Almost Periodic Sequence Solutions of a Discrete Predator-Prey System with Beddington-DeAngelis Functional Response

Abstract

This paper considers a discrete predator-prey system with Beddington-DeAngelis functional response. Sufficient conditions are obtained for the existence of the almost periodic solution which is uniformly asymptotically stable by constructing a Lyapunov function.

1. Introduction

In the past decades, the predator-prey competition models have been extensively studied by many authors (see [1–7]). Many excellent works have been done for the predator-prey model with functional response, such as Zhu and Wang [8] who considered a Volterra model with modified Leslie-Gower Holling-type II schemes. Cai et al. [9] studied the positive periodic solution for a multispecies competition-predator system with the Holling III functional and time delays. A well-known model of such systems is the predator-prey model with a Beddington-DeAngelis functional response which was originally proposed by Beddington [10] and DeAngelis et al. [11], independently. The dynamics of this model is described by the following differential equations: ̇(𝑡𝑥(𝑡)𝑥=𝑟𝑥)1−𝐾−𝑏𝑥(𝑡)𝑦(𝑡)(𝑡)(𝑡),̇(𝑡(𝑡)1+𝑛𝑥+𝑚𝑦𝑦=𝑦)−𝑑+𝑓𝑏𝑥(𝑡)(𝑡),1+𝑛𝑥+𝑚𝑦 (1) where 𝑥 and 𝑦 represent prey and predator densities, respectively. Usually, the constant 𝑟 is called intrinsic growth rate of the prey; 𝐾 is the carrying capacity of the prey; 𝑏 and 𝑛 are positive constants that describe the effects of capture rate and handling time, respectively, on the feeding rate; 𝑑 is the death rate of the predator; 𝑓 is the birth rate of the predator; 𝑚≥0 (units: 1/predator) is a constant describing the magnitude of interference among predators. Many excellent works have been done for the predator-prey system with a Beddington-DeAngelis functional response, such as Fan and Kuang [12], Liu et al. [13], and Baek [14].

It has been found that discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. There are some existing results on discrete predator-prey systems [15–17]. For example, in [15], by using the method of upper and lower solutions and the degree theory the authors have studied the existence of periodic positive solutions for a competitive system with two parameters. Zhang and Wang [16] studied the following discrete model: 𝑥()(𝑘)𝑎(𝑘)(𝑘)𝑥(𝑘)−𝑐(𝑘)𝑦(𝑘)𝑘+1=𝑥exp−𝑏𝛼(𝑘)(𝑘)𝑥(𝑘)(𝑘)𝑦(𝑘),𝑦()(𝑘)(𝑘)+𝑓(𝑘)𝑦(𝑘)+𝛽+𝛾𝑘+1=𝑦exp−𝑑𝛼(𝑘)(𝑘)𝑥(𝑘)(𝑘)𝑦(𝑘).+𝛽+𝛾 (2) The authors used the Mawhin’s coincidence degree theory to obtain some sufficient conditions for the existence of positive solutions.

But, nowadays, models with almost periodic coefficient have drawn more attention. Li and Chen [18] considered the almost periodic solutions of the following discrete almost periodic logistic equation: 𝑥()(𝑛)𝑟(𝑛)𝑥(𝑛)𝑛+1=𝑥exp1−𝐾(𝑛). (3)

Niu and Chen [19] proposed the following model: 𝑥₁(𝑛+1)=𝑥₁(𝑛)×exp𝑟₁(𝑛)×𝑥1−¹(𝑛)𝐾₁(𝑛)−𝜇₂(𝑛)𝑥₂(𝑛)−𝑏₁(𝑛)𝜇₁(𝑛),𝑥₂(𝑛+1)=𝑥₂(𝑛)×exp𝑟₂(𝑛)×𝑥1−²(𝑛)𝐾₂(𝑛)−𝜇₁(𝑛)𝑥₁(𝑛)−𝑏₂(𝑛)𝜇₂(𝑛),Δ𝜇₁(𝑛)=−𝛼₁(𝑛)𝜇₁(𝑛)+𝛽₁(𝑛)𝑥₁(𝑛),Δ𝜇₂(𝑛)=−𝛼₂(𝑛)𝜇₂(𝑛)+𝛽₂(𝑛)𝑥₂(𝑛). (4) By constructing a suitable Lyapunov function, they obtained the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.

Li et al. [20] studied the following system: 𝑥()(𝑘)𝑎(𝑘)(𝑘)𝑥(𝑘)(𝑘)(𝑘)()𝑦(𝑘)𝑘+1=𝑥×exp−𝑏−𝑝𝑘,𝑥,𝑦,𝑥𝑘−𝜇,𝑦𝑘−𝜈𝑥(𝑘),𝑦()(𝑘)𝑐(𝑘)−𝑑(𝑘)𝑦(𝑘)𝑘+1=𝑦exp𝑥.𝑘−𝜇 (5) With the method of the theory of difference inequality and constructing a suitable Lyapunov function, they obtained the permanence and the almost periodic solution of the system.

Inspired by the above papers, in this paper, we consider the following discrete predator-prey system: 𝑥₁(𝑛+1)=𝑥₁(𝑛)×exp𝑟₁(𝑛𝑥)1−¹(𝑛)𝐾₁(𝑛)−𝑏(𝑛)𝑥₂(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛),𝑥₂(𝑛+1)=𝑥₂(𝑛)×exp−𝑟₂(𝑛)+𝑐(𝑛)𝑥₁(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛), (6) where 𝑥₁(𝑛) and 𝑥₂(𝑛) represent prey and predator densities at time 𝑡.

