Concepts of Stability and Resilience in Predator-Prey Models

Abstract

(1) The persistence of populations in the real world depends on their ability to withstand perturbations imposed by either exogenous or endogenous factors. (2) This paper illustrates the appropriate mathematical concepts for describing persistence by means of a simple predator-prey model framed in difference equations. (3) Perturbations may be visualized either as a change in the perameters of the model or as alterations to the densities of the component populations. (4) A local stability analysis is performed to demarcate regions of parameter space in which the model displays locally stable equilibrium behaviour. Within these regions different rates of return to equilibrium occur, some so small that perturbations to populations do not damp within a biologically feasible time-span. (5) Each point in locally stable parameter space has associated with it a well-defined domain of attraction in phase space outside which perturbed populations do not return to equilibrium. There is considerable variation in the size of these domains. Thus some locally stable equilibria are robust to large displacements of their populations while some parameter combinations define equilibria which are fragile to any but the smallest perturbations. (6) Ideas of persistence are also applicable in locally unstable parameter space, where the domains of attraction centre on stable limit cycles or chaotic behaviour rather than on stable equilibria. (7) Analysing the local stability properties of a model is merely the first step in understanding persistence in the real world. Large regions of locally stable parameter space may not be occupied by real populations because of long characteristic return times or very small domains of attraction in phase space. Furthermore, parameter combinations giving rise to locally unstable equilibria may characterize populations showing stable limit cycle behaviour. (8) There would seem to be no single number, or small set of numbers, capable of describing `the stability' or `the resilience' of a population. Characteristic return times are a partial measure of the resilience of a population to perturbation, but a full understanding of persistence requires in addition a description of the non-local stability properties of the interaction. (9) The qualitative insights which the model provides into two fundamental ecological problems are considered. We ask what happens at the edge of a species' range, and we consider the success and failure of biological control programmes involving predators and parasites. (Numerical response)