Uniformly Asymptotic Stability of Positive Almost Periodic Solutions for a Discrete Competitive System

Abstract

This paper is devoted to the study of almost periodic solutions of a discrete two-species competitive system. With the help of the methods of the Lyapunov function, some analysis techniques, and preliminary lemmas, we establish a criterion for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the system. Numerical simulations are presented to illustrate the analytical results.

1. Introduction

In recent years, many works have been done for the difference system (see [1–14] and the references cited therein) since the discrete time models governed by the difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world. In particular, Qin et al. [1] introduced the following discrete Lotka-Volterra competitive system:𝑥₁(𝑛+1)=𝑥₁(𝑛)exp𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁(𝑛)−𝑐₂(𝑛)𝑥₂(𝑛)1+𝑥₂(𝑛),𝑥₂(𝑛+1)=𝑥₂(𝑛)exp𝑟₂(𝑛)−𝑎₂(𝑛)𝑥₂(𝑛)−𝑐₁(𝑛)𝑥₁(𝑛)1+𝑥₁(𝑛),𝑛=0,1,2…, (1) where 𝑥_𝑖(0)>0, 𝑥_𝑖(𝑛) stand for the densities of species 𝑥_𝑖 at the 𝑛th generation, 𝑟_𝑖(𝑛) represent the natural growth rates of species 𝑥_𝑖 at the 𝑛th generation, 𝑎_𝑖(𝑛) are the intraspecific effects of the 𝑛th generation of species 𝑥_𝑖 on own population, and 𝑐_𝑖(𝑛) measure the interspecific effects of the 𝑛th generation of species 𝑥_𝑖 on species 𝑥_𝑗(𝑖,𝑗=1,2;𝑖≠𝑗). They investigated the permanence and global asymptotic stability of positive periodic solutions of system (1).

Notice that the investigation of almost periodic solutions for difference equations is one of most important topics in the qualitative theory of difference equations due to the applications in biology, ecology, neural network, and so forth (see [10–14] in detail), and few work has been done previously on an almost periodic version which is corresponding to periodic system (1). In this paper, we will further investigate the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the above almost periodic version. To this end, we assume that the coefficients of system (1) {𝑟_𝑖(𝑛)},{𝑎_𝑖(𝑛)} and {𝑐_𝑖(𝑛)} are bounded nonnegative almost periodic sequences.

For the sake of simplicity and convenience in the following discussion, the notations below will be used throughout this paper: 𝑓^𝑈=sup_𝑛∈ℤ+𝑓(𝑛),𝑓^𝐿=inf_𝑛∈ℤ+𝑓(𝑛), (2) where {𝑓(𝑛)} is a bounded sequence and ℤ⁺={0,1,2,…}.

The remaining part of this paper is organized as follows. In the next section, we introduce some notations, definitions, and lemmas which are available for our main results. In Section 3, sufficient conditions for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of system (1) are given. Numerical simulations are carried out to substantiate the above analytical results in Section 4. Finally, we give some proofs of theorems in the appendices for convenience in reading this paper.

2. Preliminaries

In this section, we will need some preparations and give some notations, definitions, and lemmas which will be useful for our main results.

Denote by ℝ, ℝ⁺, ℤ, and ℤ⁺ the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. ℝ² and ℝ^𝑘 denote the cone of 2-dimensional and 𝑘-dimensional real Euclidean space, respectively.

Definition 1 (see [13]). A sequence 𝑥∶ℤ→ℝ^𝑘 is called an almost periodic sequence if the following 𝜀-translation set of 𝑥𝐸{𝜀,𝑥}∶={𝜏∈ℤ∶|𝑥(𝑛+𝜏)−𝑥(𝑛)|<𝜀,∀𝑛∈ℤ} (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 discrete interval of length 𝑙(𝜀) contains a 𝜏=𝜏(𝜀)∈𝐸{𝜀,𝑥} such that |𝑥(𝑛+𝜏)−𝑥(𝑛)|<𝜀,∀𝑛∈ℤ. (4)𝜏 is called the 𝜀-translation number of 𝑥(𝑛).

Definition 2 (see [13]). Let 𝑓∶ℤ×𝔻→ℝ^𝑘, where 𝔻 is an open set in ℝ^𝑘. 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 an integer 𝜏 for which 𝑓(𝑛+𝜏,𝑥)−𝑓(𝑛,𝑥)<𝜀, (5) for all 𝑛∈ℤ and all 𝑥∈𝕊. 𝜏 is called the 𝜀-translation number of 𝑓(𝑛,𝑥).

