Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 508686, 5 pages

http://dx.doi.org/10.1155/2013/508686

Research Article

Average Conditions for the Permanence of a Bounded Discrete Predator-Prey System

Yong-Hong Fan and Lin-Lin Wang

School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, China

Received 21 June 2013; Accepted 18 July 2013

Academic Editor: Antonia Vecchio

Copyright © 2013 Yong-Hong Fan and Lin-Lin Wang. 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.

Average conditions are obtained for the permanence of a discrete bounded system with Holling type II functional response , The method involves the application of estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.

1. Introduction

In this paper, we will study the permanence of the following discrete system:

where the sequences , , , , , are all assumed to be bounded and , , , are all positive for .

If all the coefficients of the previous system (1) are periodic sequences with period , in [1], the authors obtained the following.

Theorem 1 (see [1]). Assume that

hold; then the periodic system (1) is permanent.

In [2], by a standard comparison argument, they proved the following.

Theorem 2. Assume that

hold; then the bounded system (1) is permanent.

In the previous two theorems, we used the denotation as follows. For a bounded sequence , we define

And for a given periodic sequence with period , its average value is defined as

Throughout this paper, we always assume that ,  ,  ,  .

If all the coefficients of system (1) are periodic sequences with period , then it is a special form of the bounded coefficients of system (1), but from Theorem 2, we cannot obtain Theorem 1; that is to say, there is a gap between Theorems 1 and 2. In this paper, we attempt to fill in this gap.

In order to illustrate our main results, similar to the corresponding definitions of the bounded continuous function in [3], we first introduce some notations.

For a bounded sequence , we define the lower average of by

Some remarks:

(a)For a bounded sequence , define the upper average of by replacing inf with sup in (6).(b)If is -periodic, then

(c)The following inequalities hold true:

(d)For any , the lower average satisfies

Proof. We only prove that (b) hold; (c) and (d) can be proved similarly as that in [3]. Setting , where , in the following, we assume that is sufficiently large; then

from the previous equality, we have

therefore

which completes the proof.

During the study of the permanence for the bounded system, in view of the property (b), one can usually use the lower average or upper average instead of the sup and inf values. And we call the condition obtained by using the method of lower average or upper average as “average conditions.” For the permanence results with “average conditions," one can refer to [4–7], and so forth.

For the permanence of system (1), we have the following.

Theorem 3. Assume that

then the bounded system (1) is permanent.

Obviously, Theorem 3 includes both Theorems 1 and 2. Therefore, this theorem is a bridge that combines the bounded system and the periodic system.

2. Preliminaries

In order to prove Theorem 3, we need some lemmas below. The first lemma could be found in [8].

Lemma 4 (see [8, Corollary ]). Let be a positive solution of the following inequality:

if and ; then

We should point out that when , the conclusion of the previous lemma is not true. That is, is a necessary condition. We give an example to illustrate it.

Example 5. Consider the following inequality:

Obviously, is a solution of it, but .

Lemma 6. Let be a solution of the following inequality:

and bounded above; if and

then there exists some positive constant such that

To prove this lemma, we give two claims in what follows. First, by using mathematical induction, we can easily obtain the following.

Claim 1. If is a solution of (17), then

In what follows, we use contradiction to prove the lemma.

Claim 2. Assume that is a solution of (17) and bounded above by a positive constant ; if (19) does not hold, then there exist positive integer sequences and such that

Proof of the claim. Notice that

where is a constant.

If (19) does not hold, then from Claim 1, , thus, for any positive integer , there exist such that

In addition, there exists a number such that , and for . In the following, we only need to prove that . From the first equation of (17), we have

which implies that . This completes the proof of Claim 2.

Proof of Lemma 6. From the first equation of (17), we have

by Claim 2, we obtain that if (19) does not hold, then for any , we have

which implies that

Notice that

thus, by (27), we have

This is in contradiction to (18); the proof is complete.

From Lemmas 4 and 6, we have the following.

Theorem 7. Let be a solution of the following inequality:

if

then there exist some positive constants and such that

From Theorem 7, we can easily obtain the following.

Corollary 8. Let be a solution of the following inequality:

for any , , . If

then the conclusion of Theorem 7 also holds true, where is a positive integer.

3. Permanence

In this section, we give some applications of Theorem 7. First we use it to prove Theorem 3.

Proof of Theorem 3. From the first equation of (1), we have

by Theorem 7 and the condition (13), we can obtain that there must exist some positive constants and such that

for any solution () of (1) with positive initial conditions and .

From the second equation of (1), we have

by Theorem 7 and condition (13), we can obtain that there exists a positive constant such that

Set ; then from the second equation of (1), we can obtain

for sufficiently large ; by Theorem 7 and (13), we have

By (36), (38), and (40), we complete the proof.

Through some similar analysis as in [9], we have the following.

Corollary 9. Assume that any positive solution of the periodic equation satisfies

where () are all -periodic sequences; if

then the periodic equation has at least one -periodic positive solution.

We should point out that the previous corollary can be generalized to the -dimensional situation. As a direct application of the previous corollary, we have the following.

Theorem 10. Assume that

hold; then the periodic system (1) (the coefficients of the system (1) are all periodic sequences with a common period ) has at least one -periodic positive solution.

This theorem generalized Theorem  3.1 in [10].

Acknowledgments

This work was supported by the Natural Science Foundation of China (11201213), the Natural Science Foundation of Shandong Province (ZR2010AM022), and the Outstanding Young and Middle-Aged Scientists Research Award Fund of Shandong Province (BS2011SF004).

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