Journal of Applied Mathematics

Volume 2013 (2013), Article ID 932607, 6 pages

http://dx.doi.org/10.1155/2013/932607

Research Article

Global Behavior of a Discrete Survival Model with Several Delays

Meirong Xu1,2 and Yuzhen Wang¹

¹School of Control Science and Engineering, Shandong University, Jinan 250061, China

²School of Mathematical Sciences, University of Jinan, Jinan 250022, China

Received 21 September 2013; Revised 28 October 2013; Accepted 11 November 2013

Academic Editor: Mohamad Alwash

Copyright © 2013 Meirong Xu and Yuzhen 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.

The difference equation is studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.

1. Introduction

The delay differential equation

was first proposed by Wazewska-Czyzewska and Lasota [1] as a model for the survival of red blood cell in an animal. Here, denotes the number of red blood cells at time , is the probability of death of red blood cells, and are positive constants which are related to the production of red blood cells, and is the time which is required to produce a red blood cell. The oscillation and global attractivity of (1) were studied by Győri and Ladas [2] and Li and Cheng [3], while the bifurcation and the direction of the stability were investigated by Song et al. [4]. Xu and Li [5] and Liu [6] considered its generalization with several delays and obtained sufficient conditions for the global stability of survival blood cells model with several delays and piecewise constant argument.

Research on the oscillation and global stability of the discrete analogue of (1), that is, for the equation

where

was proposed by Kocić and Ladas [7] as an open problem.

Kubiaczyk and Saker [8] investigated the oscillation of (2) about its positive equilibrium point , where is the unique solution of the equation

and showed that every solution of (2) oscillates about if

Meng and Yan [9] investigated the global attractivity of the positive equilibrium point and showed that is a global attractor of all positive solutions of (2) if

where .

Zeng and Shi [10] established another condition for global attractivity of   and showed that is a global attractor of all positive solutions of (2) if

Obviously, the condition (7) improves (6).

Kubiaczyk and Saker [8] also considered (2) when and proved that is a global attractor of all positive solutions of (2) provided that

Ma and Yu [11] proved that is a global attractor of all solutions of (2) if

By (2), we have

So, if (7) holds, then we have , which implies that . Hence, (8) is satisfied. But, the converse is not true. So, the condition (8) improves (7).

In addition, we can also easily see that the conditions (7) and are equivalent to the condition .

For the system with delay, many authors deemed that arbitrary finite number of discrete delays is more appropriate than the single discrete delay; see [12–14] and the references cited therein.

Stemming from the above discussion, the difference equation in the following form will be studied in this paper:

where

Besides, we denote that , , .

Obviously, the case is the form of (2). Besides, if , , then, the corresponding solution of (10) is positive, and (10) has a unique positive equilibrium point , which satisfies

The aim of this paper is to investigate the oscillation and the global asymptotic stability of (10).

2. Some Lemmas

Lemma 1 (see [7, page 6]). Assume that and with , . Let be sequences of positive numbers such that

Suppose that the linear difference inequality

has an eventually positive solution. Then, the difference equation

has a positive solution.

Lemma 2 (see [7, page 5]). Consider the linear homogeneous difference equation

where is a nonnegative integer and , . Then, the following statements are equivalent:

(a)every solution of (16) oscillates;(b)the characteristic equation of (16)

has no positive roots.

Lemma 3 (see [7, page 12]). Assume that and is a nonnegative integer. Then, is a sufficient condition for the asymptotic stability of the difference equation

Lemma 4. Assume that (11) holds, and is a solution of (10) with positive initial conditions . Then,

Proof. Clearly, we have , for . So by (10), we can find that

Define a sequence by

Obviously,

So, we have

Lemma 5. Assume that (11) holds, and

Then, the positive equilibrium of (10) is locally asymptotically stable.

Proof. To prove that the positive equilibrium is locally asymptotically stable, it suffices to prove that the zero solution of the linear equation of (10) is locally asymptotically stable. The linearized equation associated with (10) about positive equilibrium is

which satisfies

Then, by Lemma 3, the positive equilibrium solution of (10) is locally asymptotically stable.