Lemma 3 (see [13]). {𝑥(𝑛)} is an almost periodic sequence if and only if for any sequence {ℎ^′_𝑘}⊂ℤ there exists a subsequence {ℎ_𝑘}⊂{ℎ^′_𝑘} such that 𝑥(𝑛+ℎ_𝑘) converges uniformly on 𝑛∈ℤ as 𝑘→∞. Furthermore, the limit sequence is also an almost periodic sequence.

Consider the following almost periodic difference system: 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛)),𝑛∈ℤ⁺, (6) where 𝑓∶ℤ⁺×𝕊_𝐵→ℝ^𝑘, 𝕊_𝐵={𝑥∈ℝ^𝑘∶‖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 (6) is the following system: 𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛)),𝑦(𝑛+1)=𝑓𝑛,𝑦(𝑛), (7) and Zhang [14] obtained the following lemma.

Lemma 4 (see [14]). Suppose that there exists a Lyapunov function 𝑉(𝑛,𝑥,𝑦) defined for 𝑛∈ℤ⁺, ‖𝑥‖<𝐵, ‖𝑦‖<𝐵 satisfying the following conditions:(i)𝑎(‖𝑥−𝑦‖)≤𝑉(𝑛,𝑥,𝑦)≤𝑏(‖𝑥−𝑦‖), where 𝑎,𝑏∈𝐾 with 𝐾={𝑎∈𝐶(ℝ⁺,ℝ⁺)∶𝑎(0)=0 and 𝑎 is increasing};(ii)|𝑉(𝑛,𝑥₁,𝑦₁)−𝑉(𝑛,𝑥₂,𝑦₂)|≤𝐿(‖𝑥₁−𝑥₂‖+‖𝑦₁−𝑦₂‖), where 𝐿>0 is a constant;(iii)Δ𝑉_(2.2)(𝑛,𝑥,𝑦)≤−𝛽𝑉(𝑛,𝑥,𝑦), where 0<𝛽<1 is a constant and Δ𝑉_(2.2)𝑛,𝑥,𝑦=𝑉𝑛+1,𝑓(𝑛,𝑥),𝑓𝑛,𝑦−𝑉𝑛,𝑥,𝑦. (8)Moreover, if there exists a solution 𝜑(𝑛) of system (6) such that ‖𝜑(𝑛)‖≤𝐵^∗<𝐵 for 𝑛∈ℤ⁺, then there exists a unique uniformly asymptotically stable almost periodic solution 𝑝(𝑛) of system (6) which satisfies ‖𝑝(𝑛)‖≤𝐵^∗. In particular, if 𝑓(𝑛,𝑥) is periodic of period 𝜔, then there exists a unique uniformly asymptotically stable periodic solution of system (6) of periodic 𝜔.

Lemma 5 (see [1]). Any positive solution (𝑥₁(𝑛),𝑥₂(𝑛)) of system (1) satisfies limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀=_𝑖^defexp𝑟^𝑈_𝑖−1𝑎^𝐿_𝑖,𝑖=1,2. (9)

Lemma 6 (see [1]). Suppose that system (1) satisfies the following assumptions: 𝑟^𝐿₁>𝑐^𝑈₂,𝑟^𝐿₂>𝑐^𝑈₁. (10) Then, any positive solution (𝑥₁(𝑛),𝑥₂(𝑛)) of system (1) satisfies liminf_𝑛→+∞𝑥_𝑖(𝑛)≥𝑚=_𝑖^def𝑟^𝐿_𝑖−𝑐^𝑈_𝑗𝑎^𝑈_𝑖exp𝑟^𝐿_𝑖−𝑎^𝑈_𝑖𝑀_𝑖−𝑐^𝑈_𝑗,𝑖≠𝑗;𝑖,𝑗=1,2. (11)

3. Main Result

From (9) and (11), we denote by Ω the set of all solutions (𝑥₁(𝑛),𝑥₂(𝑛)) of system (1) satisfying 𝑚_𝑖≤𝑥_𝑖(𝑛)≤𝑀_𝑖, 𝑖=1,2 for all 𝑛∈ℤ⁺. According to Lemma 4, we first prove that there is a bounded solution of system (1), and then structure a suitable Lyapunov function for system (1).

Theorem 7. If the assumptions in (10) hold, then Ω≠Φ.

The proof of Theorem 7 is given in Appendix A.

