Global stability analysis and Hopf bifurcation due to memory delay in a novel memory‐based diffusion three‐species food chain system with weak Allee effect

Abstract

This paper presents a detailed study on the dynamics of a three‐species food chain system with memory‐based delay under weak Allee effect and middle predator refuge. The local stability analyses of the non‐diffusive and memory‐based diffusion systems are investigated. Moreover, we give a priori estimates and obtain the existence of the positive constant steady state by applying the comparison theorem. Sufficient conditions for the global stability are established by Barbǎlat lemma and the Lyapunov function. The theoretical results suggest the joint effect of cross‐diffusion and memory‐based delay can lead to Hopf bifurcation, which cannot appear in the system with self‐diffusion or a small cross‐diffusion coefficient. Numerical results verify the validity of the theoretical analysis.