Global stability of May cooperative system with feedback controls
- Rongyu Han^{1},
- Fengde Chen^{1}Email author,
- Xiangdong Xie^{2}Email author and
- Zhanshuai Miao^{1}
https://doi.org/10.1186/s13662-015-0657-6
© Han et al. 2015
Received: 12 June 2015
Accepted: 30 September 2015
Published: 24 November 2015
Abstract
In this paper, a May cooperative system with feedback controls is proposed and studied. The dynamic behaviors of the system are discussed by using the Lyapunov function method. If \(b_{i}\neq0\), \(i=1,2\), we show that feedback control variables have no influence on the global stability of the unique positive equilibrium of the system, which means that feedback control variables only change the position of the positive equilibrium and retain its global stability property. If \(b_{i}= 0\), \(i=1,2\), we can make the system which has a unique globally stable equilibrium or has unboundedly large solutions become globally stable. Some examples are given to illustrate the feasibility of the main results.
Keywords
MSC
1 Introduction
The rest of the paper is organized as follows. We will state and prove the main results in next section. In Section 3, numerical simulations are presented to illustrate our results. We end this work by a brief conclusion.
2 Main results
Lemma 2.1
System (1.10) admits a unique positive equilibrium \(P(x_{1}^{*},x_{2}^{*}, u_{1}^{*},u_{2}^{*})\).
Proof
Before we state and prove the global stability of this work, we need to state a definition and a useful lemma.
Definition 2.2
[16]
- (1)
all of the eigenvalues of the matrix A have positive real parts;
- (2)
the order principal minor of matrix A is positive;
- (3)
matrix A is nonsingular and \(A^{-1}\geq0\);
- (4)
there exists a vector \(x>0\) such that \(Ax>0\);
- (5)
there exists a vector \(y>0\) such that \(A^{T}y>0\).
Lemma 2.3
[17]
If A is an M matrix, then there exists a positive diagonal matrix \(D=\operatorname{diag}(d_{1},d_{2},\ldots,d_{n})\), \(d_{i}>0\), \(i=1,\ldots,n\), such that matrix \(B= \frac{1}{2}(DA+A^{T}D)\) is positive definite.
Theorem 2.4
The unique positive equilibrium \(P(x_{1}^{*},x_{2}^{*},u_{1}^{*},u_{2}^{*})\) of system (1.10) is globally stable.
Proof
Lemma 2.5
System (2.6) admits a unique equilibrium \(P_{1}(x_{10},x_{20},u_{10},u_{20})\).
Proof
Theorem 2.6
The unique positive equilibrium \(P_{1}(x_{10},x_{20},u_{10},u_{20})\) of system (2.6) is globally stable.
3 Examples
The following three examples show the feasibility of our main results.
Example 3.1
Example 3.2
Example 3.3
4 Conclusion
In this paper, we propose and study May cooperative system with feedback controls. In Theorem 2.4, by constructing a suitable Lyapunov function, we show that feedback control variables have no influence on the global stability of the system. Our result improve the corresponding result of Chen et al. [10]. In [16], Chen and Chen have a conjecture that the condition \(a_{2}b_{1}\geq a_{1}r_{2}\) is not needed to ensure the global stability of the unique interior equilibrium. In this paper, corresponding to a May cooperative system, we give a strict proof of an affirmative answer which is without any conditions. Compared with Chen et al. [13, 14], the authors showed that the feedback control variables have no influence on the permanence of the cooperation system. We have a further insight.
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. Also, the research was supported by the Natural Science Foundation of Fujian Province (2015J010121, 2015J01019).
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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