Theorem 8. If the assumptions in (10) are satisfied, furthermore, 0<𝛽<1, where 𝛽=min{𝑠₁,𝑠₂}, and 𝑠₁=2𝑎^𝐿₁𝑚₁−𝑎₁^𝑈2𝑀²₁−1+𝑎^𝑈₁𝑀₁𝑐^𝑈₂𝑀₂1+𝑚₂²−1+𝑎^𝑈₂𝑀₂𝑐^𝑈₁𝑀₁1+𝑚₁²−𝑐₁^𝑈2𝑀²₁1+𝑚₁⁴,𝑠₂=2𝑎^𝐿₂𝑚₂−𝑎₂^𝑈2𝑀²₂−1+𝑎^𝑈₂𝑀₂𝑐^𝑈₁𝑀₁1+𝑚₁²−1+𝑎^𝑈₁𝑀₁𝑐^𝑈₂𝑀₂1+𝑚₂²−𝑐₂^𝑈2𝑀²₂1+𝑚₂⁴, (12) then there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by Ω for all 𝑛∈ℤ⁺.

The proof of Theorem 8 is given in Appendix B.

4. Numerical Simulations

In this section, we give the following example to check the feasibility of the assumptions of Theorem 8.

Example 9. Consider the following discrete system: 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp1.20−0.02sin√−√2𝑛𝜋1.05+0.01sin2𝑛𝜋𝑥₁(𝑛)−0.025+0.002cos√2𝑛𝜋𝑥₂(𝑛)1+𝑥₂(𝑛),𝑥₂(𝑛+1)=𝑥₂(𝑛)exp1.15−0.02cos√−√2𝑛𝜋1.02+0.02cos2𝑛𝜋𝑥₂(𝑛)−0.035+0.005sin√3𝑛𝜋𝑥₁(𝑛)1+𝑥₁(𝑛). (13) A computation shows that 𝑟^𝐿₁−𝑐^𝑈₂=1.1530>0,𝑟^𝐿₂−𝑐^𝑈₁=1.0900>0, (14) and moreover, we have 𝑠₁=0.3515,𝑠₂=0.2502, (15) that is, 0<𝛽=min{𝑠₁,𝑠₂}=0.2502<1. It is easy to see that the assumptions of Theorem 8 are satisfied. Hence, in system (13) there exists a unique uniformly asymptotically stable positive almost periodic solution. From Figure 1, it is easy to see that there exists a positive almost periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), and the 2-dimensional and 3-dimensional phase portraits of almost periodic system (13) are revealed in Figure 2, respectively. Figure 3 shows that any positive solution (𝑥₁(𝑛),𝑥₂(𝑛)) tends to the almost periodic solution (𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)).

(a)

(b)

(c)

(d)

Figure 1 

Positive almost periodic solution of system (13). (a), (c) Time-series 𝑥∗1(𝑛) and 𝑥∗2(𝑛) with initial values 𝑥∗1(0)=1.19, 𝑥∗2(0)=1.18 for 𝑛∈[0,100], respectively. (b), (d) Time-series 𝑥∗1(𝑛) and 𝑥∗2(𝑛) with the above initial values for 𝑛∈[700,800], respectively.

(a)

(b)

(c)

(d)

Figure 2 

Phase portrait. (a), (b) 2-dimensional phase portrait of almost periodic system (13). Time-series 𝑥∗1(𝑛) and 𝑥∗2(𝑛) with initial values 𝑥∗1(0)=1.19, 𝑥∗2(0)=1.18 for 𝑛∈[0,100] and 𝑛∈[700,800], respectively. (c), (d) 3-dimensional phase portrait of almost periodic system (13). Time-series 𝑥∗1(𝑛) and 𝑥∗2(𝑛) with the above initial values for 𝑛∈[0,100] and 𝑛∈[700,800], respectively.

(a)

(b)

(c)

(d)

Figure 3 

Uniformly asymptotic stability. (a), (c) Time-series 𝑥∗1(𝑛) and 𝑥∗2(𝑛) with initial values 𝑥∗1(0)=1.19, 𝑥∗2(0)=1.18 and 𝑥1(𝑛) and 𝑥2(𝑛) with initial values 𝑥1(0)=1.06, 𝑥2(0)=1.03 for 𝑛∈[0,100], respectively. (b), (d) Time-series 𝑥∗1(𝑛), 𝑥∗2(𝑛), 𝑥1(𝑛), and 𝑥2(𝑛) with the above initial values for 𝑛∈[700,800], respectively.

