Abstract

A new set of sufficient conditions for the permanence of a discrete -species cooperation system with delays and feedback controls are obtained. Our result shows that feedback control variables have no influence on the persistent property of the discrete cooperative system, thus improves and supplements the main result of F. D. Chen (2007).

1. Introduction

The aim of this paper is to investigate the permanent property of the following nonautonomous discrete -species cooperation system with time delays and feedback controls of the form:

where is the density of cooperation species , is the control variable (see [1, 2]).

Throughout this paper, we assume the following.

() are all bounded nonnegative sequences such that

Here, for any bounded sequence and , and

() are all nonnegative integers.

Let ; we consider (1.1) together with the following initial conditions:

It is not difficult to see that the solutions of (1.1)–(1.3) are well defined for all and satisfy

where is the set of integer numbers.

Recently, Chen [3] proposed and studied the permanence of system (1.1). Set

Using the comparison theorem, he obtained the following result.

Theorem 1 A (see [3]). Assume that () and () hold, and assume further that
holds, then system (1.1) is permanent.

However, as was pointed out by Fan and Wang [4], “if we use the method of comparison theorem, then the additional condition, in some extent, is necessary. But for the system itself, this condition may not necessary.’’ In [4], by establishing a new difference inequality, Fan and Wang showed that feedback control has no influence on the permanence of a single species discrete model. Their success motivated us to consider the persistent property of system (1.1). Indeed, in this paper, we will develop the analysis idea of [3] and apply the difference inequality obtained by Fan and Wang [4] to prove the following result.

Theorem 1.1. Assume that () and () hold, then system (1.1) is permanent.

Remark 1.2. Theorem 1.1 shows that feedback control variables have no influence on the permanent property of system (1.1). It is natural to ask whether the feedback control variables have the influence on the stability property of the system or not. At present, we had difficulty to give an affirm answer to this problem, and we will leave this in our future study.

We will prove Theorem 1.1 in the next section. For more works on cooperative system and feedback control ecosystem, one could refer to [1–23] and the references cited therein.

2. Proof of Theorem 1.1

Now we state several lemmas which will be useful for the proof of our main result.

Lemma 2.1 (see [5, page 125]). Consider the first-order difference equation where and are positive constants. Assume that for any initial value , there exist a unique solution of (2.1) which can be expressed as follows: where Thus, for any solution of system (2.1), one has

Lemma 2.2 (see [5, page 241] (Comparison theorem)). Let For any fixed , is a nondecreasing function, and for the following inequalities hold: If then for all

Lemma 2.3 (see [6, Theorem  2.1]). Consider the following single species discrete model: where and are strictly positive sequences of real numbers defined for and Any solution of system (2.4) with initial condition satisfies where

Lemma 2.4 (see [7]). Assume that satisfies and , where and are nonnegative sequences bounded above and below by positive constants and Then

Lemma 2.5 (see [4]). Assume that and . Further suppose that(i) then for any integer , Especially, if and is bounded above with respect to , then ;(ii) then for any integer , Especially, if and is bounded below with respect to , then

Lemma 2.6. Let be any positive solution of system (1.1), there exists a positive constant , which is independent of the solution of system (1.1), such that

Proof. Let be any positive solution of system (1.1); similarly to the proof of Theorem in [3], we have where are defined by (1.5). In fact, from the th equation of (1.1), it follows that Let , then (2.11) is equivalent to Summing both sides of (2.12) from to leads to We obtain that and hence, Substituting (2.14) to the th equation of (1.1), it immediately follows that By applying Lemmas 2.2 and 2.3 to (2.15), we have For any small enough , it follows from (2.16) that there exists enough large such that This, together with th equation of (1.1), leads to And so, Notice that ; it follows from (2.19) and Lemmas 2.1 and 2.2 that Let in above inequality, then Set The conclusion of Lemma 2.6 holds. The proof is complete.

Lemma 2.7. Let be any positive solution of system (1.1), there exists a positive constant , which is independent of the solution of system (1.1), such that

Proof. Let be a solution of system (1.1) satisfying the initial condition (1.3). From Lemma 2.6, there exists a such that for all , . Thus, for , from the th equation of system (1.1), it follows that Obviously, is a negative constant. Let , the above inequality is equivalent to Summing both sides of (2.23) from to leads to and so, therefore, Specially, we have Substituting the first inequality into the th equation of system (1.1) leads to where Then Lemma 2.5 and (2.24) imply that, for any integer , Note that and for enough large , which satisfy , then and Thus, for and , Then, there exists a positive integer such that for any positive solution of system (1.1), for all and In fact, we could choose where Fix , for , we get And so, for , we have Substituting (2.28) and (2.29) into the th equation of system (1.1), this together with (2.25) leads to (note that ) where
By applying Lemma 2.4 to (2.30), it immediately follows that where
From (2.31), we know that there exists enough large such that This together with the th equation of (1.1) leads to And so, Noticing that and applying Lemmas 2.1 and 2.2 to (2.34), we have Setting the conclusion of Lemma 2.7 follows. This ends the proof of Lemma 2.7.