Discrete Dynamics in Nature and Society

Volume 2011, Article ID 872738, 18 pages

http://dx.doi.org/10.1155/2011/872738

## Feedback Control in a Periodic Delay Single-Species Difference System

^{1}Key Laboratory of Biological Resources Protection and Utilization of Hubei Province, Hubei University for Nationalities, Enshi, Hubei 445000, China^{2}Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China

Received 19 October 2010; Accepted 8 January 2011

Academic Editor: Li Xian Zhang

Copyright © 2011 Yi Yang and Zhijun Liu. 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

A periodic delay single-species difference system with feedback control is established. With the help of analysis method and Lyapunov function, a good understanding of the permanence and global attractivity of the system is gained. Numerical simulations are presented to verify the validity of the proposed criteria. Our results show that feedback control has no influence on the permanence while it has influence on the global attractivity of the system.

#### 1. Introduction

In 1978, Ludwig et al. [1] considered a single-species system which is modeled by where is the density of species at time , is the intrinsic growth rate, is the competing rate, and -term represents predation. To be specific, Murray [2] took in the form of , and the dynamic behavior of is then governed by where is an -shaped function and , , , and are positive constants. For the relevant ecology sense of system (1.2), we refer the readers to [1, 2] and the references cited therein.

It seems reasonable to assume that the reproduction of species will not be instantaneous, but mediated by a delay required for gestation of species . Thus, a revised version is where the constant delay is positive. It is evident that all the coefficients of system (1.3) are assumed to be constant. However, in the real world, the coefficients are not fixed constants owing to the variation of environments. The influence of a varying environment is important for evolutionary theory as the selective forces on systems in such a fluctuating medium differ from those in a stable environment. In addition, as we know, ecosystems are often disturbed by unpredictable forces and, so, species population may experience changes. In ecology, an interesting issue is whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control, we call the disturbance functions control variables. For more discussion on this direction, we refer the readers to [3–8].

Considering the possible effects of fluctuating environment and feedback control on system (1.3), we obtain the following periodic system: where is the control variable. We assume that coefficients , , , , , , and are continuous and bounded above and below by positive constants and .

Following the same idea and method in [9–12], one can easily derive the discrete analogue of system (1.4), which takes the form of where , is a positive integer, all the coefficients , , , , , , and are positive bounded sequences and . is the first forward difference operator . For biological reasons, we only consider the solution of system (1.5) with initial value , . The principle aim of this paper is to explore the permanence and global attractivity of system (1.5). To the best of our knowledge, no work has been done for system (1.5).

For the sake of simplicity and convenience, the notations and definitions below will be used through this paper: and denote the set of nonnegative integers and the greatest integer function, respectively. We denote , for any bounded nonnegative sequence . Meanwhile, we denote the product of from to by with the understanding that for all .

*Definition 1.1. *System (1.5) is said to be permanent if there exist positive constants , and , such that

*Definition 1.2. *The positive solutions of system (1.5) are globally attractive if any two positive solutions and of system (1.5) satisfy

The organization of this paper is as follows. In the next Sections 2 and 3, two main results on permanence and global attractivity of system (1.5) are given, respectively. Numerical simulations are present to illustrate the validity of our main results in Section 4, and a brief conclusion is provided to summarize the paper in the final section.

#### 2. Permanence

This section is concerned with the permanence of system (1.5). We first introduce the following lemmas which are useful for establishing our result.

Lemma 2.1 (see [4]). *Assume the constant and , and further suppose that*(1)*if **
then for any integer ,
**
Especially, if and is bounded above with respect to , then
*(2)*If **
then for any integer ,
**
Especially, if and is bounded below with respect to , then
*

Lemma 2.2 (see [10]). *Assume that satisfies and
**
for , where is a positive constant and . Then
*

Lemma 2.3 (see [10]). *Assume that satisfies and
**
for , where is a positive constant such that and . Then
*

Now, we state the permanence of system (1.5).

