12.1 INTRODUCTION

It is not difficult to sell the notion, especially to ecologists, that the world is a complex place, and that the modelling of any ecological system automatically brings one into the realm of complex systems. Indeed, one can advance this idea without ever having to define what is meant by a complex system, a definition that one may feel must be complex itself. Nonetheless, for progress to be made in modelling complex systems, the definitional problem must be confronted.

    There is more than one pathway to complexity. Indeed, systems may be complex because their structures are complex, replete with details and interlocking effects; or they may be complex because their dynamics are complex, which may be true even for very simple models. These two notions obviously are not the same; yet either may be implied when one refers to complex systems.

   The study of how complicated dynamics can arise from simple models is one of the central areas of research in science today (Gleick, 1988); it is also a vibrant area of mathematical research. Physicists focus on the problem of how turbulence arises from instabilities in fairly regular flows; population biologists and epidemiologists examine similar problems in the dynamics of interacting populations (May, 1974, 1976; Schaffer and Kot, 1985; Schaffer et al., 1985). The stimulus for this revolution came from the work of Edward Lorenz, a meteorologist interested in the (structurally) complex models used to predict the weather. Working on a primitive electronic computer, the Royal McBee, Lorenz in 1960 created a very simplified, three-component model that captured the essential features of the giant weather models. Working with this simple model, Lorenz was surprised to find difficulty in obtaining reproducible predictions of weather patterns. The slightest change in the conditions under which he initiated his simulations would produce dramatically different predictions (Lorenz, 1963, 1964).

   This property, extreme sensitivity to initial conditions, is a fundamental characteristic of the phenomenon that has been termed chaos, and obviously frustrates our ability to make precise predictions. Chaotic systems typically do not exemplify the fairly regular long-term (asymptotic) dynamics seen in most classical ecological models. The equilibrium points and limit cycles of those models give way to quasiperiodic solutions, and worse yet to structures aptly named strange attractors, which wander through space without apparent pattern, making abrupt and unpredictable digressions. Often, these patterns can be understood as the result of competition among distinct and in-compatible periods in the system's dynamics, as can occur for example when an inherently oscillatory system is driven (forced) by a process (e.g. seasonal demography) with a distinct periodicity. Yet chaotic patterns can arise in completely autonomous systems, that is in systems that are not driven from outside; and it is this feature that has captured the imagination of large groups of theoretical physicists, mathematicians, and other scientists.

   Chaotic systems are not without their regularities. Although their time traces may appear indistinguishable from the time-series that would arise from random processes, they exhibit characteristic statistical properties. Modern methods in dynamical systems theory can be used to retrieve the deterministic relations underlying the observed relations (Ruelle and Takens, 1971). Methods of this sort also have been used on ecological and evolutionary data sets in an attempt to extract patterns (Schaffer et al., 1985).

   Fractals represent a second active area of research in science and mathematics, with strong relationships to the theory of chaos. Fractals are the creation of Benoit Mandelbrot, a mathematician at IBM, and refer to the complex spatial structures that seem to be everywhere in nature, and that can be generated from very simple theoretical relationships. Fractals turn our attention to the importance of scale, through their elucidation of the fact that the problem of measurement is inextricably tied up with the problem of scale. What has been most seductive here is the observation that, in many applications, there are puzzling regularities in the way measurements vary across scales.

   A case in point, perhaps the best-known example, is exhibited by Richardson's data on the lengths of coastlines. As Mandelbrot (1977) discusses, the notion of the length of coastline makes no sense without reference to the precision one applies to the measurement process. Typically, coastline length increases as increasing precision picks up smaller and smaller embayments; the process does not asymptote, as it would for a theoretically derived Euclidean curve, but continues to increase in an inverse logarithmic relationship with the length of the measure unit. What is striking is not that there is a relationship, but that it is as regular as it is. This is a manifestation of the fact that the tortuosity of the coastline seems to be scale-invariant; photographic enlargements of very small areas look, to the naked eye, like their larger siblings. Its ineluctable tortuosity on finer and finer scales confines the coastline to a nether-world intermediate between the smooth Euclidean curves of our schoolbooks and truly two-dimensional objects; hence, they have a fractional or fractal dimension intermediate between 1 and 2. This key attribute, self-similarity across a range of scales, is familiar to statistical physicists interested in critical phenomena, and also may be a property of many ecological processes.

   The considerations raised by these examples are ones that must be addressed when we attempt to describe or simulate the dynamics of ecological systems. In particular, the problem of scale is one of the most basic we must face, since the description of pattern cannot be separated from the observational scale chosen. The hope must be that there are regularities underlying the different descriptions processes generate on different scales, and that those relationships are discoverable theoretically or by analysis of data. Without such regularities we can have no basis for extrapolating from one system to another, or one scale to another. Ecosystem experiments can be designed to elucidate these issues by providing information simultaneously on multiple scales. The rest of this short chapter is directed to these issues.