Turing–Hopf Bifurcation Analysis for a Diffusive Prey–Predator Model with a Chemotactic Term

Classical reaction–diffusion systems exhibit diverse dynamical behaviors, which become more complex particularly in the context of bifurcation analysis and normal form theory when chemotactic terms are incorporated. This paper investigates the stability of a positive constant equilibrium and the Turing–Hopf (TH) bifurcation in a diffusive predator–prey model with searching, handling predators, and chemotaxis. A computational method is developed for deriving the normal form of the TH bifurcation on the center manifold in general three-dimensional chemotactic reaction–diffusion systems with Neumann boundary conditions. The method enables an explicit dynamical classification near the bifurcation point. Combined theoretical and numerical analyses reveal that TH interactions significantly enrich spatiotemporal dynamics.