Double Hopf bifurcation analysis for a diffusive prey-predator model with a chemotactic term

Interactions between chemotaxis-driven Hopf bifurcation and other bifurcations can lead to a remarkably rich array of spatiotemporal dynamics. In this paper, the stability of the positive constant equilibrium and the existence of double Hopf bifurcation for a diffusive predator-prey model with chemotactic terms are studied. For the first time, we have formulated computational methods for determining the normal forms associated with double Hopf bifurcation in a reaction-diffusion system that features a chemotactic term and adheres to Neumann boundary conditions. Especially, the developed normal form method is applicable for classifying the dynamics near the double Hopf bifurcation point. Finally, numerical simulations are provided to substantiate the results.