Almost Periodic Solution of a Discrete Schoener’s Competition Model with Delays

Abstract

We consider an almost periodic discrete Schoener’s competition model with delays. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result.

1. Introduction

In 2009, Wu et al. [1] had studied a discrete Schoener’s competition mode with delays: 𝑥₁(𝑛+1)=𝑥₁(𝑛)𝑎exp¹⁰(𝑛)𝑥₁𝑛−𝜏₁₀+𝑘₁(𝑛)−𝑎₁₁(𝑛)𝑥₁𝑛−𝜏₁₁−𝑎₁₂(𝑛)𝑥₂𝑛−𝜏₁₂−𝑐₁(𝑛,𝑥₂)(𝑛+1)=𝑥₂(𝑛)𝑎exp²⁰(𝑛)𝑥₂𝑛−𝜏₂₀+𝑘₂(𝑛)−𝑎₂₁(𝑛)𝑥₁𝑛−𝜏₂₁−𝑎₂₂(𝑛)𝑥₂𝑛−𝜏₂₂−𝑐₂(𝑛), (1) where {𝑘_𝑖(𝑛)},{𝑎_𝑖𝑗(𝑛)}, and {𝑐_𝑖(𝑛)} are real positive bounded sequences and 𝜏_𝑖𝑗 are positive integers, 𝑖=1,2;𝑗=0,1,2. Sufficient conditions which guarantee the permanence and the global attractivity of positive solutions for system (1) are obtained.

By the biological meaning, the system (1) is considered together with the following initial condition: 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠)≥0,𝜑_𝑖(0)[]{}>0,𝑠∈−𝜏,0∩𝑍=−𝜏,−𝜏+1,…,0,𝑖=1,2, (2) where 𝜏=max_{1≤𝑖≤2,0≤𝑗≤2}{𝜏_𝑖𝑗}. Let (𝑥₁(𝑛),𝑥₂(𝑛)) be any solution of system (1) with the initial condition (2). One could easily see that 𝑥_𝑖(𝑛)>0, 𝑖=1,2 for all 𝑛∈𝑁.

Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [1–6] and the references cited therein). Recently, a few papers investigate the global stability of the pure-delay model (see [7–13]).

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 [14–21] and the references cited therein). But to the best of the author’s knowledge, to this day, still no scholars have studied the almost periodic version which is corresponding to system (1). Therefore, with stimulation from the works of [12, 19], the main purpose of this paper is to derive a set of sufficient conditions ensuring the existence of a unique strictly positive almost periodic solution of system (1) which is globally attractive.

Denote by 𝑍 and 𝑍⁺ the set of integers and the set of nonnegative integers, respectively. For any bounded sequence {𝑓(𝑛)} defined on 𝑍, define 𝑓^𝑢=sup_𝑛∈𝑍𝑓(𝑛),𝑓^𝑙=inf_𝑛∈𝑍𝑓(𝑛).

Throughout this paper, we assume the following.(H1)𝑘_𝑖(𝑛),𝑎_𝑖𝑗(𝑛), and 𝑐_𝑖(𝑛) are bounded positive almost periodic sequences such that 0<𝑘^𝑙_𝑖≤𝑘_𝑖(𝑛)≤𝑘^𝑢_𝑖,0<𝑎^𝑙_𝑖𝑗≤𝑎_𝑖𝑗(𝑛)≤𝑎^𝑢_𝑖𝑗,0<𝑐^𝑙_𝑖≤𝑐_𝑖(𝑛)≤𝑐^𝑢_𝑖,𝑖=1,2,𝑗=0,1,2. (3)

The remaining part of this paper is organized as follows. In Section 2, we will introduce some definitions and several useful lemmas. In Section 3, by applying the theory of difference inequality, we present the permanence results for system (1). In Section 4, we establish the sufficient conditions for the existence of a unique globally attractive almost periodic solution of system (1). The main result is illustrated by an example with a numerical simulation in the last section.

2. Preliminaries

In this section, we give the definitions and lemmas of the terminologies involved.

Definition 1 (see [22]). A sequence 𝑥∶𝑍→𝑅 is called an almost periodic sequence if the 𝜀-translation set of 𝑥𝐸{𝜀,𝑥}={𝜏∈𝑍∶|𝑥(𝑛+𝜏)−𝑥(𝑛)|<𝜀,∀𝑛∈𝑍} (4) 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(𝑛+𝜏)−𝑥(𝑛)|<𝜀,∀𝑛∈𝑍. (5)𝜏 is called an 𝜀-translation number of 𝑥(𝑛).

Definition 2 (see [23]). Let 𝐷 be an open subset of 𝑅^𝑚,𝑓∶𝑍×𝐷→𝑅^𝑚. 𝑓(𝑛,𝑥) is said to be almost periodic in 𝑛 uniformly for 𝑥∈𝐷 if for any 𝜀>0, and any compact set 𝑆⊂𝐷, there exists a positive integer 𝑙=𝑙(𝜀,𝑆) such that any interval of length 𝑙 contains an integer 𝜏 for which 𝑓(𝑛+𝜏,𝑥)−𝑓(𝑛,𝑥)<𝜀,∀(𝑛,𝑥)∈𝑍×𝑆. (6)𝜏 is called an 𝜀-translation number of 𝑓(𝑛,𝑥).

