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 spatiotemp oral 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.