Appendices

A. Proof of Theorem 7

Clearly, by an inductive argument we have from system (1) that 𝑥₁(𝑛)=𝑥₁(0)exp^𝑛−1_𝑙=0𝑟₁(𝑙)−𝑎₁(𝑙)𝑥₁(𝑙)−𝑐₂(𝑙)𝑥₂(𝑙)1+𝑥₂(𝑙),𝑥₂(𝑛)=𝑥₂(0)exp^𝑛−1_𝑙=0𝑟₂(𝑙)−𝑎₂(𝑙)𝑥₂(𝑙)−𝑐₁(𝑙)𝑥₁(𝑙)1+𝑥₁(𝑙). (A.1) According to Lemmas 5 and 6, for any solution (𝑥₁(𝑛),𝑥₂(𝑛)) of system (1) and an arbitrarily small constant 𝜀>0, there exists 𝑛₀ sufficiently large such that 𝑚₁−𝜀≤𝑥₁(𝑛)≤𝑀₁+𝜀,𝑚₂−𝜀≤𝑥₂(𝑛)≤𝑀₂+𝜀,∀𝑛≥𝑛₀. (A.2) Set {𝜏_𝑘} be any positive integer sequence such that 𝜏_𝑘→+∞ as 𝑘→+∞, we can show that there exists a subsequence of {𝜏_𝑘} still denoted by {𝜏_𝑘}, such that 𝑥_𝑖(𝑛+𝜏_𝑘)→𝑥^∗_𝑖(𝑛), 𝑖=1,2 uniformly in 𝑛 on any finite subset 𝐶 of ℤ⁺ as 𝑘→+∞, where 𝐶={𝑎₁,𝑎₂⋯𝑎_𝑚}, 𝑎_ℎ∈ℤ⁺(ℎ=1,2⋯𝑚), and 𝑚 is a finite number.

As a matter of fact, for any finite subset 𝐶⊂ℤ⁺, 𝜏_𝑘+𝑎_ℎ>𝑛₀, ℎ=1,2⋯𝑚, when 𝑘 is large enough. Therefore, 𝑚_𝑖−𝜀≤𝑥_𝑖(𝑛+𝜏_𝑘)≤𝑀_𝑖+𝜀, 𝑖=1,2; that is, {𝑥_𝑖(𝑛+𝜏_𝑘)} are uniformly bounded for 𝑘 large enough.

Now, for 𝑎₁∈𝐶, we can choose a subsequence {𝜏_𝑘⁽¹⁾} of {𝜏_𝑘} such that {𝑥₁(𝑎₁+𝜏_𝑘⁽¹⁾)} and {𝑥₂(𝑎₁+𝜏_𝑘⁽¹⁾)} uniformly converge on ℤ⁺ for 𝑘 large enough.

Analogously, for 𝑎₂∈𝐶, we can also choose a subsequence {𝜏_𝑘⁽²⁾} of {𝜏_𝑘⁽¹⁾} such that {𝑥₁(𝑎₂+𝜏_𝑘⁽²⁾)} and {𝑥₂(𝑎₂+𝜏_𝑘⁽²⁾)} uniformly converge on ℤ⁺ for 𝑘 large enough.

Repeating the above process, for 𝑎_𝑚∈𝐶, we get a subsequence {𝜏_𝑘^(𝑚)} of {𝜏_𝑘^(𝑚−1)} such that {𝑥₁(𝑎_𝑚+𝜏_𝑘^(𝑚))} and {𝑥₂(𝑎_𝑚+𝜏_𝑘^(𝑚))} uniformly converge on ℤ⁺ for 𝑘 large enough.

Now, we choose the sequence {𝜏_𝑘^(𝑚)} which is a subsequence of {𝜏_𝑘} denoted by {𝜏_𝑘}; then, for all 𝑛∈𝐶, we obtain that 𝑥_𝑖(𝑛+𝜏_𝑘)→𝑥^∗_𝑖(𝑛), 𝑖=1,2 uniformly in 𝑛∈𝐶 as 𝑘→+∞. Hence, the conclusion is valid by the arbitrary of 𝐶.

Recall the almost periodicity of {𝑟_𝑖(𝑛)},{𝑎_𝑖(𝑛)} and {𝑐_𝑖(𝑛)}, 𝑖=1,2, for the above sequence {𝜏_𝑘}, 𝜏_𝑘→+∞ as 𝑘→+∞, there exists a subsequence denoted by {𝜏_𝑘} such that 𝑟_𝑖𝑛+𝜏_𝑘⟶𝑟_𝑖(𝑛),𝑎_𝑖𝑛+𝜏_𝑘⟶𝑎_𝑖(𝑛),𝑐_𝑖𝑛+𝜏_𝑘⟶𝑐_𝑖(𝑛), (A.3) as 𝑘→+∞ uniformly on ℤ⁺.

