Introduction

If it was ... straightforward, then simple laws operating in simple circumstances would always lead to simple patterns, while complexlaws operating in complex circumstances would always lead to complex patterns. ... This no longer looks correct, but it's taken time to find out because we seem to be predisposed toward such a principle.

A central goal in population biology and ecology is to understand temporal fluctuations in population abundance. Such fluctuations, however, often appear to be erratic and random, with levels of variation ranging from small percentages to several orders ofmagnitude.1 In the 1970s Lord (Robert) May of Oxford put forth a bold new hypothesis concerning the possible explanation of the complex dynamic patterns so often observed in biological populations [122, 124, 125], The prevailing point of view had been that complex patterns have complex causes and simple causes have simple consequences. May's hypothesis implies, on the other hand, that complex patterns can result from simple rules. To a few mathematicians and scientists this thesis had been known at least since the pioneering work of Henri Poincare in the late 19th century [2, 3, 12]. However, to most ecologists the assertion was novel; it raised the intriguing possibility that (at least some of) the complexity of nature might arise from simple laws.

The complexity about which May wrote is a result of nonlinearity. Although the classical models of theoretical ecology from the first half of the 20th century are nonlinear, the theories derived from them were centered on equilibrium dynamics. The famous logistic differential equation and the Lotka-Volterra equations for competition and predation are the

1 For example, one literature survey has found that annual adult recruitment could vary by factors of over 30 in terrestrial vertebrates, 300 in plants, 500 in marine invertebrates, and 2000 in birds [78], prototypical examples. Fundamentally, the mind set at the heart of these theories encompassed the notion of a "balance of nature" in which ecological systems are inherently at equilibrium and the erratic fluctuations and complexity observed in data are due to "random disturbances" or "noise." From this point of view, ecosystems are "stochastic perturbations" of underlying equilibrium configurations (in which processes are in some kind of optimal efficiency). The point of view suggested by May, however, was not based on "noisy equilibrium" states. As he put it, the fact that a simple, deterministic equation can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rigidly deterministic population growth relationship... [124].

May's hypothesis opened the door to new ways of thinking about population dynamics and ecological systems—ways that bring nonlinearity to the forefront and make it a major role player. Unexplained noise will always be present in ecological data. However, May's insight provided a new point of view: fluctuation patterns observed in the abundances of some population systems might be explained, to a large extent, by relatively low-dimensional nonlinear effects as predicted by simple mathematical models.

Despite the fact that mathematical and theoretical ecology developed and expanded profusely during the decades following May's seminal work, his hypothesis has proved both controversial and elusive to test [85, 146, 147]. Mathematicians have invented a plethora of ecological models and proved complicated theorems about them. Theoretical ecol-ogists have applied methods from dynamical systems theory to ecological problems and drawn implications from the results. Nonetheless, as a whole, the community of empirical ecologists remains unconvinced. They are unconvinced that one can effectively reason about ecological systems using mathematical models, that there are reliable ecological "laws" available for such an enterprise, and that mathematical ecology is anything but peripheral to real populations in real ecosystems.

This point of view of deep skepticism is not surprising, given the lack of evidence. Where are the examples in which mathematical models provide convincing explanations of real biological systems and accurate predictions for actual populations or ecosystems? Despite the optimism generated by famous experiments carried out by such notable figures as G. F. Gause [70], T. Park [139,141], and P. H. Leslie [114] decades before

May's work (and still cited in most ecology texts), theorists admit there are few if any such examples. Does this mean there is some inherent property of the ecological world that precludes the application of the methods that have proved so successful in other scientific disciplines? Must ecology remain primarily a descriptive endeavor, in which mathematical reasoning cannot hope to provide quantitatively accurate predictions, and theoretical issues remain purely semantic? Can ecology ever take its place among the "hard" sciences?

We will not attempt to answer such sweeping philosophical questions in this book. However, we will address some issues lying at the core of these problems, one of the most fundamental of which is a serious gap or "disconnection" between theory (mathematical models) and data. We will do this in the context of a study in complex nonlinear dynamics that we have conducted over the past decade — a study motivated by and designed to address May's hypothesis. Prerequisite to this project is the establishment of a strong connection between population data obtained from a particular biological organism and a mathematical model that describes the population's dynamics. By showing that simple, low-dimensional nonlinear models can "work"—that is to say, can provide quantitatively accurate descriptions and predictions of the dynamics of a real biological population—we will then be in a position to explore nonlinear phenomena in a rigorous way. The results of these explorations will document a variety of nonlinear effects (including chaos) whose occurrence, although well known in theoretical models, was mere speculation in real populations. However, beyond these specific phenomena, the project will provide an unequivocal example of how nonlinearity is absolutely central to the understanding of the dynamics of a real biological population. In some cases, in fact, we will see how surprisingly subtle nonlinear effects are required to obtain a more complete understanding of observed patterns — effects whose observation in real population data would be thought highly unlikely.

We hope that these studies will supply new insights into how nonlinearity, particularly when coupled with stochasticity, can provide a new level of explanatory and predictive power in population dynamics. The studies will focus on a particular biological system which, in the tradition of experimental science, we are able to control, manipulate, replicate, and accurately measure. However, as in other scientific disciplines, that tradition provides the insight into fundamental concepts, the understanding of principles, and the verification of hypotheses that can then serve as guidelines for the investigation of other systems and other situations and circumstances. In this way, we hope our studies provide a small step toward raising the explanatory and predictive power of ecological science.

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