In 1958, when ecology was a young science and mathematical models for ecological systems were in their infancy, Elton wrote of the neolithic days of animal ecology, that is to say about twenty-five years ago. Acknowledging the influence of Lotka and Volterra, he noted, Being mathematicians, they did not attempt to contemplate a whole food-chain with all the complications of five stages. They took two: a predator and its prey.
Today, in the era of computational ecological modeling, deterministic systems for two variablesand even a whole food chainappear like simple idealizations well removed from the complexity of nature. We now consider predatorprey interactions as consumerresource interactions embedded within the large ecological networks that underlie biodiversity () . Consequently, the scale of the problems we model has grown to reflect the world as we now need to observe it. For example, the interplay between ecosystem dynamics and the physical environment that influences global change occurs over a tremendous range of spatial and organizational scales (e.g., ). Similarly, the population dynamics of the transmission of infectious diseases often involve spatial or social networks with large numbers of individuals, but the interactions of each individual involve only a subset of the network and can span from local to global distances (e.g., ).
These examples illustrate the current view of ecological systems as complex adaptive systems ,. Complex adaptive systems are distinguished not only by the multiplicity of components within them, but also by interactions that can be local or distributed among these components and whose rates vary as nonlinear functions of the state of the system itself. One obvious role of computation in the science of complex systems is simply one of synthesis: to reconstruct the whole from the parts as we learn more and more about the components and their interactions. There are obvious limitations to this approach, evident in the famous image of those imperial cartographers who produced a map of the empire of the same size as the empire itself . I argue here that an alternative and more useful role of computation is to address questions on the relationship between dynamics at different temporal, spatial, and organizational scales, that is, to address the importance of variability at small, local scales to the dynamics of aggregated quantities measured at large, global scales. If small-scale details matter, we need to ask how much complexity we need to incorporate into large-scale models if we seek to both understand and predict the dynamics of global quantities.
Is it possible to incorporate the effect of small-scale variability without resorting to the brute force approach of using higher and higher resolution? I start with examples from theoretical ecology that illustrate problems and approaches related to these scaling questions; I then present more specific examples related to global change and ecosystem dynamics, and end with a series of related problems on the dynamics of large food webs, the ultimate networks of ecological interactions.