Global stability for a new predator–prey model with cross-dispersal among patches based on graph theory

In this paper, cross-dispersal is considered in a predator–prey model with a patchy environment. A new predator–prey model with cross-dispersal among patches is constructed. A new cross-dispersal matrix is established by the coupling relationship between vertices. First, an existence theorem of the positive equilibrium for the new model is obtained. Secondly, based on the idea of constructing Lyapunov functions and a graph-theoretical approach for coupled systems, sufficient conditions that the positive equilibrium of the new model is globally asymptotically stable in R2n + are derived on a network with strongly connected graphs. Thirdly, based on the theory of asymptotically autonomous systems, Lyapunov functions method and graph theory, a stability theorem for the positive equilibrium of the new model is established on a complex network without strongly connected graphs. Finally, two examples are given to illustrate main results.

prey systems with self-dispersal of predators between two patches were studied in [4]. Predator-prey models with n patches and self-dispersal for both predators and prey were investigated in [5,6].
Because of the close relationship among species in different patches, cross-dispersal should be considered in the real environment. In fact, predator dispersal can affect prey density, while prey dispersal also can affect predator density. Hence, it is necessary to study the dynamics of the predator-prey model with cross-dispersal among all patches. Recently, cross-dispersal was used to study multi-group models (see [7]). Based on graph theory and Lyapunov functions method, the dynamics of a general multi-group model with cross-dispersal were given in [7]. To the best of the author's knowledge, few researchers have focused on the dynamics of predator-prey models with cross-dispersal. Hence, it is important to discuss the problem in this paper.
Based on graph theory, a systematic approach to construct global Lyapunov functions for coupled systems was developed by the authors of [2]. Much work had been done in order to apply this method to many areas [1-3, 6, 8-11]. The systematic approach was based on the assumption that the network was strongly connected. However, to the best of the author's knowledge, graphs without strong connectedness are universal in reality. Due to dealing with large-scale complex networks without strong connectedness, a hierarchical method and a hierarchical algorithm were proposed in [12]. Based on the hierarchical algorithm and the theory of asymptotically autonomous systems, stability theorems for a new fractional-order coupled system on a network without strong connectedness were obtained in [13].
Although predator-prey models based on ordinary differential equations (ODE) have been discussed for many years, to the best of the author's knowledge, cross-dispersal was not considered in the predator-prey model based on ODE by researchers. In fact, crossdispersal is reasonable and applicable for patchy environment. The construction for the new predator-prey model with cross-dispersal in a patchy environment is interesting and can be widely applied to the ecological field. In order to fill this gap, a new predatorprey model with cross-dispersal is constructed in this paper. To the best of the author's knowledge, the new predator-prey model with cross-dispersal constructed herein has not been proposed in any other literature. The new cross-dispersal matrix presented here has not been established by any other researcher. Based on the method of graph theory and Lyapunov theory, global stability theorems of the positive equilibrium are established. Innovative points are listed as follows: 1. Cross-dispersal is introduced into the predator-prey model in a patchy environment.
A new predator-prey model is established. 2. A new cross-dispersal matrix is established by the coupling relationship between vertices. 3. Based on the idea of graph theory, a global stability theorem for the positive equilibrium is established on a network with strongly connected graphs. 4. Based on the theory of asymptotically autonomous systems and graph theory, a global stability theorem for the positive equilibrium is established on a network with strongly connected components, but without strongly connected graphs. This paper is organised as follows. Preliminary results are introduced in Sect. 2. In Sect. 3, the main results are obtained, and examples are presented in Sect. 4. Finally, conclusions and outlook are outlined in Sect. 5.

Preliminaries
In this section, some definitions and theorems are listed that will be used in the later sections (see [2,12,13] Here, F ij (x i , x j ), 1 ≤ i, j ≤ n, are arbitrary functions, Q is the set of all spanning unicyclic graphs of (G, A), w(Q) is the weight of Q, and C Q denotes the directed cycle of Q.

