Hopf Bifurcation Analysis of a Two-Delayed Diffusive Predator–Prey Model with Spatial Memory of Prey and Delay-Dependent Parameters

In this paper, we consider the memory delay of prey and construct a diffusive predator–prey model with memory-based diffusion of prey and gestation delay of predator. Sufficient conditions for the local stability of the positive equilibrium of the model without delays are given. For the model with delays, we discuss the existence of bifurcation in the case of no diffusions by taking gestation delay as the bifurcation parameter. The joint effect of two delays on spatiotemporal dynamics of the model with delays and diffusions is analyzed by applying the method of stability switching curves. The model experiences a Hopf bifurcation when the two delays transition from the stable region, as defined by the stability switching curves, to the unstable region. Further, we calculate the normal form of Hopf bifurcation induced by the memory delay to determine the properties of Hopf bifurcation. Finally, several numerical simulations are performed to illustrate the theoretical results.