O (2)-Equivariant Hopf Bifurcation and Wave Selection in a Delayed Cross-Diffusive Predator-Prey Model: Theory and Simulation

This study develops a modeling and simulation framework for wave selection in an O(2)-equivariant predator-prey model with general local kinetics and two delayed cross-diffusion mechanisms. We derive the mode-dependent characteristic equation and construct stability switching curves in the two-delay parameter plane, which provide computable criteria for delay-induced loss of stability. Center manifold reduction and O(2)-equivariant Hopf bifurcation theory are applied to obtain a normal form for the critical nonzero spatial mode, and the resulting coefficients determine the existence, bifurcation direction, and stability of the rotating and standing waves. A concrete fear-memory predator-prey model is used to illustrate the analytical results. The stability switching curves, normal form coefficients, and direct numerical simulations consistently identify the emergence of rotating- and standing-wave branches. The numerical scheme combines finite-difference spatial discretization, delayed-state interpolation, and a splitting time integrator. Moreover, simulations with multimode and random perturbations reveal mixedmode patterns associated with nonlinear interactions among spatial modes. These results show that behavioral memory, cross-diffusion, and spatial symmetry jointly organize stability switching, wave selection, and spatiotemporal pattern formation in delayed ecological systems.