Dynamics of A Single Population Model with Memory Effect and Spatial Heterogeneity

TL;DR

This paper investigates a population model with memory effects and environmental heterogeneity, revealing how their interaction can lead to stable states or complex spatial-temporal patterns through bifurcations.

Contribution

It demonstrates the joint influence of memory effects and spatial heterogeneity on population dynamics, including conditions for stability and pattern formation.

Findings

Global existence of nonhomogeneous steady states

Hopf bifurcation leading to periodic solutions

Memory effects induce spatial-temporal patterns

Abstract

In this paper, a single population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of upper and lower solutions, which is asymptotically stable for relatively small memorized diffusion. However, after the memorized diffusion rate exceeding a critical value, spatial inhomogeneous periodic solution can be generated through Hopf bifurcation, if the integral of intrinsic growth rate over the domain is negative. Such phenomenon will never happen, if only memorized diffusion or spatially heterogeneity is presented, and therefore must be induced by their joint effects. This indicates that the memorized diffusion will bring about spatial-temporal patterns in the overall hostile environment. When the integral of intrinsic growth rate over the domain is positive, it turns out that the steady state is still asymptotically stable. Finally, the possible dynamics of the model is also discussed, if the boundary condition is replaced by Dirichlet condition.