Definition 3 (see [24]). The hull of 𝑓, denoted by 𝐻(𝑓), is defined by 𝐻𝑓=𝑔(𝑛,𝑥)∶lim_𝑘→∞𝑓𝑛+𝜏_𝑘(),,𝑥=𝑔𝑛,𝑥uniformlyon𝑍×𝑆 (7) for some sequence {𝜏_𝑘}, where 𝑆 is any compact set in 𝐷.

Definition 4. Suppose that 𝑋(𝑛)=(𝑥₁(𝑛),𝑥₂(𝑛)) is any solution of system (1). 𝑋(𝑛) is said to be a strictly positive solution in 𝑍 if 0<inf_𝑛∈𝑍𝑥_𝑖(𝑛)≤sup_𝑛∈𝑍𝑥_𝑖(𝑛)<∞,𝑖=1,2, (8) for 𝑛∈𝑍.

Lemma 5 (see [25]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)≤𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)} (9) for 𝑛∈𝑁, where 𝑎(𝑛) and 𝑏(𝑛) are nonnegative sequences bounded above and below by positive constants. Then limsup_𝑛→+∞𝑥(𝑛)≤1𝑏^𝑙exp𝑎^𝑢−1. (10)

Lemma 6 (see [25]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛+1)≥𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)},𝑛≥𝑁₀,limsup_𝑛→+∞𝑥(𝑛)≤𝑥^∗, (11) and 𝑥(𝑁₀)>0, where 𝑎(𝑛) and 𝑏(𝑛) are nonnegative sequences bounded above and below by positive constants and 𝑁₀∈𝑁. Then liminf_𝑛→+∞𝑥(𝑛)𝑎≥min^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑥^∗,𝑎^𝑙𝑏^𝑢. (12)

3. Permanence and Global Attractivity

Now we state several lemmas which will be useful in proving our main result.

Proposition 7 (see [1]). Any solution (𝑥₁(𝑛),𝑥₂(𝑛)) of system (1) with the initial condition (2) is positive and ultimately bounded; that is, limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀_𝑖, (13) where 𝑀_𝑖=1𝑎^𝑙_𝑖𝑖𝑎exp^𝑢𝑖0𝑘^𝑙_𝑖𝜏_𝑖𝑖+1−1,𝑖=1,2. (14)

Proposition 8. Assume that ()𝑎𝐻2^𝑙10𝑀₁+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂−𝑐^𝑢₁𝑎>0,_𝑙20𝑀₂+𝑘^𝑢₂−𝑎^𝑢₂₁𝑀₁−𝑐^𝑢₂>0 (15) holds; then for any solution (𝑥₁(𝑛),𝑥₂(𝑛)) of system (1) with the initial condition (2), one has liminf_𝑛→+∞𝑥_𝑖(𝑛)≥𝑚_𝑖,𝑖=1,2, (16) where 𝑚₁ and 𝑚₂ are defined by (24) and (26), respectively.

Proof. For any small positive constant 𝜀>0, according to Proposition 7, there exists a 𝑁₁>0 such that, for all 𝑛>𝑁₁ and 𝑖=1,2, 𝑥_𝑖(𝑛)≤𝑀_𝑖+𝜀. (17)
It follows from (17) and the first equation of system (1), for 𝑛≥𝑁₁+𝜏, 𝑥₁(𝑛+1)≥𝑥₁(𝑛)𝑎exp^𝑙10𝑀₁+𝜀+𝑘^𝑢₁−𝑎^𝑢₁₁𝑀₁+𝜀−𝑎^𝑢₁₂𝑀₂+𝜀−𝑐^𝑢₁≜𝑥₁(𝑛)exp𝐴^𝜀₁, (18) where 𝐴^𝜀₁=(𝑎^𝑙₁₀/((𝑀₁+𝜀)+𝑘^𝑢₁))−𝑎^𝑢₁₁(𝑀₁+𝜀)−𝑎^𝑢₁₂(𝑀₂+𝜀)−𝑐^𝑢₁.
Thus, by using (18) we obtain 𝑥₁𝑛−𝜏₁₁≤𝑥₁(𝑛)exp−𝜏₁₁𝐴^𝜀₁. (19) Substituting (19) into the first equation of system (1), for 𝑛≥𝑁₁+𝜏₁₁, it follows that 𝑥₁(𝑛+1)≥𝑥₁(𝑛)𝑎exp^𝑙10𝑀₁+𝜀+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂+𝜀−𝑐^𝑢₁−𝑎₁₁(𝑛)exp−𝜏₁₁𝐴^𝜀₁𝑥₁(𝑛). (20)
When 𝜀 is an arbitrary small positive constant, it follows from condition (H2) 𝑎^𝑙₁₀𝑀₁+𝜀+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂+𝜀−𝑐^𝑢₁>0. (21) Thus, as a direct corollary of Lemma 6, according to (14) and (20), one has liminf_𝑛→+∞𝑥₁(𝑛)≥min𝐵_1𝜀,𝐵_2𝜀, (22) where 𝐵_1𝜀=1𝑎^𝑢₁₁𝑎^𝑙₁₀𝑀₁+𝜀+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂+𝜀−𝑐^𝑢₁exp𝜏₁₁𝐴^𝜀₁,𝐵_2𝜀=𝐵_1𝜀𝑎exp^𝑙10𝑀₁+𝜀+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂+𝜀−𝑐^𝑢₁−𝑀₁𝑎^𝑢₁₁exp−𝜏₁₁𝐴^𝜀₁. (23)
Letting 𝜀→0, it follows that liminf_𝑛→+∞𝑥₁(𝑛)≥12min𝐵₁,𝐵₂≜𝑚₁>0, (24) where 𝐵₁=1𝑎^𝑢₁₁𝑎^𝑙₁₀𝑀₁+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂−𝑐^𝑢₁exp𝜏₁₁𝐴₁,𝐵₂=𝐵₁𝑎exp^𝑙10𝑀₁+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂−𝑐^𝑢₁−𝑀₁𝑎^𝑢₁₁exp−𝜏₁₁𝐴₁,𝐴₁=𝑎^𝑙₁₀𝑀₁+𝑘^𝑢₁−𝑎^𝑢₁₁𝑀₁−𝑎^𝑢₁₂𝑀₂−𝑐^𝑢₁. (25)
Similar to the analysis of (18)–(24), by applying (17), from the second equation of system (1), we also have that liminf_𝑛→+∞𝑥₂(𝑛)≥12min𝐷₁,𝐷₂≜𝑚₂>0, (26) where 𝐷₁=1𝑎^𝑢₂₂𝑎^𝑙₂₀𝑀₂+𝑘^𝑢₂−𝑎^𝑢₂₁𝑀₁−𝑐^𝑢₂exp𝜏₂₂𝐶₁,𝐷₂=𝐷₁𝑎exp^𝑙20𝑀₂+𝑘^𝑢₂−𝑎^𝑢₂₁𝑀₁−𝑐^𝑢₂−𝑀₂𝑎^𝑢₂₂exp−𝜏₂₂𝐶₁,𝐶₁=𝑎^𝑙₂₀𝑀₂+𝑘^𝑢₂−𝑎^𝑢₂₂𝑀₂−𝑎^𝑢₂₁𝑀₁−𝑐^𝑢₂. (27) The proof is completed.

