Dynamics of a diffusive two-species interaction model with memory effect

In this paper, we investigate a two-species interaction model with the memory-based diffusion, which is used to describe the movement of one species depending on the past density of another species. The global existence and uniqueness are proved, and uniform boundedness of the solution is shown in the case of either predator-prey or competition interaction. Using the memory-based diffusion rate as varying parameter, it is found that either Turing or Hopf bifurcation will take place, generating inhomogeneous steady states or sptially inhomogeneous time-periodic solutions under assumptions that guarantee the constant steady state is locally stable when memory-based diffusion rate is zero. These results are then applied to a diffusive predator-prey model with the Beddington-DeAngelis functional response and a Lotka-Volterra competition model, finding that memory-based diffusion rate have a great effect on their local dynamics. More complex spatiotemporal patterns are also observed in numerical simulations for parameters chosen near the values for occurrence of higher co-dimensional bifurcations.