Global Stability for a Predator-Prey Model with Dispersal among Patches

and Applied Analysis 3 each vertex dynamics is described by a system of differential equations u 󸀠 i = f i (t, u i ) , (5) where u i ∈ Rmi and f i : R × Rmi → Rmi . Let g ij : R × Rmi × Rmj → Rmi represent the influence of vertex j on vertex i, and let g ij ≡ 0 if there exists no arc from j to i in G. Then we obtain the following coupled system on graph G: u 󸀠 i = f i (t, u i ) + n


Introduction
In the literature of predator-prey population systems, both continuous reaction-diffusion systems and discrete patchy models are used to study the spatial heterogeneity [1,2]; patchy models are often used to describe directed movement of population among niches or migration among habitats. It is naturally interesting problem to consider how the dispersal or migration of predator and prey influences the global dynamics of the interacting ecological system; thus patchy predator-prey model received lots of attentions [1,[3][4][5][6].
Since the discrete patchy models usually involve highdimensional 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 [3] or on the permanence and existence of periodic solutions [4][5][6].
Recently, Li and Shuai [7] considered the following predator-prey model with dispersal for prey among -patch: Here, , denote the densities of prey and predators on the patch , respectively. The parameters , and , in the model are nonnegative constants. What is more, the parameters and in the model are positive constants. Constant is the dispersal rate of the prey from patch to patch and constants can be selected to represent different boundary conditions in the continuous diffusion case.
In [7], the authors studied the global stability of the coexistence equilibrium of system (1), by considering (1) as a coupled predator-prey submodels on networks. Using results from graph theory and a developed systematic approach that allows one to construct global Lyapunov functions for largescale coupled systems from building blocks of individual vertex systems, Li and Shuai [7] obtain the following sharp results for (1).
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 (1) assumes no dispersal for predator, which is not realistic in many cases [1,3]. Thus it is interesting for us to consider the global stability of the positive equilibrium for predator-prey model with dispersal for both predator and prey.
Motivated by the above work in [7], in this paper we generalize model (1) into the following predator-prey model with dispersal for both predator and prey: Here, the parameters , , , , , and are defined the same as those in (1). The nonnegative constants , , and are the dispersal rate of the predators from patch to patch , and represents the different boundary conditions in the continuous diffusion case. Clearly, when = 0 for all , = 1, . . . , , model (2) directly reduces to (1); thus our model (2) directly extends model (1) in [7].
The main purpose of this paper is to obtain the global stability for the coexistence equilibrium of (2). We will engage the techniques of constructing Lyapunov function based on graph-theory which were well developed by Li et al. in [7][8][9]; we refer to [10][11][12] for recent applications. Our study seems to be the first attempt in applying the network method for coupled network systems of differential equations to address the predator-prey system with dispersal for both predator and prey among patches. Networked method has been extensively investigated in the several fields. For example, multiagent systems can be seen as complicated network systems. A lot of researchers take their interest in flocking and consensus of the multiagent systems [13][14][15][16][17]. What is more, neural network systems can be seen as complicated network systems. Over the past few decades, various neural network models have been extensively investigated [18][19][20].
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 with vertices for the system (2) can be constructed as follows. Each vertex represents a patch and ( , ) ∈ ( ) if and only if , > 0. At each vertex of , the vertex dynamics is described by a predator-prey system. The coupling among these predator-prey systems is provided by dispersal of predator and prey among patches.
This paper is organized as follows. In the next section, we introduce preliminaries results on graph-theory based on coupled network models. In Section 3, we obtain the main result of system (2). This is followed by a brief conclusion section.

