PERMANENCE AND GLOBAL STABILITY OF POSITIVE SOLUTIONS OF A NONAUTONOMOUS DISCRETE RATIO-DEPENDENT PREDATOR-PREY MODEL

In the theoretical ecology, permanence and global stability of the population model are very important. There are extensive literature related to these topics for differential equation models (see [3, 6, 7, 8, 9, 12] and the references cited therein). Recently, there has been a tendency for some researchers in the field of difference equations to develop some new methods which are analogous to those used in the study of differential equations. (See, e.g., [1, 2, 4, 5, 10, 11] and the references therein.) In [5], Fan and Wang considered the following discrete periodic ratio-dependent predator-prey model:


Introduction
In the theoretical ecology, permanence and global stability of the population model are very important.There are extensive literature related to these topics for differential equation models (see [3,6,7,8,9,12] and the references cited therein).Recently, there has been a tendency for some researchers in the field of difference equations to develop some new methods which are analogous to those used in the study of differential equations.(See, e.g., [1,2,4,5,10,11] and the references therein.) In [5], Fan and Wang considered the following discrete periodic ratio-dependent predator-prey model: and establish sufficient conditions for the existence of a positive periodic solution of the periodic system (1.1).In this paper, we will establish sufficient conditions for the permanence of system (1.1) and also obtain some conditions which ensure that a positive solution of the model is stable and attracts all positive solutions.First, we present two definitions.
Definition 1.1.System (1.1) is defined to be permanent if there are positive constants M and m such that each positive solution {x 1 (k),x 2 (k)} of system (1.1) satisfies 136 Discrete ratio-dependent predator-prey model Definition 1.2.System (1.1) is defined to be globally asymptotically stable if a positive solution of system (1.1) is stable and this solution attracts all positive solutions.
Throughout this paper, we will assume that a(k), b(k), c(k), d(k), m(k), and f (k) are bounded nonnegative sequences, and use the following notations: for any bounded sequence {u(k)}, For biological reasons, we only consider solution The organization of this paper is the following.In the next section, we establish the permanence of system (1.1).In Section 3, we obtain the sufficient conditions which ensure that a positive solution of system (1.1) is stable and attracts all positive solutions.

Permanence
In this section, we establish a permanence result for system (1.1).
Lemma 2.1.For every solution {x 1 (k),x 2 (k)} of (1.1), where Proof.Clearly, x 1 (k) > 0 and x 2 (k) > 0 for k ≥ 0. We first prove that To prove (2.3), we first assume that there exists an l 0 ∈ N such that x 1 (l 0 + 1) ≥ x 1 (l 0 ).Then, Hence, It follows that (2.7) We claim that By way of contradiction, assume that there exists a p 0 > l 0 such that x 1 (p 0 ) > B 1 .Then p 0 ≥ l 0 + 2. Let p 0 ≥ l 0 + 2 be the smallest integer such that x 1 ( p 0 ) > B 1 .Then x 1 ( p 0−1 ) < x 1 ( p 0 ).The above argument produces that x 1 ( p 0 ) ≤ B 1 , a contradiction.This proves the claim.Now, we assume that x 1 (k + 1) < x 1 (k) for all k ∈ N. In particular, lim k→∞ x 1 (k) exists, denoted by x 1 .We claim that x 1 ≤ a M /b L .By way of contradiction, assume that x 1 > a M /b L .Taking limit in the first equation in system (1.1) gives which is a contradiction since (2.10) This proves the claim.Note that a M /b L ≤ B 1 .It follows that (2.3) holds.
Next, we prove that At first, we assume that there exists an n 0 ∈ N such that x 2 (n 0 + 1) ≥ x 2 (n 0 ).Then Hence, (2.13) It follows that We claim that x 2 (k) ≤ B 2 for k ≥ n 0 .By way of contradiction, assume that there exists a q 0 > n 0 such that x 2 (q 0 ) > B 2 .Then q 0 = n 0 + 2. Let q 0 ≥ n 0 + 2 be the smallest integer such that x 2 ( q 0 ) > B 2 .Then x 2 ( q 0−1 ) < x 2 ( q 0 ).The above argument produces that x 2 ( q 0 ) ≤ B 2 , a contradiction.This proves the claim.Now, we assume that x 2 (k + 1) < x 2 (k) for all k ∈ N. In particular, lim k→∞ x 2 (k) exists, denoted by x 2 .We claim that x 2 ≤ f M B 1 /m L d L .By way of contradiction, assume that x 2 > f M B 1 /m L d L .Taking limit in the second equation in system (1.1) gives which is a contradiction since (2.17)

