Abstract
We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse among different 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 equilibrium of this model is unique and globally asymptotically stable if it exists.
1. Introduction
Coupled systems on networks are used to describe a wide variety of physical, natural, and artificial complex dynamical systems, such as neural networks, biological systems, and the spread of infectious diseases in heterogeneous populations (see [1–6] and the references therein). A mathematical description of a network is a directed graph consisting of vertices and directed arcs connecting them. At each vertex, the dynamics are given by a system of differential equations called vertex system. The directed arcs indicate interconnections and interactions among vertex systems.
Diffusion in patchy environment is one of the most prevalent phenomena of nature. Since the spatiotemporal heterogeneity incurs great impacts on the species’ diversity, structure, and genetical polymorphism, scientists of biology, ecology, and biomathematics paid great attention on the population dynamics with diffusion. The stability of equilibrium is the precondition of applications of dispersal models in practice. Therefore, there are a great amount of literatures on this topic (see [7–10] and the references therein). In [10], Kuang and Takeuchi considered a predator-prey model in which preys disperse among two patches and proved the uniqueness and global stability of a positive equilibrium by constructing a Lyapunov function.
Recently, a graph theoretic approach was proposed to construct Lyapunov functions for some general coupled systems of ordinary differential equations on networks, and the global stability was explored in [11, 12]. We refer to [13, 14] for recent applications.
In [12], Li and Shuai considered the following predator-prey model where prey species disperse among patches : where . They provided a systematic method for constructing a global Lyapunov function for the coupled systems on networks and then gave some sufficient conditions of stability for system (1). In fact, there may be more species in some habitats and they can construct a food chain; in this case it is more realistic to consider a multiple species predator-prey system. Based on this fact, in this paper, we investigate the following three-species food chain model in a patchy environment: where , , and denote the densities of prey species, mid-level predator species, and top predator species, respectively; all the parameters are nonnegative constants, and , , , and are positive. Constant is the dispersal rate of prey species from patch to patch , constant is the dispersal rate of mid-level predator species from patch to patch , and constant is the dispersal rate of top predator species from patch to patch . We refer the reader to [10, 15] for interpretations of predator-prey models and parameters.
This paper is organized as follows. In Section 2, we introduce some preliminaries on graph theory which will be used in Section 3. In Section 3, the global stability of the positive equilibrium of system (2) is proved. Finally, a conclusion is given in Section 4.
2. Preliminaries
Since the coupled system considered in this paper is built on a directed graph, the following basic concepts and theorems on graph theory can be found in [12].
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 . Here 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.
Given a weighted digraph with vertices, define the weight matrix whose entry equals the weight of arc if it exists and 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 defined as Let denote the cofactor of th diagonal element of . The following results are listed as follows from [12].
Proposition 1 (see [12]). Assume . Then, where is the set of all spanning trees of that are rooted at vertex and is weight of . In particular, if is strongly connected, then for .
Theorem 2 (see [12]). Assume . Let be given in Proposition 1. Then the following identity holds: where , , are arbitrary functions.
3. Main Results
In this section, the stability for the positive equilibrium of a three-species food chain model in a patchy environment is considered.
Theorem 3. Assume that a positive equilibrium exists for system (2) and the following assumptions hold.(H1)Dispersal matrixes , , and are irreducible.(H2)There exist nonnegative constants and such that and for all .
Then the positive equilibrium of system (2) is unique and globally asymptotically stable in .
Proof. From equilibrium equations of (2), we obtain
Next, we show that is globally asymptotically stable in , and thus it is unique. Let
Note that for and equality holds if and only if . Differentiating along the solution of system (2), we obtain
where
Consider a weight matrix with entry , and denote the corresponding weighted digraph as . Let
Note that is the Laplacian matrix of . Since is irreducible, we know that is irreducible; by Proposition 1, we obtain that for all . Then, from Theorem 2, the following identity holds:
Set
Using (8) and (11), we obtain
for all . Therefore, as defined in Theorem 3.1 of [12] is a Lyapunov function for the system (2); namely, for all ; implies that , , and for all . By LaSalle Invariance Principle [16], is globally asymptotically stable in ; this also implies that is unique in . This completes the proof of Theorem 3.
Remark 4. , , and are dispersal matrices; a typical assumption we impose on these matrices is that they are irreducible. In biological terms, this means individuals in each patch can disperse between any two patches directly or indirectly.
Applying the similar proof as that for Theorem 3, we have the following corollaries.
Corollary 5. Consider the model where all the parameters are nonnegative constants, , , , and are positive, and the dispersal matrix is irreducible. Then, if a positive equilibrium exists in (14), it is unique and globally asymptotically stable in the positive cone .
Corollary 6. Consider the model where all the parameters are nonnegative constants, , , , and are positive, and the dispersal matrix is irreducible. Then, if a positive equilibrium exists in (15), it is unique and globally asymptotically stable in the positive cone .
4. Conclusion
In this paper, we generalize the model of the -patch predator-prey model of [12] to a three-species food chain model where prey species, mid-level predator species, and top predator species can disperse among different patches . Our proof of global stability of the positive equilibrium utilizes a graph-theoretical approach to the method of Lyapunov function.
Biologically, Theorem 3 implies that if a three-species food chain system is dispersing among strongly connected patches (which is equivalent to the irreducibility of the dispersal matrixes) and if the system is permanent (which guarantees the existence of positive equilibrium), then the numbers of prey species, mid-level predator species, and top predator species in each patch will eventually be stable at some corresponding positive values given the well-coupled dispersal (condition of Theorem 3).
Corollaries 5 and 6 imply that if a three-species food chain system is dispersing among strongly connected patches (only one species can disperse among different patches ) and if the system is permanent, then the numbers of prey species, mid-level predator species, and top predator species in each patch will eventually be stable at some corresponding positive values.
Theorem 3 requires the extra condition ; the global stability for the positive equilibrium of system (2) without condition is still unclear. It remains an interesting problem for a three-species food chain model in patchy environment.
Conflict of Interests
The authors declare that there is no conflict of interests.
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant nos. 11371287, 11361059), the International Science and Technology Cooperation Program of China (Grant no. 2010DFA14700), Xinjiang Introduction Plan Project of High Level Talents, the Natural Science Foundation of Xinjiang (Grant no. 2012211B07), and the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2013I03).