Stability analysis for coupled systems with timevarying coupling structure
 Jiqiang Feng^{1}Email author and
 Xiaojia Luan^{2}
https://doi.org/10.1186/s136620171217z
© The Author(s) 2017
Received: 27 November 2016
Accepted: 22 May 2017
Published: 3 July 2017
Abstract
The stability of coupled systems with timevarying coupling structure (CSTCS) is considered in this paper. The graphtheoretic method on a digraph with constant weight has been successfully generalized into a digraph with timevarying weight. In addition, we construct a global Lyapunov function for CSTCS. By using the graph theory and the Lyapunov method, a Lyapunovtype theorem and some sufficient criteria are obtained. Furthermore, the theoretical conclusions on CSTCS can successfully be applied to the predatorprey model with timevarying dispersal. Finally, a numerical example of CSTCS is given to illustrate the effectiveness and feasibility of our results.
Keywords
1 Introduction
Generally speaking, the coupling structure of CSs is not constant. For example, in biomathematics, the dispersal rate of some species among different groups changes over time. When a natural disaster occurs, such as earthquakes and flood, some species will migrate to a safe place, then the dispersal rate of them will improve significantly. In epidemiology, the transmission rate of infectious diseases is also timevarying. For instance, population migration across regions increases greatly during the high season, and then the transmission rate of infectious diseases will be higher. However, when some regions discontinue with each other due to some reasons, the transmission rate of infectious diseases will be lower. All these facts can illustrate that timevarying coupling structure should not be neglected. Thus, we should take this timevarying coupled behavior into account, which can help us investigate the persistence of species and control infectious diseases. Consequently, it is essential for us to investigate the coupled systems with timevarying coupling structure (CSTCS). Unfortunately, as far as we know, few papers mentioned in the existing literature consider CSTCS.
Until now, many researchers, including us, have given some results of global stability of CSs. However, in our previous work we do not consider the timevarying coupling structure. In this paper, in order to make up for the defect of our previous work and characterize CSs reasonably, we will devote ourselves to the investigation of CSTCS. Generally, compared with system (1), \(a_{kh}(t)H_{kh}(x_{k}(t),x_{h}(t))\) may more reasonably be used to describe the interactions within a group or among different groups in the course of the dispersal. However, it is intricate to study the dynamics of CSTCS, since constructing a global Lyapunov function and estimating the symbol of its derivative are complex and technical under the circumstances of the timevarying coupling structure.
Motivated by the above discussions, in this paper, by introducing the timevarying coupling structure, we present a class of novel CSs that is CSTCS. Based on the graph theory and the Lyapunov method, a systematic method is established to construct a global Lyapunov function for CSTCS. Moreover, sufficient criteria ensuring the stability of CSTCS can be obtained. Furthermore, we consider a predatorprey model with timevarying dispersal. Meanwhile, stability criteria for it are presented, respectively. Then a numerical example is given to illustrate the effectiveness of our results.
The paper is outlined as follows. In Section 2, we introduce several preliminaries and give the formulation of the CSTCS. In Section 3, sufficient criteria of the stability for CSTCS are obtained. The stability results for predatorprey model with timevarying dispersal in Section 4. Finally, a numerical example is given in Section 5.
2 Preliminaries and model formulation
2.1 Mathematical preliminaries
For the sake of simplicity, the following notations are used in this paper. Let \(\mathbb{Z}^{+}=\{1,2,\ldots\}\), \(\mathbb{L}=\{1,2,\ldots,l\} \), \(\mathbb{R}^{1}_{+}=[0,+\infty)\), \(n=\sum_{k=1}^{l}n_{k}\) for \(n_{k}\in\mathbb{Z}^{+}\), and \(\mathbb{R}^{n}\) denote ndimensional Euclidean space. And we write \(C^{1,1}(\mathbb{R}^{n}\times\mathbb {R}^{1}_{+};\mathbb{R}^{1}_{+})\) for the family of all nonnegative functions \(V(x,t)\) on \(\mathbb{R}^{n}\times\mathbb{R}^{1}_{+}\) that are continuously once differentiable at x and t.
In this paper, use \((\mathcal{G},A)\) to represent the digraph with l vertices. Let \(V(\mathcal{G})\) denote the set of vertices in \(\mathcal {G}\); without loss of generality, let \(V(\mathcal{G})=\mathbb{L}\).
In what follows, we show an important lemma in [14] which will be used in the proof of our main results.
Lemma 1
2.2 Model formulations
In the proofs of our main theorems, a Lyapunov function for system (3) is constructed by combining the Lyapunov functions of vertices with the coupling structure. In the sequel, the definition of a vertex Lyapunov function set is given as follows.
Definition 1
 V1.:

