- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

Volume 2009 (2009), Article ID 306425, 10 pages

http://dx.doi.org/10.1155/2009/306425

## Feedback Control Variables Have No Influence on the Permanence of a Discrete -Species Cooperation System

^{1}Ministry of Science Training, Fujian Institute of Education, Fuzhou, Fujian 350001, China^{2}Department of Mathematics, Ningde Teachers College, Ningde, Fujian 352100, China^{3}College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China

Received 14 March 2009; Revised 28 June 2009; Accepted 27 July 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Liujuan Chen 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

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.

*Proof of Theorem 1.1. *Lemmas 2.6 and 2.7 show that under the assumptions () and (), for any positive solution of system (1.1), one has
where and are independent of the solution of system (1.1), thus, system (1.1) is permanent. This ends the proof of Theorem 1.1.

#### 3. Conclusions

Stimulated by the works of Fan and Wang [4], in this paper, we revisit the model proposed by Chen [3]. We showed that condition () in [3] is not necessary to ensure the permanence of the system, which means that feedback control variables have no influence on the persistent property of system (1.1).

#### Acknowledgments

The authors are grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. Also, this work was supported by the Foundation of Education Department of Fujian Province (JA08253).

#### References

- M. Fan, K. Wang, P. J. Y. Wong, and R. P. Agarwal, “Periodicity and stability in periodic $n$-species Lotka-Volterra competition system with feedback controls and deviating arguments,”
*Acta Mathematica Sinica*, vol. 19, no. 4, pp. 801–822, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-F. Huo and W.-T. Li, “Positive periodic solutions of a class of delay differential system with feedback control,”
*Applied Mathematics and Computation*, vol. 148, no. 1, pp. 35–46, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence of a discrete $N$-species cooperation system with time delays and feedback controls,”
*Applied Mathematics and Computation*, vol. 186, no. 1, pp. 23–29, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-H. Fan and L.-L. Wang, “Permanence for a discrete model with feedback control and delay,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 945109, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Wang and M. Q. Wang,
*Ordinary Difference Equation*, Xinjiang University Press, Xinjiang, China, 1991. - Z. Zhou and X. Zou, “Stable periodic solutions in a discrete periodic logistic equation,”
*Applied Mathematics Letters*, vol. 16, no. 2, pp. 165–171, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence for the discrete mutualism model with time delays,”
*Mathematical and Computer Modelling*, vol. 47, no. 3-4, pp. 431–435, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at MathSciNet - F. Chen, X. Liao, and Z. Huang, “The dynamic behavior of $N$-species cooperation system with continuous time delays and feedback controls,”
*Applied Mathematics and Computation*, vol. 181, no. 2, pp. 803–815, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Weng, “Existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls,”
*Computers & Mathematics with Applications*, vol. 40, no. 6-7, pp. 747–759, 2000. View at MathSciNet - Y. Xiao, S. Tang, and J. Chen, “Permanence and periodic solution in competitive system with feedback controls,”
*Mathematical and Computer Modelling*, vol. 27, no. 6, pp. 33–37, 1998. View at Zentralblatt MATH · View at MathSciNet - K. Gopalsamy and P. X. Weng, “Feedback regulation of logistic growth,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 16, no. 1, pp. 177–192, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Global asymptotic stability in $n$-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control,”
*Applied Mathematics and Computation*, vol. 170, no. 2, pp. 1452–1468, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Yin and Y. Li, “Positive periodic solutions of a single species model with feedback regulation and distributed time delay,”
*Applied Mathematics and Computation*, vol. 153, no. 2, pp. 475–484, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence in nonautonomous multi-species predator-prey system with feedback controls,”
*Applied Mathematics and Computation*, vol. 173, no. 2, pp. 694–709, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence for the discrete mutualism model with time delays,”
*Mathematical and Computer Modelling*, vol. 47, no. 3-4, pp. 431–435, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Liao, S. Zhou, and Y. Chen, “Permanence and global stability in a discrete $n$-species competition system with feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 4, pp. 1661–1671, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lu, “On the existence of positive periodic solutions to a Lotka Volterra cooperative population model with multiple delays,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 6, pp. 1746–1753, 2008. View at Zentralblatt MATH · View at MathSciNet - Z. Liu, R. Tan, Y. Chen, and L. Chen, “On the stable periodic solutions of a delayed two-species model of facultative mutualism,”
*Applied Mathematics and Computation*, vol. 196, no. 1, pp. 105–117, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Liu and L. Chen, “Periodic solutions of a discrete time nonautonomous two-species mutualistic system with delays,”
*Advances in Complex Systems*, vol. 9, no. 1-2, pp. 87–98, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Li and H. Zhang, “Existence of periodic solutions for a periodic mutualism model on time scales,”
*Journal of Mathematical Analysis and Applications*, vol. 343, no. 2, pp. 818–825, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Wu, Y. Xia, and M. Lin, “Existence of positive periodic solution of mutualism system with several delays,”
*Chaos, Solitons & Fractals*, vol. 36, no. 2, pp. 487–493, 2008. View at Zentralblatt MATH · View at MathSciNet - Y. Xia, J. Cao, and S. S. Cheng, “Periodic solutions for a Lotka-Volterra mutualism system with several delays,”
*Applied Mathematical Modelling*, vol. 31, no. 9, pp. 1960–1969, 2007. View at Publisher · View at Google Scholar