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

Abstract

We consider a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls. Sufficient conditions are obtained for the permanence and global attractivity of the system. Furthermore, by using an almost periodic functional Hull theory, we show that the almost periodic system has a unique globally attractive positive almost periodic solution.

1. Introduction

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The predator-prey models have been extensively studied by many scholars [1–4]. Under the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food, Leslie [5] proposed the famous Leslie-Gower predator-prey model where the carry capacity of the predator’s environment is proportional to the number of prey: 𝑥̇₁=𝑥₁𝑏−𝑎𝑥₁−𝑐𝑥₁𝑥₂,𝑥̇₂=𝑥₂𝑥𝑔−𝑓²𝑥₁, (1.1) where 𝑥₁ and 𝑥₂ represent prey and predator densities at time 𝑡, respectively. 𝑐𝑥₁ is the predator’s rate of feeding upon prey, that is, the so-called predator’s functional response. If we assume that the predator consumes the prey according to the functional response 𝑝(𝑥₁), then system (1.1) formulates as the following: 𝑥̇₁=𝑥₁𝑏−𝑎𝑥₁−𝑝𝑥₁𝑥₂,𝑥̇₂=𝑥₂𝑥𝑔−𝑓²𝑥₁, (1.2) where 𝑝(𝑥₁) is prey-dependent functional responses. Owing to its theoretical and practical significance, system (1.2) and its various generalized forms have been studied extensively and seen great progress (see, for example, [6–9]).

However, in the study of the dynamic behaviors of predator-prey system, many scholars argued that the ratio-dependent predator-prey systems are more realistic [10, 11]. A ratio-dependent predator-prey system with Leslie-Gower term takes the form of 𝑥̇₁=𝑥₁𝑏−𝑎𝑥₁𝑥−𝑝¹𝑥₂𝑥₂,𝑥̇₂=𝑥₂𝑥𝑔−𝑓²𝑥₁. (1.3)

Though much progress has been seen in the asymptotic convergence of solutions of population systems, such systems are not well studied in the sense that most results are continuous-time cases related. Already, many scholars have paid attention to the nonautonomous discrete population models, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world (see [12–23]). Furthermore, the asymptotic convergence of solutions of difference equations with delay is one of the most important topics in the study of population dynamics, and many excellent results have already been obtained and seen great progress (see [24–26] and the references cited therein).

Since time delays occur so often in nature, a number of ecological systems can be described as systems with time delays (see [14, 15, 19, 21–27]). One of the most important problems for this type of system is to analyze the effect of time delays on the stability of the system. Furthermore, as we know, ecological systems in the real world are often distributed by unpredictable forces which can result in changes in biological parameters such as survival rates, so it is reasonable to study models with control variables which are so-called disturbance functions [22, 23].

So it is very interesting to study dynamics of the following discrete ratio-dependent Leslie system with time delays and feedback controls: 𝑥₁(𝑘+1)=𝑥₁(𝑘)exp𝑏(𝑘)−𝑎(𝑘)𝑥₁𝑘−𝜏₁−𝑐(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)−𝑑(𝑘)𝑢₁𝑘−𝜎₁,𝑥₂(𝑘+1)=𝑥₂(𝑘)𝑔(𝑘)(𝑘)𝑥exp−𝑓²𝑘−𝜏₂𝑥₁𝑘−𝜏₂−𝑝(𝑘)𝑢₂𝑘−𝜎₂,Δ𝑢_𝑖(𝑘)=−𝛼_𝑖(𝑘)𝑢_𝑖(𝑘)+𝛽_𝑖(𝑘)𝑥_𝑖𝑘−𝜌_𝑖,𝑖=1,2, (1.4) where 𝑥_𝑖(𝑘), 𝑖=1,2 stand for the density of the prey and the predator at time 𝑘, respectively. 𝑢_𝑖(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 ratio-dependent 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), 𝛼_𝑖(𝑘), 𝛽_𝑖(𝑘) are all bounded nonnegative almost periodic sequences such that 0<𝑎^𝑙≤𝑎^𝑢,0<𝑏^𝑙≤𝑏^𝑢,0<𝑐^𝑙≤𝑐^𝑢,0<𝑑^𝑙≤𝑑^𝑢,0<𝑔^𝑙≤𝑔^𝑢,0<𝑓^𝑙≤𝑓^𝑢,0<𝑝^𝑙≤𝑝^𝑢,0<𝛼^𝑙_𝑖≤𝛼^𝑢_𝑖<1,0<𝛽^𝑙_𝑖≤𝛽^𝑢_𝑖. (1.5)Here, we let 𝑍, 𝑍⁺ denote the sets of all integers, nonnegative integers, respectively, and use the notations: 𝑓^𝑢=sup_𝑘∈𝑍{𝑓(𝑘)}, 𝑓^𝑙=inf_𝑘∈𝑍{𝑓(𝑘)}, for any bounded sequence {𝑓(𝑘)} defined on 𝑍.

Let 𝜏=max{𝜏_𝑖,𝜎_𝑖,𝜌_𝑖,𝑖=1,2}, we consider system (1.4) with the following initial conditions: 𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝜃∈[−𝜏,0]∩𝑍,𝜙_𝑖(0)𝑢>0,_𝑖(𝜃)=𝜑_𝑖(𝜃),𝜃∈[−𝜏,0]∩𝑍,𝜑_𝑖(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 constants 𝑚_𝑖 and 𝑀_𝑖, such that for each positive solution (𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 of system (1.4) satisfies 𝑚_𝑖≤liminf_𝑘→+∞𝑥_𝑖(k)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑛_𝑖≤liminf_𝑘→+∞𝑢_𝑖(𝑘)≤limsup_𝑘→+∞𝑢_𝑖(𝑘)≤𝑁_𝑖,𝑖=1,2. (2.1)

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∈𝑍𝑥_𝑖(𝑘)≤sup_𝑘∈𝑍𝑥_𝑖(𝑘)<∞,0<inf_𝑘∈𝑍𝑢_𝑖(𝑘)≤sup_𝑘∈𝑍𝑢_𝑖(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 𝑥𝐸{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 𝐻𝑓=𝑔(𝑘,𝑥)∶lim_𝑛→∞𝑓𝑘+𝜏_𝑛,𝑥=𝑔(𝑘,𝑥)uniformlyon𝑍×𝑆, (2.5) for some sequence {𝜏_𝑛}, 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)exp𝑟(𝑘)1−𝑎𝑦(𝑘), (2.6) for 𝑘∈[𝑘₁,+∞), where 𝑎 is a positive constant and 𝑘₁∈𝑍⁺. Then limsup_𝑘→+∞𝑦(𝑘)≤1𝑎r^𝑢exp𝑟^𝑢−1. (2.7)

Lemma 2.6 (see [20]). Assume that {𝑦(𝑘)} satisfies 𝑦(𝑘₂)>0 and 𝑦(𝑘+1)≥𝑦(𝑘)exp𝑟(𝑘)1−𝑎𝑦(𝑘), (2.8) for 𝑘∈[𝑘₂,+∞), limsup_𝑘→+∞𝑦(𝑘)≤𝑀, where 𝑎 is a constant such that 𝑎M>1 and 𝑘₂∈𝑍⁺. Then liminf_𝑘→+∞𝑦(𝑘)≥1𝑎exp𝑟^𝑢(1−𝑎𝑀). (2.9)