There exist functions \(V_{k}(x_{k},t)\), \(b_{kh}(t)\) and \(F_{kh}\) such thatwhere \(d_{k}(t)\) is the cofactor of the kth diagonal element of the Laplacian matrix of \((\mathcal{G}, (b_{kh}(t))_{l\times l})\).$$ \begin{aligned} \frac{\mathrm{d}V_{k}(x_{k},t)}{\mathrm{d}t} &\triangleq\frac{\partial V_{k}(x_{k},t)}{\partial t}+ \frac{\partial V_{k}(x_{k},t)}{\partial x_{k}} \Biggl(f_{k}\bigl(x_{k}(t),t\bigr)+\sum _{h=1}^{l}a_{kh}(t)H_{kh} \bigl(x_{k}(t),x_{h}(t) \bigr) \Biggr) \\ & \leq\sum_{h=1}^{l} b_{kh}(t)F_{kh} \bigl(x_{k}(t),x_{h}(t)\bigr)\frac {d'_{k}(t)}{d_{k}(t)}V_{k} \bigl(x_{k}(t),t\bigr), \end{aligned} $$
 V2.:

Along each directed cycle \(\mathcal{C}\) of the weighted digraph \((\mathcal{G}, (b_{kh}(t))_{l\times l})\), we havefor all \(x_{k}\in\mathbb{R}^{n_{k}}\), \(x_{h}\in\mathbb{R}^{n_{h}}\).$$ \sum_{(h,k)\in E(\mathcal{C})}F_{kh} \bigl(x_{k}(t), x_{h}(t)\bigr)\leq0, $$(4)
We always suppose that all assumptions as regards the existence and uniqueness in Theorems 1.1 and 3.1 in [31] are fulfilled, so system (3) has a unique global solution for any given initial value \(x(0)=x_{0}\), and we here denote the solution by \(x(t;x_{0})\). Assume furthermore that \(f_{k}(0,t)=0\) and \(H_{kh}(0,0)=0\). Then system (3) has a trivial solution \(x(t)\equiv0\).
3 Global stability analysis for CSTCS
In this section, we investigate the stability of the trivial solution of system (3). Based on the graph theory and the Lyapunov method, we shall establish a theoretical framework for constructing a global Lyapunov function of system (3). The method used in the proof of main results is motivated by [13, 14].
Throughout this section, we always assume that \((\mathcal {G},(b_{kh}(t))_{l\times l})\) is strongly connected for any \(t\geq0\).
Theorem 1
Proof
Corollary 1
If \((\mathcal{G},(b_{kh}(t))_{l\times l})\) is balanced for any \(t\geq 0\), the conclusion of Theorem 1 holds if (4) is replaced by (7).
Corollary 2
If \((\mathcal{G},(b_{kh}(t))_{l\times l})\) is balanced for any \(t\geq 0\), the conclusion of Theorem 1 holds if (4) is replaced by (8).
Remark 1
In Theorem 1, we apply the graph theory and the Lyapunov method to prove the stability of system (3). Recently, the stability problem for coupled systems has been widely studied. In [14], Li et al. used graph theory to explore the globalstability problem for coupled systems of differential equations on networks. In this work, if \(a_{kh}(t)=a_{kh}\), our conclusion will be consistent with the Theorem 3.1 in [14]. Hence, our work is a generalization of the previous studies of CSs with timeinvariant structure.
Remark 2
The stability result is based on the vertex Lyapunov functions set in Theorem 1. In practical applications, since finding a suitable Lyapunov function is quite difficult, the stability criterion in Theorem 1 is not very convenient to be verified for a given system. Thus it prompts us to establish a coefficienttype criterion for the stability analysis of system (3).
Theorem 2
 A1.:

There is a constant \(\alpha_{k}>0\) such that$$x_{k}^{\mathrm{T}}f_{k}(x_{k},t)\leq \alpha_{k} \vert x_{k} \vert ^{2}. $$
 A2.:

There exists a constant \(A_{kh}\) such that$$\bigl\vert H_{kh} (x_{k},x_{h} ) \bigr\vert \leq A_{kh} \bigl( \vert x_{k} \vert + \vert x_{h} \vert \bigr). $$
 A3.:

