Global Behavior of a Discrete Survival Model with Several Delays

Abstract

The difference equation 𝑦_𝑛+1−𝑦_𝑛=−𝛼𝑦_𝑛+∑^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑗𝑛−𝑘} 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 𝑑𝑁(𝑡)(𝑡)𝑑𝑡=−𝛼𝑁+𝛽𝑒^{−𝛾𝑁(𝑡−𝜏)},for𝑡≥0,(1) 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 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 𝑥_𝑛+1−𝑥_𝑛=−𝛼𝑥_𝑛+𝛽𝑒^{−𝛾𝑥𝑛−𝑘},𝑛=0,1,2,…,(2) where 𝛼∈(0,1),𝛽,𝛾∈(0,∞),𝑘∈{1,2,…},(3) 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 𝑥, where 𝑥 is the unique solution of the equation 𝛼𝑥=𝛽𝑒^−𝛾𝑥,(4) and showed that every solution of (2) oscillates about 𝑥 if 𝛽𝛾𝑒^−𝛾𝑥>𝑘𝑘+1^𝑘+1(1−𝛼)^𝑘+1.(5)

Meng and Yan [9] investigated the global attractivity of the positive equilibrium point 𝑥 and showed that 𝑥 is a global attractor of all positive solutions of (2) if 𝛽²𝛾²𝛼²𝑒^{𝛾(𝑄1+𝑥)}<1,(6) where 𝑄₁=(𝛽/𝛼)𝑒^{−𝛽𝛾/𝛼}.

Zeng and Shi [10] established another condition for global attractivity of  𝑥 and showed that 𝑥 is a global attractor of all positive solutions of (2) if 𝛽𝛾𝛼≤𝑒.(7)

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

Kubiaczyk and Saker [8] also considered (2) when 𝑘=1 and proved that 𝑥 is a global attractor of all positive solutions of (2) provided that 𝛽𝛾𝑒^−𝛾𝑥<𝛼.7^′

Ma and Yu [11] proved that 𝑥 is a global attractor of all solutions of (2) if 𝛾𝑥1−(1−𝛼)^𝑘+1≤1.(8)

By (2), we have 𝑥𝑒^𝛾𝑥=𝛽𝛼.(9) So, if (7) holds, then we have 𝛾𝑥𝑒^𝛾𝑥=𝛽𝛾/𝛼≤𝑒, which implies that 𝛾𝑥≤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 𝛾𝑥≤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 [12–14] and the references cited therein.

Stemming from the above discussion, the difference equation in the following form will be studied in this paper: 𝑦_𝑛+1−𝑦_𝑛=−𝛼𝑦_𝑛+^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑗𝑛−𝑘},(10) where 𝛼∈(0,1),𝛽_𝑗,𝛾_𝑗∈(0,∞),𝑘_𝑗∈{1,2,…},𝑗=1,2,…,𝑚;^𝑚_𝑗=1𝛽_𝑗=𝛽.(11) Besides, we denote that 𝑘=max_{1≤𝑗≤𝑚}{𝑘_𝑗}, 𝑙=min_{1≤𝑗≤𝑚}{𝑘_𝑗}, 𝛾^∗=min_{1≤𝑗≤𝑚}{𝛾_𝑗}.

Obviously, the case 𝑚=1 is the form of (2). Besides, if 𝑦_−𝑘,𝑦_−𝑘+1,…,𝑦₋₁, 𝑦₀∈[0,∞), then, the corresponding solution of (10) is positive, and (10) has a unique positive equilibrium point 𝑦, which satisfies 𝛼𝑦=^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦}.(12)

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 𝑝_𝑖∈(0,∞) and 𝑘_𝑖∈{0,1,…} with ∑^𝑚_𝑖=1(𝑝_𝑖+𝑘_𝑖)≠1, 𝑖=1,2,…,𝑚. Let {𝑝_𝑖(𝑛)} be sequences of positive numbers such that liminf_𝑛→∞𝑝_𝑖(𝑛)⩾𝑝_𝑖𝑓𝑜𝑟𝑖=1,2,…,𝑚.(13) Suppose that the linear difference inequality 𝑧_𝑛+1−𝑧_𝑛+^𝑚_𝑖=1𝑝_𝑖(𝑛)𝑧_{𝑛−𝑘𝑖}⩽0𝑓𝑜𝑟𝑛=0,1,…(14) has an eventually positive solution. Then, the difference equation 𝑥_𝑛+1−𝑥_𝑛+^𝑚_𝑖=1𝑝_𝑖𝑥_{𝑛−𝑘𝑖}=0(15) has a positive solution.