From the point of view of biology, we assumed that the initial conditions of (6) are of the form 𝑥_𝑖(0)>0,𝑖=1,2. (7) Then, it is easy to see that the solutions of (6) with the initial condition (7) are defined and remain positive for all 𝑛∈𝑍⁺. Throughout this paper, for any bounded sequence {𝑎(𝑛)}, we denote 𝑎^𝑀=sup_𝑛∈𝑁{𝑎(𝑛)} and 𝑎^𝐿=inf_𝑛∈𝑁{𝑎(𝑛)}, and we assume that (H₁){𝑟₁(𝑛)}, {𝑟₂(𝑛)}, {𝐾(𝑛)}, {𝑏(𝑛)}, {𝑐(𝑛)}, {𝛼(𝑛)}, {𝛽(𝑛)}, and {𝛾(𝑛)} are bounded nonnegative almost periodic sequences such that 0<𝑟^𝐿₁≤𝑟₁(𝑛)≤𝑟^𝑀₁,0<𝑟^𝐿₂≤𝑟₂(𝑛)≤𝑟^𝑀₂,0<𝐾^𝐿≤𝐾(𝑛)≤𝐾^𝑀,0<𝛼^𝐿≤𝛼(𝑛)≤𝛼^𝑀,0<𝛽^𝐿≤𝛽(𝑛)≤𝛽^𝑀,0<𝛾^𝐿≤𝛾(𝑛)≤𝛾^𝑀,0<𝑏^𝐿≤𝑏(𝑛)≤𝑏^𝑀,0<𝑐^𝐿≤𝑐(𝑛)≤𝑐^𝑀, (8)(H₂)𝑟^𝐿₁−(𝑏^𝑀/𝛾^𝐿)>0, (H₃)(𝑐^𝐿/𝛽^𝑀)(1−𝛼^𝑀)−𝑟^𝑀₂>0.

The organization of this paper is as follows. In Section 2, we will introduce some definitions and several useful definitions and lemmas. In Section 3, by applying the theory of difference inequality, we get the permanence of system (6). In Section 4, by constructing a suitable Lyapunov function, we obtain the existence of the almost periodic solution for system (6) which is uniformly asymptotically stable. Finally, we give some examples and numerical simulations to verify our results.

2. Preliminaries

In this section, we will introduce some basic definitions and several useful lemmas.

Definition 1. System (6) is said to be permanent if there exist positive constants 𝑀_𝑖, 𝑚_𝑖, which are independent of the solutions of the system, such that any positive solution (𝑥₁(𝑛),𝑥₂(𝑛))^𝐓 of system (6) satisfies 𝑚_𝑖≤lim_𝑛→+∞inf𝑥_𝑖(𝑛)≤lim_𝑛→+∞sup𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2. (9)

Lemma 2 (see [21]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)≤𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)},𝑛∈𝑁, (10) where 𝑎(𝑛) and 𝑏(𝑛) are nonnegative sequences bounded above and below by positive constants. Then, lim_𝑛→+∞(𝑛)≤1sup𝑥𝑏^𝐿exp𝑎^𝑀−1. (11)

Lemma 3 (see [21]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)≥𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)},𝑛≥𝑁₀, (12) and lim_𝑛→+∞sup𝑥(𝑛)≤𝑥^∗, 𝑥(𝑁₀)>0, where 𝑎(𝑛) and 𝑏(𝑛) are nonnegative sequences bounded above and below by positive constants and 𝑁₀∈𝑁. Then, lim_𝑛→+∞(𝑛)𝑎inf𝑥≥min^𝐿𝑏^𝑀exp𝑎^𝑀−𝑏^𝑀𝑥^∗,𝑎^𝐿𝑏^𝑀. (13)

Lemma 4 (see [22]). Let 𝑥(𝑛) and 𝑏(𝑛) be nonnegative sequences defined on 𝑁, and 𝑐≥0 is a constant. If 𝑥(𝑛)≤𝑐+^𝑛−1_𝑠=0𝑏(𝑠)𝑥(𝑠),𝑛∈𝑁, (14) then 𝑥(𝑛)≤𝑐^𝑛−1_𝑠=0[1+𝑏(𝑠)],𝑛∈𝑁. (15)

Definition 5 (see [23]). A sequence 𝑥∶𝑍→𝑅 is called an almost periodic sequence if the 𝜖-translation number set of 𝑥𝐸{𝜖,𝑥}={𝜏∈𝑍∶|𝑥(𝑛+𝜏)−𝑥(𝑛)|<𝜖,∀𝑛∈𝑍} (16) 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} such that |𝑥(𝑛+𝜏)−𝑥(𝑛)|<𝜖,∀𝑛∈𝑍, (17)𝜏 is called the 𝜖-translation number or 𝜖-almost period.

Definition 6 (see [23]). Let 𝑓∶𝑍×𝐷→𝑅, where 𝐷 is an open set in 𝑅; 𝑓(𝑛,𝑥) is said to be almost periodic in 𝑛 uniformly for 𝑥∈𝐷, or uniformly almost periodic for short, if, for any 𝜖>0 and any compact set 𝑆 in 𝐷, there exists a positive integer 𝑙(𝜖,𝑆) such that any interval of length 𝑙(𝜖,𝑆) contains a integer 𝜏 for which 𝑓(𝑛+𝜏,𝑥)−𝑓(𝑛,𝑥)<𝜖, (18) for all 𝑛∈𝑍 and 𝑥∈𝑆. 𝜏 is called the 𝜖-translation number of 𝑓(𝑛,𝑥).

Lemma 7 (see [23]). {𝑥(𝑛)} is an almost periodic sequence if and only if for any sequence {𝑚_𝑖}⊂𝑍 there exists a subsequence {𝑚_𝑖𝑘}⊂{𝑚_𝑖} such that 𝑥(𝑛+𝑚_𝑖𝑘) converges uniformly on 𝑛∈𝑍 as 𝑘→∞. Furthermore, the limit sequence is also an almost periodic sequence.

In [24], Zhang considered the following almost periodic difference system: 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛)),𝑛∈𝑍⁺, (19) where 𝑓∶𝑍×𝑆_𝐵→𝑅, 𝑆_𝐵={𝑥∈𝑅∶‖𝑥‖<𝐵}, and 𝑓(𝑛,𝑥) is almost periodic in 𝑛 uniformly for 𝑥∈𝑆_𝐵 and is continuous in 𝑥. The product system of (19) is the following system: 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛)),𝑦(𝑛+1)=𝑓𝑛,𝑦(𝑛), (20) and the following lemma is obtained.

