out. Not knowing what I have done. Did 1 really think 1 could get nd of it that easily? 1'11 forget about it again. It will easily lose its place among the mess. Until the next time 1 find that A blade needs sharpening. Chapter 3

HOW COMPLEX IS COMPLEX?

3.1 The vegetation -stem is cornplex

In explorations of the topic authors point to many partedness, spatiaUtempora1 heterogeneity and nonlinearity (responses, interactions) as major traits that distinguish complex systems (Gleick 1987: Nicolis and Prigogine 1989; Pilette et. al. 1990; Çarnbel 1993; Haydon 1994) in general. This notion of cornpiexity is cornpletely consistent with the prevailing views on vegetation (Whinaker 1975; Allen and Wyleto 1983; Phillips 1985; Kolasa and Pickett 199 1; Orloci 1993; Hastings and Sugihara 1993). Its quantitative definition in other than just surrogate terms have eluded for long the practitioners of the Art. While the notion may seem intuitively quite straight-foward, the operationality of the idea bears directly on the complex problem of finding an answer for Çambel's (1993) simple question: "How complex is complex?". That this is not a simple matter in ecology is quite obvious if we consider that:

1. Complexity has many manifestations (Saunders and Ho 198 1; Atlan 1988; Kolasa and Pickett 199 1). 2. Convolutions of properties are involved over several spatiaVtemporal scales (Pattee 1973; Allen and Star 1982; Maurer 1987; Wiens 1989). 3. Existing tools are insufficient (Çambel 1993).

It is true that there were attempts at measurements of complexity, but these did not reach beyond some meta-quantification state. A typical long- standing example of this is the use of community diversity, measured as entropy, as a surrogate of complexity (Margalef 1968; Woodwell and Smith 1969); Maher 197 1 ; May !979). Since I believe that the first step in studying the behaviour of the vegetation system is to recognize and embrace its complexity, 1 needed to corne up with a more satisfying, comprehensive definition of comrnunity complexity.

3.2 Kolmogorov cornplexiy

In a search for a more inclusive complexity rneasure 1 begin with reasoning that complex things are usually more difficult to grasp and describe than are simple things. The path of Halley's cornet and the path of organic evolution are extreme examples of things with very little versus very much difficulty of description. In the same way one would undoubtedly find the description of a simple corn field practically effortless compared to that of natural vegetation in recovery on abandoned agricultural land. The notion of description effort, thought of as a manifestation of complexity, was in fact formalized by Kolmogorov (1965) for physical systems. His and related work (Solomonoff 1964; Chaitin 1966) becamc the basis for the notion of dgorithmic complexity (Lofgren 1977; Packel and Traub 1987; Calude 1994). These definitions have been used in conjectures about applications to biology (Wicken, 1979; Papentin, 1980; Hinegardner, 1983). I show in this section that it is possible to develop a meaningful measure of vegetation complexity on the basis of algorithmic complexity. Two information-theoretical primitives need to be given: 1. the number of community components (populations). 2. the proportional distribution of individuds among the components.

Having framed the problem in information theoretical tems, it would be tempting to choose by Shannon's (1948) classical entropy function to measure complexity. This is given by

bits

(or Hflog2 e nats). Pursuinp the ecological terminology for interpretation of (3.1), the symbol q indicates the number of comrnunity elements (e.g., taxa, growth forms) and pj stands for the proportionate representation of element j. Note that the higher the H value, the greater is evenness, and thus higher the diversity in the community. For a given q, H is maximal when al1 elements have equal representation, the case of a perfectly even distribution, which Kullback (1959) calls "equi-distribution" and ecologists refer to as the most dispersed distribution. H is a minimum when the distribution is most contagious that is when al1 elements except one are represented by frequency 1. We consider the Rényi (196 1) generalized entropy (3.2) as a benchmark :

In this, a defines the order of entropy. The other symbols are as in (3.1). Ha is appropriately termed "generalized entropy", considering that the other entropy measures, such as richness, Shannon's H and Simpson's (Simpson 1949) diversity index cm be derived from it as special cases. In fact, a=0 corresponds to "richness", Shannon H is obtained when a is allowed to approach 1, and the log Simpson index is defined when a is set to exactly 2. We should remark that low-order Ha,such as Shannon's and the log Simpson's, has low discriminating power between cases (Orloci 199 la). In the extreme of u=O, Ha has absolutely no power of discrimination between cases for which q is identical. For us high-order entropy, say H12, is preferable over H or H2 , since it has been found (Orloci 199 1a) that at a= 12 or so, the measure has usually stabilized (the rate of change of Ha with respect to a is negligibly small). Of course, when appropriate, as with H, the consideration of entire measure profiles (e-g.. diversity 'ordering') rather than single points is best (Patil and Taille 1977,1979; Solomon 1979; Orloci 199 la; Tothmérész 1995). Notwithstanding the broad acceptance of entropy as a measure of diversity, Taille and Patil (1979) have argued the point that entropy and diversity are not completely equivalent concepts, based on their observation that sorne diversity measures fail to detect the increase in system entropy which they intuitively feel occurs. This suggests an insufficiency in the application of diversity indices. Perhaps even more misleading is the association between entropy-based diversity and complexity, and the implication that the two terms are interchangeable (McIntosh 1967; Margalef 1968; Woodwell and Smith 1969; May 1979). Furthemore, entropy is not completely consistent with our intuitive notion of cornplexity. Koppel (1987) suggests how this rnay be conceptualized. He distinguishes between "total" complexity which includes entropy and a second component, "meaningful" complexity, which captures structure. This idea is very similar to Papentin's (1980) "organized" and "unorganized" division of complexity. 1 feel that we can construct a complexity measure which includes both of these components with the help of Kolmogorov's notion and general coding theory (Abramson 1963).

The problem in coding, as understood by communication engineers, amounts to transmission of some "message" from a source to a receiver, under conditions where both parsimony and accuracy matter. The function of coding is thus to cornpress the message in order to simplify its transmission, while retaining its integrity. Abrarnson ( 1963) gives forma1 definition:

"Let the set of symbols comprising a given alphabet be called S={s1 ,s2,

.. S.Then a code is a mapping of al1 possible sequences of symbols of S into sequences of symbols [codewords] of some other alphabet X= ( x 1, x2, ... , xr 1. S is the source alphabet and X is the code alphabet".

In our case, the "message" is a vegetation record of a site (plot, quadrat). Phytosociologists refer to this record by the French word "relevé". The relevé describes a rnomentary composition of the vegetation, but if repeated, a spatiaVtempora1 chain of community states known as the "coenosere" is captured. The relevé describes composition by giving species or other population identities and quantities as seen in Table 3.2.2.1. However, the description is not in the most basic form and therefore rnay be very redundant. Nicholis (1988) put it best: "The need for a symbolic representation arises when one seeks to describe the outcome of a natural phenornenon in a way which is sufficiently compact and general to allow for a unified description of the whole classes of other phenomena similar to the observed one." To get to the very basic description, I translate the relevé into a "code" in which species quantities are replaced by "codewords". Table 3.2.2.2 gives an example of this. Table 3.2.2.1 Typical relevé from recovering vegetation on abandoned agricultural land. Cover-abundance estimates in quadrats are lower bounds of percent equivalents on the van der Maarel(1972) scale. Relative cover in last column. Site: Environmental Western Resesrch Station, London, Ontario. Year: 1994. Species names accord with Gleason (1968).

Species Cover-abundance

Solidago cclnndensis Poa pratense Acer saccharum Trifolium pratense Chyanthem~imlericanthemum Polygonum persicaria R~tmexcrisp~is Trirnrncnrn officinale Cerastium v~ilgatrim Viola pednta Table 3.2.2.2 Binary coding of the relevé in Table 3.2.2.1 by Huffman's (1952) method. The States are ordered by pj, and codewords are constructed based on the binary alphabet (0,l) and simple generating rules, such that al1 codewords are distinct. Codewords have no decimal implication. To clarify this further from the point of Abramson's definition, S is the species list and each sj represents a species identity. Each sj has an associated record of relative quantity that we represent by pj. It is the set of pj values which we translate into a code. How is the translation actually done without information loss? First off, one has to realize that there cm exist many different functional codes for a given source. Our objective is to find, arnong these, the rnost parsimonious code. Under this objective, a code with many short codewords is preferred to one with long codewords. The general criterion for assessing the parsimony of a code (Abramson 1963) is the average code length L specific to a source :

bits (3.3)

where 9 is the length of the codeword (number of symbols) of the relative quantity pj for species sj. The pj are SUC~that pl + ... + pq = 1 and al1 pj2O. Obviously, the magnitude of L depends in some complicated way not only on the number of species (q)and their relative abundance (the pj), but also on the codeword length 9. Thus both primitives of the Kolmogorov complexity notion are expressed in L, and important, an extra component exists in the sense of L = H + A. A distinguishes the average codeword length from entropy. L will depend on the coding algorithm. 1 adopt the coding rnethod of Huffman (1952) (please see Appendix 1 for a worked example) which meets Our criterion for parsimonious coding, in that low pj values are encoded into long codewords and high pj values into short codewords, somewhat analogous to Solomon's (1979) 'majorization'. What is achieved through Our rnethod is a context-independent coding. This overcomes the problem of spurious cornparison of communities.

3.2.3 Tord complexity

Coding by the Huffman method is consistent with the common sense notion that low probability events convey more information (have more surprise value in a technical sense) than do high probability events. 1 conceive of L as algorithmic cornplexity. Specifically, the higher is L, the higher the "total complexity" of the community. I have already pointed out how L is technically different from H. As coding theory is related to communications theory, we might expect that L be related to H. It tums out (Abramson 1963; see Appendix 1 for his proof) that L and H are related in the interesting way of the inequality,

In other words, entropy (or diversity of any order) is a lower bound for average code length. L may exceed H but cannot be less than H,and so to conclude, complexity includes disorder-related diversity. 3.3 Test cases and results

I examined the behaviour of L, H and A in model and natural communities. The community elements are species which rnay have linear or nonlinear responses and randomly or systematically sited mean values on a resource gradient. Six cases are considered:

Case a: 4 species model, with three variants of evenness:

spl sp2 sp3 sp4 [0.25 0.25 0.25 0.251 [0.70 O. 10 O. 10 O. 101 [0.90 0.03 0.03 0.031

Case b: 5 and 20 species model, 500 and 2000 plants in 397 and 188 1 respective systematic assortments among species from complete evenness as in Assortment 1: [100 100 100 100 1001 and [100 LOO 100 ... 100203 to minimum evenness as in Assortment 397 and 188 1: [496 1 1 1 I] and[1981 1 1 ... 1201 Al1 assortrnents are tested.

Case c: 62 species in natural Carolinian vegetation in southern Ontario in recovery stage after abandonment by agriculture. Cover-abundance records are based on 53 1 m.sq. contiguous quadrats dong transect. Two structural community properties (vertical layering and species growth-form richness) are quantified. Spatial resolution is tested from 1 m* to 10 m2 in 1 rn linear steps.

Case d: 9 species in recovering Atlantic Heathland (Lippe et al., 1985). The 18 consecutive annual records are based on percent point cover. Al1 years are tested.

Case e: 10 species model. Six different species response types almg an extreme-to-extreme resource gradient are tested: I. linear, random slope and intercept. II. rnonotonic (nonlinear), constant variance, regularly-sited means. III. rnonotonic (nonlinear), random variance and randomly-sited means. IV. unimodal, constant variance, regularly-sited means. V. unimodal, randornly-sited means and random variances. Nonlinear responses (rnonotonic and unimodal) in al1 cases conform to the Whittaker-Groenewod species response model (Groenewoud 1965; Whittaker 1975). Al1 types are tested.

Case f: 10 species in natural community (Santa Catalina Mountains. Arizona) dong moisture gradient. Natural response types are transcnbed from Figure 8 in Whittaker (1967). It is seen in Table 3.3.1 that H is equal to L when the individual plants are evenly arranged between the species. Under this arrangement H reaches the maximum value logz q, but by moving away from an even distribution, the discrepancy of H and L becomes increasingly wide. Thus any time the relevé is not at maximum disorder, entropy is insufficient to describe complexity. and the difference

A= L-H (3-5) takes on special significance. 1 note at this point that the A quantity is specific to the coding algorithm used. A temary code, for example, may not necessarily pive the sarne absolute value of A than the chosen binary code (pers. comm. A. Sziliird). It is clear from Table 3.3.1 that the level of disorder affects the magnitude of A. This behaviour is seen also in Fig. 3.3.1 which indicates that A is sensitive to changes in species numbers. As the number of species increases from 5 (Fig.3.3.1 a) to 20 (Fig. 3.3.1 b), A increases dramatically at highly even S, but less so at low evenness. Concomitant to this, H contributes increasingly less to L. Figure 3.3.2 ernphasizes that A is not a simple deviation from maximum entropy (logz q - H). Furthemore, A levels off as evenness decreases. This indicates that A cannot be simply expressed as logz q - H, and is important because it verifies that A is capturing an aspect of total complexity not accounted for by H. It indicates also that the A cornponent of complexity cannot be decreased by simple changes in the assortment of individuals among species. Table 3.3.1 Levels of evenness in a Cspecies community. H and L accord with Equations 3.2,3.3 in the text. A = (L - H). H, L and A are in bits.

Species proportions H L A [0.25 0.25 0.25 0.251 2.00 2.00 O r0.70 0.10 0.10 0.101 1.36 1.50 0.14 r0.90 0.03 0.03 0.031 0.60 1.20 0.6 1 maximum

Figure 3.3.1 The behaviour of total complexity L, Rényi's entropy H of order 12, and A in artificial data sets of 5-species (a) and 20-species (b). The abscissa displays dl possible cases of evenness, maximal at lefi, minimal at right. Ordinates accord with Eqs. 3.2 and 3.3. Note that A = L- H. The data is under Case b in the text. Figure 3.3.2 Relationships of A and H expressed as a deviation frorn its maximum in the artificial data set of 5-species under Case b. Axes as in Fig. 3.3.1. Note the leveling off of A while the deviation maxH - H monotonically increases. Since the number of species is constant, max H= log7 5.

1 present the relationship between A and the spatial scale in Figure 3.3.3. It is obvious upon inspection of this figure that A changes with position dong the transect, but only up to a certain point on the spatial scale. After that A is sustained at a constant level. The characteristic quadrat size where the leveling off occurs is about lm x 4rn. The graphs in Figure 3.3.4 emphasize further the independence of the fluctuations in A and logz q - H. transect position

Figure 3.3.3 Spatial dynamics of A in real data (Case c in text). Abscissa: quadrat position on transect. Depth: spatial scale in multiples of 1 m* quadrats. A generally fises up to quadrat size 4 after which it is stable. Position on transect matters. Tnssect position

Figure 3.3.4 Covariation of A and maxH - H in real data (Case c). Abscissa: quadrat positions 1-53. The covariation is strongly asymmetric. Unlike in Fig. 3.3.2, species number changes with position and maxH varies. Figure 3.3.5 supgests varying dependence of L, H and A on the time scale. While L and H fiuctuate in an irregular manner, A stabilizes after 5 years. This is consistent with behaviour that 1 observed in response to spatial scale changes (Figure 3.3-3). Regression analyses (Table 3.3.2) reveal further aspects of a changing A in relation to species richness, vertical layering and growth-form richness. I mention that quadratic and cubic terms did not rnaterially improve the regression fit.

Table 3.3.2 Coefficient of determination (R~)in simple linear regression of A on structural properties as identified. Nonlinear rems in regression equation did not significantly improve R'. Data source is Case c in the text.

Structural property R'

Species richness Growth-form richness Vertical layering Figure 3.3.5 Temporal dynamics of L, H and A in real data (Case d). Ordinates as in Fig. 1. Abscissa: annual steps 1- 19. Note stability of A after 5 years while L and H continue with intense fluctuations.

Cornparison of the graphs in Figure 3.3.6 reveals a behaviour dramatically different under the different response-types. Linear responses (type 1) generate a smooth A graph with long periods of constancy. This sharply contrasts with the other cases. In fact, it is clear that A captures the regularity in response types II, V and VI. The most complex type (V) generated the most unstable, irregularly fluctuating A. It is worth noting that the Santa Catalina data (VI) yields a smooth A graph, but fluctuations are considerably broad and irregular. Figure 3.3.6 Behaviour of A under species response-types (Cases e,f). Abscissa: extrerne-to-extreme gradient. Ordinate: squared A. Legend to response-types: 1. linear, random slope and intercept, II. monotonic (nonlinear), constant variance, regularly-sited means, III. monotonic (nonlinear), random variance and regularly-sited means, IV. unimodal, constant variance, regularly-sited means, V. unirnodal, random means and variance and VI. natural responses on Santa Catalina moisture gradient. General shapes of the graphs reflect species response-type. Both regular and irregular changes in complexity are detected.

3.4 Complexi~and Hiernrchv

So far, my discussion of complexity has centered on the problem of finding a context-free measure, and not considerations of organization for which we use stratification seen in Figures 3.4.1 and 3.4.2 as a simple exarnple. It is clear that complexity is related to the interaction of different levels or strata in organization in a system. My mode1 of complexity incorporates the assumption that organization is hierarchical (Pattee 1973; Allen and Starr 1982; Goodall 1986). To reflect further on this, we mention Simon (1962) who makes the point that "If there are important systems in the world that are complex without being hierarchic. they may to a considerable extent escape our observation and Our understanding". Studies applying hierarchy theory have indeed recognized the importance of the interaction of levels of organization on the perception of ecological complexity (Goodall 1974; Ziegler 1979; Allen and Wyleto 1983; Müller 1992; Pillar and Orloci 1993; Li and Müller 1995), but these work with their own definitions of complexity and hierarchy (levels). What is clear, however, is that the model to be effective cannot be a simple additive model. if for no other reason than the fact that less general levels are constrained by more general levels in the hierarchy. The interdependence of levels is what characterizes a nested hierarchy (Allen and Starr 1983; Orloci 1991). But even in this case, closer and closer inspection of a comrnunity (at lower and lower levels) may not lead simply to clearer and clearer resolution of its complexity. That is, complexity may not be cornpletely decomposable. This requiring a relaxation of the definition of "hierarchical" in line with Li and Müller ( 1995)'s discussion of partial decomposition.

Fip. 3.4.2 Vertical stratification in an idealized community (modelled after Dansereau 1957). Seven strata (A 1, A2,B 1, B2,C 1,C2,D) are recognized based upon height classes of vegetation.

3.4.1 Models of Pnrtitioning

As 1 have already pointed out, the method of hierarchical analysis of complexity, will depend on the cornplexity function, and on how the hierarchical levels are defined. Having discussed the former, I now discuss the latter. We have to accept as fact that levels emerge from observations as abstractions, after decisions about how observations are made and how systems are to be described (Allen and Wyleto 1983). But this does not clarify things in practice, because there is an infinite possibility of choices in how observations are to be made and how systems are to be described. What would be more usehl would be some reasoning why one mode1 might be more appropriate than others in a specific case. 1 now discuss the different hierarchical models of partitioning. Considering Figs. 3.4.1 and 3.4.2, obviously the simplest partitions would directly follow the material stratification of the community by layers. The most obvious of this in vegetation are the vertical strata where each strata is considered as a distinct analytical level. Partitions based on this rnodel would reveal the complexity contribution of individual strata as though the strata were unconnected. 1 know that this is not a realistic model. In fact the interactions between strata are very strong which 1 believe itself generates complexity . In a better partitioning model 1 could block strata not unlike in Greig-Smith's (1952, 1961) pattern analysis. This would be one way to find peak contributions by strata, but the exercise would be trivial, since in most cases there are so few strata to be blocked. A more realistic possibility is to follow 'directional decomposition' of the complexity function. This would reveal contributions of the community strata to total complexity under constraint of the direction in which the partition is approached. This constraint may appear at first sipht too confining, but in actud fact it can be very natural in the sense that each higher stratum has an overbearing effect upon al1 the lower strata. If we think of it in terms of the specific example pictured in Fig. 3.4.2, then the direction D to Al is set by the logic that it is straturn D in which the total effect of stratification in the stand is most confounded, through light interception, physical interactions and many other interferences. To develop the notion of the directional decomposition mode1 further, while retaining the physical structure consistent with the fact that in a community of sessile organisms every member of a stratum is also a member of al1 lower strata, as pictured in Fip. 3.4.2, 1 can formulate a directional decomposition algorithm based upon methods analogous to Orloci's (1978) ranking techniques. In the simplest case, we have the complexity level attributable to stratum Al as the complexity level in the pooled sample (A1, A2, C 1, C2, D } minus the complexity level in the reduced sample without A 1, { A2, C 1, C2, D J, and similarly for A2 after A 1 is removed from the pool, for B 1 after A 1 and A2 are removed, and so on. In the end, a perfectly additive partitioning of complexity is achieved. 1 recognize al1 of the above schemes as potentially usehl, but they rely heavily on perception of a rigid physical structure in the community. 1 consider now a nested hierarchical model. In this, the structure recognized is two steps removed from any material structure in the cornmunity. The hierarchy is analytical nested, built from information that an association matrix (2nd order structure) conveys based on the observed stratal affiliation of species (1st order structure). I examine properties which exist only at the community level, not traceable directly from individual species behaviour. Hierarchical levels are constructed, not through direct observation of stratification, but based on species association. This nested approach is realistic to the ecologist because it focuses attention on otherwise invisible interactions that must exist in a cohesive community. My modus operandi entails description of the community in terms of occupancy scores of species and stratal affiliation, computation of an association (from Euclidean distances) rnatrix, a nested hierarchical cluster analysis (Orloci 199 1b), and lastly, partitioning of complexity according to the hierarchic structure so created. 3.4.2 A ivorked example

The community in the example is a rnixed conifer-deciduous stand located in Boreal Ontario (Fig.3.4.1). The specific stratal structure is boiven in Figure 3.4.2 and Table 3.4.2.1 1 applied the two models, directional decomposition and nested decomposition, as described above. For partitioning of total complexity 1 use the weighted total complexity (SUT

whereJk is the total species abundance and Lk, the total complexity (3.3) specific to hierarchical level k, with c being the number of hierarchical levels. For partitioning the entropy-based component of complexity we use Brillouin's ( 1962) information (Sn,

SI, = f.,H,,k = 1,2,.... c

withfik and c the being the same as in (3.6) and Hk, the Shannon entropy

( 1) specific to hierarchical level k. SL and SI are the partitionable equivalents of average complexity (L), and entropy (H). The relationship SL=SI+SD , SD positive or zero, obviously holds. Table 3.4.2.1 A typicd phytosociological relevé of a IOm x 10 m plot showing strata levels. Data coilected in July, 1995 near Elk Lake in Boreal Ontario. Cover-abundance is on the Braun-Blanquet scale.

