Hopf-Hopf bifurcation and dynamical analysis of a predator-prey model with memory-based diffusion

The spatiotemporal dynamics of a memory-based diffusion predator-prey system with a ratio-dependent Holling type III response are investigated. Firstly, the existence conditions for codimension-1 (Turing, Hopf) and codimension-2 (Turing, Turing-Hopf, Hopf-Hopf) bifurcations are established, along with necessary and sufficient conditions for Turing instability. Secondly, differing from the conventional center manifold reduction approach, a key contribution of this work lies in employing the multi-timescale method to derive the normal form near a nonzero mode Hopf-Hopf bifurcation point, wherein the selection of critical modes explicitly incorporates spatial effects. Subsequently, the topological structure of the orbit distribution near the Hopf-Hopf bifurcation point is analyzed based on the obtained normal form, and the corresponding spatiotemporal patterns are identified in the original system. All theoretical predictions are validated through numerical simulations. The results reveal that weak memory-based diffusion tends to stabilize the system, whereas strong memory diffusion can induce delay-dependent spatially heterogeneous oscillations. In particular, when the memory diffusion strength lies within an intermediate range, the system undergoes transitions between steady states and periodic oscillatory behaviors.