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

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.

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][15][16][17][18][19][20][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 2 Journal of Difference Equations strictly positive almost periodic solution of system (1) which is globally attractive.
Denote by  and  + the set of integers and the set of nonnegative integers, respectively.For any bounded sequence {()} defined on , define   = sup ∈ (),   = inf ∈ ().
Throughout this paper, we assume the following.
(H1)   (),   (), and   () are bounded positive almost periodic sequences such that 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.

Preliminaries
In this section, we give the definitions and lemmas of the terminologies involved.
is called an -translation number of (, ).
Definition 3 (see [24]).The hull of , denoted by (), is defined by for some sequence {  }, where  is any compact set in .

Permanence and Global Attractivity
Now we state several lemmas which will be useful in proving our main result.
It follows from (17) and the first equation of system (1), for  ≥  1 + , where Thus, by using ( 18) we obtain Substituting (19) into the first equation of system (1), for  ≥  1 +  11 , it follows that When  is an arbitrary small positive constant, it follows from condition (H2) Thus, as a direct corollary of Lemma 6, according to ( 14) and ( 20), one has lim inf where Letting  → 0, it follows that lim inf where Similar to the analysis of ( 18)-( 24), by applying (17), from the second equation of system (1), we also have that lim inf where The proof is completed.
Note that condition (H2) of Proposition 8 is weakened compared to condition (H) in [1].

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 {  } be any integer valued sequence such that   → ∞ as  → ∞.According to Lemma 5, taking a subsequence if necessary, we have   ( +   ) →  *  (),   ( +   ) →  *  (),   ( +   ) →  *  (),  = 1, 2,  = 0, 1, 2, as  → ∞ for  ∈ .Then we get a hull equation of system (1) as follows: 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).
Repeating this procedure, for   ∈ , we can choose a subsequence { ()   } of {   () , Calculating the difference of  * () along the solution of the hull equation ( 31), one has From (40), we can see that  * () is a nonincreasing function on .Summing both sides of the above inequalities from  to 0, we have Note that  * () is bounded.Hence we have which implies that lim Define  = ∑ 2 =1     , where so lim  → −∞  * () = 0. Note that  * () is a nonincreasing function on , and then  * () ≡ 0; that is  *  () =  *  (),  = 1, 2, for all  ∈ .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.

Example and Numerical Simulation
In this section, we give the following example to check the feasibility of our result.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).

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