Note that condition (H2) of Proposition 8 is weakened compared to condition (H) in [1].

Theorem 9 (see [1]). Assume that (H2) holds; system (1) is permanent.

Proposition 10 (see [1]). Assume that (H2) holds and further that there exist positive constants 𝛼₁,𝛼₂, and 𝜂>0 such that (𝐻3)𝛼_𝑖𝐸_𝑖𝑖−𝛼_𝑖𝐹_𝑖𝑗+𝛼_𝑗𝐺_𝑗𝑖≥𝜂,𝑖,𝑗=1,2,𝑖≠𝑗, (28) where 𝐸_𝑖𝑖=min𝑎^𝑙_𝑖𝑖,2𝑀_𝑖−𝑎^𝑢_𝑖𝑖−𝑎^𝑢_𝑖0𝑘^𝑙_𝑖²1+𝜏_𝑖𝑖𝑎^𝑢_𝑖𝑖𝑀_𝑖𝐵^𝑢_𝑖−𝜏_𝑖𝑖𝑎^𝑢_𝑖𝑖²𝑀_𝑖𝐵^𝑢_𝑖,𝐹_𝑖𝑗=𝜏_𝑖𝑖𝑎^𝑢_𝑖𝑖𝐴^𝑢_𝑖𝑎^𝑢_𝑖0𝑘^𝑙_𝑖+𝑎^𝑢_𝑖𝑖𝑀_𝑖+𝑎^𝑢_𝑖𝑗𝑀_𝑗+𝑐^𝑢_𝑖,𝐺_𝑗𝑖=𝑎^𝑢_𝑗𝑖+𝜏_𝑗𝑗𝑎^𝑢_𝑗𝑗𝑀_𝑗𝐵^𝑢_𝑗𝑎^𝑢_𝑗𝑖𝐴,𝑖,𝑗=1,2;𝑖≠𝑗,_𝑖(𝑠)=exp𝜃_𝑖(𝑠)𝑎_𝑖0(𝑠)𝑥^∗_𝑖𝑠−𝜏_𝑖0+𝑘_𝑖(𝑠)−𝑎_𝑖𝑖𝑥^∗_𝑖𝑠−𝜏_𝑖𝑖−𝑎_𝑖𝑗𝑥^∗_𝑗𝑠−𝜏_𝑖𝑗−𝑐_𝑖(𝑠),𝐵_𝑖(𝑠)=exp𝜑_𝑖(𝑠𝑎)^𝑖0(𝑠)𝑥_𝑖𝑠−𝜏_𝑖0+𝑘_𝑖(𝑠)−𝑎_𝑖𝑖𝑥_𝑖𝑠−𝜏_𝑖𝑖−𝑎_𝑖𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗−𝑐_𝑖(𝑠+)1−𝜑_𝑖(𝑠𝑎)^𝑖0(𝑠)𝑥^∗_𝑖𝑠−𝜏_𝑖0+𝑘_𝑖(𝑠)−𝑎_𝑖𝑖𝑥^∗_𝑖𝑠−𝜏_𝑖𝑖−𝑎_𝑖𝑗𝑥^∗_𝑗𝑠−𝜏_𝑖𝑗−𝑐_𝑖(𝑠), (29)𝜃_𝑖(𝑠),𝜑_𝑖(𝑠)∈(0,1), 𝑖,𝑗=1,2, 𝑖≠𝑗. Then system (1) is globally attractive; that is, for any two positive solutions (𝑥₁(𝑛),𝑥₂(𝑛)) and (𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)) of system (1), we have lim_𝑛→+∞𝑥^∗_𝑖(𝑛)−𝑥_𝑖(𝑛)=0,𝑖=1,2. (30)

4. Almost Periodic Solution

In this section, by means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, we will study the existence of a globally attractive almost periodic solution of system (1) with initial condition (2) and obtain the sufficient conditions.