Cover abundance

Pinus banksiana Populus tremuloides

Abies balsarnea Picea mariana Pinus banksiana Populus tremuloides

Abies balsamea Amelanchier sp. Corylus cornuta Picea mariana Pinus banksiana Populus tremuloides Prunus pensylvanica Salix discolor Sorbus decora

Abies balsamea Amelanchier sp. Betula papyrifera Comptonia peregrina CoryIus mmuta Diemllea lonicera Kalmia angustifolia Picea mariana Pinus banksiana Populus Iremuloides Pnrnus pensylvanica Salix discolor Vaccin~umspp.

Aster sp. Betula papyrifera Cornus canadensis Cyprepedium acaule Glyceria striata Uaianthemum canadense 3ryzopsis pungens Dolygala pa ucifolia Trientalis borealis

?ornus canadensis Cpigaea repens Saultefia hispidula 5nnaea borealis Dyrola rotundifolia

3areground Sladonia rangifenna 3icranum polysetum %urozium schreben Sphagnum sp. The results from directional decomposition are presented in Table 3.4.2.2. The components are tied to dominance sorting into physicd strata, but the links to individual species or species groups are not necessarily observed. The analysis identifies stratum C 1, containing herbaceous and some chamaephytic shade tolerant species, as the most significant contributor to total complexity (SL) and interestingly the tree layers, Al and A2, as the least significant contributors. This is an interesting result because it suggests that those elements in the comrnunity which are most dominant need not be those which sustain its complexity. The second nested mode1 also revealed some interesting trends. Cluster analysis of species associations yielded the analytical structure in Fig. 3.4.2.1. This third-order structure reflects the effect of dominance sorting on complexity in species-groups which expand their membership as the hierarchic level increases. The nested partitioning of complexity is presented in Table 3.4.2.3. On this basis we can evaluate contributions of the species groups to cornplexity. The species grouping contributing most to total complexity occurs at level H2 where, interestingly, woody, herbaceous and brioid species are separated. That a high order analytical technique finds that the community is best described through a sharp recognition of these three life-form-based groups is reassuring to the ecologist observer. Table 3.4.2.2 Partitioning of quantities through directional decomposition. H is Shannon's entropy hnction (Equation 3. l), SI is Brïllouin information (Equation 3.7)' L is average codeword length (Equation 3) and SL is a weighted L (Equation 3.6). The deviations colurnns give strata-specific contributions of SI and SL in absolute (dev) and percentage of total (%) terms (e.g., for Strata D, dev(SI) = - SID)

dev dev ('m (W Level D 44.0964 66 (8.67) (6.36) 74.4 159 139 ( 14.52) ( 13.4) 182.955 469 (35.7 1 ) (45.2) 90.43 230 (17.65) (22.1 ) 46.502 77 (9.03) (7-43 ) 41.153 22 (8 .O3) (2.12) Fig. 3.4.2.1 Analyticai nested hierarchy emerging from cluster analysis of association matnx. Four hierarchical levels (H 1-H4) are obsewed with 3 1, 5, 3. 2 groups respectively. Please refer to Table 3.4.2.1 for species names corresponding to numbers 1 to 3 1. Table 3.4.2.3 Partitioning of quantities through anaiytical nested decomposition. H is Shannon's entropy function (Equation 3. i), SI is Bdlouin information (Equation 33,L is average codeword length (Equation 3.3) and SL is a weighted L (Equation 3.7). The deviations columns give strata-specific contributions of SI and SL sirnilar to Table

dev (W 40 (3.86) 86 (8.29) 796 (76.76)

3.4.3 Concirrsions Porn hiernrchicnl pnrtitioning

While a context-free measurement of community complexity is achievable using L, information can be gained by examining the behaviour of the complexity function at various levels of the system. My hierarchical analysis reveals that complexity is, in general, partitionable arnong hierarchical levels and that there is variation in complexity over different levels. Cornparisons can be made between levels to detect which level contributes most to total complexity. 1 presented two different models showing how this can be done in practice. 1 showed by the first model, directional decornposition. that the most transient and unassuming strata, the herbaceous understory, contributes most significantly to community complexity. However, 1 made the argument that attributing complexity to specific community properties should not necessarily rely on direct observations of structure, like stratification in a forest stand. Thus, 1 acknowledge the limitations of this model. Higher-order analytical structures may capture the interactions and thus generate complexity, not obvious from direct inspection. 1 found this to be the case from Our results. I found our second model to not only be more sound conceptually, but also surprisingly reflective of the data of Our example. In this way the analysis provides hard evidence for the idea that a complex systern exhibits properties not detectable through the a summation of its parts.

Dawkins ( 1986), Nicolis and Prigogine ( 1989) and Çambel (1 993) discuss complex things and point to characteristics that are clearly emergent under the effect of synergistic interactions and exchanges, beyond and in addition to disorder. Excepting disorder, emergent properties cannot be accounted for by entropy alone, confirming the suspicions of Papentin (1980), Hinegardner (1983) and Koppel (1987). Yet in the equation L = H + A maximum H is setting the upper bound. That "complexity" cannot rise above what a random assortment of States can create is indeed a paradox that A cannot resolve. But we hardly can imagine natural vegetation in this disorded state, nor cm we reconcile the need for operationality of the notion of complexity with the idea that a "corn" field should be regarded "complex" by virtue of its dependence on "advanced" design, sometimes several steps removed such as in the case of the machinery, to which A cannot be sensitive. In A terms, "design" and "complexity" lack a direct link. A tumed out to be relatively large in the examples. What does this signify? We know that by virtue of the mathematics, A arises when at least one species exerts dominance in the cornmunity. But the relevant question is why should the information captured by L exceed the information captured by H, or equivalently, why should a A arise in addition to H in the first place? We recall the fact that, by consequence of the coding algorithm, L is very sensitive to the presence of high and low probabilities. Very low probability events cause L to greatly increase, and very high probability events cause it to greatly decrease. This is not the case with H and for this reason A is not simply related to the decline of H from its maximum value. It is then logical to look for interpretations of A in the causes of differential probabilities. Intuitively, these causes change the community from a perfect chance arrangement of plants among populations to greater differentiation. I suggest that these causes are likely to appear as emergent fuzzy properties in the categories of structure and organization. Clearly, there can be many sources of these and therefore A may not have a unique, universal biological interpretation. It has to be examined on a case-by-case basis. From my results 1 am tempted to refer to A as "structural complexity" (cf. Greig-Smith 1986), to distinguish it from "disorder-based complexity", for which we use the term "diversity". Watkins and Wilson (1994) have in fact shown that, in general, plant community structure complexity are related. 1 believe that the intuitive notion that complexity is dependent on the number of species and also related to emergent characteristics has a concrete, quantitative basis in L. This fact has implications in quantitative evaluations of changes in community composition in the sense that species losses or gains have to be evaluated for their potential consequences based on their specific effects on al1 components of complexity. I believe that effects on structural complexity are potentially as important as effects on diversity. We know that these two effects corne hand in hand, but they are not simply related, beyond the fact of being in some ways complementary. Stmctural complexity and disorder-based diversity are not substitutes for one another and should be considered as autonomous, equally important properties. We have seen that as the number of species is increased, A is most affected. This may be signaling that as species are added to a community, the pressure for organization and establishment of structure is increased and the sustainment of chance arrangements becomes increasingly difficult. So it is through the demise of chance arrangements that we see the emergence of structural complexity. But what is especially noteworthy, is that this process may work in a counter way if species are removed which may lead to the collapse of structure and organization. Considerations of the scaling dynamics afTecting A lead me to suggest that at small spatial-temporal scales, structural complexity tends to be both scale dependent and locally variable. The fact that scale-dependence dissipates past a critical spatial and temporal scale has both practical and conceptual significance. Practically, it suggests that to stick with the 1 m.sq. quadrat size andor short-time spans in the analysis of Carolinian vegetation recovery would prevent detection of existent higher level complexity. Combining the 1 m2 into larger units, and longer time spans are necessary. But to go beyond the lm x 4m plot size or 5-year time steps for the purpose of capturing hl1 numerical complexity would be redundant. Similar practical considerations apply for the time scale in the Atlantic Heathland. Conceptually, the observation of spatial and temporal scale-invariance suggest that structural complexity is a "stable" property, perhaps a fractal (Mandelbrot 1977; Scheuring 199 1). It is clear from what has been discussed that the relationship between complexity and nonlinearity is difficult to generalize. This has two main reasons. First, there are an infinite number of possible cases of nonlinearity, and second, complexity is not always exhibited in nonlinear systems (Nicolis and Prigogine 1989). However, cornparisons revealed that A is sensitive to nonlinearities of species response in particular and different ways. Thus, as 1 see it, the intuitive notion that nonlinear species response generates detectable complexity is tenable. However, unlike other stmctural characteristics, the effect of response-type on A cannot be expressed as a simple correlation. Instead, the effect of response type has to be perceived as a pattern in the behaviour of A over a range of localized points (a process). Simple and regular response types result in processes of regular, symmetrical, or repeating patterns. This kind of behaviour contains no surprises and its states are senaily correlated. Complex species-response types generate aberrant, unstable chaotic fluctuations in A, whether small or large. This kind of behaviour is unpredictable. The behaviour of A is indeed a signature of the convoluted effects of species response types. It has been suggested that plant species responses are normally nonlinear and 1 expect this to be the rule rather than the exception. Exploration of complexity in the vegetation system, leads me to the hypothesis that enhanced understanding of the dynarnics of the system will require a recognition of the nontrivial relationship between nonlinearity, unpredictability, hierarchy and complexity. These properties are not accommodated by classical mathematical or statistical analysis. My intuition at this point was in the application of the chaos-theoretical (May 1986; Gleick 1987; Çambel 1993) tools. The reason for this will become obvious through later discussions. One fine day, she bst her way. Points in space and time lost connection. The map was toa creased from inspection. To calm her fear, she checked the minor. That never failed for protection. She saw a maned mess of introspection. She held her sigh, and steaded her thigh. She could do neroier of these by election. She just couldn't resist the defection. That fine day, she lost her way. Herein lies some affection. She just pretended to be lost for your direction. Chapter 4

CONTROVERSY OF HISTORIC PROPORTIONS AND CALLS FOR ITS RESOLUTION

4.1 Foundation, edifice, superstructure

In 1926 Cooper said that "a periodic inspection of foundations is most desirable for any edifice, and particularly so where the superstructure is being continually added to, as in the development of bodies of scientific knowledge". By "foundation" he meant "the fundamental facts upon which the science is built, such as the universality of change, the nature, direction and rate of change, and the causes thereof." The "edifice" was "dynamic ecology". The "superstructure" was "al1 those more or less arbitrary and subjective concepts which we find it convenient to impose upon the facts". He includes "succession" among these "superstructures". Given the "edifice" as the vegetation system described in the previous two chapters, 1 now tum to a question of its very "foundations". The fundamental question 1 ask is "what is the basic nature of vegetation dynamics?" In doing so, 1 open up the possibility of needing to redefine the "superstmctures". 1 make but one premise: the simple definition that "vegetation dynamics" is specific change in the community with time under impetus of some perturbation, nothing more, nothing less. Traditionally, this change is called "succession". We find "succession theoryf' is widely discussed in ecology texts. It is this theory which 1 am calling into question. In his text, Krebs (1994) discusses succession under the general chapter heading of "community change". He defines succession as "the development of the community by the action of vegetation on the environment leading to the establishment of new speciesf' and States that it is "the universal process of directional change in vegetation during ecological time". Despite the broad nature of this definition, 1 hesitate to use the term "successsion" because of the textbook connotations that it implies. Miles (1980) suggests that the concept of succession may find a useful place in textbook ecology because it appears simple, but within the field it is more like Medawar (1969)'s description of biology as 'cornplex, messy and richly various'. Mchtosh (1980) stated that "succession is one of the oldest, most basic, yet still in some ways, most confounded of ecological concepts. Since its formalization ...in the early 1900s, thousands of descriptions of, cornmentaries about and interpretations of succession have been published and extended inconclusive controversy has been generated. Withal, no effective synthesis of divergent observations.. .has produced a body of laws and theories which ecologists, generally, have embraced. Repeated symposia .. . have not produced notable convergence of thoughtf'. Pickett and McDonnell(1989) present a highly conceptual discussion of the 'components of a contemporary theory of community dynamics'. They clairn that contemporq usage of the term "succession" is nearly synonymous with "vegetation dynamics" and that "while some degree of directionality is implied, neither monotonic change, progress nor stable termination is necessary". In final desperation they make the plea: "We hope, too, that the broad frarnework reviewed here, centered on differential species performance but showing the place and role of other factors, caii be used to help uni@ the diverse and growing body of information on one of the universal phenornena of the natural world." Obviously, many ecologists are discontent with the claim of textbook succession theory that dynamics niles the "natural world" Iike that called for by a 'textbook' succession theory. Krebs (1994) clearly admits this in the end by stating that "No single mode1 explains an entire successional sequence". But does this mean that the search for a universal theory should stop? Miles (1980) ended his comprehensive review on vegetation dynamics with the following statement: "Hopefully the quest for uniQing, generalized theories of succession has been abandoned". While this is discouraging, Mclntosh's (1980) suggestion holds out sorne hope for ambitious ecologists. Despite the general pessirnism he clairns that there is "no reason to forgo that search." Indeed the search goes on; the present dissertation is a proof of this and it is not in solitude. Many regard that it is time for a unified theory (e.g.- Roberts 1987; Niering 1989; Pickett and McDonnell 1989). 1 believe that rnuch of the dissatisfaction cm be connected to the inability of ecology hitherto to resolve the classical Clements-Gleason controversy. 4.2 Clements-Gleason

Clearly, from Chapter 2, it is seen that Clements and Gleason held dramatically different views on the fundamental nature of vegetation dynamics. They were elaborated at extremes and involved scales that were considered incomparable. Gleason's dynamics may be seen as the underlying turbulence owing mainly to chance events and Clements' as some broad determi nistic wave. The theoretical ecologist, whose interest is to generate models, should be tempted towards Clementsian determinism, because it offers regularity and thus a hope for mathematical tractability. These indeed may be considered prerequisite for modelling vegetation dynamics -- a reasonable suggestion considering that under the Gleasonian alternative, modelling would be a futile exercise. Should one really try to look for something which one has to assume non-existent? After a century of study, it is clear that Clements' view of intense determinism has dominated vegetation ecology. Numerous and various attempts at modelling vegetation dynamics, Le., searching for regularity is evidence of this. But in each case, dynamics was labeled deterministic or random (and thus Clementsian or Gleasonian) based on which extreme dominated the trend. Drury and Nisbet (1973), McIntosh (1 9801, Finegan (1984) van der Maarel (1988). and van Hulst (1992) provide reviews of the effort and. in my judgment, a chronicle of perpetuation, if not deepening of the philosophical dichotomy. However, several studies vehemently refute the power of these models. Conne11 and Slatyer (1977) questioned the 'facilitation' models and Dmry and Nisbet (1973), Walker (1970) and Matthews (1979) questioned the climax convergence of successional pathways. Glenn-Lewin (1980) gives examples which he feels cal1 into question the predictability of plant comrnunity change over time. So, where to from here? 1 hope that the task which 1 embark on at this point will not suffer the fate of those just enumerated. 1 view the Clements-Gleason dichotomy as the controversy of determinism and disorder, holism and reductionism, and universality and individualism-- things which pervade E.C. Pielou's "soft" science (Pielou 1980; Orloci 1993) of ecology. My focus on this dichotomy is an attempt to advance a uniQing theory. I claim that neither of the classical views captures vegetation dynarnics accurately, but at the sarne tirne that neither is wrong, as they stand. Both views are realistic, but incomplete, and surprising as it may sound. and notwithstanding Gleason's intentions to refute Clements. I propose that their two views are not mutually exclusive. This resolution is the key towards a unified theory of vegetation dynamics.

4.3 Classics emnrined anew

Upon closer inspection of the classics, we should realize that both Clements and Gleason had a lot more to Say about the nature of vegetation dynamics than conventional interpretations let us believe. For exarnple, while Clements viewed the process as being completely deterministic, he adrnitted that "the most stable association is never in complete equilibrium". Despite his emphasis on chance as a mechanism and disorder as its manifestation, Gleason (19 17) acknowledged the "developrnent of vegetation" manifested as "continued and progressive change". 1 believe that a dichotomization of these two views ignored the intentions of Clements and Gleason. 1 believe that these arnbiguities tacitly ask central ecological questions, which have yet to be resolved.

The analogy of Cooper (1926) provides a good view on the complexity of vegetation dynamics in which the Clements-Gleason dichotomy finds some resolution: "Regarded from the broadest possible viewpoint, the vegetation of the earth presents itself as a flowing strearn, undergoing constant change. It is not a simple Stream. but a "braided" one, of enormous complexity, with its origin in the far distant past. Its more or less separate and definite elements branch, interweave, anastomose, disappear, reappear. We ourselves watch its advancing front, just as one may watch the advancing front of a rnountain torrent, born of a cloudburst, as it travels down its canyon bed. Vegetation as we see it today is thus a mere cross section of the complex strearn. The sarne is tnie of any point in past time. As to the future we may, from study of the present and past, prophesy a linle distance onward with some degree of certainty, but more remote progress remains absolutely in the dark. Ln the cross section which gives us our view of present-day vegetation we make out various more or les definite groupings of individuals. Many of these are very distinct and repeat themselves in space. Traced back into the past, we find that they are the advancing termini of the sueamlet-elements which make up the great vegetational Stream. As section at any point in past time would likewise show such groupings, but no two sections, even if but an instant apart, could possibly be exactly alike. "

The point W. S. Cooper makes is clear. The point I make is that acknowledgment of the complexity of vegetation is not new. However, ecology has not succeeded as yet in embracing the complexity and its contradictory interpretation into a comprehensive theory. We may question whether the two views are in fact contradictory. But more important is the fact that case studies of the vegetation system have shown that rarely is dynamics well described by either side of the dichotomy (e.g., T. Poulson, pers. cornm.). If we are allowed no flexibility, inaccuracies even on this scale are a mvial matter. Of course, in reality we cannot look for a perfect fit, but rather work within reasonable (stochastic) bounds. This is the tradition in statistical analyses of cornplex cases, where one teases out trends and acknowledges deviations from them. The question to be asked is how valid can this approach be to modelling vegetation dynamics? i feel that there are situations where deviations from a trend line or state cannot be simply disrnissed as aberrations, and that a predisposition towards simple (usually linear) detenninistic models may not be justified. What remains to be shown is whether vegetation dynarnics cm fit into a theoretical constnict representing a convolution of 'order' and 'chaos' (using the terms colloquially), which these two theories individually imply. I explore this possibility in the next chapter. take this kiss 9 canrtcontain it is free take this wish make in vain e ternitg take this tmst to make me tame keep the key take Mis iust to keep me sane one, two.thv@ take this givlmd don 'tcornplain when she is me take the log ic of .ter bain when in fear take the cornfort or hev caue my dear take the we~ghtof her gaiu to stay sincevn tcrke the mernwy of heu name and hoid it hue take th& wonlm and her strai~ when ~heis near Chapter 5

MODELLING DETERMINISM

5.1 The role of models

A theory of vegetation, as al1 theories, must include models. Jeffers (1982) identifies the need for models to reduce arnbiguity, and to describe complexity with maximum parsimony. In fact, Pickett and Kolasa (1989) define models as "conceptual constructs that indicate the objects and phenomena of interest, show the relationships and causal interactions between them, and specify the states the system rnay take." Hill (1973) clairns that the îùnction of modelling, and a major aim of vegetation science, is to describe data in a way which is suggestive of their underlying structure. Attempts towards modelling vegetation dynarnics implies predictability, which in tum, asserts the presence of some non-random pattern. Whether the assumption of predictability is plausible cmbe questioned, however, it is a requirement for modelling attempts of any kind. Goodall (1977) states that knowledge of the changes is in some degree generalizable, and thus should be predictable. If this condition is not satisfied, modelling could hardly claim to be a part of science. Once a mode1 is formulated, it can be tested and re-tested to see if a reproducible property (pattern) of the behaviour (processes) of the system (vegetation) emerges. The recent trend has been towards mathematical models, since they are mostly easily tested (Goodall 1972). Pielou (1981) defines (mathematical) modelling as "mentally constructing a plausible symbolic representation of the functioning of an (eco)system in the form of mathematical equations." There also exists, however, a group of non- mathematical models which are simply descriptive in nature (e.g., Conne11 and Slayter 1977). Nevertheless, vegetation can be modelled mathematically. The first question we must ask if we are to attempt any mathematical application is whether vegetation is quantifiable and whether forma1 hypotheses can be zuenerated. Goodall ( 1972) suggests that these are possible. Collection of data is usually done in sample areas (usually quadrats) within which the quantity of each species is recorded. This quantity can be density (the number of individual plants per unit area), biomass per unit area, cover abundance or basal area. Leps and Hadincova (1992) ask that some more accurate method of vegetation quantification te found as the present methods are highly variable. However, vegetation units cm almost always be reduced to a pattern of points (Ripley 1977). One of the major criticisms of mathematical models is their reliance upon assumptions which rnay not be valid. Assumptions, however, are necessary for models to maintain parsimony. One may even argue that, if the function of a model is to provide predictive power, and this is achieved, then, such a mode1 is valid, regardless of the assurnptions. But this goes against the idea that predictability and knowledge are synonyrnous (Peters 199 1). Furthemore, Slobodkin ( 1974) argues that it is unreasonable to suggest that one does not need a correct set of assumptions for a theoretical formulation if the conclusions are valid. He maintains that if there exists a theory based on erroneous assurnptions which lead to valid conclusions, there dso must exist a theory based on valid assumptions which lead to correct conclusions. Finding this, he warrants, may take more ingenuity, effort and intelligence and is thus a greater challenge. Wiens (1984) also acknowledges the possibility of obtaining the "'right'pattems for the wrong reasons." Models are useful and necessary (Pielou 198 l), however, they must be recognized for their limitations so that unreasonable expectations are not held. It must be stressed that they primarily test our perceptions of pattern and process, and rarely give insight into causes. In this respect, early attempts were considered a colossal failure (Yockey et al. 1958). At most, models can suggest possible mechanisms through the description of processes giving rise to patterns. This limitation exists because often mechanisms are treated as a black box, and only the output of the model are examined (McCook 1994). For example, Hom's ( 198 1) Markov model of succession suggests the nature of the process of succession and the resulting pattern, but makes no daim to explain the mechanisms at work. However, there does exist a class of 'mechanistic' models which attempt to directly Iink mechanisms to pattems. They are often mathematical in nature, however the difference is that mechanisms defne the model. These mechanistic models are usually not tnie causal models in that many of these still rely on the inference of a cause from a pattern. For example, there is a tendency to consider the correlations between certain life history traits, such as growth form or growth rate, and shade tolerance as causal connections. Assuming this correlation to be high, life history traits are then used as surrogates for the mechanisrn of cornpetition. Tilman (e.g., 1982) supports the explanation that correlations in life history traits cm explain successional patterns. Botkin et al. (1972) and Huston and Smith (1987) construct simulation models which atternpt to incorporate such causal concepts. While they may provide a closer approximation of the mechanisms driving the processes which could account for the observed patterns, correlation cannot be equated to causality. The development of modelling of vegetation processes has been reviewed by many (eg Fekete 1985; McCook 1994).

