Bifurcation in a Predator‐Prey Model With Memory‐Diffusion and Double Allee Effect

ABSTRACT In this paper, memory‐based diffusion is incorporated into a predator‐prey model with a double Allee effect. Turing instability, Hopf bifurcation, double Hopf bifurcation, and Turing‐Hopf bifurcation are analyzed by selecting the memory‐based diffusion coefficient and time delay as bifurcation parameters. Using center manifold theory and the normal form method, the paper derives a general expression for the normal form, which reveals complex spatiotemporal patterns near the Turing‐Hopf singularities. The results show that when the memory‐based diffusion coefficient satisfies certain conditions, Turing instability occurs. Additionally, the memory‐based delay can induce the spatiotemporal patterns of Hopf, double Hopf, and Turing‐Hopf bifurcations. Specifically, the system generates inhomogeneous stable periodic solutions with distinct spatial frequencies. Our results demonstrate that memory‐based diffusion can induce diverse population distribution patterns.