Bifurcations in a diffusive resource-consumer model with distributed memory

Abstract

Spatial memory is significant in modeling animal movement. For a diffusive consumer-resource model, a memory-based diffusion of consumer can result in richer and more realistic dynamics. In fact, memory-based diffusion is related to the resource distributions in past times because the memory decays over time. We originally propose a consumer-resource model with distributed memory, and then investigate the influence of the weak memory kernel on the stability of the positive constant steady state. When the memory-based diffusion coefficient is negative, the mean delay does not affect the stability of the positive constant steady state; however, when the memory-based diffusion coefficient is positive, the mean delay can lead to the spatially inhomogeneous periodic oscillation patterns. The direction and stability of Turing bifurcation induced by the memory-based diffusion coefficient are calculated by using the methods of Crandall and Rabinowitz, and the direction and stability of Hopf bifurcation induced by the mean delay are determined by the normal form theory.