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We investigate a predator-prey model with dispersal for both predator and prey among

In the literature of predator-prey population systems, both continuous reaction-diffusion systems and discrete patchy models are used to study the spatial heterogeneity [

Since the discrete patchy models usually involve high-dimensional system, it is rather mathematically challenging to study the uniqueness and stability of the positive equilibrium of the predator-prey patchy models, and the available global dynamics criteria in the literatures mainly focus on the special case of two-patch [

Recently, Li and Shuai [

In [

Assume that

Although well-improved results have been seen in the above work on dispersal predator-prey model, such models are not well studied yet in the sense that model (

Motivated by the above work in [

The main purpose of this paper is to obtain the global stability for the coexistence equilibrium of (

A mathematical description of a network is a directed graph consisting of vertices and directed arcs connecting them. At each vertex, the local dynamics are given by a system of differential equations called the vertex system. The directed arcs indicate interconnections and interactions among vertex systems.

A digraph

This paper is organized as follows. In the next section, we introduce preliminaries results on graph-theory based on coupled network models. In Section

In this section, we will list some definitions and Theorems that we will use in the later sections.

A directed graph or digraph

The weight

A connected subgraph

A tree

Given a weighted digraph

The Laplacian matrix of

Assume

Assume

Given a network represented by digraph

We assume that each vertex system has a globally stable equilibrium and possesses a global Lyapunov function

Assume that the following assumptions are satisfied.

There exist functions

Along each directed cycle

Constants

Then the function

In this section, the stability for the positive equilibrium of the

First of all, we will give a lemma for the system (

The set

The next Theorem gives the globally asymptotically stable condition for the positive equilibrium of the system (

Assume that a positive equilibrium

Dispersal matrixes

There exists nonnegative constant

Then, the positive equilibrium

Let

Next, we have two cases to consider.

For Case I, from the fact that

Furthermore, set Lyapunov functions as

If

By condition 1 and the definition of matrixes

We only consider the case that

If

If

With the similar arguments to the Case I, we can prove that

Theorem

By Theorem

Consider the model

In this paper, we generalize the model of the

Biologically, our result of Theorem

We remark that our Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The first author is supported by the Natural Science Foundation for Doctor of Daqing Normal University (no. 12ZR09). Shengqiang Liu is supported by the NNSF of China (no. 10601042), the Fundamental Research Funds for the Central Universities (no. HIT.NSRIF.2010052), and Program of Excellent Team in Harbin Institute of Technology.