Bifurcation Analysis of a Diffusive Two-Species Model with Memory-Based Cross-Diffusions

A class of reaction–diffusion models for two interacting species with two memory-based cross-diffusions is considered in this paper. Stability and bifurcation analyses at the steady states are conducted to investigate the joint effects of memory-based diffusion rates and time delays. It is found that memory-based diffusion rates will induce Turing bifurcations, and the first Turing bifurcation curves have more complicated structures than the ones for the classic reaction–diffusion equations, in that they are composed of an asymptotic curve of Turing bifurcation curves with different modes. Furthermore, using time delays as varying parameters, we also find all the stability switching curves, and provide sufficient conditions for determining the shapes of these curves. With the aid of normal form computation, it is proved that Hopf bifurcation will occur on these stability switching curves. Based on these results, predator–prey and competition models with memory-based diffusions are studied, and various temporal or spatial patterns for the models can be observed, including the spatially inhomogeneous steady-state and inhomogeneous periodic solutions.