The digraph \((\mathcal{G},H(t))\), where \(H(t)=(A_{kh}a_{kh}(t))_{l\times l}\), is strongly connected andwhere \(g_{k}(t)\) is the cofactor of the kth diagonal element of the Laplacian matrix of \((\mathcal{G},H(t))\).$$ \alpha_{k}p+2p\sum_{h=1}^{l}A_{kh}a_{kh}(t) \leq\frac{g'_{k}(t)}{g_{k}(t)}, $$(9)
Proof
Remark 3
In Theorem 2, considering condition A3 one needs to compute the derivative of determinant. Computing the derivative of highorder determinant is difficult. Thus the condition of Theorem 2 is not easy to verify. In Theorems 3, 4, we will use conditions that are described by the coefficients \(a_{kh}(t)\) to study the stability of system (3), which are easier to verify.
Theorem 3
Proof
Theorem 4
Proof
4 An application to the predatorprey model with timevarying dispersal
 \(r_{k}\)::

the intrinsic growth rate of the preys in the absence of predation in patch k,
 \(\gamma_{k}\)::

the death rate of the predators in patch k,
 \(b_{k}\)::

the intraspecific competition rate of the preys in patch k,
 \(\delta_{k}\)::

the intraspecific competition rate of the predators in patch k,
 \(\varepsilon_{k}x_{k}\)::

the proportion of preys which are eaten and become predators,
 \(e_{k}\)::

the predator response function for the predator with respect to that particular prey in the kth patch.
Function \(a_{kh}(t)\geq0\) is the dispersal rate of preys from patch h to patch k. Constant \(\alpha_{kh}\geq0\) is a boundary condition in the continuous diffusion case.
Remark 4
In existing literature, many researchers have investigated the predatorprey model with timeinvariant dispersal (see [21, 33] and the references therein). However, to our knowledge, there is little work as regards the predatorprey model with timevarying dispersal. In this paper, if \(a_{kh}(t)=a_{kh}\), system (11) turns into system (6.1) in [14]. Hence, system (11) is much more general since it considers a dispersal rate of preys which is not constant but timevarying.
Remark 5
If we simplify system (11), in other words, if we let \(r_{k}=r\), \(b_{k}=b\), \(e_{k}=e\), \(\gamma_{k}=\gamma\), \(\delta_{k}=\delta\), \(\varepsilon_{k}=\varepsilon\), and \(a_{kh}(t)=\alpha_{kh}=1\), then it is trivial to see that system (11) has a unique equilibrium: \(x^{*}=\frac{\delta r+e\gamma}{\delta b+e\varepsilon}\), \(y^{*}=\frac {\varepsilon rb\gamma}{\delta b+e\varepsilon}\). If condition \(\varepsilon r>b\gamma\) is satisfied, we can guarantee that \((x^{*},y^{*})^{\mathrm{T}}\) is a positive equilibrium. So it is reasonable to assume that there exists a positive equilibrium \(E^{*}=(x_{1}^{*},y_{1}^{*},\ldots,x_{l}^{*},y_{l}^{*})^{\mathrm{T}}\) for system (11).
Let \(D(t)=(a_{kh}(t)\varepsilon_{k}x_{h}^{*})_{l\times l}\), and \(l_{k}(t)\) be the cofactor of the kth diagonal element of the Laplacian matrix of \((\mathcal{G},D(t))\), respectively.
Theorem 5
Proof
Based on Theorem 5, by using the dispersal rate of preys \(a_{kh}(t)\), \(k,h\in\mathbb{L}\), we can get two more simple stability criteria for system (11).
Theorem 6
For any \(t\geq0\), let \((\mathcal{G},D(t))\) be strongly connected. If \(a'_{kh}(t)\leq0\), then as long as a positive equilibrium \(E^{*}\) exists, it is unique and globally asymptotically stable in the positive cone \(\mathbb{R}^{2l}_{+}\).
Proof
Theorem 7
Proof
Remark 6
In Theorem 5, we have presented the stability criterion for a predatorprey model in which the dispersal rate \(a_{kh}(t)\) is timevarying for any \(k,h\in\mathbb{L}\). Li et al. [14] discussed a predatorprey model with timeinvariant dispersal and achieved some results concerning the stability problem. In this paper, if \(a_{kh}(t)=a_{kh}\), the conclusion of Theorem 6 is consistent with Theorem 6.1 in [14], which means our work generalizes a globalstability result from timeinvariant dispersal to timevarying dispersal.
Remark 7
Biologically, timevarying dispersal, such as seasonal migration, is an extremely common phenomenon. As a result, the effect of timevarying dispersal on the species survival has been an important topic in population biology. This paper provides a possibility that researchers could use it to make the adjustment for the prey dispersal network to protect some species, which truly shows that the results in this paper are meaningful not only in theory but also in practice.
5 Numerical simulations
In this section, we will discuss a numerical example to illustrate the effectiveness and feasibility of Theorems 6 and 7.
 1.
We generalize CSs with timeinvariant coupling structure to CSTCS.
 2.
We first study the global stability for CSTCS by combining the graph theory with the Lyapunov method.
 3.
We apply these theoretical conclusions to a predatorprey model with timevarying dispersal.
Declarations
Acknowledgements
The authors would like to thank the reviewers for your very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 61401283) and Educational Commission of Guangdong Province, China (No. 2014KTSCX113).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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