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Global attractivity of a discrete Lotka-Volterra competition system with infinite delays and feedback controls
Advances in Difference Equations volume 2013, Article number: 14 (2013)
Abstract
In this paper, we propose a discrete Lotka-Volterra competition system with infinite delays and feedback controls. Sufficient conditions which ensure the global attractivity of the system are obtained. An example together with its numerical simulation shows the feasibility of the main results.
1 Introduction
For the last decades, the ecological competition systems governed by differential equations of Lotka-Volterra type have been investigated extensively. Many interesting results concerned with the global existence and attractivity of periodic solution, persistence and extinction of the population, etc. have been obtained; we refer to [1–4] and the references therein. Already, many authors [5–21] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Particularly, the persistence, permanence, extinction, local and global stability and the existence of positive periodic solutions, etc., for discrete competitive systems are studied in [5, 9, 12, 13, 15–18, 21]. Chen and Zhou [9] discussed the following discrete Lotka-Volterra competition system:
They obtained sufficient conditions which guarantee the persistence of system (1.1). Also, for the periodic case, they obtained sufficient conditions for the existence of a globally stable periodic solution.
Chen [12] studied the following nonautonomous two-species discrete competitive systems with deviating arguments:
They obtained sufficient conditions for the permanence of system (1.2).
On the other hand, feedback control is the basic mechanism by which systems, whether mechanical, electrical or biological, maintain their equilibrium or homeostasis. In the higher life forms, the conditions under which life can continue are quite narrow. A change in body temperature of half a degree is generally a sign of illness. The homeostasis of the body is maintained through the use of feedback control [22]. A primary contribution of C.R. Darwin during the last century was the theory that feedback over long time periods is responsible for the evolution of species. In 1931 Volterra [23] explained the balance between two populations of fish in a closed pond using the theory of feedback. Later, a series of mathematical models have been established to describe the dynamics of feedback control systems; see [14, 16–20, 23–25] and the references therein.
The purpose of this paper is to study the global attractivity of the following discrete Lotka-Volterra competition system with infinite delays and feedback controls:
where , , , , () are the density of the i species at time n and () are the control variables at time n. () are bounded nonnegative sequences such that .
By the biological meaning, we focus our discussion on the positive solutions of (1.3). So, it is assumed that the initial conditions of (1.3) are of the form
where . One can easily show that the solutions of (1.3) with (1.4) remain positive for all , where .
Further, assume
where , . Then system (1.3) has a unique positive equilibrium with
The aim of this paper is, by developing the analysis technique of Chen [12], Liao and Yu [14], Chen and Teng [15], to obtain a set of sufficient conditions for the global attractivity of system (1.3). The paper is organized as follows. In Section 2, as preliminaries, some useful lemmas are given. In Section 3, we study the global attractivity of positive equilibrium of system (1.3). In Section 4, the numerical simulations on the global attractivity of equilibrium are given.
2 Preliminaries
In this section, we introduce some auxiliary lemmas which will be useful in the following.
Lemma 1 (see [15])
Let the function , where α and β are positive constants. Then is nondecreasing on .
Lemma 2 (see [15])
Assume that the sequence satisfies
where α and β are positive constants and . We have
-
(i)
if , then .
-
(ii)
if , then for all  .
Lemma 3 (see [8])
Suppose that functions satisfy () for and and is nondecreasing with respect to . If sequences and are the nonnegative solutions of the following difference equations:
respectively, and (), then for all , we have
Lemma 4 (see [12])
Let be a nonnegative bounded sequence, and let be a nonnegative sequence such that , where , . Then
We further consider the following discrete linear equation:
where , . is a nonnegative sequence defined on such that and is a nonnegative bounded sequence defined on Z with
where , are nonnegative constants.
