Hopf bifurcation in a diffusive predator-prey model with a square-root singularity

Abstract

In this paper, stability and Hopf bifurcation in a diffusive predator-prey system are discussed considering prey species' group behavior. The interaction term is of Holling type II with the prey density X under the square root. The local behavior is first discussed for the corresponding homogeneous system. Then, the diffusive system's linear stability is discussed around a homogeneous equilibrium state followed by bifurcations in the infinite-dimensional system. By the linear stability analysis, we see that a Hopf bifurcation occurs in the homogeneous system. By choosing a proper bifurcation parameter, we prove that a Hopf bifurcation occurs in the nonhomogeneous system. Furthermore, the direction of the Hopf bifurcation is obtained.