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Global Stability for a Predator-Prey Model with Dispersal among Patches
We investigate a predator-prey model with dispersal for both predator and prey 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 prey among n patches. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive coexistence equilibrium of this model is unique and globally asymptotically stable if it exists.
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, 3–6].
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  or on the permanence and existence of periodic solutions [4–6].
Recently, Li and Shuai  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 , 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 large-scale coupled systems from building blocks of individual vertex systems, Li and Shuai  obtain the following sharp results for (1).
Proposition 1 (see [7, Theorem 6.1]). Assume that is irreducible. If there exists such that or , then, whenever a positive equilibrium exists in (1), it is unique and globally asymptotically stable in the positive cone .
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 , 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 for all , model (2) directly reduces to (1); thus our model (2) directly extends model in .
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 [7–9]; we refer to [10–12] 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 [13–17]. 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 [18–20].
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 . 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.
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 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. 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 such that its set of arcs is . If , we call a directed cycle.
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 .
Proposition 2 (see ). Assume . Then where is the set of all spanning trees of that are rooted at vertex , and is the weight of . In particular, if is strongly connected, then for .
Theorem 3 (see ). Assume . Let be given in Proposition 2. Then the following identity holds: where , , , are arbitrary functions, 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, , 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 each vertex dynamics is described by a system of differential equations where and . Let represent the influence of vertex on vertex , and let 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 .
Theorem 4 (see ). Assume that the following assumptions are satisfied.(1)There exist functions , and constants such that
(2)Along each directed cycle of the weighted digraph , ,
(3)Constants are given by the cofactor of the th diagonal element of .
Then the function satisfies for , ; namely, is a Lyapunov function for the system (6).
3. 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 as long as it exists.
First of all, we will give a lemma for the system (2).
Lemma 5. The set 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).
Theorem 6. Assume that a positive equilibrium exists for the system (2) and the following assumptions hold.(1)Dispersal matrixes , are irreducible; moreover there exists such that or .(2)There exists nonnegative constant such that for , , or for , .
Then, the positive equilibrium is unique and globally asymptotically stable in .
In the sequel, we have
Set Lyapunov functions as
Direct differentiating along the system (2), we have
Set , ,, and . One has
Next, we have two cases to consider.
Case I. for , .
Case II. for , .
For Case I, from the fact that and , we obtain that ; thus . Then we obtain that Let denote the cofactor of the th diagonal element of the matrix . From the irreducible character of matrix , we have .
Furthermore, set Lyapunov functions as Then differentiating along the solution of the system (2), we obtain that Let represent the directed graph associated with matrix . Then has vertices with a directed arc from to if and only if . 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 . Therefore, following from the irreducible character of matrix and Theorem 2.3 in , we obtain Combining with the fact that , therefore we have When we consider , by condition 1, there exists such that It means that or .
If and can be connected with an arc from to in , then we have and . Furthermore, Because of and . we deduce that From , or , we obtain that and or and .
By condition 1 and the definition of matrixes , , we get that are irreducible. By strong connectivity of , there exists a directed path from any to . Then we have that, for any , there must be or for any , there must be Next, we will prove that the largest compact invariant subset of is the singleton .
We only consider the case that The case that is similar to this case. So we omit it.
If , we have for any , and then we have which contradicts to the fact that
If and , we have for any , and then we have which also contradicts to the fact that Therefore, we obtain that , which means Namely, we get that the largest compact invariant subset of is the singleton . Therefore, by the LaSalle Invariance Principle (), is globally asymptotically stable in .
With the similar arguments to the Case I, we can prove that is globally asymptotically stable in for Case II. This completes the proof.
Corollary 8. Consider the model Assume that the matrix is irreducible. If there exists such that or ; then, whenever a positive equilibrium exists in (32), it is unique and globally asymptotically stable in the positive cone .
In this paper, we generalize the model of the -patch predator-prey model of  to the general model (2) that both the prey and the predator have dispersal among -patches. Based on the network method for coupled systems of differential equations developed in [7–9], 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  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 predator-prey model.
Conflict of Interests
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.
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Copyright © 2014 Yang Gao and Shengqiang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.