# Note on the Persistent Property of a Discrete Lotka-Volterra Competitive System with Delays and Feedback Controls

- Xiangzeng Kong
^{1, 2}Email author, - Liping Chen
^{1, 2}and - Wensheng Yang
^{1, 2}

**2010**:249364

**DOI: **10.1155/2010/249364

© Xiangzeng Kong et al. 2010

**Received: **26 June 2010

**Accepted: **12 September 2010

**Published: **14 September 2010

## Abstract

A nonautonomous -species discrete Lotka-Volterra competitive system with delays and feedback controls is considered in this work. Sufficient conditions on the coefficients are given to guarantee that all the species are permanent. It is shown that these conditions are weaker than those of Liao et al. 2008.

## 1. Introduction

*i*th species at time . Montes de Oca and Zeeman [6] investigated the general nonautonomous -species Lotka-Volterra competitive system

then exponentially for and where is a certain solution of a logistic equation. Teng [8] and Ahmad and Stamova [9] also studied the coexistence on a nonautonomous Lotka-Volterra competitive system. They obtained the necessary or sufficient conditions for the permanence and the extinction. For more works relevant to system (1.1), one could refer to [1–9] and the references cited therein.

where is the density of competitive species; is the control variable; ; bounded sequences , , , , and ; and are positive integer; denote the sets of all integers and all positive real numbers, respectively; is the first-order forward difference operator ; .

In [1], Liao et al. obtained sufficient conditions for permanence of the system (1.4).

They obtained what follows.

Lemma 1.1.

It was shown that in [1] Liao et al. considered system (1.4) where all coefficients , , , , , and were assumed to satisfy conditions (1.9).

Our main results are the following Theorem 1.2.

Theorem 1.2.

Assume that (1.10) holds, then system (1.4) is permanent.

Remark 1.3.

Therefore, we have improved the permanence conditions of [1] for system (1.4).

Theorem 1.2 will be proved in Section 2. In Section 3, an example will be given to illustrate that (1.10) does not imply (1.9); that is, the condition (1.10) is better than (1.9).

## 2. Proof of Theorem 1.2

The following lemma can be found in [10].

Lemma 2.1.

Following comparison theorem of difference equation is Theorem of [11, page 241].

Lemma 2.2.

Let , For any fixed is a nondecreasing function with respect to , and for , following inequalities hold: , If , then for all .

where and are strictly positive sequences of real numbers defined for and , Similarly to the proof of Propositions and in [12], we can obtain the following.

Lemma 2.3.

The following lemma is direct conclusion of [1].

Lemma 2.4.

Proposition 2.5.

Proof.

This completes the proof.

## 3. An Example

## Authors’ Affiliations

## References

- Liao X, Ouyang Z, Zhou S:
**Permanence of species in nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls.***Journal of Computational and Applied Mathematics*2008,**211**(1):1-10. 10.1016/j.cam.2006.10.084MATHMathSciNetView ArticleGoogle Scholar - Ahmad S:
**On the nonautonomous Volterra-Lotka competition equations.***Proceedings of the American Mathematical Society*1993,**117**(1):199-204. 10.1090/S0002-9939-1993-1143013-3MATHMathSciNetView ArticleGoogle Scholar - Ahmad S, Lazer AC:
**On the nonautonomous****-competing species problems.***Applicable Analysis*1995,**57**(3-4):309-323. 10.1080/00036819508840353MATHMathSciNetView ArticleGoogle Scholar - Gopalsamy K:
*Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications*.*Volume 74*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+501.View ArticleGoogle Scholar - Kaykobad M:
**Positive solutions of positive linear systems.***Linear Algebra and Its Applications*1985,**64:**133-140. 10.1016/0024-3795(85)90271-XMATHMathSciNetView ArticleGoogle Scholar - Montes de Oca F, Zeeman ML:
**Extinction in nonautonomous competitive Lotka-Volterra systems.***Proceedings of the American Mathematical Society*1996,**124**(12):3677-3687. 10.1090/S0002-9939-96-03355-2MATHMathSciNetView ArticleGoogle Scholar - Zeeman ML:
**Extinction in competitive Lotka-Volterra systems.***Proceedings of the American Mathematical Society*1995,**123**(1):87-96. 10.1090/S0002-9939-1995-1264833-2MATHMathSciNetView ArticleGoogle Scholar - Teng ZD:
**Permanence and extinction in nonautonomous Lotka-Volterra competitive systems with delays.***Acta Mathematica Sinica*2001,**44**(2):293-306.MATHMathSciNetGoogle Scholar - Ahmad S, Stamova IM:
**Almost necessary and sufficient conditions for survival of species.***Nonlinear Analysis. Real World Applications*2004,**5**(1):219-229. 10.1016/S1468-1218(03)00037-3MATHMathSciNetView ArticleGoogle Scholar - Fan Y-H, Wang L-L:
**Permanence for a discrete model with feedback control and delay.***Discrete Dynamics in Nature and Society*2008,**2008:**-8.Google Scholar - Wang L, Wang MQ:
*Ordinary Difference Equation*. Xinjiang University Press; 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.2006.03.026MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.