5.2 Clementsiniz determinism

How can Clementsian determinism be detected in the first place? To begin the discussion we have to first decide on what we mean by determinism. How should we measure it? Traditionally, deterministic models are simple and linear. In modelling vegetation dynarnics, or natural dynarnics in general for that rnatter, we reIy on dynamical system theory for methodology. The origins of the theory can be traced back to Poincaré (188 l), Gibbs (1902) and Birkhoff (1927). The applications used were quite diverse. Tsonis and Tsonis (1989) give a general definition of a dynamical system: "a system whose evolution from initial state can be determined by some rules". Thus the only assumption which 1 need to make is that there are some underlying, mathematically tractable mles governing vegetation dynamics. But recall that this assumption is implicit in al1 attempts in rnodelling vegetation dynamics. Thus 1 consider vegetation dynamics describable in terms of dynamical systems theory. This then leaves open the question: what are the rules? There are two ways to approach this: 1. to pick some known mathematical mles and see if they apply, that is, begin with the available mathematical theory, or 2. to construct a mode1 of mathematical mles from observation, that is, begin with Nature. In both approaches, constraints are involved. 1 will begin with 1. and a forma1 discussion of dynamical systerns theory and in its course, present some of the basic building blocks. There are two main questions of interest here : How does the dynamical system evolve, and where does the dynamical system go? The reader may immediately appreciate the fact that these two questions are interlinked. That is, the mles of evolution may have everything to do with destination. Yet 1 isolate the two arbitrarily mainly for simplicity of presentation, and also importantly, because oddly enough, we will see that in some peculiar cases, there may be no inherent connection between the two. For the second approach, beginning with Nature, we have to realize that we must deal with abstractions. 5.3 Abstraction of Natrwe

5.3.1 Reducing Dirnensionali~

1 define 'dimensiont in the broadest sense as the nurnber of components in the system. but 1 do not attempt now to quantify the concept, because the dimension-adding properties may not be logically related and thus not additive. This is somewhat like the distinction that a mathematician makes between "order" and "rank". Components can be variables, but also functions of the variables over time and space. So, for exarnple if variables behave similarly over dl spatial scales, then the spatial scaling-dimension is 1 (note: this is not related to the concept of fractal dimension). If the system is describable by a set of linear equations, then 1 consider it having a lower dimension than if Say some nonlinear terms were included.

5.3.2 Analogues

I use the definition given by Rosen (1970): Systems which share the same formal dynamical description but may be diverse in a structural sense are nnalogorcs. Why do we need analogues?

"There are ... some minds which can go on contemplating with satisfaction pure quantities presented to the eye by symbols, and to the mind in the form which none but mathematicians can conceive. There are others who feel more enjoyment in following geometrical forms which they draw on paper, or build up in the empty space before them. Others, again are not content unless they can project their whole physicd energies into the scene which they conjure up. For the sake of persons of these different types, scientific tmth should be presented in different forms, and should be regarded as equally scientific, whether it appears in the robust form and vivid colouring of a physical illustration, or in the tenuity and paleness of a symbolical expression."

J. C. Maxwell Presidential address on "Mathematicsand Physics" at the Liverpool meeting of the British Association 1870.

The search for analogous systems begins with the expectation that a set of dynarnical equations. once abstracted and written down, will describe not just the underlying naturai system from which it has been constructed, but rather an entire class of systems which rnay be very diverse. The fact that dynamical systems can have analogues makes it possible to study seemingly complex systems by means of conceptually different and intuitively simpler mode1 systems. "If al1 else fails, we can wait for an analogue" was Lorenz's (1963) response to limitations on his ability to understand the weather by mathematical modelling. Searching for analogies is in a way, an abstraction, and it can be a way of reducing the complexity of the problem, but it is fundamentally different from the approach of reducing the dirnensionality of the problem.

5.4 Mathemarical background

We can define a dynamical system formally if we specify the state of the system at time t by rn state variables For simplicity, I shall only consider discrete values of t, but note that t may be also taken as continuous. 1 also note that t need not necessady designate time, but may represent any other dimension, e.g., space, on which the States are ordered. 1 assume that each state variable is govemed by some function F such that:

The simplest dynamical systems are those which are differentiable, Le., those for which F is a differentiable function. In this case, following the dynamics is a matter of solving F for subsequent times t. Mathematicians point out, however, that very often, if at al1 known, F is not well-behaved in the sense that the system is not differentiable. In such 'bad' cases, the dynamical properties of the system may still be sought, but not through the functional calculus. The branch of mathematics which may then be relevant is then the geometrical approach of phase space. I elaborate on this approach. In this, we use the fact that a dynamical system possesses a unique trajectory in some open m-dimensional Euclidean space R. We cal1 R the phase space of the dynamical system. Considering t=ro as some initial time point, then any point in the phase space cmbe regarded as a possible initiai state. We rnay denote this point by

If we let time run for a duration ri, then the corresponding point in phase space will be

X(tl) = (X101 1, x201 ), X3(t!), *-• ,Xrn(tl) 1 (5.4)

The path P through phase space which connects X(to) and X(tl) may be called the trajectory of X(to). This may be done for every initial point, and thus the dynamical system induces a trajectory on Cl , in which each point X(to) in R rnoves dong its trajectory and at the end of any time tl occupies the uniquely determined point X(tl).

5.4.2 Where? : the Attrnctors

Since F is continuous and maps a closed simply connected region into itself, it possesses. according to Brouwer's fixed point theorem (Hu 1959), at least one limit point or critical point X(ti) in phase space. This is a point which the trajectory approaches closely and often. Mathematicians define X(tl) as a limit point of P if, for any e>O and any tirne tl, there exists a tirne t2(e,ti)>ti,such that IX(t2)-X(ti)l < e. The set of lirnit points rnay also be called the attractor of the trajectory. In general, dl dynamical systems possess at least one of three main types of attractors: point, cycle or strange. 1 shall illustrate the three types using simple and classical dynamical systerns, whose behaviour is well- known.

5.4.2.1 Point attractors

Example 1. The osciliating penduhm with dampening (friction).

Perhaps the simplest physical analogy to illustrate the point attractor is the movernent of a pendulum in response to some initial displacement, and friction (Fig. 5.4.2.1.1). The goveming equation of motion of this dynamical system may be written as:

* d20 de ~?d- = -cl-- mg1 sin 8 (5.5) d' t dt

There are two points of equilibrium of this system (when 1 is weightless), points where the angular velocity dûldt =O. These correspond to 0=0 (lower) and JE (upper). If the mass is slightly displaced frorn the lower equilibrium position (which gravity will do), the pendulum will oscillate back and forth with gradually decreasing amplitude until eventually re- approaching the lower equilibrium position, as the initial potential energy is dissipated by the damping force. This type of motion is called asymptotic stnbility and is typical of dissipative dynamical systems. The physical origin of asymptotic stability is the second law of thermodynamics, according to which equilibrium is attained in an irreversible fashion and corresponds to an extreme quantity known as state function or thermodynarnic potential. Dynamics in the pendulum converges on a point attractor of zero velocity and zero amplitude. What is important is that we observe convergence to this unique attractor, regardless of the initial conditions (Le., regardless of the amplitude), within certain limits. Changing initial condition simply changes the path of dynamics but not itfs endpoint. This fact is of course very important, should we think of vegetation dynamics, since there the path on which the process approaches the 'attractorf is also of interest.

Fig. 5 .M.1.1. An oscillating pendulum. A bob m is attached to one end of a rigid rod 1. The other end of the rod is supported at the origin 0, and the rod is free to rotate in the plane of the paper. The position of the pendulum is described by the angle 8 between the rod and downward vertical direction, with the counterclockwise direction taken as positive. The gravitational force mg acts downward, while the damping force, cldûldtl, where c is positive, is always opposite to the direction of motion of the pendulum. E-rample 2 Logistic equation with parumeters 2.000>r>0.

Here I present the dynarnical system which Batschelet (1971) informs us P. F. Verhulst ( 1804- 1849) introduced. This equation has been commonly applied in the study of population dynamics with much arbitrariness:.

N = poprdation size r=constant per capita growth rate K=car>ying capaci~of the environment t= time-step

May (1976) has shown that when 2.000>00 the population size Nt asymptotically approaches an equilibrium point. This point is always the same, regardless of the initial population size, the main difference being in the time to convergence.

Example 3. Stationnty Markov Chain.

When a dynamical process is pure Markov, the description of its Xt+ 1 state can be derived from the relevé of its Xt state and a known transition matrix P:

Xt+l= XtP (5.7)

Any square matrix of non-negative numbers with unit row totals will qualify as a transition matrix P. When XI and P are given, a stationary

Markov chain is completely defined (Feller 1968). A typical element phi of the transition matrix expresses the rate at which population h is replaced population i when the chah moves from one of its States to a future state. In the stationary chain, P remains constant and thus the trajectory keeps itself targeted on the same steady-state. This ensures convergence, but interestingly , and nothwithstanding contradicting declarations on this, the initial state matters. Different initial conditions will specify different paths (Fig. 5.4.2.1.2).

Fig . 5.4.2.1.2. Convergence in the stationary Markov model. The phase space is a 3-dimensional Eigenmapping. Three trajectories are shown in stereo. Each trajectory traces a pure stationary Markov chah. The transition matrices are the same. What differs is the initial conditions. The point attractor can be thus illustrated in several and variable dynarnical systems. One thing that these systems have noticably in comrnon is that the equations are simple to write down. They are few variables and few parameters. While 1 discuss applications to vegetation dynamics in later sections, 1 point out some points which have surely crossed the reader's mind: Dynarnical systems possessing point attractors will likely not hold much promise for modelling complex vegetation dynamics since the idea of a fixed inevitably unchanging vegetation state is inconceivable, and the chances of our ability to wnte down a simple goveniing equation is low.

5.4.2.2 Limit cycie

E'rrimple I. The oscillnting pendulum without dampening (friction). Consider the pendulum system in 5.5 in which the dampening coefficient c is zero. This is referred to as a conservative dynamical system. In this case, if the mass is displaced slightly from its lower equilibrium position, it will oscillate indefinitely with constant amplitude about the equilibrium position. Since there is no dissipation in the system the mass wil1 remain near the equilibrium position, but will not approach it asymptotically. This type of motion is stable, but not asymptotically stable. These ternis are discussed in more detail in section 5.4.3. Example 2. Logistic equation with parameters 2.4#9> r> 2.000. In this case, the previous equilibrium point becomes unstable and bifimates to produce two new and locally stable fixed points of period 2, between which the population oscillates with stability in a 2-point cycle.

The reader may be at this point wondering about the applicability of the limit cycle to studying vegetation dynarnics. 1 believe that we can dismiss this case as unrealistic for the sarne reasons discussed for the case of point attractors.

5.3.2.3 Strange attractor

Eiumple 1. Lorenz's 'simple' weather. Lorenz (1963) used a simple dynarnical system of three ordinary differential equations to define an idealized weather system as a hydrodynamical or fluid flow:

a the Prandtl nurnber. a = k-1v w here k represents thermal conductivity, and v represents kinematic viscosity. r = R~-f~~, where Ra is the Rayleigh number at some given constant a, Rc, it's critical (minimum) value, a quantity related to the development of convection (The criterion for the onset of convection is Ra=Rc).

He was modelling atmospheric convection currents with x being proportional to the intensity of the convective motion, y to the temperature difference between the ascending and descending currents, and z the distortion of the vertical temperature profile from linearity. Lorenz would feed his computer some initial conditions (today's atmospheric conditions (x,y,z)) to see what the predictions would be for the future. It seemed like a simple task, but in fact the results of his experiments were very unclear. He found that if he started his mode1 with initial conditions that were infinitesirnally altered, the final predictions would be totally different. Since he could not get rid of the rounding errors of the computer, his results were not reliable. When he plotted a variable against time, the pattern always ended up in chaos (unpredictability). But, what was so special about this was the fact that this unpredictability was not due to irregularity in external forces, but inherent in his completely deterministic system, Le., the rounding error, a random efTect. Lorenz's observation of sensitive dependence on the initial conditions was to be later termed the 'Butterfly Effect' and become one of the tenets of chaos theory. To try to get a cornplete picture of the behaviour of his system, Lorenz used each set of three numbers (x,y,z) as coordinates to specify the system state in three-dimensional space, and plotted the behaviour as a trajectory through time. If he was lucky, maybe eventually, such a path might lead to one place and stop., meaning that the system had settled down to a steady state. Or form loop, representing a repeating periodic pattern. But he found neither of these. His map displayed an infmite complexity and unpredictability. No pattern of points ever reoccurred. The trajectory never intersects itself, it only looks as though it does in 2 dimensions. At the same time it always stays within certain bounds. It never runs off the page, as it were, and in fact traces a distinctive shape. Lorenz described his dynamics as follows:

"When Our results concerning the instability of nonperiodic flow are applied to the atmosphere, which is ostensibly nonperiodic, they indicate that prediction of the sufficiently distant future is impossible by any method, unless the present conditions are known exactly. In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be non-existent." Lorenz called his dynamics deterministic nonperiodic flow. We now cal1 it deterministic chaos, and the Lorenz kind of attractor is cdled a strange

Although Lorenz may have been the first to discover this phenornenon, he was not the first to suspect its existence:

"...it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce a large one in the later. Prediction becomes impossible."

This quotation is from Poincaré (188 1). After Lorenz, deterministic chaos started appearing in other completely different dynamical systems. Example 2. Logistic equation with parameters ~2.570.

In this case, the population trajectory , although bounded, does not settle into any cycle. The underlying mathematical structure of the attractor is extremely intricate, but oblivious to the casual observer. Apparently erratic fluctuations in the dynamics of Say, a population need not be due to unpredictable variations in the environment or to sampling error, but simply from a strictly deteministic growth relationship expressed by the logistic equation. At first, this seems to run against biological intuition. A biological system, left unperturbed should tend towards some equilibrium, some simple set of limit points. Well, it is true that the system tends towards some equilibrium. But this is a dynarnic equilibrium in that the position of a state on the attractor is not predictable, yet still dependent on the initial conditions. May (1976) felt this discovery so fundamental that he made the foIlowing plea:

"1 would urge that people be introduced to the equation Xt+l = n Xt( 1-Xt) early in their mathematical education. This equation cm be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student's intuition about nonlinear systems. Not only in research but also in the everyday world of politics and economics, we would a11 be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties." Example 3. The oscillating pendulum with dampening (friction)and regrrlar perturbation.

Here we again consider the pendulum system, but this time with the addition of a forcing function

,d28 d'8 1nl-~=-cl--rnglsin6-Acos~t (5.9) d-t dt

where A is the amplitude of the forcing function and is the angular velocity of the forcing function. When these parameters of the forcing function reach a critical value, the pendulum dynamics becomes chaotic, sensitively dependent on initial conditions and moves dong a strange attrac tor. How realistic is the idea of the strange attractor in the view of vegetation dynamics? 1 discuss this in more detail later, but point out a few things here. The strange attractor presents us with a slightly more flexible view of the attractor. It need not be simple nor predictable, but must still be definably finite. This relaxation of the strictness of the attractors seems promising. 5.4.3 How?: thepath

Up to this point 1 have been focusing on the question of where dynamical systems end up. Now I turn to the process of how they get there. This requires a review of some of the established properties of trajectories. A trajectory is central if it passes close and often to any point through which it has previously passed and in this sense, separate sufficiently long segments of a central trajectory are statistically similar. A noncentral trajectory remains at a sufficient distance from any point through with it has previously passed, and thus approaches its attractor asymptotically. Its instantaneous distance from its closes limit point becomes increasingly small as t + m. This is so typically in the stationq Markov Chain which, for this reason, is said to have a transient property, i.e., a property which changes over the length of the trajectory. I shall term behaviour which is not transient, persistent. Stability refers roughly to the response of a dynarnical system to some kind of perturbation, more particularly to the limiting behaviour of the trajectories as t + -. A trajectory P is stable at a point X(ti) if any other trajectory passing sufficiently close to X(ti)at time t,, rernains close to P as t + W. Another way of putting this is that if the system is perturbed by modifying one or more of the state variables, the trajectory determined by the perturbed values of the state variables will retum to the set of limiting points of the original system. The movement of the pendulum is a typical case. The trajectory is asymptotically stable at a point X(ti) if any other trajectory passing sufficiently close to X(ti) at time tl returns close to P as t + m. A stable trajectory is unifomly stable if the distance within which a neighbouring trajectory must approach a point X(tl), in order to be certain of remaining close to P as t + -, itself possess a positive lower bound as ti+ W. We may also examine the system in terms of how its behaviour changes by modification of the equations of motions themselves (not just the state variables). The persistence of the shape of the trajectory with respect to sufficiently small perturbations of the equations of motion of the system implies str~ictzrralstabiliq. Since trajectories are detennined by single valued functions, any trajectory passing through a point through which it has previously passed must continue to repeat its past behaviour, and so must be periodic. A trajectory is called quasi-periodic if for some arbitrarily large time interval r, X(~+T)ultimately remains arbitrarily close to P. Periodic trajectories are of course special cases of quasi-penodic trajectories. A trajectory which is not quasi-periodic is called nonperiodic (or aperiodic). If P is nonperiodic, X(rti+r) may be arbitrarily close to X(rti) for some time ti and some arbitrarily large time interval, but if this is so, X(t+r) cannot rernain arbitrarily close to P as t + W. The reversibility of the trajectory in classical dynarnic systems is an explicit general mathematical property. Direction in tirne (forward versus backward) has no meaning. This property is however inapplicable to observed time series. Prigogine and Stengers (1984) state the obvious: "Manipulation and measurement are essentially irreversible ...A world in which ail trajectories are reversible is a strange world indeed". But how do we know that a process is irreversible? We can use the concept of entropy. The second law of thermodynamics States that systems tend toward a state of disorder or maximum entropy. The dynamics involved is spontaneous and irreversible. Moving from thermodynamics, to dynamics in general, we can use Boltzman 's interpretation of entropy in a statisticd sense. Basically, he relates high entropy to high probability, and gives the fornial definition of entropy H as

If we have a dynamical system which is describable in probabilistic terms, then we can apply this definition of entropy to examine whether or not the observed trajectory follows the second law. Does entropy increase uniformly? Does the system tend uniformly towards disorder? Let us consider again the stationary Markov process, and ask the question, 'how does the entropy of the Markov process evolve?'. Simply, we would find that as time moves on, entropy would increase unifomily. The system thus evolves from order to disorder, in the technical sense. Prigogine and Stengers (1984) suggest that this uniform increase in entropy is what gives the Markov chain a one-wayness in tirne, i.e., irreversibility. They further point out however, that the increasing of entropy is not based on an arrow of tirne present in the laws of Nature, that say. the chain may be representing, but of Our decision to use present knowledge to predict future (and not past) behaviour. That is, in defining that chain mathematically. it is of consequence that we Say,

and not,

even though in fact P-1 always exists given the constraints of P. They Say,

" it is because the future is not contained in the present and that we go from the present to the future that the arrow of tirne is associated with the transition from present to future". But probability presupposes a direction of time, and thus cannot be used to derive the arrow of time. Of course, the second law of thermodynamics applies only to closed systems and so it made sense to apply it to a theoretical dynarnical system such as the stationary Markov chain. However, its application to Nature is not so clear. For example, we would have to admit that the vegetation system is open. By this 1 mean that there is a continuous exchange of energy between the system and its environment. Vegetation is not a closed system; solar radiation is a persistent source of variable energy. The second law does not apply. The entropy time series does not apply and the vegetation system may not show a unifom tendency toward equilibrium and may even seem to move backwards in tirne. Clernents' (19 16) "retrogression" is an example of this. While Odum (1969) insisted that the vegetation system follows the second law, many (e-g., Margalef 1989) have shown that entropy does not always unifomily increase with time.

5.4.4 Constraints

The first constraint on dynamical systems theory is the assumption that al1 problems, simple or cornplex, resemble one another since they cm al1 be described in the same general form. However only the inteprable systems that have been the mode1 par excellence of dynamical systerns. Furthemore, Prigogine and Stengers (1984) make the point that "simple, integrable systems can indeed be reduced to noninteracting units, but in zoeneral, interactions cannot be eiiminated." 1 suppose there must also be some mathematical logical constraints forming a basis to the fact that the where and how usually form particular patterns inherent in the classical theory. For example, by definition, the attractor of a persistent periodic trajectory rnust be a limit cycle. A stationary Markov Chain converges to a point attractor. In these cases there is no logical distinction between the path and the destination. Knowing one is sufficient. But clearly these dynamical systems are simple and are only a small subset of the variety of systems that may be wntten down. We are faced with the question of "do transient properties exist, and if so, how important are they?" The point being that if there are no transients, then, we need not distinguish between the path and the attractor. But if there are transients, then we may inquire how is the attractor approached, and when is the attractor reached. Can we make any generalizations about the relationship between the dynamical system and its behaviour? Does the nature of the equations have anything to do with it? While most dynarnical systems can be characterized in terms of what their behaviour (how and where) will be, there still seems to be no universal classification of dynamical systems. It is still possible Say, to be given the fomulae of some unfarniliar dynarnical system and have no idea what it's attractor will look like and what its trajectory toward this attractor will be without actually tracing it ad infinitum. This is particularly the case if there are significant high-order relationships present. Can purely linear systems have complicated trajectoiies and attractors? There are indications to this. But beyond these, it is difficult to generalize further because we have seen an example where essentially the 'same' dynamical system changes its behaviour quite dramatically simply in the face of changing the value of one parameter. Which, if any. of these systems provides a reasonably realistic view of vegetation dynamics?

