Dynamics of a delayed diffusive predator-prey model with Allee effect and nonlocal competition in prey and hunting cooperation in predator

In this paper, a delayed diffusive predator-prey model with the Allee effect and nonlocal competition in prey and hunting cooperation in predators is proposed. The local stability of coexisting equilibrium and the existence of Hopf bifurcation are studied by analyzing the eigenvalue spectrum. The property of Hopf bifurcation is also studied by the center manifold theorem and normal form method. Through numerical simulation, the analysis results are verified, and the influence of these parameters on the model is also obtained. Firstly, increasing the Allee effect parameter $\beta$ and hunting cooperation parameter $\alpha$ is not conducive to the stability of the coexistence equilibrium point under some parameters. Secondly, the time delay can also affect the stability of coexisting equilibrium and induce periodic solutions. Thirdly, the nonlocal competition in prey can affect the dynamic properties of the predator-prey model and induce new dynamic phenomena (stably spatially inhomogeneous bifurcating periodic solutions).