Pattern Formation and the Spatial Scale of Interaction between Predators and Their Prey

Abstract

We study interactions of predators and prey that are characterized by a scale difference in their use of space. Prey are assumed to occupy patches, forming a metapopulation with low migration among patches. Predators are homogeneously distributed over these patches, due to broad-scale foraging behavior or long-range juvenile dispersal. The predator population thus exerts a globally uniform predation pressure on the prey subpopulations. Under these conditions a nonlinear predator functional response depending on local prey density leads to multiple equilibria that can occur for the same parameter values. These equilibria differ in the fraction of prey patches that are (nearly) empty. Equilibria with a larger fraction of empty prey patches are more stable. The system tends to approach equilibria with a sufficiently high number of empty prey patches, so that local and global population dynamics are stable. If unstable dynamics are observed at all, the fluctuations in local prey density exhibit predictable characteristics. Our main conclusion is that a nonlinear functional response of the predator to local prey density can induce the formation of static patterns in prey density and, hence, lead to stable local and global dynamics. It is shown that these results are sufficiently general to carry over to situations in which prey migration between patches does occur or the spatial domain occupied by the prey population is continuous instead of subdivided into patches.