Permanence and Almost Periodic Solutions of a Discrete Ratio-Dependent Leslie System with Time Delays and Feedback Controls

and Applied Analysis 3 where xi k , i 1, 2 stand for the density of the prey and the predator at time k, respectively. ui k , i 1, 2 are the control variables at time k. h2 is a positive constant, denoting the constant of capturing half-saturation. In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global dynamics of the discrete ratiodependent Leslie systems with feedback controls. To do so, for system 1.4 we always assume that for i 1, 2, k ∈ Z H1 a k , b k , c k , d k , g k , f k , p k , αi k , βi k are all bounded nonnegative almost periodic sequences such that 0 < a ≤ a, 0 < b ≤ b, 0 < c ≤ c, 0 < d ≤ d, 0 < g ≤ g, 0 < f ≤ f, 0 < p ≤ p, 0 < αli ≤ αui < 1, 0 < β i ≤ β i . 1.5 Here, we let Z, Z denote the sets of all integers, nonnegative integers, respectively, and use the notations: f supk∈Z{f k }, f l infk∈Z{f k }, for any bounded sequence {f k } defined on Z. Let τ max{τi, σi, ρi, i 1, 2}, we consider system 1.4 with the following initial conditions: xi θ φi θ , θ ∈ −τ, 0 ∩ Z, φi 0 > 0, ui θ φi θ , θ ∈ −τ, 0 ∩ Z, φi 0 > 0, i 1, 2. 1.6 One can easily show that the solutions of system 1.4 with initial condition 1.5 are defined and remain positive for k ∈ Z . The principle aim of this paper is to study the dynamic behaviors of system 1.4 , such as permanence, global attractivity, and existence of a unique globally attractive positive almost periodic solution of the system. To the best of our knowledge, no work has been done for the nonautonomous difference system 1.4 . The organization of this paper is as follows. In the next section, we introduce some definitions and several useful lemmas. In Section 3, we explore the permanent property of system 1.4 . We study globally attractive property of system 1.4 in Section 4 and the almost periodic property of system 1.4 in Section 5. Finally, the conclusion ends with brief remarks. 2. Preliminaries In this section, we will introduce some basic definitions and several useful lemmas. Definition 2.1. System 1.4 is said to be permanent, if there are positive constantsmi andMi, such that for each positive solution x1 k , x2 k , u1 k , u2 k T of system 1.4 satisfies mi ≤ lim inf k→ ∞ xi k ≤ lim sup k→ ∞ xi k ≤ Mi, ni ≤ lim inf k→ ∞ ui k ≤ lim sup k→ ∞ ui k ≤ Ni, i 1, 2. 2.1 4 Abstract and Applied Analysis Definition 2.2. Suppose that X k x1 k , x2 k , u1 k , u2 k T is any solution of system 1.4 . X k is said to be a strictly positive solution in Z if for k ∈ Z and i 1, 2 such that 0 < inf k∈Z xi k ≤ sup k∈Z xi k < ∞, 0 < inf k∈Z ui k ≤ sup k∈Z ui k < ∞. 2.2 Definition 2.3 see 16 . A sequence x : Z → R is called an almost periodic sequence if the ε-translation set of x E{ε, x} {τ ∈ Z : |x k τ − x k | < ε, ∀k ∈ Z} 2.3 is a relatively dense set in Z for all ε > 0; that is, for any given ε > 0, there exists an integer l ε > 0 such that each interval of length l ε contains an integer τ ∈ E{ε, x}with |x k τ − x k | < ε, ∀k ∈ Z, 2.4 τ is called an ε-translation number of x k . Definition 2.4 see 17 . The hull of f , denoted by H f , is defined by H ( f ) { g k, x : lim n→∞ f k τn, x g k, x uniformly on Z × S } , 2.5 for some sequence {τn}, where S is any compact set in D. Lemma 2.5 see 20 . Assume that {y k } satisfies y k1 > 0 and y k 1 ≤ y k exp{r k (1 − ay k )}, 2.6 for k ∈ k1, ∞ , where a is a positive constant and k1 ∈ Z . Then lim sup k→ ∞ y k ≤ 1 aru exp r − 1 . 2.7 Lemma 2.6 see 20 . Assume that {y k } satisfies y k2 > 0 and y k 1 ≥ y k exp{r k (1 − ay k )}, 2.8 for k ∈ k2, ∞ , lim supk→ ∞ y k ≤ M, where a is a constant such that aM > 1 and k2 ∈ Z . Then lim inf k→ ∞ y k ≥ 1 a exp r 1 − aM . 