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- Open Access
Feedback control effect on the Lotka-Volterra prey-predator system with discrete delays
- Chunling Shi^{1}Email author,
- Xiaoying Chen^{1} and
- Yiqin Wang^{2}
https://doi.org/10.1186/s13662-017-1410-0
© The Author(s) 2017
- Received: 25 July 2017
- Accepted: 20 October 2017
- Published: 1 December 2017
Abstract
In this paper, we study a Lotka-Volterra prey-predator system with feedback control. We establish sufficient conditions under which a unique positive equilibrium is globally stable. Further, we show that a suitable feedback control on predator species can make prey species that is on the brink of extinction become globally stable, but under the conditions of small feedback control on predator, the prey species still extinct, whereas the predator species is stable at certain values. Several examples are presented to show the feasibility of the main results.
Keywords
- Lotka-Volterra system
- discrete delays
- global stability
- extinction
- feedback controls
MSC
- 34D23
- 92D25
- 34D20
- 34D40
1 Introduction
The dynamical behavior of prey-predator system such as (1.1) has been investigated by many authors, and many excellent results concerned with permanence, extinction and persistence or uniform persistence, global stability, and almost periodic solutions are obtained (see, for example, [1–18]). In all prey-predator systems, there is a common competition between prey and predator species, and in general this competition is one sided, that is, the loss occurs only in the prey populations. In some cases, poor natural environment leads to a decrease in the birth rate of the prey population. Meanwhile, with the large predatory, the prey population is less and less and then tends to zero. Recently, many scholars have done research on the ecosystem with feedback controls (see [19–26] and the references therein). In particular, Gopalsamy and Weng [22] introduced a feedback control variable into a two-species competitive system and discussed the existence of the globally attractive positive equilibrium of the system with feedback controls. Hu et al. [26] considered the extinction of a nonautonomous Lotka-Volterra competitive system with pure delays and feedback controls, and by simulation they found that suitable feedback control variables can transform extinct species into permanent. Therefore, a natural and important question is that whether a proper control only on the predator population can make the extinct prey species become permanent, and the prey-predator can coexist in a certain pattern.
The aim of this paper is, by using the method of multiple Lyapunov functionals [22, 24] and by developing a new analysis technique [26], to obtain sufficient conditions under which a unique positive equilibrium is globally stable.
This paper is organized as follows. In the next section, as preliminaries, some assumptions and lemmas are introduced. In Section 3, the main results of this paper are stated and proved. Finally, several examples together with their numerical simulations show the feasibility of the main results and the considerable effects of feedback controls to extinction of prey species.
2 Preliminaries
Now, we state the following lemmas, which are useful in the proof of the main results.
Lemma 2.1
Lemma 2.2
Let \((x_{1}(t), x_{2}(t),u(t))^{T}\) be a solution of system (1.2) with initial condition (1.3). Then \((x_{1}(t), x_{2}(t),u(t))^{T}\) is positive and bounded for all \(t\geq0\).
Proof
3 Main results
Now, we give our main results.
Theorem 3.1
Proof
Corollary 3.1
Proof
(H_{5}) implies (H_{3}), so system (1.2) has a positive globally asymptotically stable equilibrium. □
Remark 3.1
If (H_{1}) holds, then systems (1.1) and (1.2) are globally stable. Theorem 3.1 implies that the feedback control keeps the property of stability of system (1.2) but only changes the position of the unique positive equilibrium. That is, feedback control of system (1.2) leads to the number of the prey population increased (\(x_{1}^{*}=\frac {e(r_{1}a_{22}-r_{2}a_{12})+r_{1}cd}{e(a_{11}a_{22}+a_{12}{a_{21})+cda_{11}}}>\bar {x}_{1}=\frac{r_{1}a_{22}-r_{2}a_{12}}{a_{11}a_{22}+a_{12}{a_{21}}}\)) and the number of the predator population decreased (\(x_{2}^{*}= \frac {e(r_{2}a_{11}+{r_{1}}a_{21})}{e(a_{11}a_{22}+a_{12}{a_{21}})+cda_{11}}<\bar {x}_{2}=\frac{r_{2}a_{11}+r_{1}a_{21}}{a_{11}a_{22}+a_{12}{a_{21}}}\)).
Remark 3.2
If (H_{2}) holds, system (1.1) is extinct. (H_{5}) implies (H_{2}), and system (1.2) has a unique positive equilibrium, which is globally asymptotically stable. Theorem 3.1 implies that the proper feedback control on predator species can change extinct prey species to be permanent.
Theorem 3.2
Proof
Theorem 3.3
Proof
Remark 3.3
By comparative analysis of (H_{3}) and (H_{4}) note that when the feedback control repression \(\frac{cd}{e}\) remains small, feedback control on predator species has no influence on the extinction of system (1.2).
4 Examples
In this section, we give examples to illustrate the results obtained.
Example 4.1
Example 4.2
Example 4.3
Example 4.4
Example 4.5
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of Fujian Province (2015J01012) and the Foundation of Fujian Education Bureau (JA13361).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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