For any 𝛼∈ℤ⁺, we can assume that 𝜏_𝑘+𝛼≥𝑛₀ for 𝑘 large enough. Let 𝑛∈ℤ⁺, by an inductive argument of system (1) from 𝜏_𝑘+𝛼 to 𝑛+𝜏_𝑘+𝛼, we obtain 𝑥₁𝑛+𝜏_𝑘+𝛼=𝑥₁𝜏_𝑘+𝛼exp^{𝑛+𝜏𝑘+𝛼−1}_{𝑙=𝜏𝑘+𝛼}𝑟₁(𝑙)−𝑎₁(𝑙)𝑥₁(𝑙)−𝑐₂(𝑙)𝑥₂(𝑙)1+𝑥₂(𝑙),𝑥₂𝑛+𝜏_𝑘+𝛼=𝑥₂𝜏_𝑘+𝛼exp^{𝑛+𝜏𝑘+𝛼−1}_{𝑙=𝜏𝑘+𝛼}𝑟₂(𝑙)−𝑎₂(𝑙)𝑥₂(𝑙)−𝑐₁(𝑙)𝑥₁(𝑙)1+𝑥₁(𝑙). (A.4) Thus, it derives that 𝑥₁𝑛+𝜏_𝑘+𝛼=𝑥₁𝜏_𝑘+𝛼exp^{𝑛+𝛼−1}_𝑙=𝛼𝑟₁𝑙+𝜏_𝑘−𝑎₁𝑙+𝜏_𝑘𝑥₁𝑙+𝜏_𝑘−𝑐₂𝑙+𝜏_𝑘𝑥₂𝑙+𝜏_𝑘1+𝑥₂𝑙+𝜏_𝑘,𝑥₂𝑛+𝜏_𝑘+𝛼=𝑥₂𝜏_𝑘+𝛼exp^{𝑛+𝛼−1}_𝑙=𝛼𝑟₂𝑙+𝜏_𝑘−𝑎₂𝑙+𝜏_𝑘𝑥₂𝑙+𝜏_𝑘−𝑐₁𝑙+𝜏_𝑘𝑥₁𝑙+𝜏_𝑘1+𝑥₁𝑙+𝜏_𝑘. (A.5) Let 𝑘→+∞, we have 𝑥^∗₁(𝑛+𝛼)=𝑥^∗₁(𝛼)exp^{𝑛+𝛼−1}_𝑙=𝛼𝑟₁(𝑙)−𝑎₁(𝑙)𝑥^∗₁(𝑙)−𝑐₂(𝑙)𝑥^∗₂(𝑙)1+𝑥^∗₂(𝑙),𝑥^∗₂(𝑛+𝛼)=𝑥^∗₂(𝛼)exp^{𝑛+𝛼−1}_𝑙=𝛼𝑟₂(𝑙)−𝑎₂(𝑙)𝑥^∗₂(𝑙)−𝑐₁(𝑙)𝑥^∗₁(𝑙)1+𝑥^∗₁(𝑙). (A.6) Since 𝛼 is arbitrary, we know that (𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)) is a solution of system (1) on ℤ⁺, and 0<𝑚₁−𝜀≤𝑥^∗₁(𝑛)≤𝑀₁+𝜀,0<𝑚₂−𝜀≤𝑥^∗₂(𝑛)≤𝑀₂+𝜀,∀𝑛∈ℤ⁺. (A.7) Notice that 𝜀 is an arbitrarily small positive constant; it follows that 0<𝑚₁≤𝑥^∗₁(𝑛)≤𝑀₁,0<𝑚₂≤𝑥^∗₂(𝑛)≤𝑀₂,∀𝑛∈ℤ⁺. (A.8) Thus, Ω≠Φ. This completes the proof.