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 integer 𝑚≤𝑘, 𝑦(𝑘)≤𝐴^𝑚𝑦(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗𝐵𝑘−𝑗−1. (2.11) Especially, if 𝐴<1 and 𝐵 is bounded above with respect to 𝑀, then limsup_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 integer 𝑚≤𝑘, 𝑦(𝑘)≥𝐴^𝑚𝑦(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗𝐵𝑘−𝑗−1. (2.14) Especially, if 𝐴<1 and 𝐵 is bounded below with respect to 𝑁, then liminf_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^𝑙>𝑐^𝑢/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)exp𝑏^𝑢. (3.1) It follows from (3.1) that ^𝑘−1_{𝑗=𝑘−𝜏1}𝑥₁𝑗+1𝑥₁𝑗≤^𝑘−1_{𝑗=𝑘−𝜏1}exp𝑏^𝑢≤exp𝑏^𝑢𝜏₁, (3.2) which implies that 𝑥₁𝑘−𝜏₁≥𝑥₁(𝑘)exp−𝑏^𝑢𝜏₁. (3.3) Substituting (3.3) into the first equation of (1.4), it immediately follows that 𝑥₁(𝑘+1)≤𝑥₁(𝑘)exp𝑏(𝑘)−𝑎(𝑘)𝑥₁𝑘−𝜏₁≤𝑥₁(𝑘)exp𝑏(𝑘)−𝑎(𝑘)exp−𝑏^𝑢𝜏₁𝑥₁(𝑘). (3.4) By applying Lemma 2.5 to (3.4), we have limsup_𝑘→+∞𝑥₁(𝑘)≤1𝑎^𝑙exp𝑏^𝑢𝜏₁+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)exp𝑔(𝑘)≤𝑥₂(𝑘)exp𝑔^𝑢,𝑥₂(𝑘+1)≤𝑥₂(𝑘)𝑔(𝑘)−𝑓(𝑘)exp𝑀₁𝑥+𝜀²𝑘−𝜏₂,for𝑘≥𝐾₁+𝜏. (3.7) From (3.7), similar to the argument of (3.1), one has 𝑥₂(𝑘+1)≤𝑥₂(𝑘)𝑔(𝑘)−𝑓(𝑘)exp𝑀₁+𝜀exp−𝑔^𝑢𝜏₂𝑥₂(𝑘). (3.8) By applying Lemma 2.5 to (3.8) again, we have limsup_𝑘→+∞𝑥₂(𝑘)≤𝑀₁+𝜀𝑓^𝑙exp𝑔^𝑢𝜏₂+1−1. (3.9) Setting 𝜀→0 in the above inequality, we have limsup_𝑘→+∞𝑥₂(𝑘)≤𝑀₁𝑓^𝑙exp𝑔^𝑢𝜏₂+1−1=∶𝑀₂. (3.10)
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≥𝐾₂𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀. (3.11) For 𝑘>𝐾₂+𝜏, (3.11) combining with the third and fourth equations of system (1.4) leads to Δ𝑢_𝑖(𝑘)≤−𝛼_𝑖(𝑘)𝑢_𝑖(𝑘)+𝛽_𝑖(𝑘)𝑀_𝑖+𝜀,𝑖=1,2, (3.12) that is, 𝑢_𝑖(𝑘+1)≤1−𝛼^𝑙_𝑖𝑢_𝑖(𝑘)+𝛽^𝑢_𝑖𝑀_𝑖+𝜀,𝑖=1,2. (3.13) By applying Lemma 2.7, it follows from (3.13) that limsup_𝑘→+∞𝑢_𝑖(𝑘)≤𝛽^𝑢_𝑖𝑀_𝑖+𝜀𝛼^𝑙_𝑖,𝑖=1,2. (3.14) Letting 𝜀→0 in the above inequality, we have limsup_𝑘→+∞𝑢_𝑖(𝑘)≤𝛽^𝑢_𝑖𝑀_𝑖𝛼^𝑙_𝑖=∶𝑁_𝑖,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≥𝐾₃𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀,𝑢_𝑖(𝑘)≤𝑁_𝑖+𝜀. (3.16) Thus, from (3.16) and the first equation of system (1.4), it follows that 𝑥₁(𝑘+1)≥𝑥₁(𝑘)exp𝑏^𝑙−𝑎^𝑢𝑀₁−𝑐+𝜀^𝑢2|ℎ|−𝑑^𝑢𝑁₁+𝜀=∶𝑥₁(𝑘)exp𝐷_1𝜀,for𝑘≥𝐾₃+𝜏, (3.17) where 𝐷_1𝜀=𝑏^𝑙−𝑎^𝑢𝑀₁−𝑐+𝜀^𝑢2|ℎ|−𝑑^𝑢𝑁₁+𝜀≤𝑏^𝑙−𝑎^𝑢𝑀₁≤𝑏^𝑙−𝑎^𝑢exp𝑏^𝑢−1𝑎^𝑙exp−𝑏^𝑢𝜏₁≤𝑏^𝑙−𝑏^𝑢≤0. (3.18) For any integer 𝜂≤𝑘, it follows from (3.17) that ^𝑘−1_{𝑗=𝑘−𝜂}𝑥₁𝑗+1𝑥₁𝑗≥^𝑘−1_{𝑗=𝑘−𝜂}exp𝐷_1𝜀≥exp𝐷_1𝜀𝜂, (3.19) and hence 𝑥₁𝜂≤𝑥₁(𝑘)exp−𝑘−𝜂𝐷_1𝜀. (3.20) From the third equation of system (1.4), we have 𝑢₁(𝑘+1)≤1−𝛼^𝑙₁𝑢₁(𝑘)+𝛽^𝑢₁𝑥₁𝑘−𝜌₁≤1−𝛼^𝑙₁𝑢₁(𝑘)+𝛽^𝑢₁exp−𝜌₁𝐷_1𝜀𝑥₁(𝑘)=𝐴₁𝑢₁(𝑘)+𝐵₁(𝑘), (3.21) where 𝐴₁=1−𝛼^𝑙₁ and 𝐵₁(𝑘)=𝛽^𝑢₁exp[−𝜌₁𝐷_1𝜀]𝑥₁(𝑘). Therefore, for any 𝑚≤𝑘−𝜏, by Lemma 2.7, (3.20), and (3.21), one has 𝑢₁(𝑘)≤𝐴^𝑚₁𝑢₁(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗₁𝐵₁𝑘−𝑗−1≤𝐴^𝑚₁𝑢₁(𝑘−𝑚)+^𝑚−1_𝑗=0𝐴^𝑗₁𝛽^𝑢₁exp−𝜌₁𝐷_1𝜀𝑥₁𝑘−𝑗−1≤𝐴^𝑚₁𝑢₁(𝑘−𝑚)+𝑥₁(𝑘)^𝑚−1_𝑗=0𝐴^𝑗₁𝛽^𝑢₁exp−𝑗+1+𝜌₁𝐷_1𝜀. (3.22) Since 𝛼₁(𝑘)∈(0,1), we have 0<𝐴₁<1. So 0≤𝐴^𝑚₁𝑢₁(𝑘−𝑚)≤𝐴^𝑚₁𝑁₁+𝜀→0,for𝑚→∞. (3.23)
By the conditions (𝐻₁), for any solution (𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 of system (1.4), there exists a positive integer 𝑀 such that 𝑑^𝑢𝐴^𝑚₁𝑢₁(𝑘−𝑚)<(1/2)(𝑏^𝑙−𝑐^𝑢/2|ℎ|) for 𝑚>𝑀. In fact, we can choose 𝑀=max{1,[log_𝐴1((𝑏^𝑙−𝑐^𝑢/2|ℎ|)/2𝑑^𝑢𝑁₁)]+1}. Then we get 𝑢₁(𝑘)≤𝐴^𝑀₁𝑁₁+𝜀+𝑥₁(𝑘)^𝑀−1_𝑗=0𝐴^𝑗₁𝛽^𝑢₁exp−𝑗+1+𝜌₁𝐷_1𝜀=𝐴^𝑀₁𝑁₁+𝜀+𝐹_1𝜀𝑥₁(𝑘), (3.24) where 𝐹_1𝜀=∑^𝑀−1_𝑗=0𝐴^𝑗₁𝛽^𝑢₁exp[−(𝑗+1+𝜌₁)𝐷_1𝜀].
Substituting (3.20) and (3.24) into the first equation of system (1.4), one has 𝑥₁(𝑘+1)≥𝑥₁(𝑘)exp𝑏(𝑘)−𝑎(𝑘)𝑥₁𝑘−𝜏₁−𝑐(𝑘)2|ℎ|−𝑑(𝑘)𝑢₁𝑘−𝜎₁≥𝑥₁(𝑘)𝑏(𝑘)−𝑐(𝑘)exp2|ℎ|−𝑑(𝑘)𝐴^𝑀₁𝑁₁−𝑎(𝑘)+𝜀exp−𝜏₁𝐷_1𝜀+𝑑(𝑘)𝐹_1𝜀exp−𝜎₁𝐷_1𝜀𝑥₁(𝑘)=∶𝑥₁(𝑘)exp𝑆_1𝜀(𝑘)𝑆1−^2𝜀(𝑘)𝑆_1𝜀(𝑘)𝑥₁(𝑘), (3.25) where 𝑆_1𝜀(𝑘)(𝑘)−𝑐(𝑘)=𝑏2|ℎ|−𝑑(𝑘)𝐴^𝑀₁𝑁₁,𝑆+𝜀_2𝜀(𝑘)=𝑎(𝑘)exp−𝜏₁𝐷_1𝜀+𝑑(𝑘)𝐹_1𝜀exp−𝜎₁𝐷_1𝜀. (3.26) In particular, we have 𝑆^𝑢_2𝜀𝑆^𝑙_1𝜀𝑀₁=𝑎^𝑢exp−𝜏₁𝐷_1𝜀+𝑑^𝑢𝐹_1𝜀exp−𝜎₁𝐷_1𝜀𝑏^𝑙−𝑐^𝑢/2|ℎ|−𝑑^𝑢𝐴^𝑀₁𝑁₁𝑀+𝜀¹≥𝑎^𝑢exp−𝜏₁𝐷_1𝜀𝑏^𝑙exp𝑏^𝑢𝜏₁+1−1𝑎^𝑙≥𝑎^𝑢exp−𝜏₁𝑏^𝑙𝑏^𝑙exp𝑏^𝑙𝜏₁+1−1𝑎^𝑢≥exp𝑏^𝑙−1𝑏^𝑙>1, (3.27) where we use the inequality exp(𝑥−1)>𝑥 for 𝑥>0. By applying Lemma 2.6 to (3.25), it follows that liminf_𝑘→+∞𝑥₁(𝑘)≥𝑆^𝑙_1𝜀𝑆^𝑢_2𝜀exp𝑆^𝑢_1𝜀𝑆1−^𝑢2𝜀𝑆^𝑙_1𝜀𝑀₁. (3.28) Setting 𝜀→0 in the above inequality, we have liminf_𝑘→+∞𝑥₁(𝑘)≥𝑆^𝑙₁𝑆^𝑢₂exp𝑆^𝑢₁𝑆1−^𝑢2𝑆^𝑙₁𝑀₁=∶𝑚₁. (3.29) From (3.29) we know that there exists enough large 𝐾₄>𝐾₃+𝜏 such that for 𝑘≥𝐾₄, 𝑥₁(𝑘)≥𝑚₁−𝜀. (3.30) From (3.30) and the second equation of system (1.4), it follows that 𝑥₂(𝑘+1)≥𝑥₂(𝑘)exp𝑔^𝑙−𝑓^𝑢𝑀₂+𝜀𝑚₁−𝜀−𝑝^𝑢𝑁₂+𝜀=∶𝑥₂(𝑘)exp𝐷_2𝜀,for𝑘≥𝐾₄+𝜏, (3.31) where 𝐷_2𝜀=𝑔^𝑙−𝑓^𝑢(𝑀₂+𝜀)/(𝑚₁−𝜀)−𝑝^𝑢(𝑁₂+𝜀)≤0.
This implies for any integer 𝜂≤𝑘𝑥₂𝜂≤𝑥₂(𝑘)exp−𝑘−𝜂𝐷_2𝜀. (3.32) From (3.32) and the fourth equation of system (1.4), by a procedure similar to the discussion of (3.21)–(3.24), we can verify that 𝑢₂(𝑘)≤𝐴^𝑁₂𝑁₂+𝜀+𝐹_2𝜀𝑥₂(𝑘), (3.33) where 𝐴₂=1−𝛼^𝑙₂,𝐹_2𝜀=^𝑁−1_𝑗=0𝐴^𝑗₂𝛽^𝑢₂exp−𝑗+1+𝜌₂𝐷_2𝜀,𝑁=max1,log_𝐴2𝑔^𝑙2𝑝^𝑢𝑁₂+1. (3.34) Substituting (3.32) and (3.33) into the second equation of system (1.4), one has 𝑥₂(𝑘+1)≥𝑥₂(𝑘)𝑔(𝑘)−𝑓(𝑘)exp𝑚₁𝑥−𝜀²𝑘−𝜏₂−𝑝(𝑘)𝑢₂𝑘−𝜎₂≥𝑥₂(𝑘)exp𝑔(𝑘)−𝑝(𝑘)𝐴^𝑁₂𝑁₂−𝑓(𝑘)+𝜀𝑚₁exp−𝜏₂𝐷_2𝜀+𝑝(𝑘)𝐹_2𝜀exp−𝜎₂𝐷_2𝜀𝑥₂(𝑘)=∶𝑥₂(𝑘)exp𝑆_3𝜀(𝑘)𝑆1−^4𝜀(𝑘)𝑆_3𝜀(𝑘)𝑥₂(𝑘), (3.35) where 𝑆_3𝜀(𝑘)=𝑔(𝑘)−𝑝(𝑘)𝐴^𝑁₂𝑁₂,𝑆+𝜀_4𝜀(𝑘)=𝑓(𝑘)𝑚₁exp−𝜏₂𝐷_2𝜀+𝑝(𝑘)𝐹_2𝜀exp−𝜎₂𝐷_2𝜀. (3.36) In particular, we have 𝑆^𝑢_4𝜀𝑆^𝑙_3𝜀𝑀₂=𝑓^𝑢/𝑚₁exp−𝜏₂𝐷_2𝜀+𝑝^𝑢𝐹₂exp−𝜎₂𝐷_2𝜀𝑔^𝑙−𝑝^𝑢𝐴^𝑁₂𝑁₂𝑀+𝜀²≥𝑓^𝑢/𝑚₁exp−𝜏₂𝐷_2𝜀𝑔^𝑙𝑀₁exp𝑔^𝑢𝜏₂+1−1𝑓^𝑙≥𝑓^𝑢/𝑚₁exp−𝜏₂𝑔^𝑙𝑔^𝑙𝑀₁exp𝑔^𝑙𝜏₂+1−1𝑓^𝑢≥exp𝑔^𝑙−1𝑔^𝑙>1. (3.37) By applying Lemma 2.6 to (3.35) again, it follows that liminf_𝑘→+∞𝑥₂(𝑘)≥𝑆^𝑙_3𝜀𝑆^𝑢_4𝜀exp𝑆^𝑢_3𝜀𝑆1−^𝑢4𝜀𝑆^𝑙_3𝜀𝑀₂. (3.38) Setting 𝜀→0 in the above inequality, we have liminf_𝑘→+∞𝑥₂(𝑘)≥𝑆^𝑙₃𝑆^𝑢₄exp𝑆^𝑢₃𝑆1−^𝑢4𝑆^𝑙₃𝑀₂=∶𝑚₂. (3.39) From (3.29) and (3.39) we know that there exists enough large 𝐾₅>𝐾₄+𝜏 such that for 𝑘≥𝐾₅, 𝑥_𝑖(𝑘)≥𝑚_𝑖−𝜀. (3.40) For 𝑘>𝐾₅+𝜏, (3.40) combining with the third and fourth equations of system (1.4) produces 𝑢_𝑖(𝑘+1)≥1−𝛼^𝑢_𝑖𝑢_𝑖(𝑘)+𝛽^𝑙_𝑖𝑚_𝑖−𝜀,𝑖=1,2. (3.41) By applying Lemma 2.8, it follows from (3.41) that liminf_𝑘→+∞𝑢_𝑖(𝑘)≥𝛽^𝑙_𝑖𝑚_𝑖−𝜀𝛼^𝑢_𝑖,𝑖=1,2. (3.42) Setting 𝜀→0 in the above inequality, we have liminf_𝑘→+∞𝑢_𝑖(𝑘)≥𝛽^𝑙_𝑖𝑚_𝑖𝛼^𝑢_𝑖=∶𝑛_𝑖,𝑖=1,2. (3.43)
Consequently, combining (3.5), (3.10), (3.15), (3.29), (3.39) with (3.43), system (1.4) is permanent. This completes the proof of Theorem 3.1.