5.5 Application

5.5.1 Past attempts

So far we have discussed concepts in dynarnical systems theory more or less in a purely abstract sense. We have seen that there is a considerable wealth of information in the characterization and classification of dynarnical systems, but the theory is not complete. The insuficiency of the theory becomes even more evident when we attempt to link theory to understand observed phenornena. Recall that a dynamical system in theory is defined as

The set of points X when connected, define a trajectory or process. If the States are descriptions of vegetation at a given spatial scale, they are called coenostates. A time-series of the vector of coenstates at a given time scale is called a coenosere. To study the process of vegetation dynarnics, the coenosere may be examined and classified by various properties which basically reflect the two questions of 'how do they evolve"? and 'where do they go''? What is given is a set of observations over time (a time series)

with m representing the dimension of the observed system (eg., number of species in the community) and c, the total number of time-steps (observed time-span). The question is how well can some theoretical dynamical system descnbe and predict the observed dynamics? Lorenz (1963) realized the problem in his dynarnical system: "There rernains the question as to whether Our results really apply to the atmosphere. One does not usually regard the atmosphere, as either deterministic or finite, and the lack of periodicity is not a mathematical certainty, since the atmosphere has not been observed forever. The foundation of Our principal result is the eventual necessity for any bounded system of finite dimensionality to corne arbitrarily close to acquiring a state which it has previously assumed. If the system is stable, its future development will then remain arbitrarily close to its past history, and it will be quasi-periodic ...Lack of periodicity is very common in natural systems, and is one of the distinguishing features of turbulent flow". As stated at the outset, vegetation systems are dynarnical in the sense that parameters change with tirne or space. The underlying niles aaoveming the change are still actively sought. While these systems admit a dynamical description, the description cannot be perfect and must rely on abstraction into the language of dynamicd systems theory. The goal is to write down the governing equations of the system. This has in fact been attempted using simple differential and difference equations (May 1976; Van Hulst 1979; Wedin and Tilman 1995). However, the problems which arose from these applications were in how to assess validity of these models and in the realization that limited numbers of biological variables could be accommodated. Knowing that the vegetation system (or our descriptions thereof) rarely, if ever, conforms to precise mathematical niles, we become interested in what the depamire is from these niles. Schaffer (1985) daims the "as a practical matter, ecologists will probably never be able to write down the complete governing equations for any natural system." Then the rules are usually sought through an approximative process. This brings us into the domain of statistical theory. One of the wqs classical statistics rnay be applied to this problem is what has been called "tirne series analysis". To explain the idea, Kendall and Stuart (1976) suggest that a typical time series (real observations over time) may be composed of four parts: i. A trend, or long-term movement. ii. Oscillations about the trend. iii. A seasonal effect. iv. A "random", "unsystematic" or '1 irregular"- component. They point out that the objective of traditional time-series theory entails examining the data using these components with a view to isolating them for separate study. Time-series analysis is not after a precise rule of time- dependence, but rather seeks out preconceived trends such as periodicity. Kendall and Stuart point out the fact that the decomposition of a series may be very often usefùl, but potentially misleading, asserting that this is not the ultimate objective of statistical analysis. This is a keen observation, which 1 would like the reader to keep in mind. I see this as the main justification for the conclusion that classical time-series methods can present a hindrance when dealing with complex natural dynamics. That is, natural dynamics cannot simply be decomposed into the above cornponents, and that isolation of certain undesirable effects, or 'noise' from the rest, is not an appropriate way to mode1 these dynamics. Another approach is that of stochastic models, such as those making use of the Markov chain. Markov modelling of vegetation dynamics is much discussed (Dale et. al. 1970; Waggoner and Stephens 1970; Hom 1974, 1975; van Hulst 1979; Binkely 1980; Acevedo 198 1; Usher 198 1; Lippe et. al. 1985; Orloci and Orloci 1988; Orloci et. al. 1993). albeit with varying conclusions as to its value in describing the natural process (e.g., compare results of Hom 1975; Lippe et al. 1985; Orloci et. al. 1993). Clearly, the model incorporates Clementsian determinism : predictability and convergence to a monoclimax. If the transition probabilities can be determined (for an exarnple method consult Orloci et al. (1993)),the properties of stepwise predictability and convergence to a point attractor make it a desirable choice to measure determinism. But once the transition matrix has been fixed, it remains fixed and impervious to external chance effects. Furthemore this stationary Markov model is linear and its attractor is insensitive to initial conditions. For this reason it is also impervious to interna1 chance effects. To Ian Stewart (1989) "statistics is a method for panning precious order from the sand of complexity". But to the practitioner a more commonsense definition like 'accounting for chance effects' is more clearly interpreted. While this is true, traditional statistical models like those used in time-series analysis recognize the presence of chance effects, chance is viewed as something to be isolated and avoided. Not too long ago 1 was guilty of this! "...the chances that unimportant, rarely occurring species which have appeared by chance may becorne a problem. Thus a data set which is unsummarized my actually swarnp underlying trends, making them unclear and umecognizable" (4th year thesis: "Markov Chain: a descriptive model for vegetation recovery" 1993) 5.5.2 Is chance a necessity ?

As we have seen in the exarnple of the Lorenz system, even the simplest models are susceptible to chance effects. In some mode1 systems we can't ;et rid of it, hard as we may try. But do we want to in the first place? 1s chance not only a reality but also a necessity? In this section 1 would like to re-examine the notion of 'chance' as it remains an important factor governing our view of the natural world in general, and thus in al1 Our attempts to understand it. What is 'chancef? 1s it inherent, or is it simply ignorance on Our part? A common definition of chance is in the interpretation of the intersection of two independent causal sequences (Ekeland 1993). If we cannot discem causality, then we cal1 it chance. But Poincaré was less optimistic in his statement: "A very small cause which escapes Our notice determines a considerable effect that we cannot fail to see, and then we say that effect is due to chance." Ironically, Poincaré was pointing to the property of sensitivity to initial conditions. This ties together the concepts of precision and chance. The observer will only be as precisely inforrned as the precision of his measuring instrument or keenness of the observation. In unstable, chaotic systems, even very good knowledge of the initial conditions will not guarantee predictability, and chance wiil appear to reign. If we consider entropy to be a measure of unpredictability, then we can quantify this kind of chance. For al1 practical purposes, chance becomes an inherent part of the system. Stone and Ezrati (1996) emphasize that it is only over the last century when ideas of chance and probability have been utilized to explain variability in the natural world. Preceding this was a fixation on the deterministic view emerging from Newtonian principles. They point out though that once this step was made, major disciplines were sometimes completely reappraised giving the example of the physics of quantum mechanics. Perhaps it is time for a similar reappraisal of Ecology. If chance is inherent in ecological systems, we can ask, what role does it play? The classical argument that chance is a necessity was introduced by Monod (1971). The idea was that the theory of evolution through natural selection had to acknowledge the omnipresent chance effects. The present theory of vegetation dynamics recognizes that the distribution of vegetation in time and space must be considered concurrently, and that "the composition and subsequent development of vegetation is partly stochastic and partly determined by the self-modifying properties of a dynamical system" (Roberts 1987). But it is not clear how to accommodate these effects. VanHulst ( 1992) States, for example, that "the introduction of a stochastic component in a population dynamics mode1 can give results that are often at odds with expectations derived from corresponding deterministic models (Tuljapurkar, 1989)". It is generally agreed that vegetation dynamics is neither completely deterministic, nor cornpletely random, but what remains lacking is the theoretical frarnework and methodology to deal with this resolution. A new synthetic theory is needed (Usher 198 1, Pickett and McDonnell 1989; Smith and Huston 1989; Pimm 1992; McCook 1994). Interestingly, Usher (198 1) considers the Markov model tu provide "a unifying concept in the dichotomous approach to succession". He claims that Markovian models cm be viewed as incorporating Clernents' view of orderly, predictable change, but also Gleason's views of reductionism by incorporating the fate of individual populations. However, while the fitted stationary Markov model may help to alleviate the holistic/reductionistic controversy, it clearly makes no explicit provision for Gleason's chance mechanism. In the next chapter 1 look at a case study to examine how this problem might be resolved. cold no wonder where's her coat, ncw? it was covering her costume give her a blanket, now she has work to do it might be a fever making her shiver she's usually quite waxm, they Say what was she doing outside, anyway? Chapter 6

CASE STUDIES

"...the idea that a "class" must always be enormously broad and abstract is far too limited. The reason is that Our thought makes use of an ingenious principle, which rnight be called the prototype principle: The most specific event can serve as a general exarnple of a class of events ... Everyone knows that specific events have a vividness which imprints them so strongly on the memory that they can later be used as models for other events which are like thern in some way. Thus in each specific event, there is the germ of a whole class of similar events. This idea that there is generality in the specific is of far-reaching importance."

D. R. Hofstader Godel, Escher, Bach: an etemal golden brnid.

6.1 Short-term dynarnics: Atlantic heathland recovery

My first case study is vegetation dynamics in an Atlantic heathland after severe grazing and fire. The choice was based on two simple criteria: availability (Lippe et. al. 1985) and sufficiency of data. By sufficient 1 rnean complete and detailed. The data set is shown in Table 6.1.1.1. Table 6.1.1.1 Point cover estimates. A total of 8 species, 1 species group and bare ground after severe fire and grazing. Legend: BG - bare ground; EN - Empetrrrm nigrum; CV - Calluna vulgaris; ET - Ericn tetralix; MC - Molinin caenrla; RA - Rurnex acetosella; JS - Juncus sqrrarrosris; CP - Carex pihrlifera; OS - other species. Adapted from Lippe et. al. (1985). 6.1.2 Propositions

6.1.2.1 A comm~ini~level view

The first contradiction to be resolved is whether dynamics should be viewed as the behaviour of individual populations (reductionist) or at the higher, community level (holistic). Fig. 6.1.2.1.1 shows complicated individual changes in populations in the community. To consider the dynamics at the level of individual populations we have to somehow conceptualize many kinds of dynamics simultaneously. But how do we visualize a complicated 9-dimensional process? Whether or not vegetation dynamics is driven by individual population dynamics is not the question. The question is whether interactions and feedback which are manifested at the community level should be an integral part of the reasoning. Van Hulst ( 1992) makes the point that "it is quite conceivable, for example, that unordered, largely random fluxes of different populations will underlie a behaviour that is more or less predictable at the community level, but one which is not easily predictable from knowledge of the fates of the individual populations alone, just as the behaviour of a gas is not easily derived from the behaviour of its molecules." This approach, however, must involve summarization because the medium is unworkably many-parted. It is not unusual to find that, while hundreds of species occur in the community, one, two, or perhaps three, are chosen to build a mode1 of dynamics. However, realistically, no one could or should suggest a priori importance of this kind. Summarization cannot be achieved by simply reducing the cornmunity to two or three species. It is true that multivariate statistical analyses in ecology are mainly techniques of summarization. In these studies, however, summarization is achieved through simultaneous analysis of al1 species (Orloci 1978) and not by arbitrary picking of two or three. In the example, a parsimonious summarization is achieved by projecting the 9-D configuration into 2-dimensional space using Eigenanalysis (Fig. 6.1.2.2). In doing so, we move from the population- level, to the community-level. The latter incorporates the former, and while a one-to-one correspondence is lost, the community-level mapping still retains a huge portion of the original information. Upon inspection of the total process at the comrnunity level in Fig. 6.1.2.1.2, a 2-phase structure is immediately seen, which was not obvious from the population-level view in Fig. 6.1.2.1.1. Thus it is worthwhile to adopt a community-level view. We see that in the early phase of dynamics, roughly the first eight years, a simple linear law applies. The rate of change of composition in this phase is relatively fast, which is not unusual (Shugart and Hett 1973; Orloci 1993). But then suddenly at year 8 the process slows down and continues on dong a Brownian-type path. This is the phase of violent turbulence, as it were, under increased random effects. Results on analysis of the phase structure is given in Table 6.1.2.1.1. Year

Year

Fip. 6.1.2.1.1 Population-level dynamics. Populations divided into dominant (top: bare ground; Empe tr-zim nigrum ; Call~inavnlgaris; Erica tetrczlix )and rare (bottom: Molinia caerula; Rumex acetosella; Juncus squnrrosus; Carex pilulifern; other species) types. Data in Table 6.1.1.1. Fig. 6.1.2.1.2 Two-dimensional Eigenmapping of the 19-step recovery trajectory (A) from Atlantic heathland (Lippe et. al. 1985) disturbed by fire and heavy grazing. Magnified section of last Il steps (B) involves vertical distortion. The original data represent point-cover estimates of 7 species, one species group and bare ground. Step size is one year. Nurnerals indicate years elapsed. The mapping is 99% accurate, 95% on horizontal axis (in A). Reference axes (principal components) have been removed for ease of viewing. Note linear early phase and nonlinear late phase. Break between phases coincides with point cover of bare ground dropping from 57% to less than 7%. Table 6.1.2.1.1 Partition of sums of squared deviations from centroid within and between recovery phases (Fig. 6.1.2.1.1). The dominant source is linear determinism. The separation of phases is dramatic.

Source df Sumof Mean squares square %

Within linear phase 7 43 16 617 53 Within nonlinear phase 10 493 49 6 Between phases 1 3335 3335 41

. -- - Phases pooled 18 8 144 452 100 But what does this mean? 1s it unusual that a process begins with clear linear determinism, then suddenly strikes out, ending in apparent chaos? 1s this kind of dynamics best described as Clementsian or Gleasonian? It seems at the outset as though neither theory will do justice to this case on its own, and as in such cases the determinism vs. disorder contradiction cornes into focus.

6.1.2.2 Detenninism in the phase strucrure

Ecological studies of succession entai1 a systematic search for regularity. When a phase structure is discovered, signiticance is obvious. Not surprising, ecologists saw it fit to use the phases to classify plant communities chronologically as seral or climax (Clements 19 16; Nichols 1923; McIntosh 1980; Fekete 1985). 1 believe that the seral vs. climax classification is a consequence of just how determinism and randomness trade importance in time. But it should be noted that a sirnplistic isolation of phases may be deceiving with respect to the relative contributions of determinism and randomness to the whole process. Nevertheless the first question to ask is, how much determinism is observable in the recovering heathland? We can try to apply a classical model. I chose to measure determinism as the level of concordance1 (C) with the fitted discrete-time stationary Markov model (Orloci et. al. 1993). The tight fit of model and process is observed in Fig.6.1.2.2.1 for which C is so high that it departs by 7.3 standard deviation units (s.d.2) from what would be expected if randomness were the sole deteminant of change.

l~heterm concordance describes the level of match between two structures. To define tnese structures we first compute the two c x c Euclidea~.distance matrices DX and DM. corresponding to coenostate records XI, X2, . . ., Xc and the Markov scores recoras Ml, MI, - - ., Mc such chat ME-1 = MtP with M~=XI. The coefficient C 2 1 (d~jk- dbfjk! S~(D,;%) = c

(d~jk+ dMjk12 j

6.1.2.3 Determinism ccznnot be simply decomposed

Intuitively, one would think that an overail high Markovian determinism is a residual of the early linear phase. But this is not e ctly true. In fact when 1 examine the phases independently 1 find that s.d. is 5.6 in the early phase and a meager 0.5 in the late phase (Table 6.1.2.3.1 ., Fig. 6.1.2.3.1). Observing that 5.6 + 0.5 < 7.3, deteminism in the total process is not simply related to determinism within the phases. How can this be possible? Shouldn't randornness always mess things up and make thinps less clear? Table 6.1.2.3.1 Cornparison of observed and simulated distance configurations in Atlantic Heathland recovery by phase and by phases combined. Values measure the concordance of the observed and Markov configuration in standard units assuming 100% randomness. Increasingly lower values indicate increasingly likely advent of the nonlinear turbulent phase. In these terms, the total trajectory has less evidence of turbulence than does the linear phase.

Phase Standard deviation units s.d.2 from random configuration

Linear phase (years 1-8) Nonlinear phase (years 8- 19) Phases combined (years 1- 19)

The fascinating observation that addition of what they cal1 'white noise' can actually enhance the appearance of determinism has been made by others (Schaffer et. al. 1986; Moss and Wiesenfeld 1995). Moss and Wiesenfeld (1995) called the phenornenon 'stochastic resonance' and describe it with the following three points:

1. "the key feature in any system in which stochastic resonance occurs is that the relation between its input and its output is nonlinear." 2. "random noise can benefit faint signals in nonlinear systerns by boosting them over the threshold of perception" 3. "the degree to which a signal benefits depends heavily on adding just the right amount of random noise. If too little is added, the signal is not significantly boosted. Similady, adding too much noise overwhelms the enhanced signal" There is in fact much to be said for the simple-minded wisdom behind the idea that focusing on the whole reveals more than viewing the sum of its parts (Vandermeer 1990; van Hulst 1992). This is further confirmation of the value of the holistic view. So, while the simple stationary Markov chain gives a pretty good fit, can we be satisfied with it as a mode1 of the natural process? No, for at least two reasons: Firstly, we cannot ignore nonlinear deviations from the pure linear Markov path altogether --as hard-core statistics likes to do -- when we see they are quite important, actually definitive in the Iater phase. Second, we have seen that determinism cannot simply be decomposed. Randomness seems to pervade the process in ways which are not clear. If we are dealing with stochastic resonance, we cannot ignore this.

Total ~~CBSS Years 1-0 Yearr 9-1 0 s.d. = 7.3 - s.d = 5.6 +

9 12 Fig. 6.1.2.3.1 Decomposition of determinism in sequential phases. The calculation of s.d. is discussed in the main text. 6.1-2.4 Small chance effects can produce dramatic outcomes

Looking back on "reality" in the disguise of a 2-dimensional trajectory, 1 ask: Could the transition from linear phase to nonlinear phase be connected to simple generating rules'? Can we corne up with a better model in which chance has a part to play? Knowing that detenninism in the process is well captured by the Markov model, it makes sense to try to rework it.

1 examined the generic characteristics of the trajectory of a Markov mode1 perturbed by arbitrary perturbation3 to the transition matrix. The model, defined by the natural initial state and the natural transition probabilities, passes through a large number of steps under continuous impact by on average low-level random perturbation. Long-term behaviour is of interest to discover if we should expect simply more of the same. or if in fact a nested or cyclic phase structure is native to the process. Hastings and Higgins ( 1994) have argued that long-term behaviour can be "irrelevant to the understanding of natural ecological systems because the form of dynamics changes over long time scales." Notwithstanding the well-articulated contrary views, the discovery of phase structure (transient behaviour) over any time span is not a hindrance but rather an important revelation in line with what we consider as ecological information. It is important to realize that chance cmenter into the model at various levels. We can perturb the successive states, or we cm perturb the

3?4e note chat che percentageç for distortion are upper limits and for this reason the actual randon distortion of a specific transition prcbability at arry step is between 0% ana the given upper limit. transition matrix. 1 chose to perturb the transition matrix because 1 felt that perturbations are not species-specific and should have indirect effects. This is manifested in the fact that any change to an element of the transition matrix will cause a 'domino effect' throughout the community since two underlying assurnptions, must always be held: 1. fixed canying capacity, and 2. proportionality of losses and gains. Now the initial perturbation can be global (whole matrix) or local (entire row or single elernent). The former is analogous to a change in the entire community, with each species responding in different ways. Net gain/loss of resources in community are equally apportioned to the community. In the latter, perturbation results in changing of the behaviour of a single randomly chosen species with respect to its ability to replace other species. Net gaidloss of populations in the community are still equally apportioned. We cannot impose any rules more complicated than the above, such as, for instance. perturbation and apportionment based on species- dependent functions, because it is impossible to satisfy sirnultaneously both row total and column total constraints of the transition matrix. 1 report here some of rny related findings. Firstly, 1 found that the perturbed Markov chain can definitely provide a better fit to the data (Table 6.1.2.4.1). This encouraged me to examine this model's long-term behaviour, and 1 found that dynamics in the basic (stationary) Markov mode1 could change quite dramatically in the long term under the effect of minor perturbations. Figs. 6.1 .M. 1 and 6.1.2.4.2 show examples of this behaviour to demonstrate the striking qualitative nature of change. In the phase space view, a homogeneous linear process is transformed to a two- phase process, early linear and late complex nonlinear. The distance profiles allow us to visualize the process as a conventional time series plot. Here again we see the clramatic difference between the pure Markov process (0%) which shows an expected convergence to zero distance between successive States (a point attractor). The perturbed processes show uniformly no convergence on a point attractor, or for that matter, on any cyclical attractor. If we were forced to characterize the attractors in terms of those known to dynamical systerns theory we rnight suspect that they are strange and become stranger as perturbation increases. It becomes quite reasonable to ask the question of whether the explosive behaviour that we observe in this dynamics is an inherent property of the process.

Table 6.1.2.4.1 Fit of various Markov models to observed (natural) process. Numbers reflect the concordance of the observed and given Markov configuration in standard units assuming 100% randomness as the alternative. Increasingly lower values indicate increasingly poor fit of mode1 to Nature. The best fit is by the 5% perturbed Markov chain.