Lemma 8 (see [24]). Suppose that there exists a Lyapunov functional 𝑉(𝑛,𝑥,𝑦) defined for 𝑛∈𝑍⁺, ‖𝑥‖<𝐵, ‖𝑦‖<𝐵 satisfying the following conditions: [C₁]𝑎(‖𝑥−𝑦‖)≤𝑉(𝑛,𝑥,𝑦)≤𝑏(‖𝑥−𝑦‖), where 𝑎,𝑏∈𝑇 with 𝑇=𝑎∈𝐶𝑅⁺,𝑅⁺∶𝑎(0)=0and𝑎isincreasing, (21)[C₂]|𝑉(𝑛,𝑥₁,𝑦₁)−𝑉(𝑛,𝑥₂,𝑦₂)|≤𝐿(‖𝑥₁−𝑥₂‖+‖𝑦₁−𝑦₂‖), where 𝐿>0 is a constant,[C₃]Δ𝑉₍₂₀₎≤−𝜆𝑉(𝑛,𝑥,𝑦), where 0<𝜆<1 is a constant and Δ𝑉₍₂₀₎=𝑉𝑛+1,𝑓(𝑛,𝑥),𝑓𝑛,𝑦−𝑉𝑛,𝑥,𝑦. (22)

Moreover, if there exists a solution 𝜑(𝑛) of system (20) such that ‖𝜑(𝑛)‖<𝐵^∗<𝐵 for 𝑛∈𝑍⁺, then there exists a unique uniformly asymptotically stable almost periodic solution 𝑝(𝑛) of system (20) which is bounded by 𝐵^∗. In particular, if 𝑓(𝑛,𝑥) is 𝜔-periodic function, then there exists a unique uniformly asymptotically stable 𝜔-periodic solution of (20).

3. Permanence

Theorem 9. Assume that (H₁), (H₂), and (H₃) hold, then every solution (𝑥₁(𝑛),𝑥₂(𝑛))^𝐓 of system (6) satisfies 𝑥_𝑖∗≤lim_𝑛→+∞inf𝑥_𝑖(𝑛)≤lim_𝑛→+∞sup𝑥_𝑖(𝑛)≤𝑥^∗_𝑖,𝑖=1,2. (23) That is, system (6) is permanent, where 𝑥^∗₁=𝐾^𝑀𝑟^𝐿₁exp𝑟^𝑀₁,𝑥−1_1∗=𝐾^𝐿𝑟^𝐿₁𝛾^𝐿−𝑏^𝑀𝛾^𝐿𝑟^𝑀₁exp𝑟^𝐿₁−𝑏^𝑀𝛾^𝐿−𝑟^𝑀₁𝐾^𝐿𝑥^∗₁,𝑥^∗₂2𝑐=exp^𝑀𝛽^𝐿−𝑟^𝐿₂,𝑥_2∗𝑐=min𝑇exp^𝐿1−𝛼^𝑀−𝑟^𝑀₂𝛽^𝑀𝛽^𝑀−𝑐^𝑀𝛾^𝑀𝛽^𝐿𝑥^∗₂𝛽,𝑇𝑇=^𝐿𝑐^𝐿1−𝛼^𝑀−𝛽^𝐿𝛽^𝑀𝑟^𝑀₂𝑐^𝑀𝛾^𝑀𝛽^𝑀. (24)

Proof. Let (𝑥₁(𝑛),𝑥₂(𝑛))^𝐓 be any positive solution of system (6); from the first equation of system (6), it follows that 𝑥₁(𝑛+1)≤𝑥₁(𝑛)exp𝑟₁(𝑛)−𝑟₁(𝑛)𝐾(𝑛)𝑥₁(𝑛). (25) Thus, as a direct corollary of Lemma 2, it follows that lim_𝑛→+∞sup𝑥₁(𝑛)≤𝐾^𝑀𝑟^𝐿₁exp𝑟^𝑀₁−1∶=𝑥^∗₁. (26) On the other hand, from the first equation of system (6), we obtain 𝑥₁(𝑛+1)≥𝑥₁(𝑛)exp𝑟₁(𝑛)−𝑟₁(𝑛)𝐾(𝑛)𝑥₁(𝑛)−𝑏(𝑛)𝑥₂(𝑛)𝛾(𝑛)𝑥₂(𝑛)=𝑥₁(𝑛)exp𝑟₁(𝑛)−𝑏(𝑛)𝛾(𝑛)−𝑟₁(𝑛)𝐾(𝑛)𝑥₁(𝑛). (27) According to Lemma 3 and assumption (H₂), we have lim_𝑛→+∞inf𝑥₁(𝑛)𝐾≥min^𝐿𝑟^𝐿₁𝛾^𝐿−𝑏^𝑀𝛾^𝐿𝑟^𝑀₁×exp𝑟^𝐿₁−𝑏^𝑀𝛾^𝐿−𝑟^𝑀₁𝐾^𝐿𝑥^∗₁,𝐾^𝐿𝑟^𝐿₁𝛾^𝐿−𝑏^𝑀𝛾^𝐿𝑟^𝑀₁. (28) Notice that 𝑟^𝑀₁/𝐾^𝐿𝑟^𝐿₁−𝑏^𝑀/𝛾^𝐿⋅𝑥^∗₁=𝑟^𝑀₁/𝐾^𝐿𝑟^𝐿₁−𝑏^𝑀/𝛾^𝐿⋅𝐾^𝑀𝑟^𝐿₁exp𝑟^𝑀₁=𝑟−1_𝑀1𝑟^𝐿₁⋅𝐾^𝑀𝐾^𝐿⋅exp𝑟^𝑀₁−1𝑟^𝐿₁−𝑏^𝑀/𝛾^𝐿>𝑟^𝑀₁𝑟^𝐿₁⋅𝐾^𝑀𝐾^𝐿⋅𝑟^𝑀₁𝑟^𝐿₁>1. (29) In the above inequality, using the Bernoulli inequality 𝑒^𝑥>1+𝑥, for 𝑥>−1, one has lim_𝑛→+∞inf𝑥₁(𝑛)≥𝐾^𝐿𝑟^𝐿₁𝛾^𝐿−𝑏^𝑀𝛾^𝐿𝑟^𝑀₁exp𝑟^𝐿₁−𝑏^𝑀𝛾^𝐿−𝑟^𝑀₁𝐾^𝐿𝑥^∗₁∶=𝑥_1∗. (30)
From the second equation of system (6), we can obtain 𝑥₂(𝑛+1)≤𝑥₂(𝑛)exp−𝑟₂(𝑛)+𝑐(𝑛)𝛽(𝑛). (31) Let 𝑥₂(𝑛)=exp{𝑦(𝑛)}; then 𝑦()(𝑛)+𝑐(𝑛)𝑛+1≤𝑦𝛽(𝑛)−𝑟₂(𝑛)≤^𝑛_𝑠=0𝑓(𝑠)𝑦(𝑠)+𝑐^𝑀𝛽^𝐿−𝑟^𝐿₂, (32) where 𝑓(𝑠)=0,0≤𝑠≤𝑛−1,1,𝑠=𝑛. (33) Since 𝑐^𝑀𝛽^𝐿−𝑟^𝐿₂≥𝑐^𝐿𝛽^𝑀1−𝛼^𝑀−𝑟^𝑀₂>0, (34) by applying Lemma 4, we have 𝑦()𝑐𝑛+1≤2^𝑀𝛽^𝐿−𝑟^𝐿₂. (35) Thus, lim_𝑛→+∞sup𝑥₂(𝑛)2𝑐≤exp^𝑀𝛽^𝐿−𝑟^𝐿₂∶=𝑥^∗₂. (36)
By the second equation of system (6), we can get that 𝑥₂(𝑛+1)=𝑥₂(𝑛)exp−𝑟₂(𝑛)+𝑐(𝑛)𝑥₁(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛)=𝑥₂(𝑛)exp−𝑟₂(𝑛)(𝑛)×𝑥+𝑐₁(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛)−1𝛽(𝑛)+1𝛽(𝑛)=𝑥₂(𝑛)𝑐(𝑛)exp𝛽(𝑛)−𝑟₂(𝑛)−𝑐(𝑛)𝛽(𝑛)×𝛼(𝑛)+𝛾(𝑛)𝑥₂(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛)=𝑥₂(𝑛)𝑐(𝑛)×exp𝛽(𝑛)×𝛼(𝑛)1−𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛)−𝑟₂(𝑛)−𝑐(𝑛)𝛽(𝑛)×𝛾(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑥₁(𝑛)+𝛾(𝑛)𝑥₂(𝑛)𝑥₂(𝑛)≥𝑥₂(𝑛)𝑐exp^𝐿𝛽^𝑀1−𝛼^𝑀−𝑟^𝑀₂−𝑐^𝑀𝛾^𝑀𝛽^𝐿𝑥₂(𝑛). (37) According to (H₃), applying Lemma 3 to inequality (37), one has lim_𝑛→+∞inf𝑥₂(𝑛)𝑐≥min𝑇exp^𝐿1−𝛼^𝑀𝛽^𝑀−𝑟^𝑀₂−𝑐^𝑀𝛾^𝑀𝛽^𝐿𝑥^∗₂,𝑇∶=𝑥_2∗. (38) The proof of the theorem is completed.