Lemma 2 (see [7, page 5]). Consider the linear homogeneous difference equation 𝑥_𝑛+𝑘+^𝑘_𝑖=1𝑞_𝑖𝑥_{𝑛+𝑘−𝑖}=0𝑓𝑜𝑟𝑛=0,1,…,(16) where 𝑘 is a nonnegative integer and 𝑞_𝑖∈𝑅, 𝑖=1,2,…,𝑘. Then, the following statements are equivalent:(a)every solution of (16) oscillates;(b)the characteristic equation of (16) 𝜆^𝑘+^𝑘_𝑖=1𝑞_𝑖𝜆^𝑘−𝑖=0(17)has no positive roots.

Lemma 3 (see [7, page 12]). Assume that 𝑝₁,𝑝₂,…,𝑝_𝑘∈𝑅 and 𝑘 is a nonnegative integer. Then, ∑^𝑘_𝑖=1|𝑝_𝑖|<1 is a sufficient condition for the asymptotic stability of the difference equation 𝑥_𝑛+𝑘+𝑝₁𝑥_{𝑛+𝑘−1}+⋯+𝑝_𝑘𝑥_𝑛=0𝑓𝑜𝑟𝑛=0,1,….(18)

Lemma 4. Assume that (11) holds, and {𝑦_𝑛} is a solution of (10) with positive initial conditions 𝑦_−𝑘,…,𝑦₀. Then, limsup_𝑛→∞𝑦_𝑛≤𝛽𝛼.(19)

Proof. Clearly, we have 𝑦_𝑛>0, for 𝑛=−𝑘,−𝑘+1,…,0,1,2,…. So by (10), we can find that 𝑦_𝑛+1=(1−𝛼)𝑦_𝑛+^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑗𝑛−𝑘}≤(1−𝛼)𝑦_𝑛+𝛽.(20) Define a sequence {𝜔_𝑛} by 𝜔_𝑛+1=(1−𝛼)𝜔_𝑛+𝛽,𝜔₀=𝑦₀.(21) Obviously, 𝑦_𝑛≤𝜔_𝑛=(1−𝛼)^𝑛𝜔₀+𝛽𝛼1−(1−𝛼)^𝑛.(22) So, we have limsup_𝑛→∞𝑦_𝑛≤𝛽𝛼.(23)

Lemma 5. Assume that (11) holds, and ^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝑟𝑗𝑦}<𝛼.(24)
Then, the positive equilibrium 𝑦 of (10) is locally asymptotically stable.

Proof. To prove that the positive equilibrium 𝑦 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 𝑦 is 𝑦_𝑛+1=(1−𝛼)𝑦_𝑛−^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}𝑦_{𝑛−𝑘𝑗},(25) which satisfies |1−𝛼|+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}≤1.(26) Then, by Lemma 3, the positive equilibrium solution 𝑦 of (10) is locally asymptotically stable.

Lemma 6 (see [15]). The following system of inequalities, 𝜇≤𝑒^−𝜆−1,𝜆≥𝑒^−𝜇−1,(27) with 𝜆,𝜇 being real numbers, have exactly one solution 𝜆=𝜇=0.

3. Main Results

Theorem 7. Assume that (11) holds, and ^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^−𝑙−1>1.(28) Then, every positive solution of (10) oscillates about the positive equilibrium 𝑦.