Theorem 2.4. *System (1.5) is permanent provided that
*

*Proof. *It follows from the first equation of system (1.5) that
Let , then (2.12) can be rewritten as
Summing both sides of (2.13) from to yields
which implies that
and hence
Following the above result, again from the first equation of system (1.5), we have
By applying Lemma 2.2 to (2.17), we can obtain that
For any constant , it follows from (2.18) that there exists a large enough such that
which, together with the second equation of system (1.5), leads to
that is,
Combing (2.21) with Lemma 2.1(1) and setting in (2.21), one has
For any sufficient small constant , it follows from (2.18) and (2.22) that there exists such that
Thus, by (2.23) and the first equation of system (1.5), we have
for . Let , then (2.24) is equivalent to
For convenience of exposition, we set . Summing both sides of (2.25) from to leads to
This implies that
and hence,
From the second equation of system (1.5), we have
where and . Then, for any integer , combing (2.28), (2.29) with Lemma 2.1 (1), we have
Since , we obtain that . So
By the assumption of Theorem 2.4, for any solution of system (1.5), there exists an integer such that for . In fact, we can choose . Then
Since is bounded above, let , then we have
which, together with (2.24) and (2.28), leads to
where , . Note that
where we use the inequality for . Therefore, by Lemma 2.3, it follows that
For any constant , by (2.36) we know that there exists a sufficiently large integer such that
Hence, it follows from (2.37) and the second equation of system (1.5) that
Applying Lemma 2.1(2) and setting in (2.38), we have
Consequently, combing (2.18), (2.22), and (2.36) with (2.39), system (1.5) is permanent. This completes the proof.

#### 3. Global Attractivity

On the basis of permanence, in this section we further provide sufficient conditions that guarantee the positive solutions of system (1.5) are globally attractive. To do so, we first give the following lemma.

Lemma 3.1. *For any two positive solutions and of system (1.5), one has
**
where
*

*Proof. *It follows from system (1.5) that we have
Hence,
Note that
By the mean value theorem, one has
Then we can easily obtain (3.1) by substituting (3.5) and (3.7) into (3.4). The proof is complete.

Now, we state our main result on the global attractivity of system (1.5).

Theorem 3.2. *If the assumption of Theorem 2.4 holds and, further, suppose there exist positive constants and such that
**
where , are defined by (3.27) and (3.28), respectively. Then the positive solutions of system (1.5) are globally attractive.*

*Proof. *Let and be any two positive solutions of system (1.5). To prove Theorem 3.2, we first consider the following three steps for the first equation of system (1.5).*Step 1. *Let . It follows from (3.1) that
where
By the mean value theorem, we get
that is,
where lies between and , then
According to Theorem 2.4, there exists a such that and for all . Therefore, for all , we can obtain that
*Step 2. *Let
Then
*Step 3. *Let
By a simple calculation, it derives that
Now, we can define
Then for all , it follows from (3.14)–(3.18) that
For the second equation of system (1.5), we will consider the following two steps.*Step 1. *Let . Then
*Step 2. *Let . Then
So we can define
Then it follows from (3.21) and (3.22) that
Now, we could define
where , are mentioned in (3.8).

Obviously, for all and . Therefore, combining (3.20) and (3.24), for all , we have
where
Summing both sides of (3.26) from to , it derives that
which implies
It follows from the above inequality that
that is,
and we can easily obtain that
which implies that the positive solutions of system (1.5) are globally attractive, this completes the proof.

#### 4. Numerical Simulations

To verify the feasibilities of our main results, we consider a specific example:

Obviously, , , that is, . Then the assumption of Theorem 2.4 is satisfied, which indicates that system (4.1) is permanent (see Figure 1).

We assume and , by a simple computation, we have , , , then the sufficient conditions of Theorem 3.2 are satisfied. Thus, the positive solutions of system (4.1) are globally attractive. From Figure 2, we can see that and tend to and , respectively.

#### 5. Conclusion

We conclude with a brief discussion of our results. The sufficient condition required for the result of Theorem 2.4 does not depend on the size of the feedback control. However, the sufficient conditions required for the result of Theorem 3.2 show that the feedback control has an effect on the global attractivity of system (1.5).

#### Acknowledgments

The paper is supported by the Key Project of Chinese Ministry of Education(no. #210134), the Innovation Term of Educational Department of Hubei Province of China (no. #T200804), and the Scientific Research Foundation of the Education Department of Hubei Province of China (no. #D20101902).

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