Multiple Time-Scale Analysis for Double Hopf Bifurcations inMemory-Driven Nonlocal Diffusion Systems

This paper presents a systematic investigation of the spatiotemporal dynamics of a predator–prey model incorporating memory-dependent diffusion and nonlocal interactions. First, the conditions for the emergence of Turing instability under the combined effects of memory diffusion and nonlocality are derived, thereby theoretically elucidating the intrinsic mechanisms of spatial pattern formation and the key factors influencing it.Next, rigorous existence conditions and criteria for Turing bifurcation, Hopf bifurcation, and double Hopf bifurcation are established.Moreover, a major innovation of this work lies in the development of a systematic computational framework for the normal forms of double Hopf bifurcations in reaction–diffusion systems with coupled memory diffusion and nonlocal effects, based on a multiscale method. The coefficients of the normal forms are explicitly expressed in terms of the parameters of the original system. On this basis, several typical spatiotemporal dynamical modes associated with the original system are further identified and characterized.The results demonstrate that nonlocal interactions significantly promote the formation of spatially heterogeneous structures, whereas memory diffusion effectively suppresses classical Fickian diffusion–induced Turing instability. Under the combined influence of these two mechanisms, the system can exhibit complex spatiotemporal behaviors, including stable spatially inhomogeneous periodic solutions and quasiperiodic solutions.