Proof. Assume for the sake of contradiction that (10) has a positive solution {𝑦_𝑛} which does not oscillate about 𝑦. We assume that 𝑦_𝑛>𝑦 eventually. If 𝑦_𝑛<𝑦 eventually, the proof is similar and will be omitted. So, there exists an 𝑛₀⩾0 such that 𝑦_𝑛>𝑦 for 𝑛⩾𝑛₀, and consequently 𝑦_𝑛−𝑘>𝑦 for 𝑛⩾𝑛₁, where 𝑛₁=𝑛₀+𝑘.
From Lemma 4, we have {𝑦_𝑛} as a bounded sequence. In the following, we will claim that lim_𝑛→∞𝑦_𝑛=𝑦.(29) Otherwise, let 𝜇=limsup_𝑛→∞𝑦_𝑛.(30) Then, 𝜇>𝑦 and there exists a subsequence {𝑦_𝑛𝑖} such that lim_𝑖→∞𝑦_𝑛𝑖+1=𝜇,𝑦_𝑛𝑖+1−𝑦_𝑛𝑖>0for𝑛_𝑖≥𝑛₁,𝑖=1,2,….(31) Equation (10) can be reformulated in the form 𝑦_𝑛+1=(1−𝛼)𝑦_𝑛+^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑗𝑛−𝑘}.(32) Then, from (31) and (32), we find that 𝛼𝑦_𝑛𝑖+1≤^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑛𝑖𝑗−𝑘}.(33) So, we obtain 𝛼𝜇⩽limsup^𝑚_{𝑖→∞𝑗=1}𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑛𝑖𝑗−𝑘}≤^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦}=𝛼𝑦,(34) which is a contradiction. Accordingly, (29) holds.
Set 𝑦_𝑛=𝑦+𝑥_𝑛for𝑛=−𝑘,−𝑘+1,….(35) By the assumption 𝑦_𝑛>𝑦, we have that 𝑥_𝑛 is an eventually positive solution of the difference equation 𝑥_𝑛+1−(1−𝛼)𝑥_𝑛+𝛼𝑦−^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑗𝑛−𝑘)}=0,𝑛=0,1,…,(36) which can also be rewritten in the form 𝑥_𝑛+1−(1−𝛼)𝑥_𝑛+^𝑚_𝑗=1𝑝𝑛−𝑘_𝑗𝑥_{𝑛−𝑘𝑗}=0,𝑛=0,1,…,(37) where 𝑝(𝑛−𝑘_𝑗)=(𝛽_𝑗𝑒^{−𝛾𝑗𝑦}−𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑗𝑛−𝑘)})/(𝑥_{𝑛−𝑘𝑗}), 𝑗=0,1,…,𝑚.
By some simple calculations and (29), we get lim_𝑛→∞𝑝𝑛−𝑘_𝑗=𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}>0.(38)
One can easily see that the hypothesis of Lemma 1 is satisfied and so the linear equation 𝑥_𝑛+1−(1−𝛼)𝑥_𝑛+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}𝑥_{𝑛−𝑘𝑗}=0(39) has an eventually positive solution.
Let {𝑥_𝑛} be an eventually positive solution of (39); then 𝑧_𝑛=(1−𝛼)^−𝑛𝑥_𝑛 is an eventually positive solution of 𝑧_𝑛+1−𝑧_𝑛+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^{−𝑘𝑗−1}𝑧_{𝑛−𝑘𝑗}=0,𝑛=0,1,….(40) Let 𝐹(𝜆)=𝜆^𝑛+1−𝜆^𝑛+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^{−𝑘𝑗−1}𝜆^{𝑛−𝑘𝑗}(41) be the characteristic polynomial of (40). Now, we prove that 𝐹(𝜆)>0, for 𝜆>0.
If 𝜆⩾1, then obviously 𝐹(𝜆)>0. Else if 0<𝜆<1, we have 𝐹(𝜆)=𝜆^𝑛𝜆−1+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^{−𝑘𝑗−1}𝜆^−𝑘𝑗⩾𝜆^𝑛𝜆−1+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^−𝑙−1𝜆^−𝑘𝑗⩾𝜆^𝑛𝜆−1+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^−𝑙−1≥𝜆^𝑛+1>0.(42)
Therefore, the characteristic equation of (40) 𝜆^𝑛+1−𝜆^𝑛+^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝛾𝑗𝑦}(1−𝛼)^{−𝑘𝑗−1}𝜆^{𝑛−𝑘𝑗}=0(43) 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 𝑦1−(1−𝛼)^𝑘+1≤1.(44) Then, the positive equilibrium 𝑦 of (10) is a global attractor of all positive solutions of (10).