2.19)
Proof.We first show that lim inf (2.20) According to Lemma 2.1, there exists a k * ∈ N such that Firstly, we assume that there exists an l 0 ≥ k * such that x 1 (l 0 + 1) ≤ x 1 (l 0 ).Note that, for k ≥ l 0 , (2.22) In particular, with k = l 0 , we have which implies that Then (2.26) We claim that By way of contradiction, assume that there exists a p 0 ≥ l 0 such that x 1 (p 0 ) < x 1 .Then p 0 ≥ l 0 + 2. Let p 0 ≥ l 0 + 2 be the smallest integer such that x 1 ( p 0 ) < x 1 .Then x 1 ( p 0 − 1) > x 1 ( p 0 ).The above argument produces that x 1 ( p 0 ) ≥ x 1 , a contradiction.This proves the claim.Now, we assume that x 1 (k + 1) > x 1 (k) for all k ∈ N. In particular, lim k→∞ x 1 (k) exists, denoted by x 1 .We claim that (2.28) By way of contradiction, assume that (2.29) Taking limit in the first equation in system (1.1) gives At first, we assume that there exists an n 0 ∈ N such that x 2 (n 0 + 1) ≥ x 2 (n 0 ).Note that, for k ≥ n 0 , (2.33) In particular, with k = n 0 , we get which implies that Then (2.37) We claim that x 2 (k) ≥ x 2 for k ≥ n 0 .By way of contradiction, assume that there exists a q 0 ≥ n 0 such that x 2 (q 0 ) < x 2 .Then q 0 ≥ n 0 + 2. Let q 0 ≥ n 0 + 2 be the smallest integer such that x 2 ( q 0 ) < x 2 .Then x 2 ( q 0 − 1) > x 2 ( q 0 ).The above argument produces that x 2 ( q 0 ) ≥ x 2 , a contradiction.This proves the claim.Now, we assume that x 2 (k + 1) < x 2 (k) for all k ∈ N. In particular, lim k→∞ x 2 (k) exists, denoted by x 2 .We claim that By way of contradiction, assume that (2.39) Taking limit in the second equation in system (1.1) gives which is a contradiction since (2.42) Then system (1.1) is permanent.

Global stability
In this section, we derive sufficient conditions which guarantee that the positive solution of (1.1) is globally stable.Our strategy in the proof of the global stability of the positive solution of (1.1) is to construct suitable Lyapunov functions Theorem 3.1.In addition to the assumptions made in Theorem 2.3, assume further that (i) there exist positive constant ν and positive constants n i , i = 1,2, such that for all large k, where D i and B i are given in Lemmas 2.1 and 2.2, (ii) b(k)B 1 ≤ 1 and f (k) ≤ 4 for all large k, where B 1 is given in Lemma 2.1.
Then system (1.1) is globally asymptotically stable, that is, a positive solution of (1.1) is stable and attracts all positive solutions.
Proof.Let {x * 1 (k),x * 2 (k)} be a positive solution of (1.1).We prove below that it is uniformly asymptotically stable.To this end, we introduce the change of variables 142 Discrete ratio-dependent predator-prey model System (1.1) is then transformed into which, by Taylor formula, can be rewritten as where | f i (k,u)|/ u converges, uniformly with respect to k ∈ N, to zero as u → 0. In view of system (1.1), it follows from (3.4) that H.-F. Huo and W.-T. Li 143 where | f i (k,u)|/ u converges, uniformly with respect to k ∈ N, to zero as u → 0. We define the function V by where n j are positive constants given in (i).Calculating the difference of V along the solution of system (3.5) and using (ii), we obtain x * 2 (k) , for large k. (3.7) Since | f i (k,u)|/ u converges uniformly to zero as u → 0, it follows from condition (i) and Theorem 2.3 that there is a positive constant γ such that if k is sufficiently large and u(k) < γ, By [1], we see that the trivial solution of (3.5) is uniformly asymptotically stable, and so is the solution {x * 1 (k),x * 2 (k)} of (1.1).Note that the positive solution {x 1 (k),x 2 (k)} is chosen in an arbitrary way.Proceeding exactly as in [11], we conclude that the positive solution {x * 1 (k),x * 2 (k)} of (1.1) is globally stable.The proof is complete