Research Article | Open Access

Volume 2009 |Article ID 605254 | https://doi.org/10.1155/2009/605254

Hongying Lu, "Permanence of a Discrete Nonlinear Prey-Competition System with Delays", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 605254, 15 pages, 2009. https://doi.org/10.1155/2009/605254

# Permanence of a Discrete Nonlinear Prey-Competition System with Delays

Accepted16 Sep 2009
Published27 Oct 2009

#### Abstract

A discrete nonlinear prey-competition system with m-preys and (n-m)-predators and delays is considered. Two sets of sufficient conditions on the permanence of the system are obtained. One set is delay independent, while the other set is delay dependent.

#### 1. Introduction

In this paper, we investigate the following discrete nonlinear prey-competition system with delays:

where is the density of prey species at th generation, is the density of predator species at th generation. In this system, the competition among predator species and among prey species is simultaneously considered. For more background and biological adjustments of system (1.1), we can see [15] and the references cited therein.

Throughout this paper, we always assume that for all

( are all bounded nonnegative sequences and Here, for any bounded sequence

( are bounded nonnegative integer sequences, and are all positive constants.

By a solution of system (1.1), we mean a sequence which defined for and which satisfies system (1.1) for. Motivated by application of system (1.1) in population dynamics, we assume that solutions of system (1.1) satisfy the following initial conditions:

where . The exponential forms of system (1.1) assure that the solution of system (1.1) with initial conditions (1.2) remains positive.

Recently, Chen et al. in [1] proposed the following nonlinear prey-competition system with delays:

By using Gaines and Mawhins continuation theorem of coincidence degree theory and by constructing an appropriate Lyapunov functional, they obtained a set of sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions of the system (1.3). In addition, sufficient conditions are obtained for the permanence of the system (1.3) in [2].

On the other hand, though most population dynamics are based on continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations [315]. Therefore, it is reasonable to study discrete time prey-competition models governed by difference equations.

As we know, a more important theme that interested mathematicians as well as biologists is whether all species in a multispecies community would survive in the long run, that is, whether the ecosystems are permanent. In fact, no such work has been done for system (1.1).

The main purpose of this paper is, by developing the analytical technique of [4, 8, 16], to obtain two sets of sufficient conditions which guarantee the permanence of system (1.1).

#### 2. Main Results

Firstly, we introduce a definition and some lemmas which will be useful in the proof of the main results of this section.

Definition 2.1. System (1.1) is said to be permanent, if there are positive constants and such that each positive solution of system (1.1) satisfies

Lemma 2.2 (see [8]). Assume that satisfies and for whereis a positive constant. Then

Lemma 2.3 (see [8]). Assume that satisfies and where is a constant such that . Then

For system (1.1), we will consider two cases, and respectively, and then we obtain Lemmas 2.42.6.

Lemma 2.4. Assume that Then for every positive solution of system (1.1) with initial condition (1.2), one has where

Proof. Let be any positive solution of system (1.1) with initial condition (1.2), for it follows from system (1.1) that thus Let, we can have By applying Lemma 2.2 to (2.10), we obtain so, we immediately get For any small enough, it follows from (2.12) that there exists enough large such that for all and For and , (2.13) combining with the -th equation of system (1.1) leads to thus Similarly, let , we get By using (2.16), for , according to Lemma 2.2, it follows that setting in above inequality, we have then
This completes the proof.

For convenience, we introduce the following notation.

For

For

Lemma 2.5. Assume that and hold. Then for any positive solution of system (1.1) with initial condition (1.2), one has where

Proof. Let be any positive solution of system (1.1) with initial condition (1.2). From Lemma 2.4, we know that there exists , such that for and For and , (2.25) combining with the -th equation of system (1.1) lead to thus let , we can have where According to Lemma 2.3, we obtain where Setting in (2.30) leads to therefore For any small enough, it follows from (2.33) that there exists enough large such that for all and and so, forand , it follows from system (1.1) that where by using (2.35), similarly to the analysis of (2.33), for and therefore, we easily get This ends the proof of Lemma 2.5.

Denote for

For

Lemma 2.6. Assume that and hold. Then for any positive solution of system (1.1) with initial condition (1.2), one has where

Proof. Let be any positive solution of system (1.1) with initial condition (1.2), for it follows from system (1.1) that It follows from (2.44)that and hence which, together with (2.45), produces similar to the analysis of (2.11) and (2.12), for and thus, we immediately get For any small enough, it follows from (2.50) that there exists enough large such that for all and For and , (2.51) combining with the -th equation of system (1.1) lead to from (2.53), similar to the argument of (2.44) and (2.47), for we have substituting (2.54) into (2.53), we get similar to the analysis of (2.18) and (2.19), for then, For any small enough, it follows from (2.51) and (2.57) that there exists enough large such that for all and Hence, for and , it follows from system (1.1) that from(2.59), similar to the argument of (2.44) and (2.47), for we have and this combined with (2.60) gives Similar to the argument of (2.32) and (2.33), for we obtain then For any small enough, it follows from (2.63) that there exists enough large such that for all and and so, for and , it follows from system (1.1) that
Similar to the argument of (2.61) and (2.62), for we have
then This ends the proof of Lemma 2.6.

Denote

or

Our main result in this paper is the following theorem about the permanence of system (1.1).

Theorem 2.7. Assume that and hold, then system (1.1) is permanent.

Proof. Let be any positive solution of system (1.1) with initial condition (1.2). Suppose . By Lemmas 2.42.6, if system (1.1) satisfies and , then we have The proof is completed.

In this paper, we study a discrete nonlinear predator-prey system with -preys and (-)-predators and delays, which can be seen as the modification of the traditional Lotka-Volterra prey-competition model. From our main results, Theorem 2.7 gives two sets of sufficient conditions on the permanence of the system (1.1). One set is delay independent, while the other set is delay dependent.

#### Acknowledgments

The author is grateful to the Associate Editor, Professor Xue-Zhong He, and two referees for a number of helpful suggestions that have greatly improved her original submission. This research is supported by the Nation Natural Science Foundation of China (no. 10171010), Key Project on Science and Technology of Education Ministry of China (no. 01061), and Innovation Group Program of Liaoning Educational Committee No. 2007T050.

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Copyright © 2009 Hongying Lu. 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.