Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling

This paper studies stability switching and Hopf bifurcation in a two-delay predation system with diffusion and network coupling. By using the Kronecker-product formulation and Laplacian-mode decomposition, the high-dimensional characteristic equation of the coupled system is reduced to a family of mode-dependent characteristic equations. Based on this reduction, explicit stability-switching curves are derived in the (\tau_1,\tau_2)-plane, and the corresponding stability region of the positive equilibrium is determined. Furthermore, by applying the Hassard method and center-manifold reduction, the direction of the Hopf bifurcation, the stability of the bifurcating periodic solutions, and the variation of their periods are obtained. Numerical examples are presented to verify the theoretical results and illustrate the temporal and spatiotemporal dynamics generated by the two delays under network coupling. These results provide an effective analytical framework for studying delay-induced stability switching and oscillatory behavior in networked predator-prey systems.