B. Proof of Theorem 8

Denote 𝑝₁(𝑛)=ln𝑥₁(𝑛), 𝑝₂(𝑛)=ln𝑥₂(𝑛). It follows from system (1) that 𝑝₁(𝑛+1)=𝑝₁(𝑛)+𝑟₁(𝑛)−𝑎₁(𝑛)𝑒^𝑝1(𝑛)−𝑐₂(𝑛)𝑒^𝑝2(𝑛)1+𝑒^𝑝2(𝑛),𝑝₂(𝑛+1)=𝑝₂(𝑛)+𝑟₂(𝑛)−𝑎₂(𝑛)𝑒^𝑝2(𝑛)−𝑐₁(𝑛)𝑒^𝑝1(𝑛)1+𝑒^𝑝1(𝑛). (B.1) According to Theorem 7, we can see that the system (B.1) has a bounded solution (𝑝₁(𝑛),𝑝₂(𝑛)) satisfying ln𝑚₁≤𝑝₁(𝑛)≤ln𝑀₁,ln𝑚₂≤𝑝₂(𝑛)≤ln𝑀₂,𝑛∈ℤ⁺. (B.2) Thus, |𝑝₁(𝑛)|≤𝐴, |𝑝₂(𝑛)|≤𝐵, where 𝐴=max{|ln𝑚₁|,|ln𝑀₁|}, 𝐵=max{|ln𝑚₂|,|ln𝑀₂|}. Define the norm ‖(𝑝₁(𝑛),𝑝₂(𝑛))‖=|𝑝₁(𝑛)|+|𝑝₂(𝑛)|, where (𝑝₁(𝑛),𝑝₂(𝑛))∈ℝ². Consider the product system of system (B.1) as follow: 𝑝₁(𝑛+1)=𝑝₁(𝑛)+𝑟₁(𝑛)−𝑎₁(𝑛)𝑒^𝑝1(𝑛)−𝑐₂(𝑛)𝑒^𝑝2(𝑛)1+𝑒^𝑝2(𝑛),𝑝₂(𝑛+1)=𝑝₂(𝑛)+𝑟₂(𝑛)−𝑎₂(𝑛)𝑒^𝑝2(𝑛)−𝑐₁(𝑛)𝑒^𝑝1(𝑛)1+𝑒^𝑝1(𝑛),𝑞₁(𝑛+1)=𝑞₁(𝑛)+𝑟₁(𝑛)−𝑎₁(𝑛)𝑒^𝑞1(𝑛)−𝑐₂(𝑛)𝑒^𝑞2(𝑛)1+𝑒^𝑞2(𝑛),𝑞₂(𝑛+1)=𝑞₂(𝑛)+𝑟₂(𝑛)−𝑎₂(𝑛)𝑒^𝑞2(𝑛)−𝑐₁(𝑛)𝑒^𝑞1(𝑛)1+𝑒^𝑞1(𝑛). (B.3) We assume that 𝑌=(𝑝₁(𝑛),𝑝₂(𝑛)), 𝑊=(𝑞₁(𝑛),𝑞₂(𝑛)) are any two solutions of system (B.1) defined on 𝕊; then, ‖𝑌‖≤𝐷, ‖𝑊‖≤𝐷, where 𝐷=𝐴+𝐵, and 𝕊={(𝑝₁(𝑛),𝑝₂(𝑛))∣ln𝑚_𝑖≤𝑝_𝑖(𝑛)≤ln𝑀_𝑖,𝑖=1,2,𝑛∈ℤ⁺}.

Let us construct a Lyapunov function defined on ℤ⁺×𝕊×𝕊 as follows: 𝑉(𝑛,𝑌,𝑊)=𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)². (B.4) It is obvious that the norm ‖𝑌−𝑊‖=|𝑝₁(𝑛)−𝑞₁(𝑛)|+|𝑝₂(𝑛)−𝑞₂(𝑛)| is equivalent to ‖𝑌−𝑊‖_∗=[(𝑝₁(𝑛)−𝑞₁(𝑛))²+(𝑝₂(𝑛)−𝑞₂(𝑛))²]^1/2; that is, there are two constants 𝐶₁>0, 𝐶₂>0, such that 𝐶₁‖𝑌−𝑊‖≤‖𝑌−𝑊‖_∗≤𝐶₂‖𝑌−𝑊‖, (B.5) then, 𝐶₁‖𝑌−𝑊‖²≤𝑉(𝑛,𝑌,𝑊)≤𝐶₂‖𝑌−𝑊‖². (B.6) Let 𝑎,𝑏∈𝐶(ℝ⁺,ℝ⁺), 𝑎(𝑥)=𝐶²₁𝑥², 𝑏(𝑥)=𝐶²₂𝑥²; then, condition (i) of Lemma 4 is satisfied.

Moreover, for any (𝑛,𝑌,𝑊),(𝑛,𝑌,𝑊)∈ℤ⁺×𝕊×𝕊, we have 𝑉()𝑊=𝑝𝑛,𝑌,𝑊−𝑉𝑛,𝑌,₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)²−𝑝₁(𝑛)−𝑞₁(𝑛)²−𝑝₂(𝑛)−𝑞₂(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)+𝑝₂(𝑛)−𝑝₂(𝑛+𝑞₁)(𝑛)−𝑞₁(𝑛)+𝑞₂(𝑛)−𝑞₂(𝑛𝑌+𝑊,)=𝐿𝑌−𝑊− (B.7) where 𝑌=(𝑝₁(𝑛),𝑝₂(𝑛)), 𝑊=(𝑞₁(𝑛),𝑞₂(𝑛)), and 𝐿=4max{𝐴,𝐵}. Thus, condition (ii) of Lemma 4 is satisfied.

