Impact of Spatial Memory on a Predator-Prey System with Allee Effect

Recently, spatial memory and Allee effect have been widely investigated in population models, independently. This paper introduces these two aspects to a predator–prey system, and studies the interaction of two species. Allee effect causes bistability, and the predator-free steady-state is always locally stable. Prey-taxis can play a stable role in positive constant steady-state, and spatial memory delay generates the inhomogeneous Hopf bifurcation and even stability switching. In the absence of spatial memory delay, the stronger the predator is subject to the Allee effect, the larger the prey-taxis coefficient is required to keep two species coexist in a stable spatially homogenous form. With the same prey-taxis coefficient, the critical threshold of spatial memory delay corresponding to large predator diffusion coefficient is distinctly bigger than the one corresponding to small predator diffusion coefficient. Moreover, the amplitudes of spatial patterns, which reflect the degree of inhomogeneity, oscillate as spatial memory delay varies.