Process Standard deviation units from random configuration

Pure Markov 7.3 Perturbed Markov 5% 15% 25% 50% Fig. 6.1.2.4.1 Phase space (Eigen) mappings of the Markov process. Chain length shown is 950 steps. Maximum random perturbation levels: 0% (a), 5% (b), 15% (c),25% (d), 50% (e). Fig. 6.1.2.4.2 Euclidean distance profiles of the Markov process fitted to heathland data. Distances are on vertical scale and steps (947 shown) on horizontal scale. Random perturbation levels: 0% (a). 15% (b), 25% (c), 50% (d). 6.2 Simrcluted data, deterministic chaos

6.2.1 Relevant chaos-theoretical concepts

The first point to be considered is why should we care whether or not what we have is 'deterministic chaos'? Chaos theory represents a mathematical formalization of the view that determinism and disorder, simplicity and complexity, non-random pattern and unpredictability need not and should not be considered mutually exclusive. It provides a promising framework in which to pose Our questions about vegetation dynamics where we find that these things are in fact not easy to isolate. The theory allows us to look for underlying determinism in an apparently random and explosive process. The central concepts in chaos theory rationalize the approach to modelling vegetation dynamics, and this is rnainly due to the following discoveries about general systems:

1. sensitive dependence on initial conditions (Lorenz 1963). Minute changes in the local initial conditions can accumulate and have consequential effects in the long-term. In our case study, for example we can ask: 1s it due to the accumulation of the small randorn effects which are not detectable in the early phase, but suddenly manifest themselves qualitatively as a phase transition? This suggestion would not be out of line with Gleason's statement: "In addition to the imperfections of similarity [between vegetation states] caused by chance, and largely masked by them, are other variations of a cumulative naturet'.

2. simplicity giving rise to complexity. This is probably most familiar to the reader of popular science journals in the discovery that complex and beautihl patterns can arise from very simple mathematical rules. More formal derivations are in Mandelbrot's (1983) introduction of fractal

boeometry which allows us to describe and classify these objects/patterns. In Fig. 6.1.2.4.1, we cm ask, how complex is the nonlinear phase? How can this cornplexity aise from minor perturbations away from the pure Markov path?

3. complexity giving rise to simplicity. The evolution of a fem leaf pattern is arising from a completely random generation of points (Gleick 1987) is an example. Accordingly, from Fig. 6.1.2.4.1 we can ask, is there any underlying structure in the apparently chaotic nonlinear phase?

So is it reasonable to suggest that ecological systems are chaotic? Ecological theory is clear on the existence of a feedback mechanism, through which population and community level processes change the environment, which in turn change the community itself (Clements 19 16). It is well-understood that feedback can amplify small random effects and change a simple predictable dynamics into a chaotic one (Lorenz 1963;

Çambel 1993). In Our area, biological populations have been known to show chaotic dynamics (May 1987; Berryman and Millstein 1989; Tilman and Wedin 199 1; Hastings et.al. 1993). But most have considered population variables taken one or two at a tirne. 1 believe that this is just part of the community story and ask, can a multispecies collection, the community, behave chaotically? The existence of a feedback mechanism in the presence of many small random effects should already be sufficient ground to suspect that comrnunity level chaotic dynamics exists--and 1 am not the first to suspect this (Godfray and Blythe 1990; Vandermeer 1990; Stone and Ezrati 1996). 1 now consider how to go about testing for it.

6.2 -2 Testing for chaoticiîy

Tools which allow to probe the phase space trajectory for deterministic chaos, and thus suggest responses to the questions already posed above include Lyapunov and Mandelbrot mathematics:

1. Lyapunov exponents. The purpose of this measure, explained by Wolf et. al. (1985), is to rneasure the divergence of the trajectories of initidly close points (vegetation states). If the divergence is large (exponent positive), then it is legitimate to suggest that the process is sensitively dependent on initial conditions or chaotic. In these, the Lyapunov exponents are considered a "test for chaoticity".

2. Fractal dimension. This measures structural complexity of a process and its attractor. Typically, the attnctor of a chaotic process has high fractal dimension (Peitgen et. al 1993).

To apply these tools we require far more observations than we could realistically expect from in-site ecological studies. This problem, however, may be resolved by relying on simulation. 1 briefly discuss the concepts and comment upon further methodological difficulties arising in application. 6.2-2.1 Lyapunov exponents

The Lyapunov exponent as diagnostic for chaotic systems is described by Wolf et. d.(1985) , although it is the brainchild of Lyapunov. It is a direct measure of 'sensitive dependence on initial conditions', the simplest definition of chaos. It expresses the degree of divergence of two or more trajectories in the phase space with initially close initial points. If the divergence is large, that is, if the trajectories, end up far apart, then dynarnics in the system is considered sensitively dependent on the initial conditions. Put another way, the Lyapunov exponents are the average exponential rates of divergence. How are these exponents calculated? For systems whose dynamical equations are known, the calculation is quite straightforward (Peitgen et al. 1993). The method amounts to monitoring the amplification of small errors in the specification of the initial conditions:

The letter c represents the average growth of infinitesimally small errors in the specification of the initial condition and n is the iteration number, En is the value of the error at iteration n and Eo is the initial error. If oO, that is, if the rate of growth in errors is exponential, then the system is declared unstable, Le., sensitive to initial conditions. Of course, in the perturbed Markov mode1 we no longer know the precise equation goveming Our dynamical system. In that case, the approach of foltowing error amplifications rnentioned above is not applicable. That is, we do not know the phase space in whkh the natural trajectory exists, nor do we know the precise rules of its evolution. Wolf et. al. ( 1985) solved the problem of finding Lyapunov exponents for experimental data in an ingenious way through simulation, in which the attractor is reconstructed from the observed time series. Although this involves a single trajectory, the points simulated can be considered to lie on different trajectories. Tracing the evolution of initially sirnilx points on these then becomes possible. 1 surnmarize the algorithmic steps in Appendix B.

6.2.2.3. Frçrctal dimension

In light of the importance of sale resolution in Our perception of natural patterns and processes, Our view of the vegetation is necessarily multiscale. The 'nesting' of patterns and processes that we see through the scales invokes a rather loose, but nonetheless lucid analogy: the vegetation systern is fractal. 1 refer the reader to Appendices A and B for definitions. This analogy provides common basis for a widespread qualitative ecological observation, but in order to validate the analogy, several questions must be asked, which essentially serve as pre-requisites for searching for the fractal property: 1. What constitutes a set? 2. How best the fractal dimension (D) is estimated? Ecologists have enthusiastically and almost slavishly ernbraced the work of Burrough ( 198 1) and his rnethodological 'discovery' that the fractal dimension cm be estimated directly from the semivariogram, a variance of incrernents function of the data. Palmer (1988) suggests that the fractal dimension is an index of the degree of spatial dependence of a variable. He 'discovers' that the fractal dimension of vegetation varies as a function of scale. Thus, he concludes. that while fractal, vegetation is not self-similar. However. patterns of 'homogeneity' and 'heterogeneity' cm be summarized by a plot of fractal dimension as a function of scale. Phillips (1985) applied the rnethods and concludes that the fractal dimension is a useful index of the complexity of environmental variables. Low D values indicate a domination of long-range variation. High D values indicate complex processes where short-range, local variations are important. These applications have confirmed what has already been shown in pattern ardysis of vegetation, which has always had its basis in analyses of variance (e.g., Greig-Smith 1983). It's the concept of the fractal dimension as a measure of (a certain kind of) 'complexity' which is novel. Kenkel and Walker (1993) give a good review. 1 have adopted Burroughfs method of calculating fractal dimension but would like to point out some serious drawbacks which have been ignored by ecologists. These rnay have serious implications on the appropriateness of its use. 1 refer the reader to Appendix B for algorithmic spec ifics. 6.2.3. Results

In this section 1 present the results of testing the perturbed Markov chain for chaoticity. Using the algorithm outlined above 1 was able to view dynamics in the reconstmcted attractor space (Fig. 6.2.3.1). This is a necessary step towards finding Lyapunov exponents. What is clear is that the two-phase structure which was evident in the observed data and the simulated data at up to 15% perturbations is captured quite well in the attractor space. The simulated attractor appears quite complex and its complexity appears to increase with perturbation. Furthemore, the complex nonlinear phase seems to corne about explosively. To determine whether or not the attractors were "strange" 1 calculated the fractal dimension and Lyapunov exponent for each case. These results are shown in Table 6.2.3.1. The pure Markov process has lowest fractal dimension which, according to popular wisdom. implies low structural complexity and reflects high spatial dependence of successive points. The pure Markov process also has a high. negative Lyapunov exponent which is interpreted to mean that the pure process is insensitive to initial conditions (will converge on a point attractor). This case is a quick test on how good the exponents are as a measure of sensitivity. Generally, as we increase the amount of perturbation, we increase fractal dimension and Lyapunov exponent. At the 25% level, the process starts showing definite sensitivity to initiai conditions (positive Lyapunov exponent.) Fig. 6.2.3.1 Phase space mappings of pure and perturbed Markov process (left) and reconstmcted attractors (right). Chain length is 950 steps. Distortion levels: 0% (a,b),5%(c,d),15% (e.f)., 25% (g,h), 50%(i,j). Table 6.2.3.1 Fractal dimension and Lyapunov exponent of reconstructed attractors in processes as shown. Increased fractal dimension implies increased complexity in the attractor. Positive Lyapunov exponent indicates chaoticity.

Process Fractal Lyapunov dimension exponent

Pure Markov 1.50 Perturbed Markov 5% 1.89 15% 1.89 25% 1.91 50% 1.85 100% 2.00

6.3 Long-rerm dynarnics: Postglacial Jackson Pond

6.3.1 Data description

This case study presents results for long-term records of vegetation dynamics using fossil pollen records. These came from public domain archives (Intemet). The specific data set, Jackson Pond, Kentucky USA. was chosen under the guidance of L. Orloci. The criteria were completeness of data, including proper calibration and dating of core samples, length of the time series and location. Location was important for two main reasons: 1. In the Southem sites the sediment profiles go back much farther in time than in the Northem sites and 2. adverse relapses in climate did not destroy the sedirnents (peat) as much as in the North as the Laurentide ice sheath moved back and forth before it melted back to the Arctic. The complete raw data set is given in Appendix D. Al1 94 terrestrial species were considered. The timespan is roughly 20500 years. Wilkins et. al. (1991) discusses the methods of data collection and sediment dating and other specifics about the field site.

The first problem encountered was the question of which time-step to use. Certainly a short time step such as in the Atlantic heathland records was out of the question. To make things more complicated, the timesteps available in the raw data are, in the main part. too large and inconsistent. Moving backwards through time, initially the step is a reasonable roughly 50 years, but in the older sediment the time step increases to roughly 150 years. The data set was thus reworked using a linear interpolation technique to obtain a constant timestep of 100 years. This resulted in a coenosere of 205 States. A principal components analysis revealed that the data set could be efficiently summarized by three axes accounting for a cumulative total of about 93 % of the total variation. The community trajectory through the phase space defined by these scores is shown in Fig. 6.3.2.1. present

Fig. 6-321 (a)-(d) Three-dimensional Eigenmappings of the 205-step postglacial vegetation trajectory at Jackson Pond, Kentucky (Wilkins et. al 199 1). The configuration account for 93% of the variation. The original data before interpolation to equalize steps represent pollen density estimates for 94 species in 54 cores. Plotted are the first three sets of principal component scores first scaled to size of variation (natural view) on each axis and secondly, unscaled distortion. The latter are given for easier stereo-viewing. The solitary configuration is included for labeling. Step size is 100 years. Timespan is approximately 20 500 years (a), subdivided into segments as follows: (b) 20500 BP to 12600 BP, (c) 12500 BP to 8900 BP, (d) 8800 BP to present. (b) 20500 BP to present. Notice the initial linear phase and subsequent nonlinear phase. (C j 12500 BP to 8900 BP. present

(d) 8800 BP to present. Notice the initial linear phase leading into nonlinear phase. Fig. 6.322. Euclidean distance profile of the observed postglacial vegetation dynarnics in Jackson Pond, Kentucky. Distances refer to neighbouring states from oldest (1) to present (202) .

The first observation to be made frorn Fig. 6.3.2.1 is that the natural process definitely consists of phases, alternating linear and nonlinear. In fact there is a strikingly high Markovian determinism in the entire chain (s.d.= 22.1, Fig. 6.3.2.1 (a)). But, while determinism is strong in probabilistic terms, the observed stress (o = 1.348) still remains higher than in comparative terms. Upon reexamination of the phase space trajectory this is not surprising: the process is clearly rnulti-phase and could be more realistically Iikened to a moving Markov chah Table 6.3.2.1 presents the results of Markov analyses based on decomposition of the total chah into its more or less obvious phases. What we find is that the phases interna1 structure is similar in two cases (Fig. 6.3.2.1 (b) and (d)) to what we observed in the heathland data: a linear phase followed by a strongly nonlinear phase. Dynarnics in this case study could be tested directly for chaoticity since the nurnber of timesteps is reasonably large. The reconstructed attractor is presented in Fig. 6.3.2.3. Postglacial dynamics was found to be weakly chaotic (Lyapunov exponent = 0.169) tending toward a fractal strange attractor (fractal dimension = 1.80). Table 6.3.2.1 Fit of stationary Markov chah to postglacial dynamics in Jackson pond. Step size is 100 years. When the Markov chah is fitted to the entire chain, 20500 BP to present, the obsewed stress is fairly high. But in probabilistic terms. the nul1 hypothesis of random transitions is still rejected. For the three components: 20500 BP to 12600 BP, 12500 BP to 8900 BP and 8800 BP to present the statistics are different with the early and late components showing lowest stress, but still strong determinism in probabilistic terms. Al1 cases show various degrees of the underlying determinism.

Process Stress (O ) Standard deviation units from some random configuration (s-d.)

20500 BP to present 20500 BP to 12600 BP 12500 BP to 8900 BP 8800 BP to present Fig. 6.3.2.3 Reconstmcted attractor of postglacial vegetation dynamics in Jackson Pond, Kentucky. The analyses rely on 205 component scores on each of three principal component axes. Dynamics is seen to concentrate within a limited region of the attractor space. 6.4 Recappitulation of case stzldies

In the first case study 1 asked: through what path could succession progress in the direction of chaos? 1 found that the observed path cm be re-created in the manner of a very simple mathematical function. By performing simulation experiments on the model, 1 found deterministic chaos. 1 feel encouraged to suggest, based on the results, that the attractor in simulated Atlantic heathland dynamics is "strange" and very cornplex. A very symptomatic tendency of general chaotic dynarnic systems is observed: convergence of the attractor trajectories, starting at different fiducial points, intensifies under increased randomness, and also, increased randomness increases the attractors structural complexity. Furthermore, the process shows explosive behaviour, another hallmark sign of general chaotic dynamic systems. This behaviour is manifest in the two-phase structure. There are reasons to conclude that purely deterministic models are not applicable to vegetation dynamics: 1. The deterministic iteration functions are not realistic. Ecological, or for that matter, population dynamics cannot be abstracted into purely deterministic logistic-type iteration functions. The parameters required to show chaoticity are usually ecologically meaningless. 2. Simple iteration of such a function does not seem realistic. The basic laws goveming dynamics (the equation) must change with time. 3. Stochasticity seems to be a necessary element of natural systems and cannot be avoided. Thus, rather than searching for a deterministic equation, 1 investigated the effects of simple random effects in an underlying deterministic process: the stationûry Markov chain. The finding that stochastic effects can cause chaos in a basically deterministic process has been shown in the past (Rand and Wilson 199 1; Sugihara and May 1990). This has lead to a differentiation between strictly deterministic chaos and statistical chaos (Casdagli 199 1 ; Berliner 199 1) which in essence attempt to bridge the gap between deterministic and stochastic approaches to the study of chaotic behaviour. This is a new field for which my results provide support. Chaoticity may be diagnosed directly if and only if there are sufficient data. This was the case in the second study. 1 discovered that explosive behaviour is also seen in longtenn vegetation dynamics. The process in fact showed a repeated pattern of the two-phase structure observed in the short term. Postglacial dynarnics at Jackson pond was shown to be deterministically chaotic, possessing a complex strange attrac tor. How do these case studies and results further the theory of vegetation dynamics'? 1 investigate the implications in the next chapter. Mine before 1 take it. That's how things go typically. Done before I make it, Sure before I fake it.

And when certainty was not a guarantee, I did not admit desire. What happened to me? Chapter 7

A UNIFYING THEORY OF VEGETATION DYNAMICS

"Faithful observers of Nature, even if in other things they think very differently, nevertheless agree together that dl which appears, sverything that we meet as a phrnomenon. must either mean an original division which is capable of union. or an original unit which cm be split and in that manner exhibit itself. To sever the conjoined, to unite the sevéred, that is the life of Nature ..." Goethe, Zur Farbenlèhre, vol. xxvii, 1 820

7.1 Complex ecological systerns are paradoxical

The first step towards understanding a complex system is to find some way of describing it. Mathematics is what we use in the description when we think we know what is going on precisely. The use of mathematics has always been perceived as the ex~action(or imposition) of order in a system. Statistics helps to account for chance effects. But what if the tools of conventional mathematical and/or statistical theory do not suffice? When might this be the case? 1 see two occasions: 1. application of these tools is infeasible because the natural system is simply too Iarge and has too many variables, and 2. application of these tools is unrealistic because the natural system cannot be decomposed into elements. In these cases we may remain optimitic and believe that perhaps new mathematical and/or statistical tools may be developed. It is not usual to realize though that in these situations one is forced to step out frorn within the system, as it were, to see the forest for the trees. In this, analogies cm be helpfül, and are one way of breaking out from stagnation. To rephrase Lorenz (1 963), analogues serve as tools of last resort, when al1 else fails. My original goal was to attempt to describe complex systems: vegetation dynamics in a recovering Atlantic heathland and in postglacial vegetation dynamics at Jackson Pond. 1 approached the problem with an a priori knowledge, but free of cornmitment to either of the classical vegetation dynamics theories. This required establishment of a workable definition of complexity (Chapter 3). Using this definition, 1 found that comrnunity complexity increases as the process moves through time up to a certain point but then seems to stabilize. When 1 described the community as a two-dimensional trajectory through phase space using dynamical systems theory (Chapter 6), 1 found another trend: the process seemed to have two distinct phases, dorninantly linear and subsequently dominant nonlinear. The latter phase appeared to have no relationship with time and could be labelled as turbulent or chaotic. The point of phase transition more or less coincided with the point at which stability in complexity was reached (seven or eight years). Upon first thought, these two descriptions seem contradictory: if the process moves from linearity to a chaotic nonlinearity, how can at the same time complexity be observed to be stabilizing? Furthemore, it was quite apparent that the descriptions of the heathland and postglacial dynamics did not match any of those predicted by classical theories of vegetation dynamics. Indeed dynamics was neither completely orderly and predictable, nor completely random and disorderly, Dut. in a way, both! This seemed paradoxical, but 1 remained optimistic: My conclusion from these, was that a new theory was needed to emerge about vegetation dynamics.

7.2 The necessay ingredients of a ~tnifiingtheoy

There are several:

1. Provision for the fact that vegetation dynamics is neither completely deterministic nor completely random, but a convolution of the two with one or the other dominating during phases. This essentially acknowledges the importance of what we cal1 chance effects, which 1 believe are so implicit in dynarnics that they cannot be teased out and ignored. I am aware that chance rnight be just a metaphor or euphemism for our inherent limitation of modelling long term events with precision. But until such a knowledge is gained, chance events remain non-causal.

2. An explanation for both the existence and the emergence of a two-phase structure in vegetation dynamics. One could argue that an explanation already exists with the realization that the Clements-Gleason conceptualizations are partially valid. At the reductionist level, or over long-time scales, Gleason's theory is most accurate. At the holistic level and over shorter-time scales, although not so intended, Clements' is more realistic. But this argument assumes two things. Firstly, that the two phases are easily separated sequentially, and that the two theories are mutually exclusive. But neither is! Secondly, while the classical argument suggests when the phase transition rnight occur, it does not provide any reasoning for how the phase transition might occur, or, for that matter, the mechanisms by which the phase transition occurs. In addition there is no accounting for the possibility that scale-dependence could play a role in the perception of phase transition. It is tme, that behaviour suddenly changes. many would simply Say, because the underlying mles of dynarnics have suddenly changed. In the case of recovering heathland, this could be likened to the instance when carrying capacity in the comrnunity is reached. But niles governing natural vegetation dynarnics on any scale are unlikely to undergo fundamental change quickly: A sudden unexpected qualitiative change in the comrnunity may be observed, but it does not seem realistic to suggest that this occurs in independence of the path which preceeded it. In other words, while dramatically different, these phases should somehow be related to one another.

