Analysis of Integrodifference Equations with a Separable Dispersal Kernel

Abstract

Integrodifference equations are a class of infinite-dimensional dy- namical systems in discrete-time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dis- persal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite- dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.