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Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 682769, 9 pages

http://dx.doi.org/10.1155/2014/682769
Research Article

Note on the Persistence of a Nonautonomous Lotka-Volterra Competitive System with Infinite Delay and Feedback Controls

1College of Zhicheng, Fuzhou University, Fuzhou, Fujian 350002, China

2Department of Science Research, Fuzhou Fujian Institute of Education, Fuzhou, Fujian 350001, China

Received 8 March 2014; Revised 15 May 2014; Accepted 20 May 2014; Published 24 July 2014

Academic Editor: Garyfalos Papashinopoulos

Copyright © 2014 Chunling Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. We establish a series of criteria under which a part of -species of the systems is driven to extinction while the remaining part of the species is persistent. Particularly, as a special case, a series of new sufficient conditions on the persistence for all species of system are obtained. Several examples together with their numerical simulations show the feasibility of our main results.

1. Introduction

In this paper, we consider the following nonautonomous -species Lotka-Volterra competitive system with infinite delay and feedback controls: where is the density of the th species at time and is the indirect control variable.

In particular, when the coefficients , , and for all and , the system (1) will degenerate into the following pure delay type system:

As is well known, systems such as (2) without feedback controls are very important mathematical models of multispecies populations dynamics. This is a generalization from Ahmad [1] about two-species system without delays to -species system of infinite delay. Systems without delays such as [1] have attracted the interest of many researchers (see, e.g., [25]), and systems with delays have been studied extensively in the past twenty years, and some good results on the permanence, extinction and persistence or uniform persistence, global stability, and almost periodic solution have been developed (see [618]). In [19], Montes de Oca and Pérez provided for us a very interesting work for system (2), who showed that if the coefficients are bounded and continuous and satisfy certain inequalities, then any solution with initial function of system (2) in an appropriate space will have of its components tenting to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation. And for more works about single species dynamic behaviors of infinite delay, one could refer to [20, 21].

On the other hand, as was pointed out by Fan and Wang [22], feedback control is the basic mechanism by which systems, whether mechanical, electrical, or biological, maintain their equilibrium or homeostasis. Many scholars have done works on the ecosystem with feedback controls (see, e.g., [2329] and the references cited therein). In [23], Shi et al. proposed the feedback control system (1). By using the method of multiple Lyapunov functionals and by developing a new analysis technique, Shi et al. established the sufficient conditions which guarantee part species of the -species driven to extinction. But in the paper [23], they did not discuss the survival problems for the remaining species. The main aim of this paper is to study the persistence of the remaining species of system (1). By the new method motivated by work [11, 27, 28], we will establish new sufficient conditions for which surplus species of system (1) remain persistent.

The organization of the paper is as follows. In the next section, some assumptions and lemmas are introduced. In Section 3, we state and prove our main results. Finally, several examples with their numerical simulations are presented to show the feasibility of the main results.

2. Preliminaries

Throughout this paper, for system (1), we introduce the following hypotheses. , , , , and are bounded and continuous, defined on . Furthermore, , , , and are nonnegative on , and . Here, we denote and . , , and , , , are piecewise continuous and satisfy There exists a positive constant such that for each There exist positive constants and such that for each

We will consider system (1) together with the initial conditions where , , and It is easy to verify that solutions of (1) satisfying the initial condition (6) are well defined for all and satisfy

We now introduce several lemmas which will be useful in the proofs of the main results.

We consider the following nonautonomous linear equation: where nonnegative functions and are bounded and continuous, defined on . We have the following results.

Lemma 1 (see [30]). Suppose that there exist positive constants and such that Then, there exist positive constants such that for any positive solution of (9).

Lemma 2 (see [23]). Suppose that assumptions ( )–( ) hold; then there exist constants and such that for any positive solution , of system (1).

Remark 3. If all parameters , , , , and of system (1) have the positive lower bound on , then, from Lemma 2.2 in [23], we can choose

Lemma 4 (see [6]). Let be a nonnegative and bounded continuous function, and let be an integral function satisfying . Then

3. Main Results

In this section, we discuss the persistence of part species of system (1), where integer . Let functions

Lemma 5. Suppose that assumptions ( )–( ) hold and there exists an integer such that for any there exists an integer such that Then for each we have for any positive solution , of system (1).

The proof of the extinction of part species of system (1) could be found in [23] and we hence omit it here.

On the persistence of part species of system (1), we state and prove the following results.

Theorem 6. Suppose that all assumptions of Lemma 5 hold and there exists a positive constant such that Then, for each , there exist positive constants and , with , such that for any positive solution of system (1).