4. Existence of Globally Attractive Almost Periodic Solutions

In this section, we study the existence of a globally attractive almost periodic sequence solution of system (6).

First, we denote by Ω the set of all solutions (𝑥₁(𝑛),𝑥₂(𝑛)) of system (6) satisfying 𝑥_𝑖∗≤𝑥_𝑖(𝑛)≤𝑥^∗_𝑖, 𝑖=1,2 for all 𝑛∈𝑍⁺.

In order to apply Lemma 8, we should prove that there exists a bounded solution of system (6) and then construct an adaptive Lyapunov functional for system (6).

Theorem 10. Assume that (H₁), (H₂), and (H₃) hold. Then Ω≠∅.

Proof. By the almost periodicity of {𝑟₁(𝑛)}, {𝑟₂(𝑛)}, {𝐾(𝑛)}, {𝑏(𝑛)}, {𝑐(𝑛)}, {𝛼(𝑛)}, {𝛽(𝑛)}, and {𝛾(𝑛)}, there exists an integer valued sequence 𝜏_𝑝 with 𝜏_𝑝→∞ as 𝑝→∞ such that 𝑟₁𝑛+𝜏_𝑝⟶𝑟₁(𝑛),𝑟₂𝑛+𝜏_𝑝⟶𝑟₂(𝑛),𝐾𝑛+𝜏_𝑝⟶𝐾(𝑛),𝑏𝑛+𝜏_𝑝(𝑛),𝑐⟶𝑏𝑛+𝜏_𝑝⟶𝑐(𝑛),𝛼𝑛+𝜏_𝑝(𝑛),𝛽⟶𝛼𝑛+𝜏_𝑝⟶𝛽(𝑛),𝛾𝑛+𝜏_𝑝⟶𝛾(𝑛) (39) as 𝑝→∞ uniformly on 𝑍⁺.
Let 𝜖 be an arbitrary small positive number. It follows from Theorem 9 that there exists a positive integer 𝑁₀ such that 𝑥_𝑖∗−𝜖≤𝑥_𝑖(𝑛)≤𝑥^∗_𝑖+𝜖,𝑖=1,2,∀𝑛>𝑁₀. (40) Write 𝑥_𝑖𝑝(𝑛)=𝑥_𝑖(𝑛+𝜏_𝑝), 𝑖=1,2, for 𝑛≥𝑁₀−𝜏_𝑝, 𝑝=1,2,…. For any positive integer 𝑞, it is easy to see that there exist sequence {𝑥_𝑖𝑝∶𝑝≥𝑞,𝑖=1,2} such that the sequence {𝑥_𝑖𝑝} have subsequences, denoted by 𝑥_𝑖𝑝 again, converging on any finite interval of 𝑍 as 𝑝→∞, respectively. Thus we have sequence 𝑦_𝑖(𝑛) such that 𝑥_𝑖𝑝⟶𝑦_𝑖(𝑛),𝑖=1,2for𝑛∈𝑍⁺as𝑝⟶∞. (41) Combined with 𝑥_1𝑝(𝑛+1)=𝑥_1𝑝(𝑛)exp𝑟₁𝑛+𝜏_𝑝𝑥1−^1𝑝(𝑛)𝐾𝑛+𝜏_𝑝−𝑏𝑛+𝜏_𝑝𝑥_2𝑝(𝑛)×𝛼𝑛+𝜏_𝑝+𝛽𝑛+𝜏_𝑝𝑥_1𝑝(𝑛)+𝛾𝑛+𝜏_𝑝𝑥_2𝑝(𝑛)⁻¹,𝑥_2𝑝(𝑛+1)=𝑥_2𝑝(𝑛)exp−𝑟₂𝑛+𝜏_𝑝+𝑐𝑛+𝜏_𝑝𝑥_1𝑝(𝑛)×𝛼𝑛+𝜏_𝑝+𝛽𝑛+𝜏_𝑝𝑥_1𝑝(𝑛)+𝛾𝑛+𝜏_𝑝𝑥_2𝑝(𝑛)⁻¹, (42) it gives 𝑦₁(𝑛+1)=𝑦₁(𝑛)exp𝑟₁(𝑛)𝑦1−¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑦₂(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑦₁(𝑛)+𝛾(𝑛)𝑦₂(𝑛),𝑦₂(𝑛+1)=𝑦₂(𝑛)exp−𝑟₂(𝑛)+𝑐(𝑛)𝑦₁(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑦₁(𝑛)+𝛾(𝑛)𝑦₂(𝑛). (43) We can easily see that (𝑦₁(𝑛),𝑦₂(𝑛)) is a solution of system (6); furthermore, 𝑥_𝑖∗−𝜖≤𝑦_𝑖(𝑛)≤𝑥^∗_𝑖+𝜖,𝑖=1,2,for𝑛∈𝑍⁺. (44) Since 𝜖 is an arbitrary small positive number, it follows that 𝑥_𝑖∗≤𝑦_𝑖(𝑛)≤𝑥^∗_𝑖,𝑖=1,2,for𝑛∈𝑍⁺. (45) The proof of theorem is completed.