Lemma 6 (see [15]). The following system of inequalities,

with being real numbers, have exactly one solution .

3. Main Results

Theorem 7. Assume that (11) holds, and

Then, every positive solution of (10) oscillates about the positive equilibrium .

Proof. Assume for the sake of contradiction that (10) has a positive solution which does not oscillate about . We assume that eventually. If eventually, the proof is similar and will be omitted. So, there exists an such that for , and consequently for , where .

From Lemma 4, we have as a bounded sequence. In the following, we will claim that

Otherwise, let

Then, and there exists a subsequence such that

Equation (10) can be reformulated in the form

Then, from (31) and (32), we find that

So, we obtain

which is a contradiction. Accordingly, (29) holds.

Set

By the assumption , we have that is an eventually positive solution of the difference equation

which can also be rewritten in the form

where , .

By some simple calculations and (29), we get

One can easily see that the hypothesis of Lemma 1 is satisfied and so the linear equation

has an eventually positive solution.

Let be an eventually positive solution of (39); then is an eventually positive solution of

Let

be the characteristic polynomial of (40). Now, we prove that , for .

If , then obviously . Else if , we have

Therefore, the characteristic equation of (40)

has no positive roots.

According to Lemma 2, (40) has no nonoscillatory solution.

This is a contradiction. The proof is completed.

Theorem 8. Assume that (11) holds, and

Then, the positive equilibrium of (10) is a global attractor of all positive solutions of (10).

Proof. To prove that the positive equilibrium is a global attractor of all positive solutions of (10), it suffices to show that (29) holds.

We will prove that (29) holds in each of the following two cases.

Case 1 ( is nonoscillatory).  Let be eventually positive. The case that is eventually positive is similar and will be omitted. So,there exists an such that for , and consequently for , where .

From Lemma 4, we have as a bounded sequence. Assume for the sake of contradiction that (29) is not satisfied. Let

Then, and there exists a subsequence such that

It follows from (10) that

So, we obtain

which is a contradiction. Accordingly, (29) holds.

Case 2 ( is strictly oscillatory). To show that (29) holds, it suffices to prove that holds, when is a strictly oscillatory solution of (36).

To this end, let

be the th positive semicycle of followed by the th negative semicycle

Let , be the extreme values in these two semicycles with the smallest possible indices and . Then, we claim that

In the following, we will prove that (51) holds for positive semicycles, while for negative semicycles, the proof is similar and will be omitted. Assume for the sake of contradiction that the first inequality in (51) is not true. Then, and the terms are in a positive semicycle. Because of , (36) renders

So, we have

So there exists at least a s.t. , which contradicts that is in the positive semicycle. So, (51) is true. Noting that is bounded from Lemma 4, we can let

To prove that holds, it is sufficient to show that .

From (54), it follows that if is given, then there exists such that

Equation (36) can be reformulated in the form

Multiplying (56) by and then summing up from to for being sufficiently large, we get

From (55) and , we have

So,

By using (54), is arbitrary and ; we get

From the assumption of the theorem, we have

By the same trick as in proving (61), we can prove that

Therefore, by Lemma 6, we can get ; that is, , which implies that is a global attractor of all positive solutions of (10).

By Lemma 3 and Theorem 8, we can get the following result.

Theorem 9. Suppose that (11) holds and that

Then, the positive equilibrium is globally asymptotically stable.

Remark 10. From Theorem 7, it is clear that if the condition (28) holds, then the oscillation condition for as established by Kubiaczyk and Saker [8] is already satisfied.

Remark 11. When , the condition of Theorem 8 is independent from the argument .

Acknowledgments

This work is supported by the National Natural Science Foundation of China (G61074068, G61034007, G61174036, G61374065, and G61374002), the Fund for the Taishan Scholar Project of Shandong Province, the Natural Science Foundation of Shandong Province (ZR2010FM013), and the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01).

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