Amplitude equation for the Hopf bifurcation of a reaction-diffusion predator-prey system incorporating time delay

Modeling the interactions between prey and predators has become an important method to reveal their evolutionary patterns. This paper investigated a predator-prey model incorporating Holling II functional response, considering the dynamic effects of time delay and cross-diffusion on the model. First, the existence and local stability of a positive equilibrium were proven without time delay and diffusion. Next, we selected time delay as the bifurcation parameter to study the existence of Hopf bifurcation, and determined the critical value for Hopf bifurcation. Furthermore, by utilizing multi-scale analysis, the amplitude equation was derived, thereby obtaining the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Our results show that changes in time delay or diffusion parameters can lead to periodic oscillatory solutions in time and space, corresponding to supercritical (subcritical) Hopf bifurcation, and Turing instability. Finally, the theoretical results were validated through numerical simulations.