Let {𝛿_𝑘} be any integer valued sequence such that 𝛿_𝑘→∞ as 𝑘→∞. According to Lemma 5, taking a subsequence if necessary, we have 𝑘_𝑖(𝑛+𝛿_𝑘)→𝑘^∗_𝑖(𝑛),𝑎_𝑖𝑗(𝑛+𝛿_𝑘)→𝑎^∗_𝑖𝑗(𝑛),𝑐_𝑖(𝑛+𝛿_𝑘)→𝑐^∗_𝑖(𝑛),𝑖=1,2,𝑗=0,1,2, as 𝑘→∞ for 𝑛∈𝑍. Then we get a hull equation of system (1) as follows: 𝑥₁(𝑛+1)=𝑥₁(𝑛)𝑎exp^∗10(𝑛)𝑥₁𝑛−𝜏₁₀+𝑘^∗₁(𝑛)−𝑎^∗₁₁(𝑛)𝑥₁𝑛−𝜏₁₁−𝑎^∗₁₂(𝑛)𝑥₂𝑛−𝜏₁₂−𝑐^∗₁(𝑛,𝑥₂)(𝑛+1)=𝑥₂(𝑛)𝑎exp^∗20(𝑛)𝑥₂𝑛−𝜏₂₀+𝑘^∗₂(𝑛)−𝑎^∗₂₁(𝑛)𝑥₁𝑛−𝜏₂₁−𝑎^∗₂₂(𝑛)𝑥₂𝑛−𝜏₂₂−𝑐^∗₂(𝑛). (31)

By the almost periodic theory, we can conclude that if system (1) satisfies (H2) and (H3), then the hull equation (31) of system (1) also satisfies (H2) and (H3).

By Theorem 3.4 in [26], we can easily obtain the lemma as follows.

Lemma 11. If each hull equation of system (1) has a unique strictly positive solution, then the almost periodic difference system (1) has a unique strictly positive almost periodic solution.
Now we investigate a globally attractive almost periodic solution of system (1).

Theorem 12. If the almost periodic difference system (1) satisfies (H1), (H2), and (H3), then the almost periodic difference system (1) admits a unique strictly positive almost periodic solution, which is globally attractive.