Proof. Let , be any positive solution of system (1). By Lemma 2, let ; for each , we have and . So, we only need to prove that there exists a positive constant such that and for all .

First of all, assumption (18) implies that there are positive constants and such that for all and .

By Lemmas 2 and 5, we obtain that, for any constant , there is a such that, for all ,

Now, for any , we define the Lyapunov function as follows: By assumptions ( ) and ( ) and Lemma 2, we have So we see that has definition for all . From (23), we can obtain that for any there is a positive constant , and may be dependent on the positive solution of system (1) such that Calculating the derivative of with respect to , we have Let . From (21), for all , we have Obviously, from inequality (20), we can find enough small positive constants and such that for all . So for the above , when , Consider the auxiliary equation then by (28), we obtain that where is the solution of (29) with the initial condition . If for all , then is defined on . Integrating inequality (29) from to , we obtain for all . Putting , , then, from (27) and (31), we have Letting , we have , a contradiction. Hence, there is a such that . Now, we prove that where , and the definition of implies . In fact, if (33) is not true, then there are and , , such that Choosing the integer such that , then, by (27) and (29), it follows that which is a contradiction.

From (24), (30), and (33), we can obtain that Finally, we define the constants and ; then we have Letting and , we have for all .

Further, by Lemma 4 and (38), we can choose constants and such that for all and Considering the second equation of system (1), from (39), for any , we obtain We consider the following auxiliary equation: Then by assumption ( ) and applying Lemma 1 there exists a constant such that for any positive solution of (41). Let be the solution of (41) with the initial condition ; then by the comparison theorem we have Thus, we finally obtain Let ; from (38) and (44), we obtain that and . This completes the proof of Theorem 6.

As consequences of Theorem 6 we have the following corollaries.

Corollary 7. If, in system (1), for all , then system (1) will be reduced to the following -species competitive system with infinite delay: Suppose that assumptions ( )–( ) hold and there exists an integer such that for any there exists an integer such that Furthermore, there exists a positive constant such that Then, for each , there exist positive constants such that and for each we have for any positive solution of system (45).

Proof. From the condition, And the assumptions ( )–( ) hold; from Corollary 7 in [23], for each we have Further condition holds, so we see, from Theorem 6, for each , that there exist positive constants such that

Remark 8. When , the conditions of Corollary 7 will reduce to the assumptions that ( )–( ) hold and for any such that We have that there exist positive constants such that and for each we have In comparison with the assumptions together with Proposition 2.2 given by Montes de Oca and Pérez [19], we can see that our assumptions in Corollary 7 are weaker.

Remark 9. When , from Corollary 7 we can easily obtain a criterion on the persistence of all species of system (45).

Remark 10. The conclusion of Corollary 7 improves that of Proposition 2.2 given by Montes de Oca and Pérez [19].

Corollary 11. Suppose that ( )–( ) hold and there exists a positive constant such that Then, for each , there exist positive constants and , with , such that for any positive solution , of system (1).

Remark 12. From Corollary 11 we can easily obtain a criterion on the persistence of all species of system (1).

4. Examples

In this section, we will give several examples to illustrate the conclusions of Corollary 7, Theorem 6, and Corollary 11. In the first part we will illustrate the conclusions of Corollary 7, in the second we will illustrate the conclusions of Theorem 6, and in the last we will illustrate the conclusions of Corollary 11.

Example 1. Consider the system where Obviously, we have that the period of system (59) is . By calculating, we obtain We can choose , since all conditions of Corollary 7 hold; therefore, species and in system (59) are extinct, and only species is persistent (see Figure 1). However, conditions (1.5) of Proposition 2.2 given by Montes de Oca and Pérez [19] do not apply in this example.

682769.fig.001
Figure 1: Dynamic behaviors of system (59). Here, we take the initial conditions for all .

Example 2. Consider the system where Obviously, we have that the period of system (62) is . By calculating, we obtain From Remark 3, we can choose , , , , , , , and such that All the conditions of Theorem 6 hold; therefore, species and coexist, and species in system (62) is extinct (see Figure 2).

682769.fig.002
Figure 2: Dynamic behaviors of system (62). Here, we take the initial conditions for all .

Example 3. Consider the system where and the coefficients and the other kernels are as Example 2. In this case, we can choose , , , , , , and such that All conditions of Corollary 11 hold, so all the species , , and are persistent (see Figure 3).

682769.fig.003
Figure 3: Dynamic behaviors of system (66). Here, we take the initial conditions for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (nos. JA12369 and JA13361).

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