4. Global Attractivity

Firstly, we prove two lemmas which will be useful to our main result.

Lemma 4.1. For any two positive solutions (𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 and (𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘))^𝑇 of system (1.4), one has 𝑥ln¹(𝑘+1)𝑦₁()𝑥𝑘+1=ln¹(𝑘)𝑦₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘(𝑘𝑢)−𝑑)₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘(𝑘))+𝑎^𝑘−1_{𝑠=𝑘−𝜏1}𝑃₁(𝑠)𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁𝑥₁(𝑠)−𝑦₁(𝑠)+𝑥₁(𝑠)𝑃₂(𝑠)−𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑦₁𝑠−𝜏₁−𝑑(𝑠)𝑢₁𝑠−𝜎₁−𝑣₁𝑠−𝜎₁+𝐸₁(𝑠)𝑥₁(𝑠)−𝑦₁(𝑠)+𝐸₂(𝑠)𝑥₂(𝑠)−𝑦₂(𝑠), (4.1) where 𝐸₁(𝑠)=𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)𝑥₁(𝑠)+𝑦₁(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠−𝑐(𝑠)𝑥)²(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠),𝐸₂(𝑠)=ℎ²𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)𝑥₂(𝑠)+𝑦₂(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠−𝑐(𝑠)𝑦)¹(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠),𝑃₁(𝑠)=exp𝜃₁(𝑠)𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁,𝑃₂(𝑠)=exp𝜃₂(𝑠)𝑏(𝑠)−𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑐(𝑠)𝑥₁(𝑠)𝑥₂(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠)−𝑑(𝑠)𝑢₁𝑠−𝜎₁+1−𝜃₂(𝑠)𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁,𝜃₁(𝑠),𝜃₂(𝑠)∈(0,1). (4.2)

Proof. It follows from the first equation of system (1.4) that 𝑥ln¹(𝑘+1)𝑦₁()𝑥𝑘+1−ln¹(𝑘)𝑦₁(𝑘)𝑥=ln¹(𝑘+1)𝑥₁(𝑘)𝑦−ln¹(𝑘+1)𝑦₁(𝑘)=𝑏(𝑘)−𝑎(𝑘)𝑥₁𝑘−𝜏₁−𝑐(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)−𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑏(𝑘)−𝑎(𝑘)𝑦₁𝑘−𝜏₁−𝑐(𝑘)𝑦₁(𝑘)𝑦₂(𝑘)ℎ²𝑦²₂(𝑘)+𝑦²₁(𝑘)−𝑑(𝑘)𝑣₁𝑘−𝜎₁=−𝑎(𝑘)𝑥₁𝑘−𝜏₁−𝑦₁𝑘−𝜏₁−𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁−𝑐(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)+𝑐(𝑘)𝑦₁(𝑘)𝑦₂(𝑘)ℎ²𝑦²₂(𝑘)+𝑦²₁(𝑘)=−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)−𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘(𝑘𝑥)+𝑎)₁(𝑘)−𝑦₁(𝑘)−𝑥₁𝑘−𝜏₁−𝑦₁𝑘−𝜏₁, (4.3) where 𝐸₁(𝑘)=𝑐(𝑘)𝑦₁(𝑘)𝑦₂(𝑘)𝑥₁(𝑘)+𝑦₁(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)ℎ²𝑦²₂(𝑘)+𝑦²₁(𝑘−𝑐(𝑘)𝑥)²(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘),𝐸₂(𝑘)=ℎ²𝑐(𝑘)𝑦₁(𝑘)𝑦₂(𝑘)𝑥₂(𝑘)+𝑦₂(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)ℎ²𝑦²₂(𝑘)+𝑦²₁(𝑘−𝑐(𝑘)𝑦)¹(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘). (4.4) Hence 𝑥ln¹(𝑘+1)𝑦₁()𝑥𝑘+1=ln¹(𝑘)𝑦₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)−𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘(𝑘𝑥)+𝑎)₁(𝑘)−𝑥₁𝑘−𝜏₁−𝑦₁(𝑘)−𝑦₁𝑘−𝜏₁. (4.5) Since 𝑥₁(𝑘)−𝑥₁𝑘−𝜏₁−𝑦₁(𝑘)−𝑦₁𝑘−𝜏₁=^𝑘−1_{𝑠=𝑘−𝜏1}𝑥₁(𝑠+1)−𝑥₁(𝑠)−^𝑘−1_{𝑠=𝑘−𝜏1}𝑦₁(𝑠+1)−𝑦₁(𝑠=_{𝑘−1𝑠=𝑘−𝜏1})𝑥₁(𝑠+1)−𝑦₁(𝑠+1)−𝑥₁(𝑠)−𝑦₁(𝑠,𝑥₁)(𝑠+1)−𝑦₁(𝑠+1)−𝑥₁(𝑠)−𝑦₁(𝑠=𝑥₁)(𝑠)exp𝑏(𝑠)−𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑐(𝑠)𝑥₁(𝑠)𝑥₂(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠)−𝑑(𝑠)𝑢₁𝑠−𝜎₁−𝑦₁(𝑠)exp𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁−𝑥₁(𝑠)−𝑦₁(𝑠=𝑥₁)(𝑠)−𝑦₁(𝑠)exp𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁−1+𝑥₁(𝑠)exp𝑏(𝑠)−𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑐(𝑠)𝑥₁(𝑠)𝑥₂(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠)−𝑑(𝑠)𝑢₁𝑠−𝜎₁−exp𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁. (4.6)(4.7) By the mean value theorem, one has 𝑥₁(𝑠+1)−𝑦₁(𝑠+1)−𝑥₁(𝑠)−𝑦₁(𝑠=𝑥₁)(𝑠)−𝑦₁(𝑠)𝑃₁(𝑠)𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁+𝑥₁(𝑠)𝑃₂(𝑠)−𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑦₁𝑠−𝜏₁−𝑑(𝑠)𝑢₁𝑠−𝜎₁−𝑣₁𝑠−𝜎₁−𝑐(𝑠)𝑥₁(𝑠)𝑥₂(𝑠)ℎ²𝑥²₂(𝑠)+𝑥²₁(𝑠)+𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)=𝑥₁(𝑠)−𝑦₁(𝑠)𝑃₁(𝑠)𝑏(𝑠)−𝑎(𝑠)𝑦₁𝑠−𝜏₁−𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)−𝑑(𝑠)𝑣₁𝑠−𝜎₁+𝑥₁(𝑠)𝑃₂(𝑠)−𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑦₁𝑠−𝜏₁−𝑑(𝑠)𝑢₁𝑠−𝜎₁−𝑣₁𝑠−𝜎₁+𝐸₁(𝑠)𝑥₁(𝑠)−𝑦₁(𝑠)+𝐸₂(𝑠)𝑥₂(𝑠)−𝑦₂(𝑠). (4.8) Combining (4.6) with (4.8), we can easily obtain (4.1). The proof of Lemma 4.1 is completed.