Finally, calculating the Δ𝑉(𝑛) of 𝑉(𝑛) along the solutions of system (B.3), we have Δ𝑉_(𝐵.3)(𝑛)()(𝑛)=𝑝=𝑉𝑛+1−𝑉₁(𝑛+1)−𝑞₁(𝑛+1)²+𝑝₂(𝑛+1)−𝑞₂(𝑛+1)²−𝑝₁(𝑛)−𝑞₁(𝑛)²−𝑝₂(𝑛)−𝑞₂(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=𝑝₁(𝑛)−𝑞₁(𝑛)−𝑎₁(𝑛)𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)−𝑐₂(𝑛)𝑒^𝑝2(𝑛)1+𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)1+𝑒^𝑞2(𝑛)2−𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)−𝑎₂(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)−𝑐₁(𝑛𝑒)^𝑝1(𝑛)1+𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)1+𝑒^𝑞1(𝑛)2−𝑝₂(𝑛)−𝑞₂(𝑛)²=−2𝑎₁(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)−2𝑐₂(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)𝑒^𝑝2(𝑛)1+𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)1+𝑒^𝑞2(𝑛)+2𝑎₁(𝑛)𝑐₂(𝑛)𝑒^𝑝2(𝑛)1+𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)1+𝑒^𝑞2(𝑛)𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)+𝑎²₁(𝑛)𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)2+𝑐²₂(𝑛)𝑒^𝑝2(𝑛)1+𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)1+𝑒^𝑞2(𝑛)2−2𝑎₂(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)−2𝑐₁(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)𝑒^𝑝1(𝑛)1+𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)1+𝑒^𝑞1(𝑛)+2𝑎₂(𝑛)𝑐₁(𝑛𝑒)^𝑝1(𝑛)1+𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)1+𝑒^𝑞1(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)+𝑎²₂(𝑛)𝑒^𝑝2(𝑛)−𝑒^𝑞2(𝑛)2+𝑐²₁(𝑛)𝑒^𝑝1(𝑛)1+𝑒^𝑝1(𝑛)−𝑒^𝑞1(𝑛)1+𝑒^𝑞1(𝑛)2. (B.8) By the mean value theorem, it derives that 𝑒^{𝑝𝑖(𝑛)}−𝑒^{𝑞𝑖(𝑛)}=𝜉_𝑖(𝑛)𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛),𝑒^{𝑝𝑖(𝑛)}1+𝑒^{𝑝𝑖(𝑛)}−𝑒^{𝑞𝑖(𝑛)}1+𝑒^{𝑞𝑖(𝑛)}=𝜂_𝑖(𝑛)1+𝜂_𝑖(𝑛)²𝑝_𝑖(𝑛)−𝑞_𝑖(𝑛), (B.9)𝑖=1,2, where 𝜉_𝑖(𝑛) and 𝜂_𝑖(𝑛) lie between 𝑒^{𝑝𝑖(𝑛)} and 𝑒^{𝑞𝑖(𝑛)}, respectively. Substituting (B.9) into (B.8), we get