Proof. By Lemma 11, we only need to prove that each hull equation of system (1) has a unique globally attractive almost periodic sequence solution; hence we firstly prove that each hull equation of system (1) has at least one strictly positive solution (the existence) and then we prove that each hull equation of system (1) has a unique strictly positive solution (the uniqueness).
Now we prove the existence of a strictly positive solution of any hull equation (31). By the almost periodicity of {𝑘_𝑖(𝑛)},{𝑎_𝑖𝑗(𝑛)}, and {𝑐_𝑖(𝑛)}, 𝑖=1,2, 𝑗=0,1,2, there exists an integer valued sequence {𝜏_𝑘} with 𝜏_𝑘→∞ as 𝑘→∞ such that 𝑘^∗_𝑖(𝑛+𝜏_𝑘)→𝑘^∗_𝑖(𝑛),𝑎^∗_𝑖𝑗(𝑛+𝜏_𝑘)→𝑎^∗_𝑖𝑗(𝑛),𝑐^∗_𝑖(𝑛+𝜏_𝑘)→𝑐^∗_𝑖(𝑛), 𝑖=1,2, 𝑗=0,1,2, as 𝑘→∞ for 𝑛∈𝑍. Suppose that 𝑋=(𝑥₁(𝑛),𝑥₂(𝑛)) is any solution of hull equation (31). By the proof of Propositions 7 and 8, we have 𝑚_𝑖≤liminf_𝑛→+∞𝑥_𝑖(𝑛)≤limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2. (32) And also 0<inf_𝑛∈𝑍+𝑥_𝑖(𝑛)≤sup_𝑛∈𝑍+𝑥_𝑖(𝑛)<∞,𝑖=1,2. (33)
Let 𝜀 be an arbitrary small positive number. It follows from (32) that there exists a positive integer 𝑛₀ such that 𝑚_𝑖−𝜀≤𝑥_𝑖(𝑛)≤𝑀_𝑖+𝜀, 𝑛≥𝑛₀, 𝑖=1,2. Write 𝑋_𝑘(𝑛)=𝑋(𝑛+𝜏_𝑘)=(𝑥_1𝑘(𝑛),𝑥_2𝑘(𝑛)), for all 𝑛≥𝑛₀+𝜏−𝜏_𝑘, 𝑘∈𝑍⁺. We claim that there exists a sequence {𝑦_𝑖(𝑛)}, and a subsequence of {𝜏_𝑘}, which we still denote by {𝜏_𝑘} such that 𝑥_𝑖𝑘(𝑛)→𝑦_𝑖(𝑛) uniformly in 𝑛 on any finite subset 𝐵 of 𝑍 as 𝑘→∞, where 𝐵={𝑎₁,𝑎₂,…,𝑎_𝑚}, 𝑎_ℎ∈𝑍(ℎ=1,2,…,𝑚), and 𝑚 is a finite number.
In fact, for any finite subset 𝐵⊂𝑍, when 𝑘 is large enough, 𝜏_𝑘+𝑎_ℎ−𝜏>𝑛₀, ℎ=1,2,…,𝑚. So 𝑚_𝑖−𝜀≤𝑥_𝑖𝑛+𝜏_𝑘≤𝑀_𝑖+𝜀,𝑖=1,2; (34) that is, {𝑥_𝑖(𝑛+𝜏_𝑘)} are uniformly bounded for 𝑘 large enough.
Now, for 𝑎₁∈𝐵, we can choose a subsequence {𝜏_𝑘⁽¹⁾} of {𝜏_𝑘} such that {𝑥_𝑖(𝑎₁+𝜏_𝑘⁽¹⁾)} uniformly converges on 𝑍⁺ for 𝑘 large enough.
Similarly, for 𝑎₂∈𝐵, we can choose a subsequence {𝜏_𝑘⁽²⁾} of {𝜏_𝑘⁽¹⁾} such that {𝑥_𝑖(𝑎₂+𝜏_𝑘⁽²⁾)} uniformly converges on 𝑍⁺ for 𝑘 large enough.
Repeating this procedure, for 𝑎_𝑚∈𝐵, we can choose a subsequence {𝜏_𝑘^(𝑚)} of {𝜏_𝑘^(𝑚−1)} such that {𝑥_𝑖(𝑎_𝑚+𝜏_𝑘^(𝑚))} uniformly converges on 𝑍⁺ for 𝑘 large enough.
Now pick the sequence {𝜏_𝑘^(𝑚)} which is a subsequence of {𝜏_𝑘}, which we still denote by {𝜏_𝑘}, then for all 𝑛∈𝐵, we have 𝑥_𝑖(𝑛+𝜏_𝑘)→𝑦_𝑖(𝑛) uniformly in 𝑛∈𝐵, as 𝑘→∞.
By the arbitrary of 𝐵, the conclusion is valid.
Combining with 𝑥_1𝑘(𝑛+1)=𝑥_1𝑘(𝑛)𝑎exp^∗10𝑛+𝜏_𝑘𝑥_1𝑘𝑛−𝜏₁₀+𝑘^∗₁𝑛+𝜏_𝑘−𝑎^∗₁₁𝑛+𝜏_𝑘𝑥_1𝑘𝑛−𝜏₁₁−𝑎^∗₁₂𝑛+𝜏_𝑘𝑥_2𝑘𝑛−𝜏₁₂−𝑐^∗₁𝑛+𝜏_𝑘,𝑥_2𝑘(𝑛+1)=𝑥_2𝑘(𝑛)𝑎exp^∗20𝑛+𝜏_𝑘𝑥_2𝑘𝑛−𝜏₂₀+𝑘^∗₂𝑛+𝜏_𝑘−𝑎^∗₂₁𝑛+𝜏_𝑘𝑥_1𝑘𝑛−𝜏₂₁−𝑎^∗₂₂𝑛+𝜏_𝑘𝑥_2𝑘𝑛−𝜏₂₂−𝑐^∗₂𝑛+𝜏_𝑘, (35) gives 𝑦₁(𝑛+1)=𝑦₁(𝑛)𝑎exp^∗10(𝑛)𝑦₁𝑛−𝜏₁₀+𝑘^∗₁(𝑛)−𝑎^∗₁₁(𝑛)𝑦₁𝑛−𝜏₁₁−𝑎^∗₁₂(𝑛)𝑦₂𝑛−𝜏₁₂−𝑐^∗₁(𝑛,𝑦₂)(𝑛+1)=𝑦₂(𝑛)𝑎exp^∗20(𝑛)𝑦₂𝑛−𝜏₂₀+𝑘^∗₂(𝑛)−𝑎^∗₂₁(𝑛)𝑦₁𝑛−𝜏₂₁−𝑎^∗₂₂(𝑛)𝑦₂𝑛−𝜏₂₂−𝑐^∗₂(𝑛). (36) We can easily see that 𝑌(𝑛)=(𝑦₁(𝑛),𝑦₂(𝑛)) is a solution of hull equation (31) and 𝑚_𝑖−𝜀≤𝑦_𝑖(𝑛)≤𝑀_𝑖+𝜀, 𝑖=1,2, for 𝑛∈𝑍. Since 𝜀 is an arbitrary small positive number, it follows that 𝑚_𝑖≤𝑦_𝑖(𝑛)≤𝑀_𝑖, 𝑖=1,2, for 𝑛∈𝑍; that is, 0<inf_𝑛∈𝑍𝑦_𝑖(𝑛)≤sup_𝑛∈𝑍𝑦_𝑖(𝑛)<∞,𝑖=1,2. (37) Hence each hull equation of almost periodic difference system (1) has at least one strictly positive solution.