In the following, we denote ∀𝑥∈𝑥_1∗,𝑥^∗₁,∀𝑦∈𝑥_2∗,𝑥^∗₂,𝑔(𝑛)(𝑛)(𝑛)𝑔𝑔,𝜂𝑛,𝑥,𝑦=𝛼+𝛽𝑥+𝛾𝑦,𝐻=max𝑛,𝑥,𝑦,ℎ=min𝑛,𝑥,𝑦₁=2𝑟^𝐿₁𝑥_1∗𝐾^𝑀+2𝑟^𝐿₁𝑏^𝐿𝛽^𝐿𝑥_2∗𝑥²_1∗𝐾^𝑀𝐻²−𝑟^𝑀₁𝑏^𝑀𝛼^𝑀𝑥^∗₁𝑥₂^∗2+𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥₁^∗2𝑥₂^∗2𝐾^𝐿ℎ²−𝑏^𝑀2𝛽^𝑀2+𝑐^𝑀2𝛾^𝑀2𝑥₁^∗2𝑥₂^∗2+2𝑐^𝑀𝛼^𝑀𝛾^𝑀𝑥₁^∗2𝑥^∗₂+𝑐^𝑀2𝛼^𝑀𝑥₁^∗2×ℎ⁴⁻¹−𝑐^𝑀𝛼^𝑀𝑥^∗₁+𝑐^𝑀𝛾^𝑀+2𝑏^𝑀𝛽^𝑀𝑥^∗₁𝑥^∗₂ℎ²,𝜂₂=2𝑐^𝐿𝛾^𝐿𝑥_1∗𝑥_2∗𝐻²−𝑟^𝑀₁𝑏^𝑀𝛼^𝑀𝑥^∗₁𝑥₂^∗2+𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥₁^∗2𝑥₂^∗2𝐾^𝐿ℎ²−𝛼^𝑀2𝑏^𝑀2𝑥₂^∗4+2𝑏^𝑀2𝛼^𝑀𝛽^𝑀𝑥^∗₁𝑥₂^∗4+𝑏^𝑀2𝛽^𝑀2𝑥₁^∗2𝑥₂^∗4+𝑐^𝑀𝛾^𝑀𝑥₁^∗2𝑥₂^∗2×ℎ⁴⁻¹−𝑐^𝑀𝛼^𝑀𝑥^∗₁+𝑐^𝑀𝛾^𝑀𝑥^∗₁𝑥^∗₂ℎ²,𝜂=min𝜂₁,𝜂₂. (4)

Theorem 11. Suppose the conditions (H₁), (H₂), and (H₃) are satisfied, and let 0<𝜂<1; then there exists a uniqueness uniformly asymptotically stable almost periodic solution 𝑋=(𝑥₁(𝑛),𝑥₂(𝑛)) of system (6) which is bounded by Ω for all 𝑛∈𝑍⁺.

