Dispersal and pattern formation in a discrete-time predator-prey model
There is nothing surprising in the observation that organisms track spatial and environmental variation. Indeed, there is an extensive literature on habitat selection and on the role of habitat selection in determining the dispersion of organisms. Despite the uniquity of environmental heterogeneity, patterned spatial distributions may also arise in homogeneous enviromments. Thus it is essential that we ask whether patterns can arise solely as the result of trophic interactions and dispersal. A number of scientists have investigated this question using continuous-time growth models with simple (Fickian) diffusion. The diffusion in these reaction-diffusion models is general thought of as a stabilizing influence, one that homogenizes populations and moderates temporal fluctuations. However, \textit{A. Turing} [Philos. Trans. R. Soc., London, Ser. B 237, 37-72 (1952)] demonstrated that diffusion may also combine with intra- and interspecific interactions to yield instability and spatial patterns. The conditions for diffusive instability are surprisingly severe -- at least by ecological standards. There are a number of predator-prey reaction-diffusion models that exhibit diffusive instability, but they invariably invoke tenuous biological assumptions such as the coupling of prey autocatalysis or of an Allee effect with a density dependent death state for the predator. In effect, the assumption of Fickian diffusion in both predator and prey is sufficiently restrictive that it may place an undue burden on the interaction terms. A further difficulty with reaction-diffusion equations is that they are inappropriate for the innumerable species with discrete, nonoverlapping generations. To circumvent these difficulties, we will consider discrete-time models built around contact distributions probability distributions for the distance that an organism moves. Our principal claim is that discrete-time spatial contact models -- alias integrodifference equations -- exhibit dispersal-driven instability under a broader set of ecological conditions than to reaction-diffusion models. Our purpose in this paper is to provide a general foundation for systems of integrodifference equations and to analyze and demonstrate the circumstances that lead to discrete-time dispersal-driven instability. In Section 2, we briefly formulate a system of integrodifference equations for a predator and its prey. Each equation is build around a redistribution kernel (or contact distribution). There are a variety of possible redistribution kernels (Section 3); this multiplicity facilitates the occurrence of dispersal-driven instability. We derive necessary conditions for dispersal-driven bifurcations in Section 4. A detailed example is presented in Section 5. Concluding remarks are relegated to Section 6.
scientific article; zbMATH DE number 797079