Now we prove the uniqueness of the strictly positive solution of each hull equation (31). Suppose that the hull equation (31) has two arbitrary strictly positive solutions (𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)) and (𝑦^∗₁(𝑛),𝑦^∗₂(𝑛)). We construct a Lyapunov functional 𝑉^∗(𝑛)=²_𝑖=1𝛽_𝑖𝑉^∗_𝑖1(𝑛)+𝑉^∗_𝑖2(𝑛)+𝑉^∗_𝑖3(𝑛),𝑛∈𝑍, (38) where 𝑉^∗_𝑖1(𝑛)=ln𝑥^∗_𝑖(𝑛)−ln𝑦^∗_𝑖(𝑛𝑉_∗𝑖2)(𝑛)=^𝑛−1_{𝑠=𝑛−𝜏𝑖0}𝑎_𝑖0𝑠+𝜏_𝑖0𝑘²_𝑖𝑠+𝜏_𝑖0𝑥^∗_𝑖(𝑠)−𝑦^∗_𝑖(𝑠+_{𝑛−1𝑠=𝑛−𝜏𝑖𝑗})𝑎_𝑖𝑗𝑠+𝜏_𝑖𝑗𝑥^∗_𝑗(𝑠)−𝑦^∗_𝑗(𝑠+_{𝑛−1+𝜏𝑖𝑖𝑠=𝑛})𝑎_𝑖𝑖(𝑠)×^𝑛−1_{𝑢=𝑠−𝜏𝑖𝑖}𝐴_𝑖(𝑢𝑎)^𝑖0(𝑢)𝑘_𝑖(𝑢)+𝑎_𝑖𝑖(𝑢)𝑀_𝑖+𝑎_𝑖𝑗(𝑢)𝑀_𝑗+𝑐_𝑖(𝑢×𝑥_∗𝑖)(𝑢)−𝑦^∗_𝑖(𝑢)+𝑀_𝑖𝐵_𝑖(𝑢)𝑎_𝑖0(𝑢)𝑘²_𝑖(𝑢)×𝑥^∗_𝑖𝑢−𝜏_𝑖0−𝑦^∗_𝑖𝑢−𝜏_𝑖0+𝑀_𝑖𝐵_𝑖(𝑢)𝑎_𝑖𝑖(𝑢)×𝑥^∗_𝑖𝑢−𝜏_𝑖𝑖−𝑦^∗_𝑖𝑢−𝜏_𝑖𝑖+𝑀_𝑖𝐵_𝑖(𝑢)𝑎_𝑖𝑗(𝑢)×𝑥^∗_𝑗𝑢−𝜏_𝑖𝑗−𝑦^∗_𝑗𝑢−𝜏_𝑖𝑗,𝑉^∗_𝑖3(𝑛)=𝑀_𝑖^𝑛−1_{𝑙=𝑛−𝜏𝑖0}𝑎_𝑖0𝑙+𝜏_𝑖0𝑘²_𝑖𝑙+𝜏_𝑖0𝐵_𝑖𝑙+𝜏_𝑖0×𝑥^∗_𝑗(𝑙)−𝑦^∗_𝑗(𝑙)^{𝑙+𝜏𝑖0+𝜏𝑖𝑖}_{𝑠=𝑙+𝜏𝑖0+1}𝑎_𝑖𝑖(𝑠)+𝑀_𝑖^𝑛−1_{𝑙=𝑛−𝜏𝑖𝑖}𝐵_𝑖𝑙+𝜏_𝑖𝑖𝑎_𝑖𝑖𝑙+𝜏_𝑖𝑖×𝑥^∗_𝑖(𝑙)−𝑦^∗_𝑖(𝑙)^{𝑙+2𝜏𝑖𝑖}_{𝑠=𝑙+𝜏𝑖𝑖+1}𝑎_𝑖𝑖(𝑠)+𝑀_𝑖^𝑛−1_{𝑙=𝑛−𝜏𝑖𝑗}𝐵_𝑖𝑙+𝜏_𝑖𝑗𝑎_𝑖𝑗𝑙+𝜏_𝑖𝑗×𝑥^∗_𝑗(𝑙)−𝑦^∗_𝑗(𝑙)^{𝑙+𝜏𝑖𝑗+𝜏𝑖𝑖}_{𝑠=𝑙+𝜏𝑖𝑗+1}𝑎_𝑖𝑖(𝑠),𝑖,𝑗=1,2,𝑖≠𝑗. (39)
Calculating the difference of 𝑉^∗(𝑛) along the solution of the hull equation (31), one has Δ𝑉^∗(𝑛)≤−𝜂²_𝑖=1𝑥^∗_𝑖(𝑛)−𝑦^∗_𝑖(𝑛),𝑛∈𝑍. (40) From (40), we can see that 𝑉^∗(𝑛) is a nonincreasing function on 𝑍. Summing both sides of the above inequalities from 𝑛 to 0, we have 𝜂⁰²_{𝑘=𝑛𝑖=1}𝑥^∗_𝑖(𝑘)−𝑦^∗_𝑖(𝑘)≤𝑉^∗(0)−𝑉^∗(𝑛+1),𝑛<0. (41) Note that 𝑉^∗(𝑛) is bounded. Hence we have ⁰²_{𝑘=−∞𝑖=1}𝑥^∗_𝑖(𝑘)−𝑦^∗_𝑖(𝑘)<+∞, (42) which implies that lim_{𝑛→−∞}𝑥^∗_𝑖(𝑛)−𝑦^∗_𝑖(𝑛)=0,𝑖=1,2. (43)
Define 𝑄=∑²_𝑖=1𝛽_𝑖𝑄_𝑖, where 𝑄_𝑖=1𝑚_𝑖+𝜏_𝑖0𝑎^𝑢_𝑖0𝑘^𝑙_𝑖²1+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑖𝜏_𝑖𝑗+𝜏_𝑖𝑗𝑎²_𝑖𝑗+𝜏_𝑖𝑖𝑎^𝑢_𝑖𝑖+𝜏_𝑖𝑗𝑎^𝑢_𝑖𝑗𝑀_𝑖𝐵^𝑢_𝑖𝜏_𝑖𝑖𝑎^𝑢_𝑖𝑖+𝜏²_𝑖𝑖𝑎²_𝑖𝑖𝐴^𝑢_𝑖𝐴^𝑢_𝑖0𝐾^𝑙_𝑖+𝐴^𝑢_𝑖𝑖𝑀_𝑖+𝐴^𝑢_𝑖𝑗𝑀_𝑗+𝑐^𝑢_𝑖+𝐵^𝑢_𝑖𝑀_𝑖𝑎^𝑢_𝑖0𝐾^𝑙_𝑖²+𝑎^𝑢_𝑖𝑖+𝑎^𝑢_𝑖𝑗,𝑖,𝑗=1,2,𝑖≠𝑗. (44)
Let 𝜀 be an arbitrary small positive number. It follows from (43) that there exists a positive integer 𝑛₁>0 such that |𝑥^∗_𝑖(𝑛)−𝑦^∗_𝑖(𝑛)|<𝜀/𝑄,𝑛<−𝑛₁,𝑖=1,2. Therefore, for 𝑛<−𝑛₁,𝑖,𝑗=1,2,𝑖≠𝑗, 𝑉^∗_𝑖1(𝑛)≤1𝑚_𝑖𝑥^∗_𝑖(𝑛)−𝑦^∗_𝑖(𝑛≤1)𝑚_𝑖𝜀𝑄,𝑉^∗_𝑖2(𝑛)≤𝜏_𝑖0𝑎^𝑢_𝑖0𝑘^𝑙_𝑖²max_𝑝≤𝑛𝑥^∗_𝑖𝑝−𝑦^∗_𝑖𝑝+𝜏_𝑖𝑗𝑎^𝑢_𝑖𝑗max_𝑝≤𝑛𝑥^∗_𝑗𝑝−𝑦^∗_𝑗𝑝+𝜏²_𝑖𝑖𝑎^𝑢_𝑖𝑖𝐴^𝑢_𝑖𝑎^𝑢_𝑖0𝑘^𝑙_𝑖+𝑎^𝑢_𝑖𝑖𝑀_𝑖+𝑎^𝑢_𝑖𝑗𝑀_𝑗+𝑐^𝑢_𝑖×max_𝑝≤𝑛𝑥^∗_𝑖𝑝−𝑦^∗_𝑖𝑝+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖0𝑘^𝑖_𝑖²max_𝑝≤𝑛𝑥^∗_𝑖𝑝−𝑦^∗_𝑖𝑝+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑖max_𝑝≤𝑛𝑥^∗_𝑖𝑝−𝑦^∗_𝑖𝑝+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑗max_𝑝≤𝑛𝑥^∗_𝑗𝑝−𝑦^∗_𝑗𝑝≤𝜏_𝑖0𝑎^𝑢_𝑖0𝑘^𝑙_𝑖²+𝜏_𝑖𝑗𝑎^𝑢_𝑖𝑗+𝜏²_𝑖𝑖𝑎^𝑢_𝑖𝑖𝐴^𝑢_𝑖𝑎^𝑢_𝑖0𝑘^𝑙_𝑖+𝑎^𝑢_𝑖𝑖𝑀_𝑖+𝑎^𝑢_𝑖𝑗𝑀_𝑗+𝑐^𝑢_𝑖+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖0𝑘^𝑖_𝑖²+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑖+𝑀_𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑗𝜀𝑄,𝑉^∗_𝑖3(𝑛)≤𝑀_𝑖𝜏_𝑖0𝜏_𝑖𝑖𝑎^𝑢_𝑖0𝐵^𝑢_𝑖𝑘^𝑙_𝑖²𝑎^𝑢_𝑖𝑖max_𝑝≤𝑛𝑥^∗_𝑗𝑝−𝑦^∗_𝑗𝑝+𝑀_𝑖𝜏²_𝑖𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑖²max_𝑝≤𝑛𝑥^∗_𝑖𝑝−𝑦^∗_𝑖𝑝+𝑀_𝑖𝜏_𝑖𝑗𝜏_𝑖𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑗𝑎^𝑢_𝑖𝑖max_𝑝≤𝑛𝑥^∗_𝑗𝑝−𝑦^∗_𝑗𝑝≤𝑀_𝑖𝜏_𝑖0𝜏_𝑖𝑖𝑎^𝑢_𝑖0𝐵^𝑢_𝑖𝑘^𝑙_𝑖²𝑎^𝑢_𝑖𝑖+𝑀_𝑖𝜏²_𝑖𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑖²+𝑀_𝑖𝜏_𝑖𝑗𝜏_𝑖𝑖𝐵^𝑢_𝑖𝑎^𝑢_𝑖𝑗𝑎^𝑢_𝑖𝑖𝜀𝑄,𝑖,𝑗=1,2,𝑖≠𝑗. (45)
It follows from (38) and above inequalities that 𝑉^∗(𝑛)≤²_𝑖=1𝛽_𝑖𝑄_𝑖𝜀𝑄=𝜀,𝑛<−𝑛₁, (46) so lim_{𝑛→−∞}𝑉^∗(𝑛)=0. Note that 𝑉^∗(𝑛) is a nonincreasing function on 𝑍, and then 𝑉^∗(𝑛)≡0; that is 𝑥^∗_𝑖(𝑛)=𝑦^∗_𝑖(𝑛), 𝑖=1,2, for all 𝑛∈𝑍. Therefore, each hull equation of system (1) has a unique strictly positive solution.
In view of the above discussion, any hull equation of system (1) has a unique strictly positive solution. By Propositions 7–10 and Lemma 11, the almost periodic difference system (1) has a unique strictly positive almost periodic solution which is globally attractive. The proof is completed.

