Spatially inhomogeneous bifurcating periodic solutions induced by nonlocal competition in a predator–prey system with additional food

The nonlocal competition in prey is incorporated into a diffusive predator–prey model with additional food in predator and time delay. The local stability of the coexisting equilibrium is studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is also investigated by using time delay as bifurcation parameter. Some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions are given by utilizing the normal form method and center manifold theorem. Our results suggest that nonlocal competition together with time delay can induce spatially inhomogeneous bifurcating periodic solutions in the diffusive predator–prey model.