Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect

Abstract

To incorporate spatial memory and nonlocal effect of animal movements, we propose and investigate the spatiotemporal dynamics of the single population model with memory-based diffusion and nonlocal reaction. We first study the stability of a positive equilibrium and the steady state bifurcation induced by diffusion and nonlocality. We then investigate the impact of the averaged memory period on stability and bifurcation, and show that the combination of the averaged memory period and the diffusion can lead to the occurrence of Turing-Hopf and double Hopf bifurcations. The paper originally derives the normal form theory for Turing-Hopf bifurcation in the general reaction-diffusion equation with memory-based diffusion and nonlocal reaction. This novel algorithm can be widely used to classify the spatiotemporal dynamics near the Turing-Hopf bifurcation point. Finally, we apply the obtained results to a model proposed by Britton and numerically illustrate the spatiotemporal patterns induced by Hopf, Turing-Hopf and double Hopf bifurcations. Stable spatially homogeneous/nonhomogeneous periodic solutions, homogeneous/nonhomogeneous steady states and the transition from one of these solutions to another are provided in this paper. We additionally acquire the coexistence of two stable spatially nonhomogeneous steady states or two spatially nonhomogeneous periodic solutions near the Turing-Hopf bifurcation point.