Proof. To prove that the positive equilibrium 𝑦 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 ({𝑦_𝑛} is nonoscillatory).  Let {𝑦_𝑛} be eventually positive. The case that {𝑦_𝑛} is eventually positive is similar and will be omitted. So,there exists an 𝑛₀⩾0 such that 𝑦_𝑛>𝑦 for 𝑛⩾𝑛₀, and consequently 𝑦_𝑛−𝑘>𝑦 for 𝑛⩾𝑛₁, where 𝑛₁=𝑛₀+𝑘.
From Lemma 4, we have {𝑦_𝑛} as a bounded sequence. Assume for the sake of contradiction that (29) is not satisfied. Let 𝜇=limsup_𝑛→∞𝑦_𝑛.(45) Then, 𝜇>𝑦 and there exists a subsequence {𝑦_𝑛𝑖} such that lim_𝑖→∞𝑦_𝑛𝑖+1=𝜇,𝑦_𝑛𝑖+1−𝑦_𝑛𝑖>0for𝑛_𝑖≥𝑛₁,𝑖=1,2,….(46) It follows from (10) that 𝛼𝑦_𝑛𝑖+1≤^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑛𝑖𝑗−𝑘}.(47) So, we obtain 𝛼𝜇⩽limsup^𝑚_{𝑖→∞𝑗=1}𝛽_𝑗𝑒^{−𝛾𝑗𝑦𝑛𝑖𝑗−𝑘}≤^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦}=𝛼𝑦,(48) which is a contradiction. Accordingly, (29) holds.
Case 2 ({𝑦_𝑛} is strictly oscillatory). To show that (29) holds, it suffices to prove that lim_𝑛→∞𝑥_𝑛=0 holds, when {𝑥_𝑛} is a strictly oscillatory solution of (36).
To this end, let 𝑥_𝑝𝑖+1,𝑥_𝑝𝑖+2,…,𝑥_𝑞𝑖(49) be the 𝑖th positive semicycle of {𝑥_𝑛} followed by the 𝑗th negative semicycle 𝑥_𝑞𝑖+1,𝑥_𝑞𝑖+2,…,𝑥_𝑠.(50) Let 𝑥_𝑀𝑖, 𝑥_𝑚𝑖 be the extreme values in these two semicycles with the smallest possible indices 𝑀_𝑖 and 𝑚_𝑖. Then, we claim that 𝑀_𝑖−𝑝_𝑖⩽𝑘+1,𝑚_𝑖−𝑞_𝑖⩽𝑘+1.(51) 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, 𝑀_𝑖−𝑝_𝑖>𝑘+1 and the terms 𝑥_{𝑀𝑖−𝑘−1},𝑥_{𝑀𝑖−𝑘},…,𝑥_𝑀𝑖−1 are in a positive semicycle. Because of 𝑥_𝑀𝑖>𝑥_𝑀𝑖−1, (36) renders 𝛼𝑥_𝑀𝑖+𝛼𝑦≤^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑀𝑖𝑗−𝑘−1)}.(52) So, we have 𝑥_𝑀𝑖⩽1𝛼^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑀𝑖𝑗−𝑘−1)}−𝑦=1𝛼^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑀𝑖𝑗−𝑘−1)}−^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦}=1𝛼^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦}𝑒^{−𝛾𝑗𝑥𝑀𝑖𝑗−𝑘−1}−1.(53) So there exists at least a 𝑗 s.t. 𝑥_{𝑀𝑖−𝑘𝑗−1}<0, which contradicts that 𝑥_{𝑀𝑖−𝑘𝑗−1} is in the positive semicycle. So, (51) is true. Noting that {𝑦_𝑛} is bounded from Lemma 4, we can let 𝜆=liminf_𝑛→∞𝑥_𝑛=liminf_𝑖→∞𝑥_𝑚𝑖,𝜇=limsup_𝑛→∞𝑥_𝑛=limsup_𝑖→∞𝑥_𝑀𝑖.(54)