3. An explanation for how stability cm be considered to increase when the dynarnics appears to breakdown into apparent chaos. Essentially, the problem to be resolved is whether or not the late 'chaotic' phase is inevitable -- that is, whether it is an 'attractor'. If this is in fact the case, then what are the properties of this 'attractor'? Could there be some order in chaotic vegetation dynarnics? 7.3 What the simulated data show

Simulation is used as an approximative tool when knowledge of reality is incornplete. This is the rule rather than the exception in ecological research -- especially in visionary ecological research where one is forced to move out of the system at hand. The simulated data showed that a realistic behaviour could be reproduced using a simple realistic model. 1 used the perturbed Markov chain to simulate vegetation dynamics because I felt that it was more realistic in describing natural dynamics than the stationary Markov chain which has been used in the past by others. 1 was interested in its long-tem behaviour because it is not trivial. Unlike the stationary Markov chain, there is no possibility of convergence to a single point attractor, only to a region in phase space within which the elusive target oscillates. In application of the chaos-theoretical tools to simulated data, 1 found that the process was measurably "chaotic" (positive Lyapunov exponent and high fractal dimension). It is suggested by the model that the path of vegetation dynamics is sensitive to initial conditions, and by the accumulation of small but sustained random feedback effects , exhibits explosive behaviour, always tending towards a complex " strange" attractor which is fiactal. This attractor itself is moving on a deterministic path imposed on the process by a highest order environmental constraint, the longtem (postglacial in this case) evolution of the Earth's climate. In this section 1 draw on an analogy to try to suggest what may be causing the explosive chaotic behaviour seen in the real and simulated data. I use concepts from the physics of nonequilibrium phenornena. This theory has been previously linked with the mathematical theory of dynamical systems to ay to understand complex systems (Nicholis 1988; Proccacia 1988). The analogy is rooted in the fact that the dynarnics in the case studies were of a particular class of dynarnics -- those resulting from major perturbation of vegetation from its steady state. in the Atlantic heathland, the perturbation was fire and grazing. In the longterm dynamics at Jackson pond, the perturbation was climatic shift. The analogy 1 intend to make is that the vegetation dynamics is a self-organizing complex system. To develop the idea, 1 follow Nicholis and Prigogine's (1977) discussions of self-organization and nonequilibrium thermodynamics. A system may be considered to be at equilibrium, when it has reached its steady-state or 'attractor' (Prigogine and Stenger 1984). By definition, the systern will remain within the bounds of this attractor indefinitely. Small perturbations will not change this basic behaviour of the system. The rate at which the system reaches this 'equilibrium' is of interest, because it defines the path, that is the nature of the dynamics. Nicholis (1988) realized that equilibrium is a state of homeostasis characterized by uniqueness and stability. and it is for this reason he suggested that it could not serve as a mode1 for understanding the complexity of biological processes. The complex system has great compositional inertia, but if it is pushed far from equilibrium by increasing the rate of dynamics, then strange things begin to happen. The system is no longer stable in the sense that small influences rnay have consequential effects. Nicholis (1988) States this another way, " .. .nonequilibrium reveals the potentiali ties hidden in the nonlinearities which remain 'dormant' at or near equilibrium." Phenomena of this kind are well known in the field of hydrodynamics and fluid flow. For example, it has long been known that once a certain flow rate of flux has been reached, fluid flow moves from larninar/linear fiow to turbulence. For a long time this turbulence was identified with disorder or white noise. But it was discovered that, while the turbulent motion appeared irregular or chaotic on the macroscopic scale, it was actually highly organized (Prigogine and Stenger 1984). The idea of applying non- equilibrium thermodynamic theory to the study of succession has been suggested by Bmelisauer et. al. (1996). According to E. C. Clements and much later E. Odum the 'climax' vegetation is considered to be the mature state. This is analogous to Prigogine's 'equilibriurn'. Once the mature state is reached, dynamics of the vegetation system slows down. Which climax concept we choose to adopt is of no real concem at this point, but 1 mention that the climax equilibrium may be represented by a single point or multiple points (climax-pattern). Ideally, small perturbations to the climax should not result in consequential effects. That is, the system should quickly return to equilibrium state. We see this in the heathland example, but 1 urge the reader to keep uppermost in rnind the scale at which we observe this dynamics. For something appearing stable at one scale may not appear so at another. This is the law of dialectics of social science (Jackson 1971). However, clearly there is a limit to resilience. If the initial perturbation is large enough, not only should the path of dynamics change, but alas the destination. Also clearly, the system need not return to its original equilibrium state (climax), but may approach a new steady state. This is clear from the lack of backtracking or intersection of the trajectory in the Jackson Pond dynamics. Prigogine's concept of "dissipative structures" provides a framework for an explanation of vegetation dynamics that leads towards the organization and complexity which 1 saw develop in the case studies. Bmelisauer et. al. (1996) describe it: "Higher organization and complexity are achieved by minimizing dissipation from the system. This is realized by interna1 cycling of the available energy leading to an increasing finer network of functional interactions accompanied by increased specialization and diversity." A thermodynamic interpretation helps to explain how order (minimum dissipation) is extracted out of chaos (increased complexity, nonequilibrium state), and thus how chaohc behaviour rnay in fact be adaptive in ecological systems, especially those which have been pushed 'far from equilibrium'. Berryman and Millstein (1989) have gone so far as to suggest that ecological systems are not usually chaotic, but rnay be 'made' chaotic by human action, in the form of a large enough perturbation to move them initially far from equilibrium. The cause of the phase transition and self-organization remains to be elucidated. Gleason (1 91 7) wrote, "...as soon as the ground is occupied cornpetition resmcts it (the species, community, pattern) to its proper proportions". Indeed is it just coincidence that the explosion into turbulence seen in the heathland case study happens to occur at the point at which bare ground is reduced substantially from 57% to 7%? In the case of the postglacial dynamics the phase transition is more dependent on large scale environmental changes and small scde effects are blurred. My discovery that long-term vegetation dynamics has a strange attractor follows closely in line with the discovery by physicists that climate has a strange attractor. In fact the idea has corne to me first from reading Nicholis and Nicholis (1984) after 1 came across the genius of Lorenz (1 963). Thus 1 suggest that the patterns in the observed heathland process, the observed postglacial dynamics and the simulated process based on the heathland process are al1 characteristic of a much broader class of phenornena, narnely, self-organizing complex systems. These systems display chaotic behaviour and are consistent with nonequilibrium thermodynamic theory.

7.5 Unifi ing theory a first approximation

When 1 interlink initial condition, linear and nonlineadturbulent phase structure and space-time scale dependence, i.e. fractal nature, and by the demonstration that the convolution of determinism and randormess is a force shaping comrnunity dynamics, 1 find at the core the central notion chaos upon which to construct a general, uniQiug theory of vegetation dynamics. This theory even accommodates the realistic view that the vegetation system is a complex system and the implications of "self organization" which has origins in nonequilibrium thermodynamics theory. The advantages of this theory are intuitively suaight forward: both Clementsian and Gleasonian theories are supported. Clementsian (Clements 19 16) detenninism is supported by recognition of the existence of a cleariy two-phase structure dominated by linear determinism. The Gleasonian (Gleason 1926) hypothesis is supported by incorporating the importance of initial conditions, which define a stochastic range for the attractor, and by showing the significant role randornness plays in the process. Vegetation dynamics is individualistic, Le., not exactly repeatable, and also site- specific by being dependent on initial conditions. Indeed Gleason's remarks that "there is no exact repetition of the same vegetation from one community to the next" and that "observed heterogeneity may well be due to chance" are confirmed. Clements' ideas in the striking and rarely quoted statement, "in the thorough analysis of succession it becomes evident that the use of the term in both a concrete and an abstract sense tends to inexactness and uncertainty", also finds residence in the chaos-theoretical frarnework. Chaos theory provides just that flexible "dynamic connection" which Gleason found lacking in the classification of series of plant communities, forcing him to the conclusion that the plant community must be an individualistic phenornenon. Furthemore, the climax pattern view of Whittaker (1953) is also supported in that the attractor is not a fixed point, but a ballistic target, as it were, with a defined stochastic range. Clearly, short-term dynamics, such as the heathland recovery process, occurs withm and inseparable from the frarnework of long-term biogeoclirnatic dynarnics (thousands of years). 1 showed that similar patterns of vegetation dynamics, for exarnple, rapid movement to a 'strange' attractor, occurs also in the phases on the postglacial process scales. Thus it is tempting to suggest that vegetation dynamics is a self- similar, hierarchically nested process. Isolation of the components of this process may be analytically justified but definitely arbitrary. This dynamics is thus predictable in the very short-term and in the very long- term, but its fractured transient path, so important to humanity, are totally unpredictable and not exactly repeatable. These findings should be satisfying to ecologists who face the restrictions of the classical viewpoints and methodology. It is clear, neverthless that the classical theories are not only reconcilable but realistic as special cases. My supervisor, Acadernician L. Orloci suggested, when 1 started, that "chaos" would be a unifying notion, an "equalizer", in a manner of speaking, among theories in the development of a new theory of vegetation dynarnics -- a prediction which 1 believe indeed has proven correct. Chaos should thus not defeat us but rather allow us to focus on the search for appropriate modelling tools. These include the usage of appropriate complexity measures to classify time-variant vegetation processes. Blind iteration of simple analytic f'unctions with a fixed number of parameters, fixed parameter values and fixed initial conditions are fundamentally predictively dysfunctional for modelling adaptive vegetative processes. The vegetation path is uansient. There are phases and they proceed from catastrophe to catastrophe. This kind of behaviour defeats continuous models, but we have alternatives: these include pattern developrnent analysis, threshold functions, fiactals, threshold logic, fuzzy logic, adaptive programming techniques, as well as advanced (non- parametric) statistical methods, to name a few. Since chaos is a reality, I recornmend that more research be done in these areas. 1 want to wnte, but 1st night 1 emptied out my head and spit out my hem.

1 want to recite, but lut night you swalIowed my tongue whole and left not a single bit for me. The rain felt like rain. 1 was simply, hot. 1 was simply, yours. 1 lost poetry. I want to uy another. but 1 don't bother. It wouId bè a lie, and that would be bad. Appendix A

DEFINITIONS

butterfly effect when smaii differences in the specification of the initiai input lead to dramatically different outcornes. The name cornes from the thought experirnent in which a tiny butterfly in China decides to Bap its wings before settling on a dandelion, and the resulting air movement seeds a tiny twister which is fmt picked up by the trade winds, then arnplified by the ocean gradient into a huge hunicane that touches down on Kansas and wipes out the city. Had the butterfly decided to settle on the dandelion without doing any wing flaps, Kansas would have enjoyed just a gentle, joyful summer breeze. chaos the irregular and unpredictable, yet detemùnistic time evolution of many non-monotonie systerns. Non- monotonicity is a necessary and sufficient condition for chaos. An infinite series of non re-enforcing fractures of an exponential growth process tends to create chaos. Trend- free, irregular changes that de@ description are called chaotic. Examples: Brownian motion, the last digit of the Dow's Industrial Average on the New York Stock Exchange, the paths of points in a force-field of densely conflicting forces. chaos theory al1 that is known about chaotic systems. cornplexi ty here, algorithmic complexity which depends on the ease/dficulty of descnbing a system. But 1 recognize three kinds of aigonthmic complexity: Complexity of synthesis is the total cost of ali the operations that are required to create the paner. Complexity of analysis is the cost of veribing that statements about the pattern are me. Asymptotic complexity has to do with making general statements about an infinite class of patterns rather than just a single pattern. dynamical systern

A system which changes with time; mathematicaliy speaking, (i) the set of States representable as a cloud of points R in Euclidean n-space En, (ii) the set of n real-valued functions {il,..., fn) which uniquely define the properties of R, and (iii) a system of n function relations xi(t+h)=fi(x ~(t),..., xn(t),h), i= 1 to n, which represent rates of change. fractal a property indicating an infinite recursion of scaled embedding of components: mathematically a set for which the Hausdorff Besicovitch dimension sirictiy exceeds the topological dimension. fractal dimension (D) D = log a / log l/s where a is the nurnber of pieces into which a structure can be divided and s is a reduction factor. See also Appendix B.

Lyapunov exponent A quantification of the average growth of infinitesimal small errors in the specification of the initial condition. A test of the 'butterfly effect'. See also Appendix B. orderly predictable, non-nndom; this is however not a simply binq value, since there may be degrees of orderliness if order is slightly hidden or totally hidden or even encrypted. pattern an element of pattem space, where a pattem space is a function space whose elernents are functions that map the n- dimensional Reals Rn into a space of attribute vectors. random a charactenstic property of a non-redundant sequence, the type of sequence on which data compression techniques are useless, as the sequence is incompressible; due to chance: unpredictable based on known facts. recovery the attainment of a new steady-state. scale Iirnits in space ancilor time; the penod of time or space over which signais (performances, responses) are integrated or smoothed in the message (unit record set, phytosociologicd relevé). self-sinii lar syrnmetry across scales; recursion, pattem inside pattern irnplied.

Markov chain al1 States derivable from the initial state (Xi) by multiplication of the transition matrix (P) by recunive application of: Xj+l=XjP stochast ic assaiiated with a probability law; chance variation generated by mechanisms driven by a probability law. strange attractor the stable state which evolves by slow randorn and explosive Linear change; mathematically a bounded subset A of the plane for the transformation T for which there exists a set R with the following properties: (i) A is an attractor, (ii) A is chaotic, (iii) A has a fractal structure, (iv) A carmot be split into two different attractors. Precisely, R is a minimal fixed point of T such that R is infinite and bounded, it is computationaily infeasible to determine whether a given point is in R, and the boundary of R is fractal. Appendix B

ALGORITHMS

B. 1 General description

The computational tasks were consolidated into two main programs -- one that L. Orloci made available to me for which the description appears in Orloci, et. al. (1993) and another by me written in the TruBasic programming language. In this Appendix 1 mainly describe the latter in layman terms and offer compiled code to inquirers. 1 focus in the description on the input required, processing, and relevant output.

B.2 Input and prelhinary calculations

The input, in general, is the description of a coenosere, that is, cornmunity-level species abundance or other consistent numerical records over time. Specifically a text file where species abundance values are presented in a single colurnn, entered by coenostate (or core sample) is required. The data is adjusted to equal time steps. Euclidean distance, angles, Shannon and Rényi entropy of order 12 and total complexity (as described in Chapter 3 and Appendix C) are computed. In the heathland case, testing of chaos was done based on the simulated coenosere, and the input arnounted to a perturbed Markov chain. In the case of postglacial dynamics, the core samples were directly tested.

A.3 Chaos algorithms

A.3.1 L~pzinovexponents

Reconstructed attractors and Lyapunov exponents are determined for Euclidean distance or other profiles based on the Wolf et. al. (1985) algorithm. I summarize this below:

Given: One-dimensional time series data: X = Xi, X2, .. . , Xc 7 where c is the number of timesteps Select: Parameters necessary to reconstruct the attractor -- 1. Dimension of the phase-space (m) (3 used for easy visualization.) 2. Reconstruction time-lag (7) 3. Time-step in data Number 3 is always initially set to 1, be the real time step in the data 1 year, 100 years or else.

1 mention that although there are intuitive choices for par ameters, 1 wm the novice user that the actual time step has a significant effect on the outcome of the analysis. Also 1 found that experimenting with different time-lags is prudent. By doing so, the Lyapunov exponent stabilizes, in my case at a critical lag, which 1 eventually chose. But this was different for different time-series, and thus the choice of time lag is unique to each senes analyzed. This should not hinder the ability to compare exponents among series. There lies a strength in this manipulation of time-lag: a nested scale is introduced and complex structural properties revealed. A three-dimensional phase portrait (attractor) is reconstructed with delay coordinates. The latter implies that points on the attractor correspond to

( X(t), X(t + 'c), .. ., X( t+ (m- 1)~)}

The nearest neighbour to the fiducial (initial) point is found. The subsequent trajectory of the nearest neighbour (evolved trajectory) is then compared to that of the fiducial trajectory over a lirnited, presumably stable region of the attractor, i.e. over a selected evolution time. When the distance between the trajectories becomes too large, this is the point when a new phase of the attractor has been reached and a thus new point for the evolved trajectory (replacement) is sought. This point should be close (in the Euclidean sense) to where we are at on the fiducial trajectory, and at the same time, lie in the same direction (distance vectors, angular deviation zero or close to it) as the previous evolved trajectory. The procedure is repeated until the fiducial trajectory has passed through the entire data set. At this point, the Lyapunov exponent h is calculated: 1 M L' (tk) A=- C log2 tnr-tOk=1 L(*-1) where M is the total nurnber of replacement steps. If b0 then two initially close trajectories will diverge, and the system is deemed chaotic. Now. 1 rnust stress that sensitivity to initial conditions does not automatically lead to chaos. For example, clearly in the linear transformation:

any slight deviation in initial condition will magnify after several iterations. The system is sensitive, but definitely not chaotic. However, what we can safely Say is that dlsystems which are chaotic are sensitively dependent on initial conditions. Unpredictability results from inherent nonlinearities.

B.3.2 Fractd dimension

By definition, a fiactal is a set for which the Hausdorff Besicovitch dimension (fractional dimension) smctly exceeds the topological dimension (Mandelbrot 1977). Topologicd dimensions are what we traditionally understand dimensions to be (O for a point, 1 for a line, 2 for a plane etc.). A fractional dimension may be in-between topological dimensions (e.g. 1.7) and is estimated for a set by different methods. The method usually depends on the nature of the data or point set. Fractals cannot be measured in practical terms. They only exist as mathematical abstractions. Fractals are said to have infinite resolution. This is because they continue to display detail as they are perceived at finer and finer scales. Many (but not dl) fractals are invariant under change of scale. These are termed scding fiactals. When a fractal is invariant under ordinary geometric sirnilarity, it is self-similar. According to Bmough, given the following equation

where V(t) is the variance of increments function

for W(t) and W(t+h) values of any fractal function W at points t and t+h, separated by a distance h, then D is the fractal dimension of the set defined by the curve obtained by plotting logV(h) against log(h). The function dlogv(h)/dlog(h) is simply the slope of the curve which is usually approximated by some regression method. A direct consequence of Burrough's reasoning is that for any spatial or temporal serial data set (e.g. gradients, successions in vegetation studies), a variance of increments function V(h) can be easily calculated. The closest relative of V(h) is the semivariance: where z(i + h) is a value of any function z at a point separated from point i by distance h, and N(h) is the number of pairs of points separated by distance h. A plot of y(h) as a function of t is called a semivariogram. The relationship of V(h) to y(h) is simple:

if and only if z is a fractal function as specified by Burrough. However, V(h) can also be expressed in terms of other, more conventional variance functions. For example,

where sl is the variance and p(h) is the autocorrelation function. Another example is the relationship between block variance szk, of block length k, and V(h): k s2k= 2C (k-h+l)*V(h) k(k+l) h=l

In fact an\. variance function (other examples are power spectrum and covariance), can be transformed to the function Va). Thus it appears that the classical analyses of variance has the ability to produce a new and seemingly more informative measure -- the fiactal dimension. To understand the drawbacks 1 asked the simple question, where did this relationship between fractal dimension and variance function corne from? Burrough (1981) bases his application on the results of Berry and Lewis (1980), who investigated the properties of the Weierstrass-Mandelbrot function . The function takes the following form:

143~2,pl, $.= arbitrary phases.

Since the function is continuous everywhere, but differentiable nowhere, and thus hûs no scale, it is termed a fractal function. By this definition, there exist many kinds of fractal functions. The fractal dimension of taken to be D, which is based on experimental results (no forma1 proof yet exists). Berry and Lewis (1980) have shown that the variance of increments function of W(t) itself has the form of a Weierstrass function:

and obeys the scaling law, It is from this generalization that Burrough justifies his method. The validity of this relation, however, is based upon how generalizable this really is to cases where the underlying function is not Weierstrass. The ecologist authors did not pose this question, yet it is a very important one. In a later work, Burrough (1983) extends the application of fractal concepts by drawing on the propemes of another class of fractd functions, namely the fractional brownian functions (Mandelbrot and Van Ness 1968). These functions were defined based on one-dimensional ordinary Brownian motion. It is known that in random processes ( X(t) ) with Gaussian incremen ts.

var ( X(t + h) -X(t)) p lh12H where H=0.5 (i.e. in ordinary Brownian motion). When H is allowed to Vary such that O < H < 1, then the resulting functions are fiactional Brownian functions. It has been shown that the fractal dimension (D) of brownian functions is 2-H. Thus, 2H=4-2D. This is exactly the exponent in the semivariance relation. It thus seems to me at this point, that the variance of increments function of fractal functions can be used to calculate the fractal dimension of the fractal functions (or the variance of increments function). But the practice of so many ecologists to generalize to the point of including al1 functions, fracta1 or not, appears far-fledged, to Say the least. While 1 am thus quite skeptical of the assumptions underlying the methods of calculating fractal dimension using Burrough's methods, it is the only one available to me now.

The Markov analysis gives several things of interest. First and foremost is a transition matrix and the fitted Markov coenosere. The mode1 to which the obsewed coenosere is fitted varies from a stationary Markov chain (0% perturbation), to perturbed Markov chains (the user may define the maximum percentage of perturbation; 1 used levels of 596, 15%, 25%. and 50%). Also of interest from the Markov analysis is how good the fit is. This is expressed both in terms of an obsewed stress value (type of divergence from Markovity) and in probabilistic terms. The explanation of these is made in Chapter 5 and in Orloci et. al. (1993). Thrre are several things of interest from the chaos analysis. First, the Euclidean distance profiles are plotted for a visual inspection of dynarnics as a time-series plot. Then there is the qualitative nature of the reconstructed attractor. This is represented as a trajectory through a newly defined phase space. A quantitative analysis of the times series gives Lyapunov exponent and fractal dimension, and accordingly, the presence of chaotici ty may be judged. Appendix C

CODING-THEORETICAL BACKGROUND

Let the set of symbols comprising a given alphabet be called S = [ SI,s2,

**., sql Then a code is an invemble mapping of al1 possible sequences of the symbols of S into sequences of symbols of some other alphabet X ={ xl, xz, *.- . x,}

S symbolizes the source alphabet and X the code alphabet.

A block code is a code which maps each of the symbols of the source alphabet S into a fixed sequence of symbols of the code alphabet X. These fixed sequences of the code alphabet (sequences of xi) are called code words and are denoted by Xi.

The nth extension of a block code which maps the symbols si into the code words Xi is the block code which maps the sequences of source symbols (sil,sizy...,si,J into the sequences of code words (Xil, Xi2, ..., Xin].

A block code is said to be nonsingular if al1 the words of the code are distinct. A block code is uniquely decodable if, and only if the nth extension is nonsingular for every finite n. A uniquely decodable code is said to be instantaneous if it is possible to decode each word in a sequence without reference to succeeding code symbols.

C.2 Average Length L

For a given source alphabet and a given code alphabet, we cm constmct many instantaneous and uniquely decodable codes. The natural criterion for selection among these from the standpoint of economy of expression is length. We are interested in the uniquely decodable code with the smallest possible average length.

Let a block code transform the source symbols { SI, sz, ..., sq} into the code words (Xr,Xz, ..., Xq}. Let the probabilites of the source symbols be { pl, pz, ..., pq)and let the lengths of the code words be { Ir, 12, ..., Iq} Then L is the average length of the code and is given by

4 L = C pjZj bits C. 1 j=l

Consider a uniquely decodable code which maps the symbols fiom a source S into code words composed of symbols fiom a binary code alphabet. This code is compact if its average length L is less than or equal to the average length L of al1 other uniquely decodable for the same source and the sarne code alphabet. C.3 Relationship between H and L

We make use of the Kraft Inequality (Kraft 1949),

where r is the number of different symbols in the code and il, Zz, ..., Zq the code word lengths of an instantaneous code. For simplicity we will consider codes with r=2 [binary].

Recall that

4 such that Qi20 for al1 i and Q = 1 i=l

It is not difficult to prove (Abrarnson 1963) that

with equality if and only if pi = Qi for al1 i. Relation C.4 is valid for any set of nonnegative numbers Qiwhich sum to 1. We may choose therefore,

and obtain

The Kraft inequality tells us that argument of the second logarithm on the right of C.6 must be less than or equal to 1. The logarithm is therefore less than or equal to 0, and C.4 Hufian coding

We require a description of the plant cornmunity at a desired spatial scale commonly known in ecology as a 'coenostate' or 'relevé'. This includes each species (sj) and its relative abundance (pj). For the sake of facile illustration, suppose the relevé is as below. The example and its explanation is adapted from Abramson (1963) and Hamrning (1980).