Let 𝜏_𝑖𝑗=0, 𝑖=1,2, 𝑗=0,1,2. Like in the proof of Theorem 12, we have the following corollary.

Corollary 13. Let 𝜏_𝑖𝑗=0, 𝑖=1,2, 𝑗=0,1,2. Assume that 𝑎^𝑙₁₀𝑀₁+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂−𝑐^𝑢₁𝑎>0,_𝑙20𝑀₂+𝑘^𝑢₂−𝑎^𝑢₂₁𝑀₁−𝑐^𝑢₂>0 (47) hold and further that there exist two positive constants 𝛼₁ and 𝛼₂, such that 𝛼_𝑖min𝑎^𝑙_𝑖𝑖,2𝑀_𝑖−𝑎^𝑢_𝑖𝑖−𝑎^𝑢_𝑖0𝑘^𝑙_𝑖²−𝛼_𝑗𝑎^𝑢_𝑗𝑖>0,𝑖,𝑗=1,2,𝑖≠𝑗, (48) where 𝑀_𝑖=(1/𝑎^𝑙_𝑖𝑖)exp{(𝑎^𝑢_𝑖0/𝑘^𝑙_𝑖)−1}. Then the almost periodic difference system (1) admits a unique strictly positive almost periodic solution, which is globally attractive.

