Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics

Abstract

In spite of their complex behaviour the dynamics of some ecological systems may be explained by deterministic laws of motion. This is clearly theoretically possible, because deductive approaches to the dynamics of ecological systems lead to nonlinear models and even very simple model ecosystems generate typically a rich spectrum of dynamics, ranging from coexisting periodic regimes to chaotic behaviour. The present study analyses the complex interplay between deterministic nonlinear dynamics and demographic stochasticity using two examples. Firstly, the search for chaos in ecology has drawn much attention to the analysis of recurrent outbreaks of childhood epidemics—in particular of measles infections—in large population centres. The second example is a compartmental model of a three-species food chain, where sudden population disappearances can be observed due to transient chaos. Our analysis lends support to the plausibility of chaos in population dynamics. However, for decreasing population size (increasing fluctuations) the distinction between chaos and stochasticity becomes more and more problematic. The interplay between deterministic dynamics and stochastic fluctuations, e.g. due to the integer structure of populations or a noisy environment, suggests an analysis combining methods from the theory of nonlinear dynamical systems as well as stochastic processes.