Proof. Let 𝑥₁(𝑛)=exp{𝑝₁(𝑛)}, 𝑥₂(𝑛)=exp{𝑝₂(𝑛)}. From system (6),we have 𝑝₁(𝑛+1)=𝑝₁(𝑛)+𝑟₁(𝑛)𝑒1−^𝑝1(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛),𝑝₂(𝑛+1)=𝑝₂(𝑛)−𝑟₂(𝑛)+𝑐(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛). (47) From Theorem 11, we know that the system (47) have bounded solution 𝑌(𝑛)=(𝑝₁(𝑛),𝑝₂(𝑛)) satisfying ln𝑥_𝑖∗≤𝑝_𝑖(𝑛)≤ln𝑥^∗_𝑖,𝑖=1,2,𝑛∈𝑍⁺. (48) Hence, |𝑝_𝑖(𝑛)|≤𝐴_𝑖, where 𝐴_𝑖=max{|ln𝑥_𝑖∗|,|ln𝑥^∗_𝑖|}, 𝑖=1,2. For (𝑝₁(𝑛),𝑝₂(𝑛))∈𝑅², we define the norm ‖(𝑝₁(𝑛),𝑝₂(𝑛))‖=|𝑝₁(𝑛)|+|𝑝₂(𝑛)|.
Consider the product system of system (47) 𝑝₁(𝑛+1)=𝑝₁(𝑛)+𝑟₁(𝑛)𝑒1−^𝑝1(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛),𝑝₂(𝑛+1)=𝑝₂(𝑛)−𝑟₂(𝑛)+𝑐(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛),𝑞₁(𝑛+1)=𝑞₁(𝑛)+𝑟₁(𝑛)𝑒1−^𝑞1(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛),𝑞₂(𝑛+1)=𝑞₂(𝑛)−𝑟₂(𝑛)+𝑐(𝑛)𝑒^𝑞1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛). (49) Suppose 𝑋=(𝑝₁(𝑛),𝑝₂(𝑛)), 𝑌=(𝑞₁(𝑛),𝑞₂(𝑛)) are any two solutions of system (49) defined on 𝑍⁺×𝑆^∗×𝑆^∗; then ‖𝑋‖≤𝐴, ‖𝑌‖≤𝐴, where 𝐴=𝐴₁+𝐴₂, 𝑆^∗=𝑝₁(𝑛),𝑝₂(𝑛)∣ln𝑥_𝑖∗≤𝑝_𝑖(𝑛)≤ln𝑥^∗_𝑖,𝑖=1,2,𝑛∈𝑍⁺. (50)
Consider a Lyapunov function defined on 𝑍⁺×𝑆^∗×𝑆^∗ as follows: 𝑉(𝑛,𝑋,𝑌)=²_𝑖=1𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛)². (51) It is easy to see that the norm ‖𝑋−𝑌‖=∑²_𝑖=1|(𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛))| and the norm ‖𝑋−𝑌‖_∗=∑²_𝑖=1(𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛))² are equivalent; that is, there exist two constants 𝐶₁>0, 𝐶₂>0, such that 𝐶₁‖𝑋−𝑌‖≤‖𝑋−𝑌‖_∗≤𝐶₂‖𝑋−𝑌‖; (52) then 𝐶₁‖𝑋−𝑌‖²≤𝑉(𝑛,𝑋,𝑌)≤𝐶₂‖𝑋−𝑌‖². (53)
Let 𝑎∈𝐶(𝑅⁺,𝑅⁺), 𝑎(𝑥)=𝐶²₁𝑥², 𝑏∈𝐶(𝑅⁺,𝑅⁺), and 𝑏(𝑥)=𝐶²₂𝑥²; thus the condition [C₁] in Lemma 8 is satisfied.
In addition, 𝑉()𝑌=𝑛,𝑋,𝑌−𝑉𝑛,𝑋,_2𝑖=1𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛)²−²_𝑖=1𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛)²≤𝑝₁(𝑛)−𝑞₁(𝑛)²−𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)²−𝑝₂(𝑛)−𝑞₂(𝑛)²=𝑝₁(𝑛)−𝑞₁(𝑛)+𝑝₁(𝑛)−𝑞₁(𝑛×𝑝₁)(𝑛)−𝑞₁(𝑛)−𝑝₁(𝑛)+𝑞₁(𝑛+𝑝₂)(𝑛)−𝑞₂(𝑛)+𝑝₂(𝑛)−𝑞₂(𝑛×𝑝₂)(𝑛)−𝑞₂(𝑛)−𝑝₂(𝑛)+𝑞₂(𝑛≤𝑝₁)(𝑛)+𝑞₁(𝑛)+𝑝₁(𝑛)+𝑞₁(𝑛×𝑝₁)(𝑛)−𝑝₁(𝑛)+𝑞₁(𝑛)−𝑞₁(𝑛+𝑝₂)(𝑛)+𝑞₂(𝑛)+𝑝₂(𝑛)+𝑞₂(𝑛×𝑝₂)(𝑛)−𝑝₂(𝑛)+𝑞₂(𝑛)−𝑞₂(𝑛𝑝)≤𝐿₁(𝑛)−𝑝₁(𝑛)+𝑞₁(𝑛)−𝑞₁(𝑛+𝑝₂)(𝑛)−𝑝₂(𝑛)+𝑞₂(𝑛)−𝑞₂(𝑛𝑋+𝑌,)=𝐿𝑋−𝑌− (54) where 𝐿=4max{𝐴₁,𝐴₂}. Hence, the condition [C₂] of Lemma 8 is satisfied.
Finally, calculate the Δ𝑉 of 𝑉(𝑛) along the solutions of (49); we can obtain Δ𝑉₍₄₉₎()()()=𝑛,𝑋,𝑌=𝑉𝑛+1,𝑋,𝑌−𝑉𝑛,𝑋,𝑌_2𝑖=1𝑝_𝑖(𝑛+1)−𝑞_𝑖(𝑛+1)²−²_𝑖=1𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛)²=𝑝₁(𝑛+1)−𝑞₁(𝑛+1)²−𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛+1)−𝑞₂(𝑛+1)²−𝑝₂(𝑛)−𝑞₂(𝑛)². (55) In the view of (49), we get 𝑝₁(𝑛+1)−𝑞₁(𝑛+1)²−𝑝₁(𝑛)−𝑞₁(𝑛)²=𝑝₁(𝑛)−𝑞₁(𝑛)−𝑟₁(𝑛)𝐾(𝑛)𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)(𝑛)𝑒−𝑏^𝑝2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)2−𝑝₁(𝑛)−𝑞₁(𝑛)²=𝑝₁(𝑛)−𝑞₁(𝑛)−𝑟₁(𝑛)𝐾(𝑛)𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)(𝑛)𝑒−𝑏^𝑝2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)−𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)+𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)−𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)2−𝑝₁(𝑛)−𝑞₁(𝑛)²=𝑝₁(𝑛)−𝑞₁(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)−𝑟₁(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)2−𝑝₁(𝑛)−𝑞₁(𝑛)²=𝑏²(𝑛)𝑒^2𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)2𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)2+𝑟₁(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)2×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)2𝑏(𝑛)𝑒−2^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)𝑟−2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)𝑟+2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛),𝑝₂(𝑛+1)−𝑞₂(𝑛+1)²−𝑝₂(𝑛)−𝑞₂(𝑛)²=𝑝₂(𝑛)−𝑞₂(𝑛)(𝑛)×𝑒+𝑐_𝑝1(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)−𝑒^𝑞1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)2−𝑝₂(𝑛)−𝑞₂(𝑛)²=𝑝₂(𝑛)−𝑞₂(𝑛(𝑛)𝑒)+𝑐^𝑝1(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)−𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)+𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)−𝑒^𝑞1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)2−𝑝₂(𝑛)−𝑞₂(𝑛)²=𝑝₂(𝑛)−𝑞₂(𝑛)+𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)−𝑐(𝑛)𝛾(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)2−𝑝₂(𝑛)−𝑞₂(𝑛)²=𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)2𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)2+𝑐²(𝑛)𝛾²(𝑛)𝑒^2𝑝1(𝑛)𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)2𝑐(𝑛)𝛼(𝑛)(𝑛)𝛾(𝑛)𝑒+2+𝑐^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒−2^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)𝑐(𝑛)𝛼(𝑛)(𝑛)𝛾(𝑛)𝑒−2+𝑐^𝑝2(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒^𝑝1(𝑛)𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛). (56) Substituting (56) into (55), we have