Population Dynamic Study of Prey-Predator Interactions with Weak Allee Effect, Fear Effect, and Delay

In this study, a predator-prey model with the Allee effect and fear effect is established. We use the comparison principle to prove boundedness. The zero equilibrium point and nonzero equilibrium point of the model are calculated, and the local stability conditions are obtained. Next, according to the Sotomayor theorem, the cross-sectional conditions of transcritical bifurcation and Hopf bifurcation are obtained. The conditions for the Hopf bifurcation to be supercritical or subcritical can be calculated by the normal form theory. Then, to make the model more realistic, we introduce the gestation delay in the proposed mathematical model. Stability and Hopf bifurcation are also analyzed. Finally, several numerical simulations are presented to verify the conclusions. Our results demonstrate that the Allee effect, fear effect, and delay play significant roles in population dynamics. The Allee effect and delay destabilize the originally stable model, after which Hopf bifurcation occurs. However, the fear effect can enhance stable coexistence.