Main results
Based on cross-dispersal, the new predator-prey model is constructed as follows: Here, x i , y i denote the densities on patch i for various prey and predators, respectively. Model parameters b i , δ i , e i , ε i are all positive constants. r i and γ i are non-negative constants. k xy ij denotes the dispersal rate of predators from patch j to patch i. s ij x i denotes functional response of predators that disperse from patch j to patch i. k yx ij denotes the dispersal rate of various prey from patch j to patch i. p ij x j denotes the functional response of predators on patch i to various prey that disperse from patch j to patch i. z ij denotes the conversion rate of various prey that come from patch j and are preyed on by predators on patch i. The meanings of the above parameters are listed as Table 1.
Based on the Volterra predator-prey model [2], the new model constructed is reasonable. Dispersal assumptions are reasonable from modelling. Predator-prey systems with n patches can be studied and explained from the biological viewpoint, any prey dispersed to this patch can be preyed on by predators on this patch. Furthermore, any prey can be preyed on not only by predators on this patch, but also by predators dispersed to this patch. In other words, any prey have a positive effect on the predator when prey disperse to a predator's population. Prey dispersed from patch j can be preyed on by predators on patch i. Furthermore, the predator population on patch i will increase, while predators have a negative effect on prey when predators disperse to a prey's population. Predators dispersal rate of prey from patch j to patch i s ij x i functional response of predators that disperse from patch j to patch i p ij x j functional response of predators on patch i to prey that disperse from patch j to patch i z ij conversion rate of prey that come from patch j and are preyed on by predators on patch i dispersed from patch j can prey on patch i. In addition, the prey population on patch i will decrease. When i = j, we assume k The above model (1) is transformed into the following model: (2) The model (2) is equivalent to the next model (3): Remark 3.1 Model (1) is different from the predator-prey model with dispersal that has been studied in recent years. The cross-dispersal is considered in model (1). This means that predators can disperse to a prey population, while prey can also disperse to a predator's population. (1) is that a patchy environment is formed under the influence of natural conditions or human activities. Predator populations can disperse to other patches to prey, prey species can also disperse to other patches to be preyed on by predators. For example, Eagles prey on rabbits. Rabbits can migrate across different patches, and so can eagles. Since the above model (1) is equivalent to model (2), we will discuss model (2) in the following sections.

The existence of the positive equilibrium for new model (2)
By the locally Lipschitz character of model (2)'s right-side function and the equivalent model (3), positive solutions' local existence is obvious. Now, we will construct a compact subset to prove the positive solutions' global existence [14][15][16][17][18]. (2).

Lemma 3.1 If
Here, we use the Mean Value Inequality with 2xy ≤ x 2 + y 2 . Furthermore, This means that G is positively invariant. The proof is completed.
Therefore, we have found a compact subset D = G. Now, the positive solutions' global existence for model (2) can be obtained as follows: Note that when N > N * , This means that D : (2). The next lemma is obtained naturally.

Lemma 3.3 If
Proof Two cases are considered for this lemma. One is that when N(Z 0 ) ≤ N * , G is a required compact and positively invariant set. The other is that when N(Z 0 ) > N * , D := {(x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n ) ∈ R 2n + : n i=1 ( i x i + e i y i ) ≤ N(Z 0 )} is also a required compact and positively invariant set. The proof of this lemma is similar to Lemma 3.2, hence we omit it.
Positive equilibria existence can be obtained by the next formula. Assume Consider the system of linear equations: It is reasonable to require that the unique solution be positive. Therefore, positive equilibria for system (2) exist naturally. The next theorem is obtained as follows: If the unique solution exists and is positive for the system of linear equations AZ + b = 0, the positive equilibrium for system (2) exists.
In fact, Cramer's rule can be used to prove that the unique solution exists and is positive for the system of linear equations AZ + b = 0.
In this paper, we suppose conditions (H 1 ) and (H 2 ) are satisfied for model (2) as follows: This means that the positive solution exists for model (2).