Lemma 5 Any solution of system (2.1) with satisfies
Proof From Lemma 4,
Hence, for each , there exists an enough large integer such that for ,
By system (2.1), we can obtain
Thus,
By the arbitrariness of ε, we can obtain
We can prove in a similar way. Thus, we complete the proof. □
Lemma 6 Assume . For every solution of equation (2.1), we have
By Lemma 5, the proof of Lemma 6 is obtained easily. Hence, we omit it here.
3 Global attractivity
In this section, we derive sufficient conditions which guarantee that the positive equilibrium of system (1.3) is globally attractive. The technique of proofs is to use an iteration scheme.
Theorem 1 Assume
and
Then equilibrium of system (1.3) with (1.4) is globally attractive.
Proof Let be any solution of system (1.3) with (1.4). Denote
and
We now claim that , , .
From the first equation of system (1.3), we obtain
Consider the auxiliary equation
From , by the conclusion (ii) of Lemma 2, we have that for all , where is any solution of equation (3.1) with initial value . From Lemma 1, we have is nondecreasing for .
Hence, from Lemma 3, we obtain for all , where is the solution of equation (3.1) with . Further, combining it with the conclusion (i) of Lemma 2, we obtain
From the second equation of system (1.3), we obtain
By a similar argument as that above, we have
By Lemma 4 and Lemma 5, we obtain
Then, for any constant sufficiently small, there is an integer such that if , then
Further, from Lemma 4 and the first equation of system (1.3), we have
Consider the auxiliary equation
From and the arbitrariness of , we have
By the conclusion (ii) of Lemma 2, we have that for all , where is any solution of equation (3.2) with initial value . From Lemma 1, we have
is nondecreasing for .
Hence, from Lemma 3, we have for all , where is the solution of equation (3.2) with . Combining it with the conclusion (i) of Lemma 2, we obtain
From the arbitrariness of , we conclude , where
From Lemma 4 and the second equation of system (1.3), we further have
By a similar argument as that above, we can obtain
By Lemma 4 and Lemma 5, we further obtain
Hence, for sufficiently small, there is an such that if , then
From Lemma 4 and the first equation of system (1.3), we further have
Consider the auxiliary equation
From and the arbitrariness of , we have
Similarly to the above discussion, we can obtain
From the arbitrariness of , we conclude , where
From Lemma 4 and the second equation of system (1.3), we further have
By a similar argument as that above, we can obtain
By Lemma 4 and Lemma 5, we obtain
Hence, for sufficiently small, there is an such that if ,
From Lemma 4 and the first equation of system (1.3), we have
Consider the auxiliary equation
Since
similarly to the above discussion, we obtain
From the arbitrariness of , we conclude , where
From Lemma 4 and the second equation of system (1.3), we further have
By a similar argument as that above, we can obtain
Continuing the above process, we can obtain four sequences , , such that
and
Clearly, we have
Now, by means of the inductive method, we prove is monotonically decreasing, is monotonically increasing, .
Firstly, it is clear that , , . For (), we assume and , , then we have
and
Therefore, is monotonically decreasing, is monotonically increasing, . Consequently, and both exist, . Let
From (3.3) and (3.4), we obtain
It is clear that is a unique solution of equations (3.5). Therefore,
Further, by Lemma 6, we can obtain , . Thus, we complete the proof of Theorem 1. □
4 Example
The following example shows the feasibility of the main results.
Example 1 Choose , , , , , , , , , , , , , , in system (1.3). By calculating, we have that positive equilibrium , , , . Since
the conditions of Theorem 1 hold. So, equilibrium is globally attractive.
Choose initial values , .
By the numerical simulation (see Figure 1), we find that the solution turns to equilibrium as .
Author’s contributions
The author carried out the proof of the theorem and approved the final manuscript.
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Acknowledgements
The author would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Scientific Research Programmes of Colleges in Anhui (KJ2011A197).
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Wu, D. Global attractivity of a discrete Lotka-Volterra competition system with infinite delays and feedback controls. Adv Differ Equ 2013, 14 (2013). https://doi.org/10.1186/1687-1847-2013-14
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DOI: https://doi.org/10.1186/1687-1847-2013-14