Spatial segregation and aggregation induced by memory and maturation delays in a diffusive predator-prey system

We analyze the double Hopf bifurcation arising from maturation and memory delays in a predator-prey system with memory-based diffusion, aiming to uncover their combined influence on the spatiotemporal distributions of the two interacting species. We begin by employing geometric approaches to study the linearized system with delay-dependent coefficients and identify conditions under which the constant coexistence steady state becomes unstable via double Hopf bifurcation. Next, we derive the normal form of the double Hopf bifurcation and categorize the resulting spatiotemporal dynamics in its vicinity. Combined with numerical simulations, our results reveal that maturation and spatial memory delays can give rise to abundant coexistence dynamics. Such dynamics include spatially homogeneous periodic oscillations, bistability between distinct spatially inhomogeneous periodic orbits, and transitions between spatial segregation and aggregation in a temporally periodic manner.