Lemma 4.2. For any two positive solutions (𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 and (𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘))^𝑇 of system (1.4), one has 𝑥ln²(𝑘+1)𝑦₂()𝑥𝑘+1=ln²(𝑘)𝑦₂(𝑘)−𝑓(𝑘)𝑥₁𝑘−𝜏₂𝑥₂(𝑘)−𝑦₂(𝑘+𝑓(𝑘)𝑦)²𝑘−𝜏₂𝑥₁𝑘−𝜏₂𝑦₁𝑘−𝜏₂𝑥₁𝑘−𝜏₂−𝑦₁𝑘−𝜏₂−𝑝(𝑘)𝑢₂𝑘−𝜎₂−𝑣₂𝑘−𝜎₂+𝑓(𝑘)𝑥₁𝑘−𝜏₂^𝑘−1_{𝑠=𝑘−𝜏2}𝑄₁(𝑠𝑔(𝑠)(𝑠)𝑦)−𝑓²𝑠−𝜏₂𝑦₁𝑠−𝜏₂−𝑝(𝑠)𝑣₂𝑠−𝜎₂𝑥₂(𝑠)−𝑦₂(𝑠)+𝑥₂(𝑠)𝑄₂(𝑠−𝑓(𝑠))𝑥₁𝑠−𝜏₂𝑥₂𝑠−𝜏₂−𝑦₂𝑠−𝜏₂+𝑓(𝑠)𝑦₂𝑠−𝜏₂𝑥₁𝑠−𝜏₂𝑦₁𝑠−𝜏₂𝑥₁𝑠−𝜏₂−𝑦₁𝑠−𝜏₂−𝑝(𝑘)𝑢₂𝑠−𝜎₂−𝑣₂𝑠−𝜎₂, (4.9) where 𝑄₁(𝑠)=exp𝜑₁(𝑠𝑔(𝑠)(𝑠)𝑦)−𝑓²𝑠−𝜏₂𝑦₁𝑠−𝜏₂−𝑝(𝑠)𝑣₂𝑠−𝜎₂,𝑄₂(𝑠)=exp𝜑₂(𝑠𝑔(𝑠)(𝑠)𝑥)−𝑓²𝑠−𝜏₂𝑥₁𝑠−𝜏₂−𝑝(𝑠)𝑢₂𝑠−𝜎₂+1−𝜑₂(𝑠)𝑔(𝑠)(𝑠)𝑦−𝑓²𝑠−𝜏₂𝑦₁𝑠−𝜏₂−𝑝(𝑠)𝑣₂𝑠−𝜎₂,𝜑₁(𝑠),𝜑₂(𝑠)∈(0,1). (4.10)

Proof. The proofs of Lemma 4.2 are very similar to those of Lemma 4.1. So we omit the detail here.

Now we are in the position of stating the main result on the global attractivity of system (1.4).

Theorem 4.3. In addition to (𝐻₁)-(𝐻₂), assume further that ?(𝐻₃) there exist positive constants 𝑠_𝑖, 𝜔_𝑖, 𝑖=1,2 such that 𝛼=∶min𝑠₁𝜇₁−𝑠₂𝜇₂−𝜔₁𝛽^𝑢₁,𝑠₂𝜈₁−𝑠₁𝜈₂−𝜔₂𝛽^𝑢₂,𝜔₁𝛼^𝑙₁−𝑠₁𝜚₁,𝜔₂𝛼^𝑙₂−𝑠₂𝜚₂>0 (4.11)holds, where 𝜇₁, 𝜇₂, 𝜈₁, 𝜈₂, 𝜚₁, 𝜚₂ are defined by (4.37). Then for any two positive solutions (𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 and (𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘))^𝑇 of system (1.4), one has lim_𝑘→+∞𝑥_𝑖(𝑘)−𝑦_𝑖(𝑘)=0,lim_𝑘→+∞𝑢_𝑖(𝑘)−𝑣_𝑖(𝑘)=0,𝑖=1,2. (4.12)

