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 Correspondence should be addressed to Yong-Hong Fan; fanyh 1993@sina.com Received 21 June 2013; Accepted 18 July 2013 Academic Editor: Antonia Vecchio Copyright © 2013 Y.-H. Fan and L.-L. 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. Average conditions are obtained for the permanence of a discrete bounded systemwithHolling type II functional responseu(n+1) = u(n)exp{a(n)−b(n)u(n)−c(n)V(n)/(u(n)+m(n)V(n))}, V(n+1) = V(n)exp{−d(n)+e(n)u(n)/(u(n)+m(n)V(n))}.Themethod 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.

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   > 0,   > 0,   > 0,   > 0.
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  : Z → R, 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).
(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   ∈ [0,  − 1], 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][5][6][7], and so forth.
For the permanence of system (1), we have the following.(13) 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.

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 2.5]).Let () be a positive solution of the following inequality: We should point out that when   1 = 0, the conclusion of the previous lemma is not true.That is,   1 > 0 is a necessary condition.We give an example to illustrate it.
Lemma 6.Let () be a solution of the following inequality: and bounded above; if   2 > 0 and then there exists some positive constant  such that lim inf To prove this lemma, we give two claims in what follows.First, by using mathematical induction, we can easily obtain the following.
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  > 0 is a constant.If (19) does not hold, then from Claim 1, lim inf  → ∞ () = 0, thus, for any positive integer  ≥ 1, there exist   > 0 such that In addition, there exists a number   such that 0 ≤   <   , (  ) ≥ (0)/ and () ≤ (0)/ for   <  ≤   .In the following, we only need to prove that   −   ≥  + 1.From the first equation of (17), we have which implies that   −   ≥  + 1.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  ≥ 1, we have which implies that Notice that lim thus, by (27), we have lim This is in contradiction to (18); the proof is complete.
From Lemmas 4 and 6, we have the following.