Δ𝑉_(𝐵.3)(𝑛)=−2𝑎₁(𝑛)𝜉₁(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑐²₂(𝑛)𝜂²₂(𝑛)1+𝜂₂(𝑛)⁴𝑝₂(𝑛)−𝑞₂(𝑛)²−2𝑐₂(𝑛)𝜂₂(𝑛)1+𝜂₂(𝑛)²𝑝₁(𝑛)−𝑞₁(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)+𝑎²₁(𝑛)𝜉²₁(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)²+2𝑎₁(𝑛)𝑐₂(𝑛)𝜉₁(𝑛)𝜂₂(𝑛)1+𝜂₂(𝑛)²𝑝₁(𝑛)−𝑞₁(𝑛)×𝑝₂(𝑛)−𝑞₂(𝑛)−2𝑎₂(𝑛)𝜉₂(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)²+𝑐²₁(𝑛)𝜂²₁(𝑛)1+𝜂₁(𝑛)⁴𝑝₁(𝑛)−𝑞₁(𝑛)²−2𝑐₁(𝑛)𝜂₁(𝑛)1+𝜂₁(𝑛)²𝑝₂(𝑛)−𝑞₂(𝑛)𝑝₁(𝑛)−𝑞₁(𝑛)+𝑎²₂(𝑛)𝜉²₂(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)²+2𝑎₂(𝑛)𝑐₁(𝑛)𝜉₂(𝑛)𝜂₁(𝑛)1+𝜂₁(𝑛)²𝑝₂(𝑛)−𝑞₂(𝑛)×𝑝₁(𝑛)−𝑞₁(𝑛)=−2𝑎₁(𝑛)𝜉₁(𝑛)+𝑎²₁(𝑛)𝜉²₁(𝑛)+𝑐²₁(𝑛)𝜂²₁(𝑛)1+𝜂₁(𝑛)⁴×𝑝₁(𝑛)−𝑞₁(𝑛)²+−2𝑎₂(𝑛)𝜉₂(𝑛)+𝑎²₂(𝑛)𝜉²₂(𝑛)+𝑐²₂(𝑛)𝜂²₂(𝑛)1+𝜂₂(𝑛)⁴×𝑝₂(𝑛)−𝑞₂(𝑛)²𝑎+2²(𝑛)𝑐₁(𝑛)𝜉₂(𝑛)𝜂₁(𝑛)1+𝜂₁(𝑛)²−𝑐₁(𝑛)𝜂₁(𝑛)1+𝜂₁(𝑛)²+𝑎₁(𝑛)𝑐₂(𝑛)𝜉₁(𝑛)𝜂₂(𝑛)1+𝜂₂(𝑛)²−𝑐₂(𝑛)𝜂₂(𝑛)1+𝜂₂(𝑛)²×𝑝₁(𝑛)−𝑞₁(𝑛)𝑝₂(𝑛)−𝑞₂(𝑛)≤−2𝑎^𝐿₁𝑚₁+𝑎₁^𝑈2𝑀²₁+𝑐₁^𝑈2𝑀²₁1+𝑚₁⁴𝑝₁(𝑛)−𝑞₁(𝑛)²+−2𝑎^𝐿₂𝑚₂+𝑎₂^𝑈2𝑀²₂+𝑐₂^𝑈2𝑀²₂1+𝑚₂⁴𝑝₂(𝑛)−𝑞₂(𝑛)²+𝑎^𝑈₂𝑐^𝑈₁𝑀₁𝑀₂1+𝑚₁²+𝑐^𝑈₁𝑀₁1+𝑚₁²+𝑎^𝑈₁𝑐^𝑈₂𝑀₁𝑀₂1+𝑚₂²+𝑐^𝑈₂𝑀₂1+𝑚₂²×𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)²=−2𝑎^𝐿₁𝑚₁−𝑎₁^𝑈2𝑀²₁−𝑐₁^𝑈2𝑀²₁1+𝑚₁⁴−𝑎^𝑈₂𝑐^𝑈₁𝑀₁𝑀₂1+𝑚₁²−𝑐^𝑈₁𝑀₁1+𝑚₁²−𝑎^𝑈₁𝑐^𝑈₂𝑀₁𝑀₂1+𝑚₂²−𝑐^𝑈₂𝑀₂1+𝑚₂²×𝑝₁(𝑛)−𝑞₁(𝑛)²−2𝑎^𝐿₂𝑚₂−𝑎₂^𝑈2𝑀²₂−𝑐₂^𝑈2𝑀²₂1+𝑚₂⁴−𝑎^𝑈₂𝑐^𝑈₁𝑀₁𝑀₂1+𝑚₁²−𝑐^𝑈₁𝑀₁1+𝑚₁²−𝑎^𝑈₁𝑐^𝑈₂𝑀₁𝑀₂1+𝑚₂²−𝑐^𝑈₂𝑀₂1+𝑚₂²×𝑝₂(𝑛)−𝑞₂(𝑛)²=−𝑠₁𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑠₂𝑝₂(𝑛)−𝑞₂(𝑛)²≤−𝛽𝑝₁(𝑛)−𝑞₁(𝑛)²+𝑝₂(𝑛)−𝑞₂(𝑛)²=−𝛽𝑉(𝑛), (B.10) where 𝛽=min{𝑠₁,𝑠₂}. By the conditions of Theorem 8, we have 0<𝛽<1, and hence, condition (iii) of Lemma 4 is satisfied. So, it follows from Lemma 4 that there exists a unique uniformly asymptotically stable almost periodic solution (𝑝^∗₁(𝑛),𝑝^∗₂(𝑛)) of system (B.1) which is bounded by 𝕊 for all 𝑛∈ℤ⁺; that is, there exists a unique uniformly asymptotically stable almost periodic solution (𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)) of system (1) which is bounded by Ω for all 𝑛∈ℤ⁺. This completed the proof.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (nos. 210134, 212111), the Project of Key Laboratory of Biological Resources Protection and Utilization of Hubei Province, and the Key Subject of Hubei Province (Forestry).