Proof. For two arbitrary nontrivial solutions (𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 and (𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘))^𝑇 of system (1.4), we have from Theorem 3.1 that there exist positive constants 𝑘₀ and 𝑀_𝑖, 𝑁_𝑖, 𝑚_𝑖, 𝑛_𝑖(𝑖=1,2) such that for all 𝑘≥𝑘₀ and 𝑖=1,2𝑚_𝑖≤𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑛_𝑖≤𝑢_𝑖(𝑘)≤𝑁_𝑖,𝑚_𝑖≤𝑦_𝑖(𝑘)≤𝑀_𝑖,𝑛_𝑖≤𝑣_𝑖(𝑘)≤𝑁_𝑖. (4.13) Firstly, we define 𝑉₁₁(𝑘)=ln𝑥₁(𝑘)−ln𝑦₁(𝑘). (4.14) From (4.1), we have 𝑥ln¹(𝑘+1)𝑦₁()≤𝑥𝑘+1ln¹(𝑘)𝑦₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘(𝑘))+𝑎^𝑘−1_{𝑠=𝑘−𝜏1}𝑃₁(𝑠)𝐺₁(𝑠)+𝑥₁(𝑠)𝑃₂(𝑠)𝐸₁(𝑠)𝑥₁(𝑠)−𝑦₁(𝑠)+𝑥₁(𝑠)𝑃₂(𝑠)𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑦₁𝑠−𝜏₁+𝑑(𝑠)𝑢₁𝑠−𝜎₁−𝑣₁𝑠−𝜎₁+𝐸₂(𝑠)𝑥₂(𝑠)−𝑦₂(𝑠), (4.15) where 𝐺₁(𝑠)=𝑏(𝑠)+𝑎(𝑠)𝑦₁𝑠−𝜏₁+𝑐(𝑠)𝑦₁(𝑠)𝑦₂(𝑠)ℎ²𝑦²₂(𝑠)+𝑦²₁(𝑠)+𝑑(𝑠)𝑣₁𝑠−𝜎₁. (4.16) By the mean value theorem, we have 𝑥₁(𝑘)−𝑦₁(𝑘)=expln𝑥₁(𝑘)−expln𝑦₁(𝑘)=𝜉₁(𝑘)𝑥ln¹(𝑘)𝑦₁(𝑘), (4.17) that is, 𝑥ln¹(𝑘)𝑦₁(𝑘)=1𝜉₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘), (4.18) where 𝜉₁(𝑘) lies between 𝑥₁(𝑘) and 𝑦₁(𝑘). So, we have 𝑥ln¹(𝑘)𝑦₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘=𝑥)ln¹(𝑘)𝑦₁(𝑘)−𝑥ln¹(𝑘)𝑦₁(𝑘)+𝑥ln¹(𝑘)𝑦₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘=𝑥)ln¹(𝑘)𝑦₁(𝑘)−1𝜉₁(𝑘)−1𝜉₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘). (4.19) For 𝑘>𝑘₀+𝜏, it follows that Δ𝑉₁₁1≤−𝜉₁(𝑘)−1𝜉₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘(𝑘𝑢)+𝑑)₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘(𝑘))+𝑎^𝑘−1_{𝑠=𝑘−𝜏1}𝑃₁(𝑠)𝐺₁(𝑠)+𝑀₁𝑃₂(𝑠)𝐸₁(𝑠)𝑥₁(𝑠)−𝑦₁(𝑠)+𝑀₁𝑃₂(𝑠)𝑎(𝑠)𝑥₁𝑠−𝜏₁−𝑦₁𝑠−𝜏₁+𝑑(𝑠)𝑢₁𝑠−𝜎₁−𝑣₁𝑠−𝜎₁+𝐸₂(𝑠)𝑥₂(𝑠)−𝑦₂(𝑠). (4.20) Secondly, we define 𝑉₁₂(𝑘)=^𝑘−1_{𝑠=𝑘−𝜎1}𝑑𝑠+𝜎₁𝑢₁(𝑠)−𝑣₁(𝑠+_{𝑘−1+𝜏1𝑠=𝑘})𝑎(𝑠)^𝑘−1_{𝑢=𝑠−𝜏1}𝑃₁(𝑢)𝐺₁(𝑢)+𝑀₁𝑃₂(𝑢)𝐸₁(𝑢)𝑥₁(𝑢)−𝑦₁(𝑢)+𝑀₁𝑃₂(𝑢)𝑎(𝑢)𝑥₁𝑢−𝜏₁−𝑦₁𝑢−𝜏₁+𝑑(𝑢)𝑢₁𝑢−𝜎₁−𝑣₁𝑢−𝜎₁+𝐸₂(𝑢)𝑥₂(𝑢)−𝑦₂(𝑢). (4.21) Then Δ𝑉₁₂=𝑑𝑘+𝜎₁𝑢₁(𝑘)−𝑣₁(𝑘)−𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+^𝑘+𝜏1_𝑠=𝑘+1𝑎(𝑠)𝑃₁(𝑘)𝐺₁(𝑘)+𝑀₁𝑃₂(𝑘)𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝑀₁𝑃₂(𝑘)𝑎(𝑘)𝑥₁𝑘−𝜏₁−𝑦₁𝑘−𝜏₁+𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘(𝑘))−𝑎^𝑘−1_{𝑢=𝑠−𝜏1}𝑃₁(𝑢)𝐺₁(𝑢)+𝑀₁𝑃₂(𝑢)𝐸₁(𝑢)𝑥₁(𝑢)−𝑦₁(𝑢)+𝑀₁𝑃₂(𝑢)𝑎(𝑢)𝑥₁𝑢−𝜏₁−𝑦₁𝑢−𝜏₁+𝑑(𝑢)𝑢₁𝑢−𝜎₁−𝑣₁𝑢−𝜎₁+𝐸₂(𝑢)𝑥₂(𝑢)−𝑦₂(𝑢). (4.22) Thirdly, we define 𝑉₁₃(𝑘)=𝑀₁^𝑘−1_{𝑙=𝑘−𝜏1}𝑃₂𝑙+𝜏₁𝑎𝑙+𝜏₁𝑥₁(𝑙)−𝑦₁(𝑙)^𝑙+2𝜏1_{𝑠=𝑙+𝜏1+1}𝑎(𝑠)+𝑀₁^𝑘−1_{𝑙=𝑘−𝜎1}𝑃₂𝑙+𝜎₁𝑑𝑙+𝜎₁𝑢₁(𝑙)−𝑣₁(𝑙)^{𝑙+𝜎1+𝜏1}_{𝑠=𝑙+𝜎1+1}𝑎(𝑠). (4.23) By a simple calculation, it follows that Δ𝑉₁₃=^𝑘+2𝜏1_{𝑠=𝑘+𝜏1+1}𝑎(𝑠)𝑀₁𝑃₂𝑘+𝜏₁𝑎𝑘+𝜏₁𝑥₁(𝑘)−𝑦₁(𝑘−_{𝑘+𝜏1𝑠=𝑘+1})𝑎(𝑠)𝑀₁𝑃₂(𝑘)𝑎(𝑘)𝑥₁𝑘−𝜏₁−𝑦₁𝑘−𝜏₁+^{𝑘+𝜎1+𝜏1}_{𝑠=𝑘+𝜎1+1}𝑎(𝑠)𝑀₁𝑃₂𝑘+𝜎₁𝑑𝑘+𝜎₁𝑢₁(𝑘)−𝑣₁(𝑘−_{𝑘+𝜏1𝑠=𝑘+1})𝑎(𝑠)𝑀₁𝑃₂(𝑘)𝑑(𝑘)𝑢₁𝑘−𝜎₁−𝑣₁𝑘−𝜎₁. (4.24) We now define 𝑉₁(𝑘)=𝑉₁₁(𝑘)+𝑉₁₂(𝑘)+𝑉₁₃(𝑘). (4.25) Then for all 𝑘>𝑘₀+𝜏, it follows from (4.14)–(4.24) that Δ𝑉₁1≤−𝜉₁(𝑘)−1𝜉₁(𝑘)−𝑎(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝑑𝑘+𝜎₁𝑢₁(𝑘)−𝑣₁(𝑘)+𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘+_{𝑘+𝜏1𝑠=𝑘+1})𝑎(𝑠)𝑃₁(𝑘)𝐺₁(𝑘)+𝑀₁𝑃₂(𝑘)𝐸₁(𝑘)𝑥₁(𝑘)−𝑦₁(𝑘)+𝑀₁𝑃₂(𝑘)𝐸₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘+_{𝑘+2𝜏1𝑠=𝑘+𝜏1+1})𝑎(𝑠)𝑀₁𝑃₂𝑘+𝜏₁𝑎𝑘+𝜏₁𝑥₁(𝑘)−𝑦₁(𝑘+_{𝑘+𝜎1+𝜏1𝑠=𝑘+𝜎1+1})𝑎(𝑠)𝑀₁𝑃₂𝑘+𝜎₁𝑑𝑘+𝜎₁𝑢₁(𝑘)−𝑣₁(𝑘). (4.26) Similarly, we define 𝑉₂(𝑘)=𝑉₂₁(𝑘)+𝑉₂₂(𝑘)+𝑉₂₃(𝑘), (4.27) where 𝑉₂₁(𝑘)=ln𝑥₂(𝑘)−ln𝑦₂(𝑘,𝑉₂₂)(𝑘)=^𝑘−1_{𝑠=𝑘−𝜏2}𝑀₂𝑚²₁𝑓𝑠+𝜏₂𝑥₁(𝑠)−𝑦₁(𝑠)+^𝑘−1_{𝑠=𝑘−𝜎2}𝑝𝑠+𝜎₂𝑢₂(𝑠)−𝑣₂(𝑠+_{𝑘−1+𝜏2𝑠=𝑘})𝑓(𝑠)𝑚₁^𝑘−1_{𝑢=𝑠−𝜏2}𝑄₁(𝑢)𝐺₂(𝑢)𝑥₂(𝑢)−𝑦₂(𝑢)+𝑀₂𝑄₂(𝑢𝑓(𝑢))𝑚₁𝑥₂𝑢−𝜏₂−𝑦₂𝑢−𝜏₂+𝑀₂𝑚²₁𝑓(𝑢)𝑥₁𝑢−𝜏₂−𝑦₁𝑢−𝜏₂+𝑝(𝑢)𝑢₂𝑢−𝜎₂−𝑣₂𝑢−𝜎₂,𝑉₂₃(𝑘)=𝑀₂𝑚₁^𝑘−1_{𝑙=𝑘−𝜏2}𝑄₂𝑙+𝜏₂𝑓𝑙+𝜏₂𝑥₂(𝑙)−𝑦₂(𝑙)^𝑙+2𝜏2_{𝑠=𝑙+𝜏2+1}𝑓(𝑠)𝑚₁+𝑀²₂𝑚²₁^𝑘−1_{𝑙=𝑘−𝜏2}𝑄₂𝑙+𝜏₂𝑓𝑙+𝜏₂𝑥₁(𝑙)−𝑦₁(𝑙)^𝑙+2𝜏2_{𝑠=𝑙+𝜏2+1}𝑓(𝑠)𝑚₁+𝑀₂^𝑘−1_{𝑙=𝑘−𝜎2}𝑄₂𝑙+𝜎₂𝑝𝑙+𝜎₂𝑢₂(𝑙)−𝑣₂(𝑙)^{𝑙+𝜎2+𝜏2}_{𝑠=𝑙+𝜎2+1}𝑓(𝑠)𝑚₁,𝐺₂(𝑠)(𝑠)(𝑠)𝑦=𝑔+𝑓²𝑠−𝜏₂𝑦₁𝑠−𝜏₂+𝑝(𝑠)𝑣₂𝑠−𝜎₂. (4.28)(4.29)(4.30)(4.31) Thus for all 𝑘>𝑘₀+𝜏, it follows from (4.27)–(4.30) that Δ𝑉₂1≤−𝜉₂(𝑘)−1𝜉₂(𝑘)−𝑓(𝑘)𝑥₁𝑘−𝜏₂𝑥₂(𝑘)−𝑦₂(𝑘+𝑀₂)𝑚²₁𝑓𝑘+𝜏₂𝑥₁(𝑘)−𝑦₁(𝑘)+𝑝𝑘+𝜎₂𝑢₂(𝑘)−𝑣₂(𝑘+_{𝑘+𝜏2𝑠=𝑘+1})𝑓(𝑠)𝑚₁𝑄₁(𝑘)𝐺₂(𝑘)𝑥₂(𝑘)−𝑦₂(𝑘+_{𝑘+2𝜏2𝑠=𝑘+𝜏2+1})𝑓(𝑠)𝑚₁𝑀₂𝑚₁𝑄₂𝑘+𝜏₂𝑓𝑘+𝜏₂𝑥₂(𝑘)−𝑦₂(𝑘+_{𝑘+2𝜏2𝑠=𝑘+𝜏2+1})𝑓(𝑠)𝑚₁𝑀²₂𝑚²₁𝑄₂𝑘+𝜏₂𝑓𝑘+𝜏₂𝑥₁(𝑘)−𝑦₁(𝑘+_{𝑘+𝜎2+𝜏2𝑠=𝑘+𝜎2+1})𝑓(𝑠)𝑚₁𝑀₂𝑄₂𝑘+𝜎₂𝑝𝑘+𝜎₂𝑢₂(𝑘)−𝑣₂(𝑘). (4.32) For 𝑖=1,2, we define 𝑉^′_𝑖(𝑘)=𝑢_𝑖(𝑘)−𝑣_𝑖(𝑘)+^𝑘−1_{𝑠=𝑘−𝜌𝑖}𝛽_𝑖𝑠+𝜌_𝑖𝑥_𝑖(𝑠)−𝑦_𝑖(𝑠). (4.33) By calculation, it derives that Δ𝑉^′_𝑖≤−𝛼_𝑖(𝑘)𝑢_𝑖(𝑘)−𝑣_𝑖(𝑘)+𝛽_𝑖𝑘+𝜌_𝑖𝑥_𝑖(𝑘)−𝑦_𝑖(𝑘). (4.34) We now define a Lyapunov function as 𝑉(𝑘)=𝑠₁𝑉₁(𝑘)+𝑠₂𝑉₂(𝑘)+𝜔₁𝑉^′₁(𝑘)+𝜔₂𝑉^′₂(𝑘). (4.35) It is easy to see that 𝑉(𝑘)>0 and 𝑉(𝑘₀+𝜏)<+∞. Calculating the difference of 𝑉 along the solution of system (1.4), we have that for 𝑘≥𝑘₀+𝜏, Δ𝑉≤𝑠₁−min𝑎^𝑙,2𝑀₁−𝑎^𝑢+𝐸^𝑢₁+𝜏₁𝑎^𝑢𝑃^𝑢₁𝐺^𝑢₁+𝑀₁𝑃^𝑢₂𝐸^𝑢₁+𝑀₁𝑃^𝑢₂𝑎^𝑢+𝑠₂𝑀₂𝑚²₁𝑓^𝑢1+𝜏₂𝑀₂𝑚₁𝑓^𝑢𝑄^𝑢₂+𝜔₁𝛽^𝑢₁𝑥₁(𝑘)−𝑦₁(𝑘+𝑠₂)𝑓−min^𝑙𝑀₁,2𝑀₂−𝑓^𝑢𝑚₁+𝜏₂𝑓^𝑢𝑚₁𝑄^𝑢₁𝐺^𝑢₂+𝑀₂𝑚₁𝑄^𝑢₂𝑓^𝑢+𝑠₁𝐸^𝑢₂1+𝜏₁𝑎^𝑢𝑀₁𝑃^𝑢₂+𝜔₂𝛽^𝑢₂𝑥₂(𝑘)−𝑦₂(𝑘+)−𝜔₁𝛼^𝑙₁+𝑠₁𝑑^𝑢1+𝜏₁𝑎^𝑢𝑀₁𝑃^𝑢₂𝑢₁(𝑘)−𝑣₁(𝑘+)−𝜔₂𝛼^𝑙₂+𝑠₂𝑝^𝑢1+𝜏₂𝑓^𝑢𝑚₁𝑀₂𝑄^𝑢₂𝑢₂(𝑘)−𝑣₂(𝑘𝑠)≤−₁𝜇₁−𝑠₂𝜇₂−𝜔₁𝛽^𝑢₁𝑥₁(𝑘)−𝑦₁(𝑘)−𝑠₂𝜈₁−𝑠₁𝜈₂−𝜔₂𝛽^𝑢₂𝑥₂(𝑘)−𝑦₂(𝑘−𝜔₁)𝛼^𝑙₁−𝑠₁𝜚₁𝑢₁(𝑘)−𝑣₁(𝑘)−𝜔₂𝛼^𝑙₂−𝑠₂𝜚₂𝑢₂(𝑘)−𝑣₂(𝑘)≤−𝛼²_𝑖=1𝑥_𝑖(𝑘)−𝑦_𝑖(𝑘)+𝑢_𝑖(𝑘)−𝑣_𝑖(𝑘), (4.36) where 𝜇₁=min𝑎^𝑙,2𝑀₁−𝑎^𝑢−𝐸^𝑢₁−𝜏₁𝑎^𝑢𝑃^𝑢₁𝐺^𝑢₁+𝑀₁𝑃^𝑢₂𝐸^𝑢₁+𝑀₁𝑃^𝑢₂𝑎^𝑢,𝜇₂=𝑀₂𝑚²₁𝑓^𝑢1+𝜏₂𝑀₂𝑚₁𝑄^𝑢₂𝑓^𝑢,𝜈₁𝑓=min^𝑙𝑀₁,2𝑀₂−𝑓^𝑢𝑚₁−𝜏₂𝑓^𝑢𝑚₁𝑄^𝑢₁𝐺^𝑢₂+𝑀₂𝑚₁𝑄^𝑢₂𝑓^𝑢,𝜈₂=𝐸^𝑢₂1+𝜏₁𝑎^𝑢𝑀₁𝑃^𝑢₂,𝜚₁=𝑑^𝑢1+𝜏₁𝑎^𝑢𝑀₁𝑃^𝑢₂,𝜚₂=𝑝^𝑢1+𝜏₂𝑓^𝑢𝑚₁𝑀₂𝑄^𝑢₂. (4.37) Summing both sides of (4.36) from 𝑘₀+𝜏 to 𝑘, it derives that ^𝑘_{𝑠=𝑘0+𝜏}[𝑉(𝑠+1)−𝑉(𝑠)]≤−𝛼^𝑘_𝑠=𝑘0²_+𝜏𝑖=1𝑥_𝑖(𝑠)−𝑦_𝑖(𝑠)+𝑢_𝑖(𝑠)−𝑣_𝑖(𝑠). (4.38) It then follows from (4.38) that for 𝑘>𝑘₀+𝜏, 𝑉(𝑘+1)+𝛼^𝑘_𝑠=𝑘0²_+𝜏𝑖=1𝑥_𝑖(𝑠)−𝑦_𝑖(𝑠)+𝑢_𝑖(𝑠)−𝑣_𝑖(𝑠)≤𝑉𝑘₀+𝜏, (4.39) that is ^𝑘_𝑠=𝑘0²_+𝜏𝑖=1𝑥_𝑖(𝑠)−𝑦_𝑖(𝑠)+𝑢_𝑖(𝑠)−𝑣_𝑖(𝑠≤𝑉𝑘)⁰+𝜏𝛼. (4.40) Then ^∞_𝑘=𝑘0²_+𝜏𝑖=1𝑥_𝑖(𝑠)−𝑦_𝑖(𝑠)+𝑢_𝑖(𝑠)−𝑣_𝑖(𝑠≤𝑉𝑘)⁰+𝜏𝛼<+∞, (4.41) from which we see that lim_𝑘→+∞𝑥_𝑖(𝑘)−𝑦_𝑖(𝑘)=0,lim_𝑘→+∞𝑢_𝑖(𝑘)−𝑣_𝑖(𝑘)=0,𝑖=1,2. (4.42) This completes the proof of Theorem 4.3.

