# Permanence of a Discrete -Species Schoener Competition System with Time Delays and Feedback Controls

- Xuepeng Li
^{1}and - Wensheng Yang
^{1}Email author

**2009**:515706

**DOI: **10.1155/2009/515706

© X. Li and W. Yang 2009

**Received: **4 March 2009

**Accepted: **3 September 2009

**Published: **27 September 2009

## Abstract

## 1. Introduction

In 1974, Schoener [1] proposed the following competition model:

where are all positive constants.

May [2] suggested the following set of equations to describe a pair of mutualists:

where are the densities of the species at time , respectively. are positive constants. He showed that system (1.2) has a globally asymptotically stable equilibrium point in the region .

Both of the above-mentioned works are considered the continuous cases. However, many authors [3–5] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Bai et al. [6] argued that the discrete case of cooperative system is more appropriate, and they proposed the following system:

On the other hand, as was pointed out by Huo and Li [7], ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. During the last decade, many scholars did excellent works on the feedback control ecosystems (see [8–11] and the references cited therein).

Chen [11] considered the permanence of the following nonautonomous discrete N-species cooperation system with time delays and feedback controls of the form

where is the density of cooperation species , is the control variable ([11] and the references cited therein).

Motivated by the above question, we consider the following discrete -species Schoener competition system with time delays and feedback controls:

where is the density of competitive species at th generation; is the control variable; is the first-order forward difference operator .

Throughout this paper, we assume the following.

Here, for any bounded sequence ,

Let , we consider (1.5) together with the following initial conditions:

It is not difficult to see that solutions of (1.5) and (1.7) are well defined for all and satisfy

The aim of this paper is, by applying the comparison theorem of difference equation, to obtain a set of sufficient conditions which guarantee the permanence of the system (1.5).

## 2. Permanence

In this section, we establish a permanence result for system (1.5).

Definition 2.1.

for any solution of system (1.5).

Now, let us consider the first-order difference equation

where are positive constants. Following Lemma is a direct corollary of Theorem of L. Wang and M. Q. Wang [12, page 125].

Lemma 2.2.

Following comparison theorem of difference equation is Theorem of [12, page 241].

Lemma 2.3.

Now let us consider the following single species discrete model:

where and are strictly positive sequences of real numbers defined for and . Similarly to the proof of Propositions and [13], we can obtain the following.

Lemma 2.4.

Proposition 2.5.

Proof.

This completes the proof of Proposition 2.5.

Now we are in the position of stating the permanence of system (1.5).

Theorem 2.6.

then system (1.5) is permanent.

Proof.

This ends the proof of Theorem 2.6.

Now let us consider the following discrete -species Schoener competition system with time delays:

where is the density of species . Obviously, system (2.44) is the generalization of system (1.5). From the previous proof, we can immediately obtain the following theorem.

Theorem 2.7.

then system (2.44) is permanent.

## Declarations

### Acknowledgments

This work is supported by the Foundation of Education, Department of Fujian Province (JA05204), and the Foundation of Science and Technology, Department of Fujian Province (2005K027).

## Authors’ Affiliations

## References

- Chen L, Song X, Lu Z:
*Mathematical Models and Methods in Ecology*. Sichuan Science and Technology Press, Chengdu, China; 2003.Google Scholar - May RM:
*Theoretical Ecology, Principles and Applications*. Sounders, Philadelphia, Pa, USA; 1976.Google Scholar - Agarwal RP:
*Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 228*. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar - Murry JD:
*Mathematical Biology*. Springer, New York, NY, USA; 1989.View ArticleGoogle Scholar - Wang W, Lu Z:
**Global stability of discrete models of Lotka-Volterra type.***Nonlinear Analysis: Theory, Methods & Applications*1999,**35**(7):1019-1030.MATHGoogle Scholar - Bai L, Fan M, Wang K:
**Existence of positive periodic solution for difference equations of a cooperative system.***Journal of Biomathematics*2004,**19**(3):271-279.MathSciNetGoogle Scholar - Huo H-F, Li W-T:
**Positive periodic solutions of a class of delay differential system with feedback control.***Applied Mathematics and Computation*2004,**148**(1):35-46. 10.1016/S0096-3003(02)00824-XMathSciNetView ArticleMATHGoogle Scholar - Chen FD, Chen XX, Cao JD, Chen AP:
**Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control.***Acta Mathematica Sinica*2005,**21**(6):1319-1336. 10.1007/s10114-005-0585-6MathSciNetView ArticleMATHGoogle Scholar - Chen F:
**Positive periodic solutions of neutral Lotka-Volterra system with feedback control.***Applied Mathematics and Computation*2005,**162**(3):1279-1302. 10.1016/j.amc.2004.03.009MathSciNetView ArticleMATHGoogle Scholar - Chen F:
**Permanence in nonautonomous multi-species predator-prey system with feedback controls.***Applied Mathematics and Computation*2006,**173**(2):694-709. 10.1016/j.amc.2005.04.047MathSciNetView ArticleMATHGoogle Scholar - Chen F:
**Permanence of a discrete****-species cooperation system with time delays and feedback controls.***Applied Mathematics and Computation*2007,**186**(1):23-29. 10.1016/j.amc.2006.07.084MathSciNetView ArticleMATHGoogle Scholar - Wang L, Wang MQ:
*Ordinary Difference Equation*. Xinjiang University Press, Xinjiang, China; 1991.Google Scholar - Chen F:
**Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems.***Applied Mathematics and Computation*2006,**182**(1):3-12. 10.1016/j.amc.2006.03.026MathSciNetView ArticleMATHGoogle Scholar

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