Source S

Huffman (1952) developed the method which we use for constmcting compact codes for the case of a binary code alphabet. We use his method. How do we translate this 'source' into a parsimonious code by Huffman coding?

Consider the source S with q symbols and symbol probabilitie~p~, pz ...,pP In the example, q=5. Let the symbols be ordered so that PI>=P~>=,...,P~as in the exarnple. By regarding the last two symbols of S as combined into one symbol [by adding the correspond pi's], we may obtain a new source from S containing only q-1 symbols. This new source is referred to as a reduction of S. The symbols of this reduction may be reordered, and again we may combine the two least probable symbols to form a reduction of this reduction of S. By proceeding in this manner, we consûuct a sequence of sources, each containing one fewer symbols than the previous one, until we arrive at a source with only two symbols. For the example above, the sequence of reduced sources is as follows: Construction of this sequence of reduced sources is the first step in Huffman coding. The second step entails the recognition that a binary compact instantaneous code for the last reduced source [only two symbols] is trivial: it consists of two words, O and 1. Working backwards, in general, the compact instaneous code for Sj-1 is formed from Sj by the following rule:

"One of the symbols of Sj, say sa, is fomed from two symbols of the preceeding source S,.]. We cal1 these two symbols sdand s,l. Each of the other symbols of S, corresponds to one of the remaining symbols of S,J. Then the compact instaneous code for Sjdrfrom the code for Sj as follows: We assign to each symbol of (except sd and sal.) the code word used by the corresponding symbol of Si . The code words used by sa and sal. are fomed by adding a O and 1, respectively, to the code word used for sa."(Abramson 1963) The compact Huffman code for the example is illustrated below.

The general case should be obvious fkom the example. When al1 p, are of the form (f y', where a, is an integer, the coding alpithm is simplified, the code word lengths Zj taken a priori as Q,- and the codes constructed by sirnpiy creating unique combinations of the code alphabet [O, 11. Appendix D

JACKSON POND DATA

D.1 Reference infornarion

This data is available from the National Oceanic and Atrnospheric Administration (NOAA), U.S.A. at the following internet address url: http://www .ngdc.noaa.gov/paleo/ftp-pollen.Search may be accomplished by specifying principal investigator, Wilkins, GR.

NAPD ASCII Format # 16 Jul 95 # Raw Counts - Al1 Pollen Types # Site narne: Jackson Pond Place: USA:Kentucky Latitude: 37-27.00 Longitude: -85.43.00 Elevation(m): 2 12 Sigle: JACKSON Entity name: core 83-B Contact person: Wilkins. GR. Radiocarbon dates: Depth Thk Age SDup SDlo Lab no. Basis Material 133.0 10 120 50 50 A-3870 U fibrous peat 160.5 11 940 80 80 A-3871 U organic silty clay 220-5 i 1 10040 190 190 A-3872 U organic silty clay 270-0 14 1 1860 250 250 A-3873 U organic silty clay 510.0 10 17750 270 270 A-3874 U silty clay 705.0 10 20330 630 630 A-3875 U siltyclay Chron name: Wilkins et al. 199 1 Age model: linear interpolation

Age basis: Depth Thk Age AgeUp RCode 93.0 O -33 -33 TOP 133.0 IO 120 170 C 14 160.5 11 940 1020 Cl4 220.5 11 10040 10230 Cl4 Chron notes: Ages extrapolated beyond last C-14 date based on deposition tirne between last two C 14 dates. Publications: Wilkins, G-R.,P.A. Delcourt, H.R, Delcourt, F.W. Harrison, and M.R. Turner. 1991. Pdeoecology of central Kentucky sincr the last glacial maximum. Quatemary Research 36:224-239. Wilkins, G.R. 1985. Late-quatemary vegetational history at Jackson Pond, Lame County, Kentucky. Thesis. The University of Tennessee, Knoxville, Tennessee, USA- Variable list

There are a total of 1 15 variables. 109 are based on fossil pollen and tissue plant material and are, in the most part, species specific. Two classes, indeterminable and unidentifiable exist. The last 4 are parameters dealing with the extraction technique. The first 94 variables (terrestrial species) were used in the analysis.

1 Abi A Abies 2 Ace.n A Acer negundo 3 Accp A Acer pensylvanicum 4 Ace.r A Acer rubrum 5 Ace.i A Acer saccharinum 6 Ace.s A Acer saccharum 7 Ace. t A Acer spicatum 8 A1n.i-t A Alnus incana-type 9 A1n.v A Alnus viridis 10 Arne- t A Arne lanc hier- type 11 Bet A Betula 12 Car A Carya 13 Cas A Castanea 14 CeMa A CeltisIMaclura 15 Cep A Cephalanthus 16 Cm-c-t A Cornus canadensis-type 17 Crn-f A Cornus Clorida 18 Cor A Corylus 19 Cupae A Cupressaceae 20 Fag A Fagus 2 1 Fra.n-t A Fraxinus nigra-type 22 Fra-p-t A Fraxinus pennsylvanica-type 23 Fra-ud A Fraxinus undiff. 24 Ilx A Ilex 25 Ite A Itea 26 Juge A Juglans cinerea 27 Jug.n A Juglans nigra 28 Jug-ud A Juglans undiff. 29 Lar A Larix 30 Liq A Liquidambar 3 1 Lir A Liriodendron 32 Mag A Magnolia 33 Nys A Nyssa 34 OslCa A OstrydCarpinus 35 Pic A Picea 36 Pin.Pi A Pinus subg. Pinus 37 Pin.St A Pinus subg. Strobus 38 Pla A Platanus 39 Pop A Populus 40 Pru A Prunus 41 Que A Quercus 42 Rub A Rubus 43 Slx A Salix 44 Sam A Sambucus 45 Spi A Spiraea 46 Ti1 A Tilia 47 Tsu A Tsuga 48 Ulm A Ulmus 49 Vib.0-t A Viburnum opulus-type 50 Ach- t B Achillea-type 5 1 Amb-t B Arnbrosia-type 52 Apiae B Apiaceae 53 Art B Artemisia 54 Astae.Ci B Asteraceae subf. Cichorioideae 55 Bid-t B Bidens-type 56 Cal-t B Caltha-type 57 Cryae B Caryophyllaceae 58 ChJAm B Chenopodiaceae/Amaranthaceae 59 Ci/Ca-t B Cirsium/Carduus-type 60 Cypae B Cyperaceae 61 Da1 B Dalea 62 Ga1 B Galium 63 Hu/Ca.s B Humulus/Cannabis sativa 64 Hyx B Hypoxis 65 1va.a-t B Iva annua-type 66 1va.x-t B Iva xanthifolia-type 67 Lamae B Lamiaceae 68 Malae B Malvaceae 69 Ongae B Onagraceae 70 Pln.1 B Plantago lanceolata 7 1 P1n.m B Plantago major 72 Poaae-ud B Poaceae undiff. 73 Po1.i-t B Polygonum aviculare-type 74 Po1.h-t B Polygonum hydropiper-type 75 Po1.v B Polygonurn viviparum 76 Ptl-t B Potentilla-type 77 Ranae-ud B Ranunculaceae undiff. 78 Rum B Rumex 79 San.c B Sanguisorba canadensis 80 Tha B Thalictrum 8 1 Urt-t B Urtica- type 82 Xan B Xanthium 83 2ea.m B Zea mays 84 Bot F Botrychium 85 Cys F Cystopteris 86 Dph.c- t F Diphasiastmm complanatum- type 87 Drp-t F Dryoptaris-type 88 Equ F Equisetum 89 Lycae.ud F Lycopodiaceae undiff. 90 Ly1.i F Lycopodiella inundata 9 1 Lyc.0 F Lycopodium obscurum 92 0su.c-t F Osmunda cinnarnomea-type 93 0su.r-t F Osmunda replis-type 94 Plyae F Polypodiacaae 95 Ptr F P~eridiurn 96 Spg M Sphagnum 97 Iso Q Isoetes 98 Men.t Q Menyanthes trifoliata 99 Myp.a Q Mynophyllum altemiflorum LOO Myp-i Q Myriophyllum sibiricum-type 101 Nup Q Nuphar 102 Nym Q Nymphaea 103 Pot.Co Q Potamogeton subg. Coleogeton 104 Pot.Po Q Potamogeton subg. Potamogeton 105 Sag Q Sagittaria 106 Spa-t Q Sparganium-type 1O7 Typ.1.m Q Typha latifolia (monads) 108 Typ.l.ud Q Typha latifolia undiff. 109 Utr Q Utricularia 110 ind X Indeterminable Il 1 unk X Unknown 1 12 Euc-glt -Concentration of Eucalyptus tablets (grainsltablei) 113 Euc-spik -Eucalyptus spike (number counted) 1 14 Euc.ta.n -Number of Eucalyptus tablets added 1 15 sa.vo.ml -Sample volume (ml)

D.3 Raiv core data, calibrated and dated

The header values, x, y represents core depth and approximate years before present (BP) respectively. The data matrix below each represents the 115 variables described in C.2. Only the first 94 variables were used in my analysis and, as explained in Chapter 6, the core depth dates were interpolated for consistent timestep. The 54 of 58 given paleocoenostates (core depths), upon which interpolation analysis was performed, are given below.

REFERENCES

Abrarnson, N. 1963. Information theory and coding. McGraw-Hill, Inc., New York. Acevedo, M. F. L. 198 1. On Hom's Markovian mode1 of forest dynarnics with particular reference to tropical forests. Theoretical Population Biology. 19: 230-250. Alechin, W. W. 1926. Was isr eine Pflanzengesellschafr? Ihr Wesen und ihr Wert als Ausdruck des sozialen Lebens der Pflanzen. Fedde Repert. spec. novar. reg. veg. Beih. 37: 1-50. Allen, T. F. H. and Hoekstra, T. W. 1990. The confusion between scale- defined levels and conventional levels of organization in ecology. Journal of Vegetation Science. 1 : 5- 12. Allen, T .F. H. and Starr, T. B. 1982. Hierarchy: Perspectives for Ecological Complexity. The Unive rsity of Chicago Press, Chicago. Allen, T. F. H. and Wyleto, E. P. 1983. A hierarchical mode1 for the complexity of plant communities. Journal of theoretical Biology. 10 1: 529-540. Anderson, O. J. and Kikkewa, J. 1986. Development of concepts. In: Cornmunity Ecology: Pattern and Process. (Kikkewa, J. and Anderson, 0. J., eds.) Blackwell Scientific Publishing, Melbourne. Anger, G. 1985. On the relationship beîween mathematics and its application: a citical analysis by means of inverse problems. Inverse Problerns. 1: L7-L 12. Anger, G. 1993. Basic principles of inverse problems. In: Inverse Problems: Principles and Applications in Geophysics, Technology, and Medicine. (Anger, G., R. Gorenflo, H., Jochmann, H. Moritz, and W. Webers., eds.) Alcademie Verlag, Berlin. Atlan, H. (1988). Measures of biologically meaningful complexity. In: Measures of Complexity (Peliti, Land Vulpiani,A., eds.) pp. 1 12- 127. Spnnger- Verlag, Berlin. Austin, M. P. and Smith, M. P. 1989. A new mode1 for the continuum concept. Vegetatio. 83: 35-47. Barnsley, M. 1988. Fractals everywhere. Academic Press, Inc., Boston. Batschelet, E. 197 1. Introduction to mathematics for life scientists. New York: Springer-Verlag. Benzecri, J. P. 1969. Statistical analysis as a tool to make patterns emerge from data. In: Methodologies of pattern recognition. (Watanabe, S., ed.) Academic Press, New York. Berliner, L. M. 1992. Statistics, probability and chaos. Jo~rmalof the Royal Statistical Society 7:69- 122. Berry, T. 1988. The Dream of the Earth. Sierra Club Books, San Franscisco. Berryman, A. A. and Millstein, J. A. 1989. Are ecological systems chaotic- and if not, why not? Trends in Ecology and Evolution. 4: 26- 28. Binkely, C. S. 1980. 1s succession in hardwood forests a stationary Markov process? Forest Science. 26: 566-570. Birkhoff, G. 0. 1927. Dynamical -rems. American Mathematics Society, Colloquia Publication, New York. Botkin, D. B., Janak, J. F and Wallis, J. R. 1972. Some ecological consequences of a cornputer mode1 of forest growth. Journal of Ecology 60: 849-872. Bouxin, G. 199 1. The measurement of horizontal patterns in vegetation: a review and proposais for models. In: Haridbook of Vegetation Science: Compter assisted vegetation analysis. (Feoli, E. and Orloci L., eds) Kluwer Academic Publishers, Dordrecht. Braun-Blanquet, J. 1965. Pflanzensoziologie. Springer-Verlag, Vienna. Bdlouin, L. 1962. Science and Information Theory. Academic Press, New York. Bruelisauer, A. R., Bradfield, G. E. and Maze, J. 1996. A thermodynamic interpretation of succession. Coenoses. 11 : 73-76 Burrough P. A. 198 1. The fractal dimensions of landscape and other data. Nature . 294: 240-242. Burrows, C. J. 1990. Processes of Vegetation Change. Unwin Hyman Ltd., Boston. Cain, S. A. 1939. The climax and its cornplexities. American Midland Naturalist 2 1: 147- 158. Cale, W. G. and Odell, P. L. 1980. Conceming aggregation in ecosystem modeling. In: Theoretical systems ecology. (Halfon, E., ed.) Acadernic Press, New York. Cale, W. G., O'Neill, R. V., and Gardner, R. H.. 1983. Aggregation error in non-linear ecological models. Journal ofTheoretical Biology. 100: 539-555. Cale, W. G., Henebry, G. M. and Yeakley, J. A. 1989. Infemng process from pattern in natural communities. Bioscience. 39: 600-605. Calude, C. 1994. Information and Randomness: an Algorithrnic Perspective. Springer-Verlag, Berlin. Çambel A. B. 1993. Applied Chaos Theoy: a paradigm for complexit-y Academic Press, New York. Cannon, J. R. and Hornung, U. 1986. Inverse Problems. Birkhauser Verlag, Basel. Carpenter, S. R. and Chaney, J. E. 1983. Scale of spatial pattern: four methods compared. Vegetatio. 53: 153- 160. Casdagli, M. 199 1. Chaos and deterministic versus stochastic non-linear modelling. Jorlrnal of the Royal Statistical Society. B 2:303-328. Chaitin, G. J. 1966. On the length of prograrns for computing binary sequences. Journal of the Association of Comp~rtationalMachines. 13: 547. Clements F. E. 19 16. Plant Succession: an analysis of the development of vegetation, Publ. No. 242 Carnegie Institution, Washington. Colinvaux, P. 1973. introduction to Ecoiogy. John Wiley and Sons, Inc., New York. Conne11 S. H. and Slatyer, R. 0. 1977. Mechanisms of succession in natural communities and their role in comrnunity stability and organization. American Naturalist. 1 11 : 11 19- 1 144 Cooper, W. S. 1926. The fundarnentals of vegetation change. Ecology. 7: 391-413. Copi, 1. M. 1978. Inrroduction to Logic. Macmillan Publishing Co., Inc., New York. Cowles, H. C. 1899. The ecological relations of the vegetation on the sand dunes of Lake Michigan. Part. 1. -- geographical relations of the dune floras. Botanical Gazette. 27: 95- 1 17. Dale, M. B. 1980. A syntactic basis of classification. Vegetatio 42: 93-98. Dale, M. B., MacNaugton-Smith, P., Williams, W. T. and Lance, G.N. 1970. Numerical classification of sequences. Australian Cornputer Journal. 2: 9- 13. Dale, M. R. T. and MacIssac, D. A. 1989. New methods for the analysis of spatial pattern in vegetation. Journal of Ecology. 77: 78-9 1. Dale, M. R. T., and Blundon, D. 1. 1990. Quadrat variance analysis and pattern development during primary succession. Journal of Vegetation Science. 1: 153-164. Dansereau, P. 1957. Biogeography: An ecological perspective. The Ronald Press Company, New York. Darwin, C. 1859. On the origin of species. Murray, London. Dawkins, R. 1986. The Blind Watchmaker. Penguin Books, Suffolk. Delcourt, H. R., Delcourt, P. A. and Thompson III, W. 1983. Dynamic plant ecology: the spectrum of vegetational change in space and time. Quaternary Science Reviews 1: 153- 175. Digby, P. G. N. and Kempton, R. A. 1987. Muftivariate analysis of ecological communities. Chapman and Hall, London. Drake, J. A. 1990. The mechanisms of community assembly and succession. Journal of theorerical Biology l47:2 13-233. Drury, W. H., and Nisbet, 1. C. T. 1973. Succession. Journal of the Arnold Arboretum of Harvard University. 54: 33 1-68. Eckland, 1. 1993. The Broken Dice and other Mathematical Tales of Chance. The University of Chicago Press, Chicago. Fekete, G. 1985. A terresztns vegetacib szukcesszi6ja: elméletek, modellek, val6siig. In Paradigms of coenological succession, (G. Fekete, ed.) (in Magyar) Akademiai Kiad6, Budapest Feller, W. 1957. An introduction to probability theory and its applications. Vol. I John Wiley & Sons, Inc., London Finegan, B. 1984. Forest succession. Nature 3 12: 109- 114. Ford, E. D. and Renshaw, E. 1984. The interpretation of process from pattern using two-dimensional spectral analysis: modelling single species patterns in vegetation. Vegetatio. 56: 1 13- 123. Freud, S. 196 1. Civilization and its Disconrents. (translation) W.W. Norton, New York. Gardner, R. H., Cale, W. G.and O'Neill, R. V. 1982. Robust analysis of aggregation error. Ecology. 63: 177 1- 1779. Gauch, H. G. and Whittaker, R. H. 198 1. Hierarchical classification of community data. Joumal of Ecology 69: 537-557. Gibbs, J. W. 1902. Elernentaq principles in statistical mechanics. Scrïbner, New York. Gilpin, M. E. and Case, T. 1. 1976. Multiple domains of attraction in cornpetition communities. Nature. 40-42. Gleason, H. A. 19 17. The structure and development of the plant association. Bulletin of theTorrey Botanical Club. 44: 463-48 1. Gleason H. A. 1926. The individualistic concept of the plant association. Bulletin of the Torrey Botanical Club. 53: 7-26. Gleason, H. A. 1968. The New Britton and Brown lllustratedflora of the Northeastem United States and Adjacent Canada. Hafher Publishing Company, Inc., New York. Gleick, J. 1987. Chaos. Making a New Science. Penguin Books, New York. Glenn-Lewin, D. C. 1980. The individualistic nature of plant cornmunity development. Vegetatio. 43: 14 1- 146. Godfray H. C. J.and Blythe, S. P. 1990. Complex dynarnics in multispecies communities.Philosophical Transactions of the Royal Society of London B. 330: 22 1-233. Goodall, D. W. 1972. Building and testing ecosystem models. In: Mathematical models in ecology. (Jeffers, J. N. R., ed.) Blackwell Scientific Publishers, Oxford. Goodall, D. W. 1977. Dynarnic changes in ecosystems and their study: the roles of induction and deduction. Journal of Environmentcil Management 5: 309-3 17. Goodall, D. W. 1986. Classification and ordination: their nature and role in taxonomy and comrnunity studies. Coenoses. 1: 3-9. Greig-Smith, P. 1952. The use of random and contiguous quadrats in the study of the structure of plant cornmunities. Annals of Botany 16: 293- 3 16. Greig-Smith, P. 196 1. Data on pattern within plant communities. 1. The analysis of pattern. Journal of Ecology. 49: 695-702. Greig-Smith, P. 1970. Statistical plant ecology. Annual Review of Ecology and Systematics 1 :99- 1 24. Greig-Smith, P. 1979. Pattem in vegetation. Journal of Ecology 67: 755- 779. Greig-Smith, P. 1986. Chaos or order -- organization. In: Comrnuni~ Ecology: Pattern and process. (Kikkawa, J and Anderson, D. J., eds.) Blackwell Scientific Publications, Melbourne. Groenewoud, H. van. 1965. Ordination and classification of Swiss and Canadian coniferous forests by various biometnc and other methods. Berïchte geobotanica Inst. Zurich. 36: 28- 102. Hadamard, J. 1902. Sur les problèmes derivées partielles et leur signification physique. Princeton University Bulletin 20:49-52. Halvorson, J. 1994. Geostatistics and ecology. ESA Bulletin. 75: 36-38. Hamming, R. W. 1980. Coding and Infornation Theory. Prentice-Hall, Inc., New Jersey. Hastings, H. M. and Sugihara, G. (eds.), 1993. Fractals. A User's guide for the natrrral sciences-- Case studies: pattern and process in vegetation ecosystems. Oxford University Press, Oxford. Hastings A., Hom, C. L., Ellner, S., Turchin, P. and Godfray, H. C. J. 1993. Chaos in ecology: is mother nature a strange attractor? Ann~tal Review of Ecology and Systematics 24: 1-33. Hastings, A. and Higgins, K. 1994. Persistence of transients in spatially stmctured ecological models. Science 263: 1 133- 1 136. Hayden, D. 1994. Pivota1 assumptions determining the relationship between stability and complexity: an analytical synthesis of the stabdity- complexity debate. American Naturalist 144: 14-29. Hill, M. 0. 1973. The intensity of spatial pattern in plant communities. Joumal of Ecology 6 1 : 225-235. Hinegardner, R. 1983. Biological complexity. Journal of theoreticai Biology 104: 7-20. Hom, H. S. 1974. The ecology of secondary succession. Annual Review of Ecology and Systernarics 4: 25-36. Hom, H. S. 1975 Markovian properties of forest succession. In: Ecology and evolution of communities (Cody, M. L. and Diamond, M.J., eds.) Harvard university Press, Cambridge Hom, H. S. 1976. Succession. In: Theoretical ecology (May, R. M., ed). Shuer Associates, Sunderland. Hom, H. S. 198 1. Some causes of variety in patterns of secondary succession. In: Forest Succession: Concepts and Application. (West, D. C., Shugart, H. H. and Botkin, D. B) Springer-Verlag, New York. Hu, S. T. 1959. Hornotopy 77zeory. Academic Press, New York. Huffman, D. A. 1952. A method for the construction of minimum redundancy codes. Proceedings of the Instirute of Radio Engineers. 40: 1098-1 101. Huisman, J., Olff, H. and Fresco, L. F. M. 1993. A hierarchical set of models for species response analysis. Journal opegetation Science. 4: 37-46. Huston, M. and Smith, T. 1987. Plant succession: life history and competition. Arnerican Naturalist 130: 168- 198. Hutchinson, G. E. 1953. The concept of pattern in ecology. Proceedings of the Academy of Natural Sciences 105: 1- 12. Huxley, T. H. 1902. Collected Essays. Vol. 9 Appleton, New York. Jackson. T. A. 197 1. Dialetics: the logic of Marxism, and ifs critics -- an essq in exploration. B. Franklin, New York. Jeffers, J. N. R. 1982. Modelling. Chapman and Hall, London. Keller, J. B. 1976. Inverse problems. Arnerican Mathematical Monthly: 83: 107-1 18. Kellert, S. H. 1993. In the wake of chaos. University of Chicago Press, Chicago. Kendall, M. G. and Stuart, A. 1976. The Advanced Theory of Statistics. Vo1.2. Hafner Publishing Company, New York. Kenkel, N. C. and Walker, D. J. 1993. Fractals and ecology. Abstracta Botanica. 1753-70. Kerner von Marilaun, A. 1863. Das Pfmzenleben der Danauldnder. Innbruck, Wagner. (Trans. by Conard, 195 1). Kershaw, K. A. 1957. The use of cover and frequency in the detection of pattern in plant cornrnunities. Ecology 38: 29 1-299. Kershaw, K. A. 1960. The detection of pattern and association. Journal of Ec010gy 48:233-242. Kershaw, K. A. 1963. Pattern in vegetation and its causality. Ecology. 44: 377-388. Kershaw, K. A. 1973. Quantitative and dynamic plant ecology. Edward Arnold Limited, London. Krebs, C. J. 1994. Ecology: the experimental analysis of distribution and abundance. HarperCollins College Publishers, New York. Kissov, V., Hadjitodovov, Sand Kuncheva, L. 1990. Using key features in pattern classification. Pattern Recognition Letters 1 1: 1-5. Kolasa, J. and Pickett, S. T. A. (eds.). 199 1. Ecological Heterogeizeity. Springer-Verlag, New York. Kolmogorov, A. N. 1965. Three approaches to the quantitative definition of information. Problems in InfonnationTransrnisstion 1: 1-7. Koppel, M. 1987. Complexity, depth and sophistication. Cornplex Systerns. 1: 1087-1091. Kraft, L. G. 1949. A device for quantking, grouping, and coding amplitude modulated pulses. MSc thesis, Electrical Engineering Department, Massachusetts Institute of Technology, March. Kullback, S. 1959. Information theory and Statistics. Wiley, New York. Kuncheva, L. 1993. 'Change-glasses' approach in pattern recognition. Pattern Recognition Letters 14: 6 19-623. Legendre, R. and Fortin, M. 1989. Spatial pattern and ecological analysis. Vegetatio. 80: 107- 138. Leps, J. and Hadincova, V. 1992. How reliable are our vegetation analyses? Journal of Vegetation Science 3: 1 19- 124. Levin, S. A. 1988. Pattern, scale, and variability: an ecological perspective. In: Communie Ecology. (Hastings, A., ed.) Springer-Verlag, Berlin. Levins, R. and Lewontin, R. 1980. Dialectics and reductionism in ecology. Synthese. 43: 47-78. Lewin, R. 1992. Complexity: life ut the edge of chaos. Macmillan Pub. Co., New York. Lewontin, R. 1966. 1s Nature probable or capricious? Bioscience. January . 25-27. Li, B. L. and Müller, F. 1995. Application of the pansystem entropy concept to hierarchical ecosystem analysis. Advances in Sysrems Science and Applications. I : L 3- 18. Lippe E., De Smidt, J.T. and Glen-Lewin, D.C. 1985. Markov models and succession: a test from a heathland in the Netherlands Journa1 of Ecology 73: 775-79 1. Lofgren, L. 1977. Complexity of descriptions of systems: a foundational study. International Journal of General Systerns. 3: 197-214. Loehle, C. 1988. Philosophical tools: potential contributions to ecology. Oikos. 57:97-104. Lorenz E. N. 1963. Deterministic nonperiodic flow. Journal of Atmospheric Sciences. 20: 130- 14 1 . Ludwig, J. A. and Reynolds, J. F. 1988. Statistical ecology: a primer on merhods and computing. Wiley, New York. McAuliffe, J. R. 1988. Markovian dynamics of simple and complex desert plant communities. Amencan Naturalist 13 1 : 459-490. McCook, L. J. 1994. Understanding ecologicd community succession: causal models and theories, a review. Vegetatio. 1 10: 1 15- 147. McIntosh, R. P. 1967. An index of diversity and the relation of certain concepts to diversity. Ecology. 48: 392-404. McIntosh, R. P. 1980. The relationship between succession and the recovery process in ecosystems. InThe Recovey Process in Damaged Ecosystems (Cairns, J, ed.) Ann Arbor Science Publishers, Ann Arbor. Maher, R. V. 197 1. Complexity analysis of vegetation patterns in an alpine meadow. M.Sc. thesis. The University of Western Ontario. Mandelbrot, B. B. 1977. Fractals: form, chance and dimension. W.H. Freeman, San Francisco. Mandelbrot, B. B. 1983. The Fractal Geometry of Nature. W.H. Freeman and Company, New York. Margalef, D. R. 1958. Information theory in ecology. General Systems. 3: 36-7 1. Margalef, R. 1968. Perspectives in Ecological Theory. Chicago: The University of Chicago Press. Margalef, R. 1989. On diversity and connectivity as historical expressions of ecosystems. Coenoses. 4: 12 1- 126. Matthews, 5. A. 1979. A study of the variability of some successionai and climax plant assemblage-types using multiple discriminant analysis. Journal of Ecology. 67: 255-272. Maurer. B.A. 1 987. Scaling of biological community structure: a systems approach to comrnunity complexity. Journal of theoretical Biology. 127: 97- 1 10. May, R. M. 1976. Simple mathematical models with very complicated dynamics. Nature. 26 1 : 459-467. May, R. M. 1977. Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature. 26947 1-477. May, R. M. 1979. The structure and dynamics of ecological cornmunities. In: Population dynamics. (Anderson, R. M. et al., eds.) Blackwell, Oxford. May, R. M. 1986. When two and two do not make four: nonlinear phenornena in ecology. Proceedings of the Royal Sociev of London series B. 228: 241-366. May, R. M. 1986. The search for patterns in the balance of nature: advances and retreats. Ecology. 67: 1 115- 1 126. May, R. M. 1987. Chaos and the dynamics of biological populations. Proceedings of the Royal. Socieq of London Series. A413: 27-44. Medawar, PB. 1969. Induction and Intuition in Scientific Thought. Methuen, London. Miles, J. 1980. Vegetation succession: past and present perceptionsh The Recovery Process in Damaged Ecosystems (Cairns, J., ed.) Ann Arbor Science Publishers Inc., Ann Arbor Monod, J. 197 1. Chance and necessity: an essay on the natural philosophy of biology. Knopf, New York. Moritz, H. 1993. Basic principles of inverse problems. In: Inverse Problems: Principles and Applications in Geophysics, Technology, and Medicine. (Anger,G., Gorenflo, R., Jochmann, H., Moritz, H. and Webers, W., eds.) Akademie Verlag, Berlin.