5. Almost Periodic Solutions

In this section, we consider the almost periodic property of system (1.4).

We assume that 𝑎^∗(𝑘)∈𝐻(𝑎(𝑘)), 𝑏^∗(𝑘)∈𝐻(𝑏(𝑘)), 𝑐^∗(𝑘)∈𝐻(𝑐(𝑘)), 𝑑^∗(𝑘)∈𝐻(𝑑(𝑘)), 𝑓^∗(𝑘)∈𝐻(𝑓(𝑘)), 𝑔^∗(𝑘)∈𝐻(𝑔(𝑘)), 𝑝^∗(𝑘)∈𝐻(𝑝(𝑘)), 𝛼^∗_𝑖(𝑘)∈𝐻(𝛼_𝑖(𝑘)), 𝛽^∗_𝑖(𝑘)∈𝐻(𝛽_𝑖(𝑘)), 𝑖=1,2. By Definitions 2.3 and 2.4, there exists an integer valued sequence 𝛾_𝑛 with 𝛾_𝑛→∞ as 𝑛→∞ such that for 𝑖=1,2, 𝑘∈𝑍𝑎𝑘+𝛾_𝑛→𝑎^∗(𝑘),𝑏𝑘+𝛾_𝑛→𝑏^∗(𝑘),𝑐𝑘+𝛾_𝑛→𝑐^∗(𝑘),𝑑𝑘+𝛾_𝑛→𝑑^∗(𝑘),𝑓𝑘+𝛾_𝑛→𝑓^∗(𝑘),𝑔𝑘+𝛾_𝑛→𝑔^∗(𝑘),𝑝𝑘+𝛾_𝑛→𝑝^∗(𝑘),𝛼_𝑖𝑘+𝛾_𝑛→𝛼^∗_𝑖(𝑘),𝛽_𝑖𝑘+𝛾_𝑛→𝛽^∗_𝑖(𝑘). (5.1) Then we have the following Hull equation of system (1.4) 𝑥₁(𝑘+1)=𝑥₁(𝑘)exp𝑏^∗(𝑘)−𝑎^∗(𝑘)𝑥₁𝑘−𝜏₁−𝑐^∗(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)−𝑑^∗(𝑘)𝑢₁𝑘−𝜎₁,𝑥₂(𝑘+1)=𝑥₂(𝑘)exp𝑔^∗(𝑘)−𝑓^∗(𝑘)𝑥₂𝑘−𝜏₂𝑥₁𝑘−𝜏₂−𝑝^∗(𝑘)𝑢₂𝑘−𝜎₂,Δ𝑢_𝑖(𝑘)=−𝛼^∗_𝑖(𝑘)𝑢_𝑖(𝑘)+𝛽^∗_𝑖(𝑘)𝑥_𝑖𝑘−𝜌_𝑖,𝑖=1,2. (5.2)

According to the almost periodic theory, we know that if system (1.4) satisfies (𝐻₁)−(𝐻₃), then Hull equation (5.2) also satisfies (𝐻₁)−(𝐻₃).

The following Lemma is Lemma?4.1 in [15].

Lemma 5.1. If each hull equation of system (1.4) has a unique strictly positive solution, then the almost periodic difference system (1.4) has a unique strictly positive almost periodic solution.

Theorem 5.2. Assume that (𝐻₁)−(𝐻₃) holds. Then the almost periodic difference system (1.4) has a unique strictly positive almost periodic solution which is globally attractive.