Δ𝑉₍₄₉₎()()()=𝑟𝑛,𝑋,𝑌=𝑉𝑛+1,𝑋,𝑌−𝑉𝑛,𝑋,𝑌₁(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)2+𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)2𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)2+𝑏²(𝑛)𝑒^2𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)2+𝑐²(𝑛)𝛾²(𝑛)𝑒^2𝑝1(𝑛)×𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^{𝑝2(𝑛)−1}×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)2𝑟+2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)−𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒^𝑝1(𝑛)𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)𝑏(𝑛)𝑒−2^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)𝑟−2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒−2^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)𝑐(𝑛)𝛼(𝑛)(𝑛)𝛾(𝑛)𝑒+2+𝑐^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛). (57) Using the mean value theorem, we get 𝑒^{𝑝𝑖(𝑛)}−𝑒^{𝑞𝑖(𝑛)}=𝜉_𝑖(𝑛)𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛),𝑖=1,2, (58) where 𝜉_𝑖(𝑛) lies between 𝑒^{𝑝𝑖(𝑛)} and 𝑒^{𝑞𝑖(𝑛)}, 𝑖=1,2. From (57) and (58), we have Δ𝑉₍₄₉₎()()()=𝑟𝑛,𝑋,𝑌=𝑉𝑛+1,𝑋,𝑌−𝑉𝑛,𝑋,𝑌₁(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)2𝜉²₁(𝑛)+𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)2𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉²₁(𝑛)×𝑟−2₁(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝜉₁(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑏²(𝑛)𝑒^2𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)2+𝑐²(𝑛)𝛾²(𝑛)𝑒^2𝑝1(𝑛)×𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^{𝑝2(𝑛)−1}𝜉²₂(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒−2^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₂×𝑝₂(𝑛)−𝑞₂(𝑛)²𝑟+2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₁(𝑛)𝜉₂(𝑛)−𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒^𝑝1(𝑛)𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝜉₁(𝑛)𝜉₂(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₂(𝑛)+𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₁(𝑛×𝑝₁)(𝑛)−𝑞₁(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛), (59) and we get Δ𝑉₍₄₉₎=𝑉₁+𝑉₂+𝑉₃, (60) where 𝑉₁(𝑛)=𝑟₁(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)2𝜉²₁(𝑛)+𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)2𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉²₁(𝑛)𝑟−2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝜉₁(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)²≤𝑏^𝑀2𝛽^𝑀2+𝑐^𝑀2𝛾^𝑀2𝑥₁^∗2𝑥₂^∗2+𝑐^𝑀2𝛼^𝑀𝑥₁^∗2+2𝑐^𝑀𝛼^𝑀𝛾^𝑀𝑥^∗₂𝑥₁^∗2×ℎ⁴⁻¹+2𝑏^𝑀𝛽^𝑀𝑥^∗₂𝑥^∗₁ℎ²−2𝑟^𝐿₁𝑏^𝐿𝛽^𝐿𝑥_2∗𝑥²_1∗𝐾^𝑀𝐻²+𝑟^𝑀₁²𝑥₁^∗2𝐾^𝐿2−2𝑟^𝐿₁𝑥_1∗𝐾^𝑀×𝑝₁(𝑛)−𝑞₁(𝑛)²,𝑉₂(𝑛)=𝑏²(𝑛)𝑒^2𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)2+𝑐²(𝑛)𝛾²(𝑛)𝑒^2𝑝1(𝑛)×𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^{𝑝2(𝑛)−1}𝜉²₂(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒−2^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₂×𝑝₂(𝑛)−𝑞₂(𝑛)²≤𝛼^𝑀2𝑏^𝑀2𝑥₂^∗4+2𝑏^𝑀2𝛼^𝑀𝛽^𝑀𝑥^∗₁𝑥₂^∗4+𝑏^𝑀2𝛽^𝑀2𝑥₁^∗2𝑥₂^∗4+𝑐^𝑀𝛾^𝑀𝑥₁^∗2𝑥₂^∗2×ℎ⁴⁻¹−2𝑐^𝐿𝛾^𝐿𝑥_1∗𝑥_2∗𝐻²×𝑝₂(𝑛)−𝑞₂(𝑛)²,𝑉₃(𝑛)𝑟=2¹(𝑛)𝐾(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛽(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₁(𝑛)𝜉₂(𝑛)−𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)𝑐(𝑛)𝛾(𝑛)𝑒^𝑝1(𝑛)𝑔²𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔²𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)×𝜉₁(𝑛)𝜉₂(𝑛)−𝑏(𝑛)𝑒^𝑝2(𝑛)𝛼(𝑛)+𝛽(𝑛)𝑒^𝑝1(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₂(𝑛)+𝑐(𝑛)𝛼(𝑛)+𝑐(𝑛)𝛾(𝑛)𝑒^𝑝2(𝑛)𝑔𝑛,𝑒^𝑞1(𝑛),𝑒^𝑞2(𝑛)𝑔𝑛,𝑒^𝑝1(𝑛),𝑒^𝑝2(𝑛)𝜉₁(𝑛×𝑝₁)(𝑛)−𝑞₁(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛𝑐)≤2^𝑀𝛼^𝑀𝑥^∗₁+𝑐^𝑀𝛾^𝑀𝑥^∗₁𝑥^∗₂ℎ²+𝑟^𝑀₁𝑏^𝑀𝛼^𝑀𝑥^∗₁𝑥₂^∗2+𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥₁^∗2𝑥₂^∗2𝐾^𝐿ℎ²×𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)². (61) Hence, Δ𝑉₍₄₉₎≤𝑏^𝑀2𝛽^𝑀2+𝑐^𝑀2𝛾^𝑀2𝑥₁^∗2𝑥₂^∗2+𝑐^𝑀2𝛼^𝑀𝑥₁^∗2+2𝑐^𝑀𝛼^𝑀𝛾^𝑀𝑥^∗₂𝑥₁^∗2×ℎ⁴⁻¹+2𝑏^𝑀𝛽^𝑀𝑥^∗₂𝑥^∗₁ℎ²−2𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥^∗₂𝑥₁^∗2𝐾^𝐿ℎ²+𝑟^𝑀₁²𝑥₁^∗2𝐾^𝐿2−2𝑟^𝐿₁𝑥_1∗𝐾^𝑀𝑝₁(𝑛)−𝑞₁(𝑛)²+𝛼^𝑀2𝑏^𝑀2𝑥₂^∗3+2𝑏^𝑀2𝛼^𝑀𝛽^𝑀𝑥^∗₁𝑥₂^∗3+𝑏^𝑀2𝛽^𝑀2𝑥₁^∗2𝑥₂^∗4+𝑐^𝑀𝛾^𝑀𝑥^∗₁𝑥₂^∗2×ℎ⁴⁻¹−2𝑐^𝐿𝛾^𝐿𝑥_1∗𝑥_2∗𝐻²×𝑝₂(𝑛)−𝑞₂(𝑛)²+𝑐^𝑀𝛼^𝑀𝑥^∗₁+𝑐^𝑀𝛾^𝑀𝑥^∗₁𝑥^∗₂ℎ²+𝑟^𝑀₁𝑏^𝑀𝛼^𝑀𝑥^∗₁𝑥₂^∗2+𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥₁^∗2𝑥₂^∗2𝐾^𝐿ℎ²×𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)²≤−2𝑟^𝐿₁𝑥_1∗𝐾^𝑀+2𝑟^𝐿₁𝑏^𝐿𝛽^𝐿𝑥_2∗𝑥²_1∗𝐾^𝑀𝐻²−𝑟^𝑀₁𝑏^𝑀𝛼^𝑀𝑥^∗₁𝑥₂^∗2+𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥₁^∗2𝑥₂^∗2𝐾^𝐿ℎ²−𝑏^𝑀2𝛽^𝑀2+𝑐^𝑀2𝛾^𝑀2𝑥₁^∗2𝑥₂^∗2+2𝑐^𝑀𝛼^𝑀𝛾^𝑀𝑥₁^∗2𝑥^∗₂+𝑐^𝑀2𝛼^𝑀𝑥₁^∗2×ℎ⁴⁻¹−𝑐^𝑀𝛼^𝑀𝑥^∗₁+𝑐^𝑀𝛾^𝑀+2𝑏^𝑀𝛽^𝑀𝑥^∗₁𝑥^∗₂ℎ²×𝑝₁(𝑛)−𝑞₁(𝑛)²−2𝑐^𝐿𝛾^𝐿𝑥_1∗𝑥_2∗𝐻²−𝑟^𝑀₁𝑏^𝑀𝛼^𝑀𝑥^∗₁𝑥₂^∗2+𝑟^𝑀₁𝑏^𝑀𝛽^𝑀𝑥₁^∗2𝑥₂^∗2𝐾^𝐿ℎ²−𝛼^𝑀2𝑏^𝑀2𝑥₂^∗4+2𝑏^𝑀2𝛼^𝑀𝛽^𝑀𝑥^∗₁𝑥₂^∗4+𝑏^𝑀2𝛽^𝑀2𝑥₁^∗2𝑥₂^∗4+𝑐^𝑀𝛾^𝑀𝑥₁^∗2𝑥₂^∗2×ℎ⁴⁻¹−𝑐^𝑀𝛼^𝑀𝑥^∗₁+𝑐^𝑀𝛾^𝑀𝑥^∗₁𝑥^∗₂ℎ²𝑝₂(𝑛)−𝑞₂(𝑛)²≤−𝜂₁𝑝₁(𝑛)−𝑞₁(𝑛)²−𝜂₂𝑝₂(𝑛)−𝑞₂(𝑛)²≤−𝜂𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)²=−𝜂𝑉(𝑛,𝑋,𝑌), (62) where 𝜂=min{𝜂₁,𝜂₂}. That is, there exists a positive constant 0<𝜂<1 such that Δ𝑉₍₄₉₎≤−𝜂𝑉(𝑛,𝑋,𝑌). (63) From 0<𝜂<1, the condition [C₃] of Lemma 8 is satisfied. So, from Lemma 8, there exists a uniqueness uniformly asymptotically stable almost periodic solution 𝑋(𝑛)=(𝑝₁(𝑛),𝑝₂(𝑛)) of (47) which is bounded by 𝑆^∗ for all 𝑛∈𝑍⁺. Which means that there exists a uniqueness uniformly asymptotically stable almost periodic solution 𝑋(𝑛)=(𝑥₁(𝑛),𝑥₂(𝑛)) of (6) which is bounded by Ω for all 𝑛∈𝑍⁺. The proof is completed.

