Global stability analysis for a two species predator-prey model with dispersal for predators among patches and holling type II functional response

In this paper, we investigate a two species predator-prey model with dispersal and Holling type II functional response for predators among n patches. Our main purpose is to extend the global stability criteria by Li and Shuai(2010) on a predator-prey model with dispersal for preys among n patches. Firstly, by using the basic reproduction number and uniform persistence theory, we obtain the threshold dynamics of the system, that is, the system is uniformly persistent and there exists at least one coexistence equilibrium if the basic reproduction number is larger than one unit. Secondly, by using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions that the positive coexistence equilibrium of the coupling model is unique and global asymptotically stable.