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.