Dynamics of two species predator-prey model with spatially nonhomogeneous diffusion strategy

Abstract

In this paper, we investigate a Holling type-II predator-prey system with spatially nonhomogeneous diffusion strategy. By employing the methods of the implicit function theorem, eigenvalue theory and bifurcation theory, we analyze the stability/instability of the positive steady state and explore the existence of a Hopf bifurcation when the diffusion rate is large. Furthermore, when the driven diffusion functions $Q_1\left(x\right)=e^{qm\left(x\right)}$ and $Q_2\left(x\right)\equiv 1$, we detailed discuss how the parameter q of the density dependent diffusion $Q_1\left(x\right)$ affect the occurrence of Hopf bifurcations and the values of Hopf bifurcations.