Global Behavior of a Discrete Survival Model with Several Delays

was first proposed by Wazewska-Czyzewska and Lasota [1] as a model for the survival of red blood cell in an animal. Here, N(t) denotes the number of red blood cells at time t, α is the probability of death of red blood cells, β and γ are positive constants which are related to the production of red blood cells, and τ 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 generalizationwith 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


Introduction
The delay differential equation  ()  = − () +  −(−) , for  ≥ 0, 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.
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  1 = (/) −/ .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  = 1 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  ≤ 1.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 (7  ) are equivalent to the condition  ≤ 1.
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][13][14] and the references cited therein.
Then, the following statements are equivalent: (a) every solution of (16) oscillates; (b) the characteristic equation of ( 16) has no positive roots.
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.

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  0 ⩾ 0 such that   >  for  ⩾  0 , and consequently  − >  for  ⩾  1 , where  1 =  0 + .
From Lemma 4, we have {  } as a bounded sequence.In the following, we will claim that lim Otherwise, let Then,  >  and there exists a subsequence 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 ( −   ) = (   −   −    −  (+ −  ) ) / ( −  ),  = 0, 1, . . ., .By some simple calculations and (29), we get lim One can easily see that the hypothesis of Lemma 1 is satisfied and so the linear equation has an eventually positive 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.
From Lemma 4, we have {  } as a bounded sequence.Assume for the sake of contradiction that (29) is not satisfied.
Then,  >  and there exists a subsequence It follows from (10) that So, we obtain which is a contradiction.Accordingly, (29) holds.
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.
So there exists at least a  s.t.   −  −1 < 0, which contradicts that    −  −1 is in the positive semicycle.So, (51) is true.Noting that {  } is bounded from Lemma 4, we can let To prove that lim  → ∞   = 0 holds, it is sufficient to show that  =  = 0.
(55) Equation ( 36) can be reformulated in the form Multiplying ( 56) by (1 − ) −−1 and then summing up from  =   to  =   − 1 for  being sufficiently large, we get From (55) and    < 0, we have Therefore, by Lemma 6, we can get  =  = 0; that is, lim  → ∞   = 0, 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.
Remark 10.From Theorem 7, it is clear that if the condition (28) holds, then the oscillation condition for  = 1 as established by Kubiaczyk and Saker [8] is already satisfied.
Remark 11.When  = 1, the condition of Theorem 8 is independent from the argument .