5. Example and Numerical Simulation

In this section, we give the following example to check the feasibility of our result.

Example 1. Consider the following almost periodic discrete Schoener’s competition model with delays 𝑥₁(𝑛+1)=𝑥₁(𝑛)√exp0.4+0.1sin2𝑛𝑥₁(𝑛−3)+10.2+0.2cos√−√2𝑛2.02+0.02sin3𝑛𝑥₁()−√𝑛−10.015+0.005sin2𝑛𝑥₂()−√𝑛−40.002+0.0005sin,𝑥5𝑛₂(𝑛+1)=𝑥₂(𝑛)√exp0.3+0.2sin3𝑛𝑥₂(𝑛−4)+10.5+0.5cos√−√3𝑛0.009+0.001cos5𝑛𝑥₁()−√𝑛−22.53+0.03cos3𝑛𝑥₂()−√𝑛−10.005+0.001sin2𝑛. (49)
By simple computation, we derive 𝑀₁≈0.2033,𝑀₂𝑎≈0.1626,_𝑙10𝑀₁+𝑘^𝑢₁−𝑎^𝑢₁₂𝑀₂−𝑐^𝑢₁𝑎≈0.0237,_𝑙20𝑀₂+𝑘^𝑢₂−𝑎^𝑢₂₁𝑀₁−𝑐^𝑢₂𝐸≈0.0018,₁₁≈1.1047,𝐸₂₂𝐹≈1.377,₁₂≈1.0075,𝐹₂₁𝐺≈1.3366,₁₂≈0.0548,𝐺₂₁≈0.0217. (50) Then 23𝐸₁₁−23𝐹₁₂+𝐺₂₁𝐸≈0.01,₂₂−𝐹₂₁+23𝐺₁₂≈0.026. (51)
Also it is easy to see that conditions (H2) and (H3) are verified. Therefore, system (49) has a unique strictly positive almost periodic solution which is globally attractive. Our numerical simulations support our results (see Figures 1, 2, 3, and 4).

Figure 1 

Dynamic behavior of the first component 𝑥1(𝑛) of the solution (𝑥1(𝑛),𝑥2(𝑛)) to system (49) with the initial conditions (𝑥1(𝑛),𝑥2(𝑛))=(0.013,0.018) and (0.032,0.025),𝑛=1,2,3,4,5 for 𝑛∈[1,100], respectively.

Figure 2 

Dynamic behavior of the second component 𝑥2(𝑛) of the solution (𝑥1(𝑛),𝑥2(𝑛)) to system (49) with the initial conditions (𝑥1(𝑛),𝑥2(𝑛))=(0.013,0.018) and (0.032,0.025),𝑛=1,2,3,4,5 for 𝑛∈[1,100], respectively.

Figure 3 

Dynamic behavior of the first component 𝑥1(𝑛) of the solution (𝑥1(𝑛),𝑥2(𝑛)) to system (49) with the initial conditions (𝑥1(𝑛),𝑥2(𝑛))=(0.013,0.018) and (0.032,0.025),𝑛=1,2,3,4,5 for 𝑛∈[500,550], respectively.

Figure 4 

Dynamic behavior of the second component 𝑥2(𝑛) of the solution (𝑥1(𝑛),𝑥2(𝑛)) to system (49) with the initial conditions (𝑥1(𝑛),𝑥2(𝑛))=(0.013,0.018) and (0.032,0.025),𝑛=1,2,3,4,5 for 𝑛∈[500,550], respectively.

Conflict of Interests

There are no financial interest conflicts between the authors and the commercial identity.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 61132008) and Scientific Research Program Funded by Shaanxi Provincial Education Department of China (no. 2013JK1098).