Research Article | Open Access
Global Behavior of a Discrete Survival Model with Several Delays
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.
The delay differential equation was first proposed by Wazewska-Czyzewska and Lasota  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  and Li and Cheng , while the bifurcation and the direction of the stability were investigated by Song et al. . Xu and Li  and Liu  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.
Kubiaczyk and Saker  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
In addition, we can also easily see that the conditions (7) and are equivalent to the condition .
Stemming from the above discussion, the difference equation in the following form will be studied in this paper: where Besides, we denote that , , .
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
Proof. Clearly, we have , for . So by (10), we can find that Define a sequence by Obviously, So, we have
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 ). The following system of inequalities, with being real numbers, have exactly one solution .
3. Main Results
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.
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 11. When , the condition of Theorem 8 is independent from the argument .
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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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.