Odum, E. P. 1977. The ernergence of ecology as a new integrative discipline. Science. 195: 1289- 1293. Olsvig-Whittaker, L. 1988. Relating small-scale vegetation patterns to the environment. In: Diversity and pattern in plant communities. (During, H. J., Werger, M. J. A. and Willems, J. H.) SPB Acadernic Publishing, The Hague. Orloci, L. 1978. Multivariate Analysis in Vegetation Research. Dr. W. Junk. bv, the Hague. Orloci, L. 198 1. Probing tirne series vegetation data for evidence of succession. Vegetatio. 46: 3 1-35. Orloci, L. 1988. Detecting vegetation patterns. ISIAtlas of Science & Plants & Animals 1: 173- 177. Orloci, L. 199 1a. Enrropy and Information. Ecological Computations Series (ECS) Vo1.3. SPB Academic Publishing bv, the Hague. Orloci, L. 199 1b. Ecological programs for Instructional Computing on the Macintosh. Ecological Computations Series (ECS) Vo1.2. SPB Academic Publishing bv, the Hague, 13 1 pp. Orloci, L. 199 1c. On character-based plant community analysis: choice, arrangement, cornparison. Coenoses 6: 103- 107 Orloci, L. 1993. Conjectures and scenarios in recovery study. Coenoses 8: 141- 148. Orloci, L. 1993. The complexities and scenarios of ecosystem analysis. In: Mulrivariate Environmental Sratistics. (Patil, G.P, and Rao, C.R., eds.). Elsevier Publishers B.V., Amsterdam. OrIoci. L., and Orloci, M. 1988. On recovery, Markov chains and canonical analysis. Ecology 69: 1260- 1265. Orloci L., Anand, M. and He, X. S. 1993. Markov chah: a realistic mode1 for temporal coenosere? Biometrie- Praxirnetrie. 33. 7-26. Packel, E. W. and Traub, J. F. 1987. Information-based complexity. Nature. 328: 29-33. Palmer, M.W. 1988. Fractal geomeûy: a tool for describing spatial pattern of plant communities. Vegetatio. 75:9 1- 102. Papentin, F. 1980. On order and complexity. 1. General considerations. Journal of theoretical Biology 87: 42 1-456. Patil, G. P. and Taille, C. 1979. An overview of diversity. In: Ecological Diversity in Theory and Practice. (Grassle, J. F., Patil, G.P., Smith, W. and Taille, C., eds.) International Co-operative Publishing House, Fairland, Maryland. Pattee, H. H. (ed.) 1973. Hierarchy Theory: The Challenge of Cornplex Systems. G. Braziller, New York. Patten, B. C. 1975. Ecosystem linearization: An evolutionary design problem. In: Proceedings of the SIAM-SIMS Conference on Ecosystem Anabis and Prediction (Levin, S. A., ed.) Philadelphia. Peitgen, H. O., Jurgens, H. and Saupe, D. 1992. Chaos and Fractals. New Frontiers of Science. Springer-Verlag, New York. Peters, R. H. 199 1. A critique for ecology. Cambridge University Press, Cambridge. Phillips, J. D. 198% Measuring complexity of environmental gradients. Vegetatio. 64: 95- 102. Pickett, S. T. A., Collins, S. L. and Armesto. J. J. 1987. A hierarchical consideration of causes and mechanisms of succession. Vegetatio 69: 109-1 14. Pickett, S. T. A. and Kolasa, J. 1989. Structure of theory in vegetation science. Vegetatio. 83: 7- 15. Pickett, S. T. A. and McDonnell, M. J. 1989. Changing perspectives in community dynamics: a theory of successional forces. Trends in Ecology and Evolution. 8:24 1-245. Pielou, E. C. 1975. Ecological Diversity. Wiley, New York. Pielou, E. C. 1977. Mathematical Ecology. John Wiley & Sons, Inc., New York. Pielou, E. C. 198 1. The usefulness of ecological models: a stock-taking. Quarterly Review of Biology 56: 17-3 1. Pilette, R., Siga., R. and Blamire J. 1990. Stability-complexity relationships within models of natural systems. BioSystems. 23: 359-370. Pillar, de Patta, V. and Orloci, L. 1993. Taxonomy and perception in vegetation analysis. Coenoses. 8: 53-66. Pimm, S. L. 1992. The Balance of Nature? The University of Chicago Press, Chicago. Podani, J. 1984. Spatial process in the analysis of vegetation: theory and review. Acta Botanicn Hungarica 30: 75-1 18 Podani, J. 1989. Comparison of ordinations and classifications of vegetation data. Vegetatio. 83: 11 1 - 128. Poincaré, H. 188 1. Mémoire sur les courbes définies par une équation différentielle. Journal de Mathematique 7:375-442. Prigogine, 1. and Stengers, 1. 1984. Order out of chaos. Bantam Books, New York. Proccacia, 1. 1988. Universal properties of dynamically complex systems: the organization of chaos. Nature. 333: 6 18-623, Rand, D. A. and Wilson, H. B. 1991. Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proceedings of the Royal Socie~ of London series B. 246: 179- 184. Raunkiaaer, C. The life fonns of plants and statistical plant geography. Clarendon Press, Oxford. Rényi, A. 196 1. On measures on entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability. (Neyman, J., ed.) University of California Press, Berkeley. Ripley, B. D. 1977. Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society Series B. 39: 172-2 12. Roberts, D. W. 1987. A dynamical systems perspective on vegetation theory. Vegetcztio. 69: 27-33. Robertson, G. P. 1987. Geostatistics in ecology: interpolating with known variance. Ecology. 68: 744-748. Root, R. B. 1975. Some consequences of ecosystem texture. In: Ecosystem Analysis and Prediction. (Levin, S. A., ed.) SIAM, Philadelphia. Rosen, R. 1970. Dynamical System Theoy in Biology. Wiley-Interscience, New York. Saunders, P. T. and Ho, M. W. 198 1. On the increase in complexity in evolution II. The relativity of complexity and the pnnciple of minimum increase. Journal of theoretical Biology 90: 5 15-530. Scales, T. A. and Gersztenkorn, A. 1988. Robust methods in inverse theory . Inverse Problems 4: 107 1 - 109 1. Schaffer, W. M. 1985. Order and chaos in ecological systems. Ecology. 66:3- 106. Schaffer W. M, S. Ellner, M. Kot. 1986. Effects of noise on some dynamical models in ecology. Journal of Mathematical Biology 24: 479. Scheiner, S. M. and Gureriter, J. 1993. Design and analysis of ecological experiments. Chapman and Hall, New York. Scheuring, 1. 199 1. The fractal nature of vegetation and the species-area relation. Theoretical Population Biology 39: 170- 177. Shannon, C. E. 1948. A mathematical theory of communication. Bell SystemsTechnology 27: 379-423. Shugart, H. H. and Hett, J. M.1973. Succession: similarities of species turnover rates. Science 180: 1379-3 1. Shugart, H. H. 1984. A theory of forest dynamics: the ecological implications of foresr succession models. Springer-Verlag, New York. Silvertown, J. and Wilson, J. B. 1994. Cornmunity structure in a desert perennial comrnunity. Ecology. 75: 409-4 17. Simon, H. A. 1962. The architecture of complexity. Proceedings of the Americm Philosophical Society 106: 467-482. Simpson. E. H. 1949. Measurement of diversity. Nature. 163: 688. Slobodkin, L. B. 1974. Comrnents from a biologist to a mathematician. In: Ecosystem Analysis and Prediction. (Levin, S. A., ed.) SIAM, Philadelphia. Smith, T. and Huston, M. 1989. A theory of the spatial and temporal dynamics of plant communities. Vegetatio. 83: 49-69. Solomon, D. L. 1979. A comparative approach to species diversity. In: Ecological Diversity in Theory and Practice. (Grassle, J. F., Patil, G. P., Smith, W. and Taillie, C.eds.) International Co-operative Publishing House, Fairland, Maryland. Solomonoff, R. J. 1964. A formal theory af inductive control. Information and Control7: 224. Southwood, T. R. E. 1980. Ecology- a mixture of pattern and probabilisrn. Synthese. 43: 1 1 1- 122. Stewart, 1. 1989. Does God Play Dice? Penguin Books, London. Stone, L. and Ezrati, S. 1996. Chaos, cycles and spatiotemporal dynamics in plant ecology. Jo~irnalof Ecology 84: 279-29 1. Sugihara, G. and May, R. M. 1990. Applications of fractals in ecology. Trends in Ecology and Evolution 5: 79-86. Taillie, C. and Patil, G. P. 1979. Diversity, entropy and the Hardy- Weinberg Equilibrium. In: Ecological Diversity in Theory and Practice. (Grassle, J. F., Patil, G.P., Smith, W. and Taillie, C., eds.) International Co-operative Publishing House, Fairland, Maryland. Tallis, J. H. 199 1. Plant Conzmuni~Histoy. Chapman and Hall, London. Tansley, A. G. 1935. The use and abuse of vegetational concepts and terms. Ecology 16: 284-307. Tikhonov A. N. and Arsnin, V. Y. 1977. Solutions of Ili-posed Problems. John Wiley and Sons, New York. Tilman, D. 1982. Resorcrce cornpetition and community structure. Princeton Univ. Press, Princeton. Tilman, D. and Wedin, D. 199 1. Oscillations and chaos in the dynamics of a perennial grass. Nature. 353: 653-655. Tothméréz, B. 1995. Cornparison of different methods of diversity ordering. Jo~cmaloflegetation Science 6:283-290. Tsonis, P. A. and Tsonis, A. A. 1989. Chaos: principles and implications in biology. CABlOS 5: 27-32. Tulj apurkar, S. 1989. An uncertain life--demography in random environments. Theoretical Population Biology 35: 227-294. Upton, G. and Fingleton, B. 1985. Spatial Data Analysis by Example. Vo1.I. John Wiley & Sons, Ltd., Chichester. Usher, M. B. 1975. Analysis of pattern in real and artificial plant populations. Journal of Ecology 63: 569-586. Usher, M. B. 198 1. Modelling ecological succession with particular reference to Markovian models. Vegetatio 46: 1 1- 18 Van Andel, J., Bakker, J. P. and Grootjans, A. P. 1993. Mechanisms of vepetation succession: a review of concepts and perspectives. Acta Botanica Neerlandica 42: 4 13-433. Vandermeer, J. 1990. Indirect and diffuse interactions: complicated cycles in a population embedded in a large community. Journal of theoretical Biology 142: 429-442. Van der Maarel, E. 1972. On the transformation of cover-abundance values in phytosociology. Report of the Botanical Laborator-. Nijmegen Van der Maarel, E. 1988. Vegetation dynamics: patterns in time and space. Vegetatio. 77: 7- 19. van Hulst, R. 1979. On the dynamics of vegetation: Markov chahs as models of succession. Vegetatio 40: 3- 14. van Hulst, R. 1992. From population dynamics to community dynamics: modelling succession as a species replacement process. 1n:Plant Succession: theory and prediction. (Glenn-Lewin, D.C., Peet, R.K. and Veblen, T.T., eds.) Chapman & Hall, London von Humbolt, L. 1850. Views of Nature. H. G. Bohn, London. Ver Hoef, J. M., Cressie, N. A. C. and Glenn-Lewin, D. C. 1993. Spatial models for spatial statistics: some unification. Journal oflegetation Science 4: 44 1-452. Van Leeuwen, C. G. 1966. A relation theoretical approach to pattern and process in vegetation. Wentia. 15: 25-46. Waggoner P. E. and Stephens, G. R. 1970. Transition probabilities for a forest. Nntzrre 225: 1160-1 161. Walker, D. 1970. Direction and rate in some British post glacial hydroseres. In: Studies in the Vegetation History of the British Isles . (Walker, D. and West, R. G., eds.), Cambridge University Press, Cambridge. Warming, E. 1895. Plantesamfund. Grundtraek of den ~kologiske Plantegeograf. Kj~benhavn,Philipsens. Watkins, A. J. and Wilson, J. B. 1994. Plant community structure, and its relation to the vertical complexity of communities: dominance/diversity and spatial rank consistency. Oikos. 70: 91-98. Watt, A. S. 1947. Pattem and process in the plant community. Journal of Ecology 35: 1-22. Webb, L. J., Tracey, I. G. and Williams, W. T. 1972. Regeneration and pattern in the subtropical rain forest. Journal of Ecology 60: 675-695. White, P. S. 1979. Pattern, process, and naiural disturbance in vegetation. Botunicul Reviews 45: 229-299. Whittaker, R. H. 1953. A consideration of climax theory: the climax as a population and pattern. Ecological Monographs 23: 4 1-78. Whittaker, R. H. 1967. Gradient analysis of vegetation. Biological Reviews 42: 207-264. Whittaker, R.H. 1970. Communities and Ecosysterns. The Macmillan Company, Collier-Macmillan Limited, London. Whittaker, R. H. 1975. Comrnunities and Ecosystems. 2nd Edition. The Macmillan Company, London. Wicken, S. 1979. The generation of complexity in evolution: a thermody narnic and information-theoretical discussion. Journal of theoretical Biology 77: 349-365. Wiegert, R. G. 1988. Holism and reductionism in ecology: hypotheses, scale and systems models. Oikos. 53: 267-273. Wiegleb, G. 1989. ExpIanation and prediction in vegetation science. Vegetario. 83: 17-34. Wiens, J. A. 1984. On understanding a non-equilibnum world: myth and reality in community patterns and processes. In: Ecological Communities. ( Strong, Jr. D. R.,Simberloff, D., Abele, L. G. and . Thistle, A. B) Princeton University Press, Princeton. Wiens, J. A. 1989. Spatial scaling in ecology. Functional Ecology 3: 385- 397. Wilkins, G. R., Delcourt, P. A., Delcourt, H. R., Harrison, F. W. and Turner, M. R. 1991. Paleoecology of Central Kentucky since the Last Glacial Minimum. Quatemas, Research. 36: 224-239. Williams, W. T., and Lambert, J. M. 1959. Multivariate methods in plant ecology. 1. Association-analysis in plant communities. Journal of Ecology. 47: 83-101. Williams, W. T., Lance, G. N., Webb, L. J., Tracey, J. G. and Connell, J. H. 1969. Studies in the numerical analysis of complex rainforest communities. Journal of Ecology. 57: 635-654. Wolf A., Swift, J. B., Swinney, H.L. and Vastano, J. A. 1985. Determining Lyapunov exponents from a time series. Physica 16D: 285-3 17. Woodwell, G. M. and Smith, H. H., (eds.) 1969. Diversity and stability in ecological systems. Brookhaven Symposium. Biology. 22. Yockey, H. P., Platzman, R. L. and Quastler, H. (eds.) 1958. Infornation Theory in Biology. Pergamon Press, New York. Zar, J. H. 1974. Biostatistical analysis. Prentice-Hall, Inc., Englewood Cliff, New Jersey. Ziegler, B. P. 1979. Multilevel multiforrnalism modeling: an ecosystem example. In: Theoretical Systems Ecology. (Halfon, E, ed.) Academic Press, New York. IMAGE EVALUATlON TEST TARGET (QA-3)

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