Spatial movement with nonlocal memory

Cognition and memory have a significant impact on the spatial movement of animals. This paper proposes a diffusive model with nonlocal memory-based diffusion to model animal movement. The stability of the positive homogeneous steady state and the bifurcation behaviors are investigated by taking the memory delay and the memory-based diffusion coefficient as bifurcation parameters. When the movement driven by the memory-based diffusion is slow, it has been shown that the memory does not affect the stability of the positive homogeneous steady state. However, when the memory-based diffusion is fast, a slight memory delay can destabilize the homogeneous steady state, while a large memory delay can strengthen the stability of the homogeneous equilibrium. Moreover, a series of Turing bifurcations and Hopf bifurcations lead to rich dynamics in the system.