Global-stability analysis for new model (2) based on strongly connected graphs
Two matrices are constructed as follows: A cross-dispersal matrix can be defined as follows: A digraph (G, A) with n vertices for system (2) can be constructed as follows. Each vertex represents a patch. At each vertex i of G, vertex dynamics are described by the following system: Let E(G) denote the set of arcs (i, j) leading from initial vertex i to terminal vertex j. We In this section, a predator-prey model with cross-dispersal is studied. By using the method of constructing Lyapunov functions based on a graph-theoretical approach for coupled systems, sufficient conditions that the positive equilibrium of coupling model (2) is globally asymptotically stable in R 2n + are derived. We obtain the main theorem as follows: Theorem 3.2 Assume the following conditions hold: 3. There exists a non-negative constant λ such that then, whenever a positive equilibrium E * = (x * 1 , y * 1 , x * 2 , y * 2 , . . . , x * n , y * n ) exists for system (2), it is unique and globally asymptotically stable in R 2n + .
Proof Let In the following, we have Set the Lyapunov functions as Directly differentiating V i along system (2), we havė Two cases are discussed as follows: Then, we obtain and Therefore, the cross-dispersal matrix is obtained as follows: In the following, we have Let c i denote the cofactor of the ith diagonal element of the matrix (a ij ) n×n . From the irreducible character of matrix (a ij ) n×n , we have c i > 0. Furthermore, a Lyapunov function is set as follows: Differentiating V along the solution of system (2), we obtaiṅ Furthermore, we obtain that sgn(p ij ) = -sgn(p ji ), sgn(f ij ) = -sgn(f ji ).
Because the cross-dispersal matrix R = (|β ij |) n×n is irreducible, the diagraph (G, A) is strongly connected. Furthermore, since diagraph (G, A) is balanced and strongly connected, we obtain that In addition, we havė Therefore, by the LaSalle Invariance Principle [2], E * is unique and globally asymptotically stable in R 2n + . Case II. λ > 1.
In this case, the cross-dispersal matrix Let c i denote the cofactor of the ith diagonal element of the matrix (b ij ) n×n . From the irreducible character of matrix (b ij ) n×n , we have c i > 0.
Furthermore, a Lyapunov function is listed as follows: After calculation, we obtaiṅ Similar to Case I, we obtain n i,j=1 Hence, Therefore, by the LaSalle Invariance Principle [2], E * is unique and globally asymptotically stable in R 2n + . From Case I and Case II, the proof is completed.
If the condition 3 of Theorem 3.2 is substituted for the formula as follows: then, we have the following corollary: Corollary 3.2 Assume the following conditions hold: There exists a non-negative constant λ such that then, whenever a positive equilibrium E * = (x * 1 , y * 1 , x * 2 , y * 2 , . . . , x * n , y * n ) exists, it is unique and globally asymptotically stable in R 2n + .

Global-stability analysis for new model (2) based on strongly connected components
Let (R hk , B hk ) denote the kth strongly connected component (SCC) of the hth layer of a network (G, A). V (R hk ) denotes the vertex set of the SCC (R hk , B hk ) and N hk denotes the number of vertices of the SCC (R hk , B hk ). Obviously, Then system (2) can be written as follows: When h = 1, system (2) is restricted on the first layer of (G, A), i.e.
Based on the theory of asymptotically autonomous systems, graph theory and Lyapunov theory, a global-stability theorem without strongly connected graphs is established in this section.

Proof
Step 1. Consider the strongly connected component (R 1k , B 1k ). The next system is obtained naturally.
A vertex Lyapunov function on the SCC (R 1k , B 1k ) is constructed as follows: Let c 1k i denote the cofactor of the kth diagonal element of the matrix L 1k . Here, L 1k is the SCC (R 1k , B 1k )'s Laplacian Matrix. As (R 1k , B 1k ) is strongly connected, we obtain c 1k i > 0 for every i ∈ V (R 1k ).
Step 2. Consider the strongly connected component (R 2k , B 2k ), we obtain According to the theory of asymptotically autonomous systems, we have Similarly, a Lyapunov function can be constructed as follows: where Let c 2k i denote the cofactor of the kth diagonal element of the matrix L 2k . Here, L 2k is the SCC (R 2k , B 2k )'s Laplacian Matrix. As (R 2k , B 2k ) is strongly connected, we obtain c 2k i > 0 for every i ∈ V (R 2k ).
Let us assume λ > 1. Choosing Similar to Step 1, we obtain thaṫ Since diagraph (G, A) is balanced, diagraph (R 2k , B 2k ) is considered to be balanced naturally. Hence, If 0 ≤ λ ≤ 1, the proof is similar. Hence, system (2) is globally asymptotically stable on the SCC (R 2k , B 2k ). The cross-dispersal matrix is listed as follows: Simple computation results in
From the construction of the graph, the relationship between vertices is shown in Fig. 1.
Example 4.2 An example is presented to illustrate Theorem 3.3. Consider the following predator-prey system with cross-dispersal: The parameters are listed as follows. The cross-dispersal matrix is listed as follows: After a calculation, we obtain the positive equilibrium for system (12) as 1, 1, 1, . . . , 1, 1).
Based on Fig. 2, we know that the diagraph (G, A) is not strongly connected. However, it has two strongly connected components with two layers. Using Corollary 3.4, we obtain that the positive equilibrium point E * of system (12) is globally asymptotically stable in R 2n + .

Conclusions and outlooks
In this paper, cross-dispersal is considered in the predator-prey model with a patchy environment. A new predator-prey model with cross-dispersal among patches is constructed.
A new cross-dispersal matrix is established by the coupling relationship between vertices. Based on a graph-theoretical approach for coupled systems and constructing Lyapunov functions, sufficient conditions that the positive equilibrium of the new model is globally