2.9 Abstract and Applied Analysis 5 Lemma 2.7 see 22 . Assume that A > 0 and y 0 > 0. Suppose that y k 1 ≤ Ay k B k , n 1, 2, . . . . 2.10 Then for any integerm ≤ k, y k ≤ Ay k −m m−1 ∑ j 0 AB ( k − j − 1). 2.11and Applied Analysis 5 Lemma 2.7 see 22 . Assume that A > 0 and y 0 > 0. Suppose that y k 1 ≤ Ay k B k , n 1, 2, . . . . 2.10 Then for any integerm ≤ k, y k ≤ Ay k −m m−1 ∑ j 0 AB ( k − j − 1). 2.11 Especially, if A < 1 and B is bounded above with respect toM, then lim sup k→ ∞ y k ≤ M 1 −A 2.12 Lemma 2.8 see 22 . Assume that A > 0 and y 0 > 0. Suppose that y k 1 ≥ Ay k B k , n 1, 2, . . . . 2.13 Then for any integerm ≤ k, y k ≥ Ay k −m m−1 ∑ j 0 AB ( k − j − 1). 2.14 Especially, if A < 1 and B is bounded below with respect toN, then lim inf k→ ∞ y k ≥ N 1 −A 2.15 3. Permanence In this section, we establish a permanent result for system 1.4 . Theorem 3.1. Assume that H1 holds; assume further that H2 b > cu/2|h| holds. Then system 1.4 is permanent. Proof. Let X k x1 k , x2 k , u1 k , u2 k T be any positive solution of system 1.4 , from the first equation of system 1.4 , it follows that x1 k 1 ≤ x1 k exp b k ≤ x1 k exp b . 3.1 6 Abstract and Applied Analysis It follows from 3.1 that k−1 ∏ j k−τ1 x1 ( j 1 ) x1 ( j ) ≤ k−1 ∏ j k−τ1 exp b ≤ exp bτ1 , 3.2 which implies that x1 k − τ1 ≥ x1 k exp −bτ1 . 3.3 Substituting 3.3 into the first equation of 1.4 , it immediately follows that x1 k 1 ≤ x1 k exp b k − a k x1 k − τ1 ≤ x1 k exp [ b k − a k exp −bτ1 x1 k ] . 3.4 By applying Lemma 2.5 to 3.4 , we have lim sup k→ ∞ x1 k ≤ 1 al exp b τ1 1 − 1 : M1. 3.5 For any ε > 0 small enough, it follows from 3.5 that there exists enough large K1 such that for k ≥ K1, x1 k ≤ M1 ε. 3.6 From the second equation of system 1.4 it follows that x2 k 1 ≤ x2 k exp [ g k ] ≤ x2 k exp[gu], x2 k 1 ≤ x2 k exp [ g k − f k M1 ε x2 k − τ2 ] , for k ≥ K1 τ. 3.7 From 3.7 , similar to the argument of 3.1 , one has x2 k 1 ≤ x2 k exp [ g k − f k M1 ε exp [−guτ2]x2 k ] . 3.8 By applying Lemma 2.5 to 3.8 again, we have lim sup k→ ∞ x2 k ≤ M1 ε fl exp [ g τ2 1 − 1 ] . 3.9 Setting ε → 0 in the above inequality, we have lim sup k→ ∞ x2 k ≤ M1 fl exp [ g τ2 1 − 1 ] : M2. 3.10 Abstract and Applied Analysis 7 For any ε > 0 small enough, it follows from 3.5 and 3.10 that there exists enough large K2 > K1 τ such that for i 1, 2 and k ≥ K2 xi k ≤ Mi ε. 3.11and Applied Analysis 7 For any ε > 0 small enough, it follows from 3.5 and 3.10 that there exists enough large K2 > K1 τ such that for i 1, 2 and k ≥ K2 xi k ≤ Mi ε. 3.11 For k > K2 τ , 3.11 combining with the third and fourth equations of system 1.4 leads to Δui k ≤ −αi k ui k βi k Mi ε , i 1, 2, 3.12 that is, ui k 1 ≤ ( 1 − αli ) ui k β i Mi ε , i 1, 2. 3.13 By applying Lemma 2.7, it follows from 3.13 that lim sup k→ ∞ ui k ≤ β i Mi ε αli , i 1, 2. 3.14 Letting ε → 0 in the above inequality, we have lim sup k→ ∞ ui k ≤ β i Mi αli : Ni, i 1, 2. 3.15 For any ε > 0 small enough, it follows from 3.11 and 3.15 that there exists enough large K3 > K2 τ such that for i 1, 2 and k ≥ K3 xi k ≤ Mi ε, ui k ≤ Ni ε. 3.16 Thus, from 3.16 and the first equation of system 1.4 , it follows that x1 k 1 ≥ x1 k exp [ b − a M1 ε − c u 2|h| − d u N1 ε ] : x1 k exp D1ε , for k ≥ K3 τ, 3.17 where D1ε b − a M1 ε − c u 2|h| − d u N1 ε ≤ b − aM1 8 Abstract and Applied Analysis ≤ b − a u exp b − 1 al exp −buτ1 ≤ b − b ≤ 0. 3.18 For any integer η ≤ k, it follows from 3.17 that k−1 ∏ j k−η x1 ( j 1 ) x1 ( j ) ≥ k−1 ∏ j k−η exp D1ε ≥ exp [ D1εη ] , 3.19