To prove that lim_𝑛→∞𝑥_𝑛=0 holds, it is sufficient to show that 𝜆=𝜇=0.
From (54), it follows that if 𝜖∈(0,𝜆) is given, then there exists 𝑛₂⩾0 such that 𝜆−𝜖⩽𝑥_𝑛⩽𝜇+𝜖for𝑛⩾𝑛₂+𝑘.(55) Equation (36) can be reformulated in the form 𝑥_𝑛+1−(1−𝛼)𝑥_𝑛=−𝛼𝑦+^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑗𝑛−𝑘)}.(56) Multiplying (56) by (1−𝛼)^−𝑛−1 and then summing up from 𝑛=𝑝_𝑖 to 𝑛=𝑀_𝑖−1 for 𝑖 being sufficiently large, we get (1−𝛼)^−𝑀𝑖𝑥_𝑀𝑖−(1−𝛼)^−𝑝𝑖𝑥_𝑝𝑖=^𝑀𝑖−1_{𝑛=𝑝𝑖}−𝛼𝑦(1−𝛼)^−𝑛−1+^𝑀𝑖−1_{𝑛=𝑝𝑖}^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑗𝑛−𝑘)}(1−𝛼)^−𝑛−1.(57) From (55) and 𝑥_𝑝𝑖<0, we have (1−𝛼)^−𝑀𝑖𝑥_𝑀𝑖⩽−𝛼𝑦^𝑀𝑖−1_{𝑛=𝑝𝑖}(1−𝛼)^−𝑛−1+^𝑀𝑖−1_{𝑛=𝑝𝑖}^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗(𝑦+𝑥𝑗𝑛−𝑘)}(1−𝛼)^−𝑛−1≤−𝛼𝑦^𝑀𝑖−1_{𝑛=𝑝𝑖}(1−𝛼)^−𝑛−1+^𝑀𝑖−1_{𝑛=𝑝𝑖}^𝑚_𝑗=1𝛽_𝑗𝑒^{−𝛾𝑗𝑦}𝑒^{−𝛾∗(𝜆−𝜀)}(1−𝛼)^−𝑛−1=−𝛼𝑦^𝑀𝑖−1_{𝑛=𝑝𝑖}(1−𝛼)^−𝑛−1+𝛼𝑦𝑒^{−𝛾∗𝑀(𝜆−𝜀)𝑖−1}_{𝑛=𝑝𝑖}(1−𝛼)^−𝑛−1=−𝛼𝑦(1−𝛼)^−𝑀𝑖−(1−𝛼)^−𝑃𝑖𝛼+𝛼𝑦𝑒^{−𝛾∗(𝜆−𝜀)}(1−𝛼)^−𝑀𝑖−(1−𝛼)^−𝑃𝑖𝛼=𝑦(1−𝛼)^−𝑀𝑖−(1−𝛼)^−𝑃𝑖𝑒^{−𝛾∗(𝜆−𝜀)}−1.(58) So, 𝑥_𝑀𝑖⩽𝑦𝑒^{−𝛾∗(𝜆−𝜀)}−11−(1−𝛼)^{𝑀𝑖−𝑃𝑖}.(59) By using (54), 𝜀 is arbitrary and 𝑀_𝑖−𝑝_𝑖⩽𝑘+1; we get 𝜇⩽𝑦𝑒^{−𝛾∗𝜆}−11−(1−𝛼)^𝑘+1.(60) From the assumption of the theorem, we have 𝜇⩽𝑒^{−𝛾∗𝜆}−1.(61) By the same trick as in proving (61), we can prove that 𝜆≥𝑒^{−𝛾∗𝜇}−1.(62) Therefore, by Lemma 6, we can get 𝜆=𝜇=0; that is, lim_𝑛→∞𝑥_𝑛=0, which implies that 𝑦 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 ^𝑚_𝑗=1𝛽_𝑗𝛾_𝑗𝑒^{−𝑟𝑗𝑦}<𝛼,𝑦1−(1−𝛼)^𝑘+1<1.(63) Then, the positive equilibrium 𝑦 is globally asymptotically stable.

Remark 10. From Theorem 7, it is clear that if the condition (28) holds, then the oscillation condition for 𝑚=1 as established by Kubiaczyk and Saker [8] is already satisfied.

Remark 11. When 𝑚=1, the condition of Theorem 8 is independent from the argument 𝛾.

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).