Stability and HOPF Bifurcation in a Delayed Diffusive Predator-Prey Model Incorporating Allee and Fear Effects

This work examines a predator-prey model that features multiple ecological mechanisms, along with Allee effect, fear effect, and time delay, with a focus on how time delay influences the model's dynamics behaviors. To begin, we analyze the Hopf bifurcation with non-delay model. Subsequently, by choosing time delay as the bifurcation parameter, we obtain the existence of Hopf bifurcation. Moreover, the direction, stability and periodicity of Hopf bifurcation are determined by the normal form theorem and the center mainfold theorem. Finally, numerical simulations are offered to further confirm these theoretical findings.