Bifurcation and mode interaction in hyperbolic reaction–diffusion systems with asymmetric inertia

This paper investigates the spatiotemporal dynamics of a hyperbolic reaction–diffusion predator–prey system with inertial effects. We first derive the critical conditions for codimension-one bifurcations (Hopf and Turing bifurcations) and codimension-two bifurcations (Turing–Turing and Turing–Hopf bifurcations). Theoretical and numerical results show that asymmetric inertial effects fundamentally alter the instability mechanism of the system: under equal diffusion rates, predator inertia alone can induce wave instability and self-organized spatiotemporal oscillations, whereas prey inertia mainly plays a stabilizing role. In addition, the multiple-scale method is successfully extended to the codimension-two bifurcation analysis of hyperbolic reaction–diffusion systems, overcoming the dimensional reduction difficulties encountered by the classical center manifold theory in dealing with higher-order time operators. The resulting normal forms accurately characterize the competition, selection, and transition of spatiotemporal patterns near the Turing–Hopf critical point. This study reveals the profound influence of inertial effects on the complex dynamics of nonlinear diffusion systems.