Stability and bifurcation on predator-prey systems with nonlocal prey competition

In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.