Double-Hopf Bifurcation and Bistable Asynchronous Periodic Orbits for the Memory-Based Diffusion System

This paper explores the impact of spatial memory on the double-Hopf bifurcation and dynamics of the memory-based diffusion system. First, employing the center manifold theory and the normal form method, the explicit formulae for the coefficients in the normal form for the double-Hopf bifurcation of the general reaction-diffusion equations with memory-based self-diffusion and cross-diffusion are derived, which are expressed in terms of the original system parameters and can be used to simplify the system to analyze the spatiotemporal dynamics revealed by the double-Hopf bifurcation. In addition, considering the effect of memory-based diffusion on the population dynamics of the predator-prey model with a Holling-Tanner-type functional response function, we improve the conditions for the occurrence of Hopf bifurcation and establish the conditions for the constant steady state to lose its stability through double-Hopf bifurcation. Further, by analyzing the normal form of the double-Hopf bifurcation, we prove that memory-based diffusion can lead to a bistable phenomenon, i.e., two stable spatially asynchronous periodic orbits with different wave numbers coexist in the model and can also lead to a spatially inhomogeneous quasi-periodic orbit, revealing that species of animals with spatial memory may survive in diverse patterns.