Proof. We divided the proof into two steps.
Step 1. We prove the existence of a strictly positive solution of each Hull equation (5.2).
By the almost periodicity of the parameters of system (1.4), there exists an integer-valued sequence 𝛾_𝑛 with 𝛾_𝑛→∞ as 𝑛→∞ such that for 𝑖=1,2,𝑘∈Z𝑎^∗𝑘+𝛾_𝑛→𝑎^∗(𝑘),𝑏^∗𝑘+𝛾_𝑛→𝑏^∗(𝑘),𝑐^∗𝑘+𝛾_𝑛→𝑐^∗(𝑘),𝑑^∗𝑘+𝛾_𝑛→𝑑^∗(𝑘),𝑓^∗𝑘+𝛾_𝑛→𝑓^∗(𝑘),𝑔^∗𝑘+𝛾_𝑛→𝑔^∗(𝑘),𝑝^∗𝑘+𝛾_𝑛→𝑝^∗(𝑘),𝛼^∗_𝑖𝑘+𝛾_𝑛→𝛼^∗_𝑖(𝑘),𝛽^∗_𝑖𝑘+𝛾_𝑛→𝛽^∗_𝑖(𝑘). (5.3)
Suppose that 𝑋(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘))^𝑇 is any positive solution of Hull equation (5.2). Since (𝐻₁), (𝐻₂) hold, according to Theorem 3.1, we obtain 𝑚_𝑖≤liminf_𝑘→+∞𝑥_𝑖(𝑘)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑛_𝑖≤liminf_𝑘→+∞𝑢_𝑖(𝑘)≤limsup_𝑘→+∞𝑢_𝑖(𝑘)≤𝑁_𝑖,𝑖=1,2. (5.4) Therefore, one has 0<inf_𝑘∈𝑍+𝑥_𝑖(𝑘)≤sup_𝑘∈𝑍+𝑥_𝑖(𝑘)<∞,0<inf_𝑘∈𝑍+𝑢_𝑖(𝑘)≤sup_𝑘∈𝑍+𝑢_𝑖(𝑘)<∞,𝑖=1,2. (5.5)
For any 𝜀>0 small enough, it follows from Theorem 3.1 that there exists enough large 𝐾₀>𝐾₅+𝜏 such that for 𝑖=1,2 and 𝑘≥𝐾₀𝑚_𝑖−𝜀≤𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀,𝑛_𝑖−𝜀≤𝑢_𝑖(𝑘)≤𝑁_𝑖+𝜀. (5.6)
Let 𝑥_𝑛(𝑘)=𝑥_𝑛(𝑘+𝛾_𝑛) and 𝑢_𝑛(𝑘)=𝑢_𝑛(𝑘+𝛾_𝑛) for 𝑘≥𝐾₀+𝜏−𝛾_𝑛, 𝑛=1,2,…. For any positive integer 𝑞 and 𝑖=1,2, it is not difficult to show that there are sequences {𝑥_𝑖𝑛(𝑘)∶𝑛≥𝑞} and {𝑢_𝑖𝑛(𝑘)∶𝑛≥𝑞} such that the sequences {𝑥_𝑖𝑛(𝑘)} and {𝑢_𝑖𝑛(𝑘)} have subsequences, denoted by {𝑥_𝑖𝑛(𝑘)} and {𝑢_𝑖𝑛(𝑘)} again, converging on any finite interval of 𝑍 as 𝑛→∞, respectively. So we have sequences {𝑦_𝑖(𝑘)} and {𝑣_𝑖(𝑘)}𝑖=1,2 such that for 𝑘∈𝑍,𝑛→∞𝑥_𝑖𝑛(𝑘)→𝑦_𝑖(𝑘),𝑢_𝑖𝑛(𝑘)→𝑣_𝑖(𝑘),𝑥₁(𝑘+1)=𝑥₁(𝑘)exp𝑏^∗𝑘+𝛾_𝑛−𝑎^∗𝑘+𝛾_𝑛𝑥₁𝑘−𝜏₁−𝑐^∗𝑘+𝛾_𝑛𝑥₁(𝑘)𝑥₂(𝑘)ℎ²𝑥²₂(𝑘)+𝑥²₁(𝑘)−𝑑^∗𝑘+𝛾_𝑛𝑢₁𝑘−𝜎₁,𝑥₂(𝑘+1)=𝑥₂(𝑘)exp𝑔^∗𝑘+𝛾_𝑛−𝑓^∗𝑘+𝛾_𝑛𝑥₂𝑘−𝜏₂𝑥₁𝑘−𝜏₂−𝑝^∗𝑘+𝛾_𝑛𝑢₂𝑘−𝜎₂,Δ𝑢_𝑖(𝑘)=−𝛼^∗_𝑖𝑘+𝛾_𝑛𝑢_𝑖(𝑘)+𝛽^∗_𝑖𝑘+𝛾_𝑛𝑥_𝑖𝑘−𝜌_𝑖,𝑖=1,2, (5.7) which implies 𝑦₁(𝑘+1)=𝑦₁(𝑘)exp𝑏^∗(𝑘)−𝑎^∗(𝑘)𝑦₁𝑘−𝜏₁−𝑐^∗(𝑘)𝑦₁(𝑘)𝑦₂(𝑘)ℎ²𝑦²₂(𝑘)+𝑦²₁(𝑘)−𝑑^∗(𝑘)𝑣₁𝑘−𝜎₁,𝑦₂(𝑘+1)=𝑦₂(𝑘)exp𝑔^∗(𝑘)−𝑓^∗(𝑘)𝑦₂𝑘−𝜏₂𝑦₁𝑘−𝜏₂−𝑝^∗(𝑘)𝑣₂𝑘−𝜎₂,Δ𝑣_𝑖(𝑘)=−𝛼^∗_𝑖(𝑘)𝑣_𝑖(𝑘)+𝛽^∗_𝑖(𝑘)𝑦_𝑖𝑘−𝜌_𝑖,𝑖=1,2. (5.8)
Thus we can easily see that 𝑌(𝑘)=(𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘))^𝑇 is a solution of Hull equation (5.2) and 𝑚_𝑖−𝜀≤𝑦_𝑖(𝑘)≤𝑀_𝑖+𝜀, 𝑛_𝑖−𝜀≤𝑣_𝑖(𝑘)≤𝑁_𝑖+𝜀, for𝑖=1,2, 𝑘∈𝑍. Since 𝜀 is an arbitrary small positive number, it follows that 0<inf_𝑘∈𝑍+𝑦_𝑖(𝑘)≤sup_𝑘∈𝑍+𝑦_𝑖(𝑘)<∞,0<inf_𝑘∈𝑍+𝑣_𝑖(𝑘)≤sup_𝑘∈𝑍+𝑣_𝑖(𝑘)<∞,𝑖=1,2. (5.9) This completes the proof of Step 1.
Step 2. We show the uniqueness of the strictly positive solution of each Hull equation (5.2).
Suppose that the Hull equation (5.2) has two arbitrary strictly positive solutions 𝑋^∗(𝑘)=(𝑥^∗₁(𝑘),𝑥^∗₂(𝑘),𝑢^∗₁(𝑘),𝑢^∗₂(𝑘))^𝑇 and 𝑌^∗(𝑘)=(𝑦^∗₁(𝑘),𝑦^∗₂(𝑘),𝑣^∗₁(𝑘),𝑣^∗₂(𝑘))^𝑇. Now we define a Lyapunov function 𝑉^∗(𝑘) on 𝑘∈𝑍 as follows: 𝑉^∗(𝑘)=²_𝑖=1𝑠_𝑖𝑉^∗_𝑖1(𝑘)+𝑉^∗_𝑖2(𝑘)+𝑉^∗_𝑖3(𝑘)+²_𝑖=1𝜔_𝑖𝑉_𝑖^′∗(𝑘), (5.10) where 𝑉^∗_𝑖1(𝑘)=ln𝑥^∗_𝑖(𝑘)−ln𝑦^∗_𝑖(𝑘𝑉),𝑖=1,2,_∗12(𝑘)=^𝑘−1_{𝑠=𝑘−𝜎1}𝑑𝑠+𝜎₁𝑢^∗₁(𝑠)−𝑣^∗₁(𝑠+_{𝑘−1+𝜏1𝑠=𝑘})𝑎(𝑠)^𝑘−1_{𝑢=𝑠−𝜏1}𝑃₁(𝑢)𝐺₁(𝑢)𝑀₁𝑃₂(𝑢)𝐸₁(𝑢)𝑥^∗₁(𝑢)−𝑦^∗₁(𝑢)+𝑀₁𝑃₂(𝑢)𝑎(𝑢)𝑥^∗₁𝑢−𝜏₁−𝑦^∗₁𝑢−𝜏₁+𝑑(𝑢)𝑢^∗₁𝑢−𝜎₁−𝑣^∗₁𝑢−𝜎₁+𝐸₂(𝑢)𝑥^∗₂(𝑢)−𝑦^∗₂(𝑢,𝑉_∗13)(𝑘)=𝑀₁^𝑘−1_{𝑙=𝑘−𝜏1}𝑃₂𝑙+𝜏₁𝑎𝑙+𝜏₁𝑥^∗₁(𝑙)−𝑦^∗₁(𝑙)^𝑙+2𝜏1_{𝑠=𝑙+𝜏1+1}𝑎(𝑠)+𝑀₁^𝑘−1_{𝑙=𝑘−𝜎1}𝑃₂𝑙+𝜎₁𝑑𝑙+𝜎₁𝑢^∗₁(𝑙)−𝑣^∗₁(𝑙)^{𝑙+𝜎1+𝜏1}_{𝑠=𝑙+𝜎1+1}𝑎(𝑠),𝑉^∗₂₂(𝑘)=^𝑘−1_{𝑠=𝑘−𝜏2}𝑀₂𝑚²₁𝑓𝑠+𝜏₂𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+^𝑘−1_{𝑠=𝑘−𝜎2}𝑝𝑠+𝜎₂𝑢^∗₂(𝑠)−𝑣^∗₂(𝑠+_{𝑘−1+𝜏2𝑠=𝑘})𝑓(𝑠)𝑚₁^𝑘−1_{𝑢=𝑠−𝜏2}𝑄₁(𝑢)𝐺₂(𝑢)𝑥^∗₂(𝑢)−𝑦^∗₂(𝑢)+𝑀₂𝑄₂(𝑢𝑓(𝑢))𝑚₁𝑥^∗₂𝑢−𝜏₂−𝑦^∗₂𝑢−𝜏₂+𝑀₂𝑚²₁𝑓(𝑢)𝑥^∗₁𝑢−𝜏₂−𝑦^∗₁𝑢−𝜏₂+𝑝(𝑢)𝑢^∗₂𝑢−𝜎₂−𝑣^∗₂𝑢−𝜎₂,𝑉^∗₂₃(𝑘)=𝑀₂𝑚₁^𝑘−1_{𝑙=𝑘−𝜏2}𝑄₂𝑙+𝜏₂𝑓𝑙+𝜏₂𝑥^∗₂(𝑙)−𝑦^∗₂(𝑙)^𝑙+2𝜏2_{𝑠=𝑙+𝜏2+1}𝑓(𝑠)𝑚₁+𝑀²₂𝑚²₁^𝑘−1_{𝑙=𝑘−𝜏2}𝑄₂𝑙+𝜏₂𝑓𝑙+𝜏₂𝑥^∗₁(𝑙)−𝑦^∗₁(𝑙)^𝑙+2𝜏2_{𝑠=𝑙+𝜏2+1}𝑓(𝑠)𝑚₁+𝑀₂^𝑘−1_{𝑙=𝑘−𝜎2}𝑄₂𝑙+𝜎₂𝑝𝑙+𝜎₂𝑢^∗₂(𝑙)−𝑣^∗₂(𝑙)^{𝑙+𝜎2+𝜏2}_{𝑠=𝑙+𝜎2+1}𝑓(𝑠)𝑚₁,𝑉_𝑖^′∗(𝑘)=𝑢^∗_𝑖(𝑘)−𝑣^∗_𝑖(𝑘)+^𝑘−1_{𝑠=𝑘−𝜌𝑖}𝛽_𝑖𝑠+𝜌_𝑖𝑥^∗_𝑖(𝑠)−𝑦^∗_𝑖(𝑠), (5.11) where 𝐸_𝑖, 𝐺_𝑖, 𝑃_𝑖, 𝑄_𝑖, 𝑖=1,2 are defined by Theorem 4.3.
