Research Article | Open Access

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

**Academic Editor:**Xue-Zhong He

#### 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 [1–5] 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 [3–15]. 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 2.4–2.6.

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 2.4–2.6, 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

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.