Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and schooling behavior

We consider a delayed diffusive predator-prey system with nonlocal competition in prey and schooling behavior in predator. We mainly study the local stability and Hopf bifurcation at the positive equilibrium by using time delay as the parameter. We also analyze the property of Hopf bifurcation by center manifold theorem and normal form method. Through the numerical simulation, we obtain that time delay can affect the stability of the positive equilibrium and induce spatial inhomogeneous periodic oscillations of prey and predator's population densities. In addition, we observe that the increase of space area will not be conducive to the stability of the positive equilibrium $(u_{\ast },v_{\ast })$, and may induce the inhomogeneous periodic oscillations of prey and predator's population densities under some values of the parameters.