Preliminaries
In this section, we will list some definitions and Theorems that we will use in the later sections.
A directed graph or digraph = ( , ) contains a set = {1, 2, . . . , } of vertices and a set of arcs ( , ) leading from initial vertex to terminal vertex . A subgraph of is said to be spanning if and have the same vertex set. A digraph is weighted if each arc ( , ) is assigned a positive weight. > 0 if and only if there exists an arc from vertex to vertex in .
The weight ( ) of a subgraph is the product of the weights on all its arcs. A directed path in is a subgraph with distinct vertices 1 , 2 , . . . , such that its set of arcs is A connected subgraph is a tree if it contains no cycles, directed or undirected.
A tree is rooted at vertex , called the root, if is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph is unicyclic if it is a disjoint union of rooted trees whose roots form a directed cycle.
Given a weighted digraph with vertices, define the weight matrix = ( ) × whose entry equals the weight of arc ( , ) if it exists, and 0 otherwise. For our purpose, we denote a weighted digraph as ( , ). A digraph is strongly connected if for any pair of distinct vertices, there exists a directed path from one to the other. A weighted digraph ( , ) is strongly connected if and only if the weight matrix is irreducible. The Laplacian matrix of ( , ) is denoted by . Let denote the cofactor of the th diagonal element of . The following results are listed as follows from [7].
Proposition 2 (see [7]). Assume ≥ 2. Then where is the set of all spanning trees T of ( , ) that are rooted at vertex , and ( ) is the weight of . In particular, if ( , ) is strongly connected, then > 0 for 1 ≤ ≤ .
Theorem 3 (see [7]). Assume ≥ 2. Let be given in Proposition 2. Then the following identity holds: where ( , ), 1 ≤ , ≤ , are arbitrary functions, Q is the set of all spanning unicyclic graphs of ( , ), ( ) is the weight of , and denotes the directed cycle of .
Given a network represented by digraph with vertices, ≥ 2, a coupled system can be built on by assigning each vertex its own internal dynamics and then coupling these vertex dynamics based on directed arcs in . Assume that Abstract and Applied Analysis 3 each vertex dynamics is described by a system of differential equations where ∈ R m i and : R × R m i → R m i . Let : R × R m i × R m j → R m i represent the influence of vertex on vertex , and let ≡ 0 if there exists no arc from to in . Then we obtain the following coupled system on graph : Here functions , are such that initial-value problems have unique solutions.
We assume that each vertex system has a globally stable equilibrium and possesses a global Lyapunov function .
(3) Constants are given by the cofactor of the th diagonal element of .

Main Results
In this section, the stability for the positive equilibrium of the -patch predator-prey model (2) is considered. We regard (2) as a coupled system on a network. Using a Lyapunov function for the -patch predator-prey model with dispersal and Theorem 4 of Section 2, we will establish that a positive equilibrium of the -patch predator-prey model (2) with dispersal is globally asymptotically stable in R 2 + as long as it exists.
First of all, we will give a lemma for the system (2).

Lemma 5.
The set R 2 + is the positive invariant set for the system (2).
The next Theorem gives the globally asymptotically stable condition for the positive equilibrium of the system (2).
(2) There exists nonnegative constant such that Then, the positive equilibrium * is unique and globally asymptotically stable in 2 + .
Proof. Let In the sequel, we have Set Lyapunov functions as  Abstract and Applied Analysis Next, we have two cases to consider.
Let denote the cofactor of the th diagonal element of the matrix . From the irreducible character of matrix , we have > 0. Furthermore, set Lyapunov functions as Then differentiating along the solution of the system (2), we obtain thaṫ Let represent the directed graph associated with matrix . Then has vertices 1, 2, . . . , with a directed arc ( , ) from to if and only if ̸ = 0. Then ( ) is the set of all directed arcs of . By Kirchhoff 's Matrix-Tree Theorem (see Proposition 2) we know that = can be expressed as a sum of weights of all directed spanning subtrees of that are rooted at vertex . Thus, each term in is the weight ( ) of a unicyclic subgraph of obtained from such a tree by adding a directed arc ( , ) from the root to vertex . Because the arc ( , ) is a part of the unique cycle of and that the same unicyclic graph can be formed when each arc of is added to a corresponding rooted tree , then the double sum can be expressed as a sum over all unicyclic subgraphs containing vertices 1, 2, . . . , .
By Theorem 6 and similar arguments to Remark 7, we directly have the following global stability theorem for the predator-prey model with discrete dispersal of predator among patches.
Assume that the matrix ( ) × is irreducible. If there exists such that > 0 or > 0; then, whenever a positive equilibrium * exists in (32), it is unique and globally asymptotically stable in the positive cone 2 + .

Discussion
In this paper, we generalize the model of the -patch predator-prey model of [7] to the general model (2) that both the prey and the predator have dispersal amongpatches. Based on the network method for coupled systems of differential equations developed in [7][8][9], we prove that the positive equilibrium of (2) is globally asymptotically stable given some conditions on the coupling (see Theorem 6). Our main theorem generalizes Theorem 6.1 in [7] and our results also cover the other case of (2) in that only the predators disperse among patches. Biologically, our result of Theorem 6 implies that if predator-prey system is dispersing among strongly connected patches (which is equivalent to the irreducibility of the dispersal matrixes of predator and prey) and if the system is permanent (which guarantees the existence of positive equilibrium), then the numbers of both predators and prey in each patches will eventually be stable at some corresponding positive values given the well-coupled dispersal (condition 2 of Theorem 6).
We remark that our Theorem 6 requires the extra condition 2 for the coupling dispersal coefficients and that the global dynamics for the coexistence equilibrium of (2) without condition 2 of Theorem 6 are still unclear. It remains an interesting future problem for the patchy dispersal predatorprey model.