Dynamics of a Predator–Prey Model with Memory-Based Diffusion.

In this paper, we propose and analyze a diffusive predator–prey model with memory-based diffusion, which is described by a delayed predator-taxis term. We first obtain the existence and uniform boundedness of solutions. Then we study the stability of positive constant equilibria. We find that the memory delay may induce Hopf bifurcation, generating spatially inhomogeneous periodic solutions. This result implies that the memory-based diffusion is beneficial to pattern formations and is contrary to the conclusion on predator–prey model only with predator-taxis. The normal form of Hopf bifurcation is also derived. The theoretical results are demonstrated through the case with the Holling II functional response. Numerical simulations also suggest the phenomenon of stability switches and the occurrence of double Hopf bifurcation. [ABSTRACT FROM AUTHOR]