Turing-Hopf bifurcation of a ratio-dependent predator-prey model with diffusion

In this paper, the Turing-Hopf bifurcation of a ratio-dependent predator-prey model with diffusion and Neumann boundary condition is considered. Firstly, we present a kind of double parameters selection method, which can be used to analyze the Turing-Hopf bifurcation of a general reaction-diffusion equation under Neumann boundary condition. By analyzing the distribution of eigenvalues, the stable region, the unstable region (including Turing unstable region), and Turing-Hopf bifurcation point are derived in a double parameters plane. Secondly, by applying this method, the Turing-Hopf bifurcation of a ratio-dependent predator-prey model with diffusion is investigated. Finally, we compute normal forms near Turing-Hopf singularity and verify the theoretical analysis by numerical simulations.