Similar to the discussion of (3.32), calculating the difference of 𝑉^∗(𝑘) along the solution of the Hull equation (5.2), one has Δ𝑉^∗≤−𝛼²_𝑖=1𝑥^∗_𝑖(𝑘)−𝑦^∗_𝑖(𝑘)+𝑢^∗_𝑖(𝑘)−𝑣^∗_𝑖(𝑘),for𝑘∈𝑍. (5.12) It follows from (5.12) we know that 𝑉^∗(𝑘) is a nonincreasing function on 𝑍. Summing both sides of the above inequalities from 𝑘 to 0, we get 𝛼⁰²_{𝑠=𝑘𝑖=1}𝑥^∗_𝑖(𝑠)−𝑦^∗_𝑖(𝑠)+𝑢^∗_𝑖(𝑠)−𝑣^∗_𝑖(𝑠)≤𝑉^∗(𝑘)−𝑉^∗(0),for𝑘<0. (5.13) So we have ∑⁰_𝑠=𝑘∑²_𝑖=1(|𝑥^∗_𝑖(𝑠)−𝑦^∗_𝑖(𝑠)|+|𝑢^∗_𝑖(𝑠)−𝑣^∗_𝑖(𝑠)|)<+∞,𝑘→−∞, thus we obtain lim_{𝑘→−∞}𝑥^∗_𝑖(𝑘)−𝑦^∗_𝑖(𝑘)=0,lim_{𝑘→−∞}𝑢^∗_𝑖(𝑘)−𝑣^∗_𝑖(𝑘)=0,𝑖=1,2. (5.14) From (5.14), for a positive integer 𝐾, we have 𝑥^∗_𝑖(𝑘)−𝑦^∗_𝑖(𝑘)<𝜀,𝑢^∗_𝑖(𝑘)−𝑣^∗_𝑖(𝑘)<𝜀,for𝑘<−𝐾,𝑖=1,2. (5.15) It follows from (5.10) that 𝑉^∗_𝑖1(𝑘)≤1𝑚_𝑖𝑥^∗_𝑖(𝑘)−𝑦^∗_𝑖(𝑘≤1)𝑚_𝑖𝑉𝜀,_∗12(𝑘)=𝜎₁𝑑^𝑢max_𝑠≤𝑘𝑢^∗₁(𝑠)−𝑣^∗₁(𝑠+𝜏₁₂)𝑎^𝑢𝑃^𝑢₁𝐺^𝑢₁+𝑀₁𝑃^𝑢₂𝐸^𝑢₁max_𝑠≤𝑘𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+𝑀₁𝑃^𝑢₂𝑎^𝑢max_𝑠≤𝑘𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+𝑑^𝑢max_𝑠≤𝑘𝑢^∗₁(𝑠)−𝑣^∗₁(𝑠)+𝐸^𝑢₂max_𝑠≤𝑘𝑥^∗₂(𝑠)−𝑦^∗₂(𝑠≤𝜎₁)𝑑^𝑢+(𝜏₁)²𝑎^𝑢𝑃^𝑢₁𝐺^𝑢₁+𝑀₁𝑃^𝑢₂𝐸^𝑢₁+𝑀₁𝑃^𝑢₂𝑎^𝑢+𝑑^𝑢+𝐸^𝑢₂𝑉𝜀,_∗13(𝑘)≤𝜏₁𝑀₁𝑃^𝑢₂𝑎^𝑢𝜏₁𝑎^𝑢max_𝑠≤𝑘𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+𝜎₁𝑑^𝑢max_𝑠≤𝑘𝑢^∗₁(𝑠)−𝑣^∗₁(𝑠)≤𝜏₁𝑀₁𝑃^𝑢₂𝑎^𝑢𝜏₁𝑎^𝑢+𝜎₁𝑑^𝑢𝑉𝜀,_∗22(𝑘)≤𝜏₂𝑀₂𝑚²₁𝑓^𝑢max_𝑠≤𝑘𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+𝜎₂𝑝^𝑢max_𝑠≤𝑘𝑢^∗₂(𝑠)−𝑣^∗₂(𝑠+𝜏₂₂)𝑓^𝑢𝑚₁𝑄^𝑢₁𝐺^𝑢₂max_𝑠≤𝑘𝑥^∗₂(𝑠)−𝑦^∗₂(𝑠)+𝑀₂𝑄^𝑢₂𝑓^𝑢𝑚₁max_𝑠≤𝑘𝑥^∗₂(𝑠)−𝑦^∗₂(𝑠+𝑀)²𝑚²₁𝑓^𝑢max_𝑠≤𝑘𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+𝑝^𝑢max_𝑠≤𝑘𝑢^∗₂(𝑠)−𝑣^∗₂(𝑠,≤𝜏₂)𝑀₂𝑚²₁𝑓^𝑢+𝜎₂𝑝^𝑢+𝜏₂²𝑓^𝑢𝑚₁𝑄^𝑢₁𝐹^𝑢₂+𝑀₂𝑄^𝑢₂𝑓^𝑢𝑚₁+𝑀₂𝑚²₁𝑓^𝑢+𝑝^𝑢𝑉𝜀,_∗23(𝑘)≤𝜏₂𝑓^𝑢𝑚₁𝑀₂𝑄^𝑢₂𝜏₂𝑓^𝑢𝑚₁max_𝑠≤𝑘𝑥^∗₂(𝑠)−𝑦^∗₂(𝑠)+𝜏₂𝑀₂𝑚²₁𝑓^𝑢max_𝑠≤𝑘𝑥^∗₁(𝑠)−𝑦^∗₁(𝑠)+𝜎₂𝑝^𝑢max_𝑠≤𝑘𝑢^∗₂(𝑠)−𝑣^∗₂(𝑠)≤𝜏₂𝑓^𝑢𝑚₁𝑀₂𝑄^𝑢₂𝜏₂𝑓^𝑢𝑚₁+𝜏₂𝑀₂𝑚²₁𝑓^𝑢+𝜎₂𝑝^𝑢𝑉𝜀,_𝑖′∗(𝑘)≤max_𝑠≤𝑘𝑢^∗_𝑖(𝑠)−𝑣^∗_𝑖(𝑠)+𝜌_𝑖𝛽^𝑢_𝑖max_𝑠≤𝑘𝑥^∗_𝑖(𝑠)−𝑦^∗_𝑖(𝑠)≤1+𝜌_𝑖𝛽^𝑢_𝑖𝜀. (5.16)
Let 𝑄=𝑠₁1𝑚₁+𝜎₁𝑑^𝑢+𝜏₁²𝑎^𝑢𝑃^𝑢₁𝐺^𝑢₁+𝑀₁𝑃^𝑢₂𝐸^𝑢₁+𝑀₁𝑃^𝑢₂𝑎^𝑢+𝑑^𝑢+𝐸^𝑢₂+𝜏₁𝑀₁𝑃^𝑢₂𝑎^𝑢𝜏₁𝑎^𝑢+𝜎₁𝑑^𝑢+𝑠₂1𝑚₂+𝜏₂𝑀₂𝑚²₁𝑓^𝑢+𝜎₂𝑝^𝑢+𝜏₂²𝑓^𝑢𝑚₁𝑄^𝑢₁𝐺^𝑢₂+𝑀₂𝑄^𝑢₂𝑓^𝑢𝑚₁+𝑀₂𝑚²₁𝑓^𝑢+𝑝^𝑢+𝜏₂𝑓^𝑢𝑚₁𝑀₂𝑄^𝑢₂𝜏₂𝑓^𝑢𝑚₁+𝜏₂𝑀₂𝑚²₁𝑓^𝑢+𝜎₂𝑝^𝑢+²_𝑖=1𝜔_𝑖1+𝜌_𝑖𝛽^𝑢_𝑖. (5.17)
It follows from (5.10) and the above inequalities that 𝑉^∗(𝑘)≤𝑄𝜀 for 𝑘<−𝐾, hence lim_{𝑘→−∞}𝑉^∗(𝑘)=0. Furthermore, 𝑉^∗(𝑘) is a nonincreasing function on 𝑍, thus 𝑉^∗(𝑘)≡0, that is, 𝑥^∗_𝑖(𝑘)?=?𝑦^∗_𝑖(𝑘), 𝑢^∗_𝑖(𝑘)?=?𝑣^∗_𝑖(𝑘), 𝑖?=?1,2 for all 𝑘∈𝑍. This ends the proof of Step 2.
By the above discussion and Theorem 4.3, we conclude that any hull equation of system (1.4) has a unique strictly positive solution. Then the almost periodic system (1.4) has a unique strictly positive almost periodic solution which is globally attractive. The proof of Theorem 5.2 is completed.

6. Concluding Remarks

In this paper, a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls is considered. By applying the difference inequality, some sufficient conditions are established, which are independent of feedback control variables, to ensure the permanence of system (1.4). By constructing the suitable Lyapunov function, we show that system (1.4) is globally attractive under some appropriate conditions. Furthermore, by using an almost periodic functional hull theory, we show that the almost periodic system (1.4) has a unique strictly positive almost periodic solution which is globally attractive.

We would like to mention here the question of whether the feedback control variables have the influence on the stability property of the system or not. It is, in fact, a very challenging problem, and we leave it for our future work.

Acknowledgments

The authors are grateful to the editor, Wolfgang Ruess, and referees for a number of helpful suggestions that have greatly improved our original submission. This work is supported by the National Natural Science Foundation of China (nos. 71171035, 71173029), China Postdoctoral Science Foundation (no. 2012M511598), and the Teaching and Research Foundation of Dongbei University of Finance and Economics (nos. YY11001, GW11003).