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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 872738, 18 pages
http://dx.doi.org/10.1155/2011/872738
Research Article

Feedback Control in a Periodic Delay Single-Species Difference System

Yi Yang1,2 and Zhijun Liu1,2

1Key Laboratory of Biological Resources Protection and Utilization of Hubei Province, Hubei University for Nationalities, Enshi, Hubei 445000, China
2Department 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 [38].

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 [912], 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).

fig1
Figure 1: Permanence of system (4.1) with and . (a) and (b) show for and for , respectively. (c) and (d) show for and for , respectively.

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.

fig2
Figure 2: Global attractivity of system (4.1) with different initial values. with , and with , . (a) and (b) show for and for , respectively. (c) and (d) show (, ) for and for , 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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