5. Application

Example 12. Consider the following discrete model: 𝑥₁(𝑛+1)=𝑥₁(𝑛)×exp(1.1+0.4sin(𝑛))1−𝑥₁(𝑛)−(0.3−0.2cos(𝑛))𝑥₂(𝑛)×(𝑛)+((𝑛𝑥0.3−0.1sin0.6+0.1cos))₁(𝑛)+𝑥₂(𝑛)⁻¹,𝑥₂(𝑛+1)=𝑥₂(𝑛)−((𝑛+((𝑛𝑥exp0.8+0.2sin))1.5−0.3sin))₁(𝑛)×(𝑛)+((𝑛𝑥0.3−0.1sin0.6+0.1cos))₁(𝑛)+𝑥₂(𝑛)⁻¹, (64) with initial conditions (0.9,0.6); then system (64) is persistent and has a globally attractive almost periodic solution.

By simple computation, we derive 𝑟^𝐿₁−𝑏^𝑀𝛾^𝐿𝑐=0.2>0,^𝐿𝛽^𝑀1−𝛼^𝑀−𝑟^𝑀₂≈0.2857>0. (65) Thus, the system (64) is persistent. Its integral curves and orbits are shown in Figure 1, respectively.

Figure 1 

The orbit of predator (𝑥2)-prey (𝑥1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their sincere thanks to the anonymous reviewers for their valuable comments and suggestions for improving the quality of the paper. This work is supported by the Natural Science Foundation of China (Grant nos. 11071205 and 11101349), PAPD of Jiangsu Higher Education Institutions, and Postgraduate training project of Jiangsu Normal University and Jiangsu Province.