Almost periodic solution of a discrete competitive system with delays and feedback controls

Abstract A discrete nonlinear almost periodic multispecies competitive system with delays and feedback controls is proposed and investigated. We obtain sufficient conditions to ensure the permanence of the system. Also, we establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. In additional, an example together with its numerical simulation are presented to illustrate the feasibility of the main result.


Introduction
The importance of species competition in nature is obvious. For example, competition may be territory which is directly related to food resources. The widely used Lotka-Volterra system is considered as a disadvantage and that is the linearity. Ayala et al. [1] presented the following competitive system: ( ) Besides, Gopalsamy [2] discussed the continuous version with discrete delays, Tan and Liao [3] established the discrete time version with discrete delays, Xue et al. [4] proposed the discrete time version with in nite delays and single feedback control. Recently, the almost periodic solutions of discrete system with feedback controls has more extensively investigated (see [5][6][7][8][9][10]). Motivated by above, we study the following system with delays and feedback controls: ( ) For any almost periodic sequence {f (k)} de ned on Z, we use the notations f l = inf k∈Z f (k) and f u = sup k∈Z f (k).
We consider the solution of system (2) with the following initial conditions: ( ) The main objective of this paper is to investigate the existence of the almost periodic solutions of system (2). The set-up of this paper is as follows. In the coming section, we present some useful de nitions and lemmas. In the rest of this paper, we systematically explore the existence of a unique positive almost periodic solution, which is uniformly asymptotically stable. An example together with its numerical simulation are presented to show the feasibility of the main results. This study reveals that the feedback controls, to some extent, will destroy the stability of the system.

Preliminaries
In this section, rst we will mention several foundational de nitions and lemmas. Denote [a, b] Z = [a, b] Z and K = [−τ, +∞) Z , where τ is de ned as before.
De nition 1 (see [11]). A sequence x : Z → R k is called an almost periodic sequence if the ε-translation set of is a relatively dense set in Z for all ε > ; that is, for any given ε > , there exists an integer l(ε) > such that each discrete interval of length l(ε) contains an integer τ = τ(ε) ∈ E{ε, x} such that De nition 2 (see [11]). Let f : is said to be almost periodic in n uniformly for φ ∈ D, if for any ε > and any compact set S ⊂ D, there exists a positive integer l = l(ε, S), such that any interval of length l = l(ε, S) contains an integer τ, for which De nition 3 (see [12]). The hull of f , denoted by H(f ), is de ned by for some sequence τ k , where S is any compact set in D. Lemma 4 (see [13]). {x(n)} is an almost periodic sequence if and only if for any integer sequence {k i }, there exists a subsequence {k i } ⊂ {k i } such that x(n + k i ) converges uniformly on n ∈ Z as i → ∞. Furthermore, the limit sequence is also an almost periodic sequence. Lemma 5 (see [14]). Assume that r(n) > , {x(n)} satis es x(n) > , and x(n + ) ≤ x(n) exp r(n)( − ax(n)) , ( ) for n ∈ [n , +∞), where a is a positive constant. Then lim sup n→+∞ x(n) ≤ ar u exp(r u − ). ( ) Lemma 6 (see [14]). Assume that r(n) > , {x(n)} satis es x(n) > , and for n ∈ [n , +∞), lim sup n→+∞ x(n) ≤ x * , and x(n ) > , where a and x * are positive constants such that ax * > .

Permanence
Theorem 11. Assume that hold, then the system (2) is permanent. i.e., there exist positive constants m i , M i , h i and H i , such that for any positive solution (x (k), · · ·, xn(k), u (k), · · ·, un(k)) of system (2), one has (2), We can choose a su ciently small ε such that for large enough K > , we have As a direction corollary of Lemma 7, one has lim sup For above ε, there exists an integer K > K such that From (23), (27) and system (2), There exists a positive integer K > K + τ such that From (32) and system (2), we have Denoting Ω = {every solution of system (2) satisfying We can choose a su ciently small ε. From Theorem 11, there exists a positive integer N such that for k > N − tp and p = , , ···. For any positive integer q, it is easy to see that there exists a sequence {x ip (k) : p ≥ q} such that the sequence {x ip (k)} has a subsequence, also denoted by {x ip (k)}, converging on any nite interval A of Z + as p → ∞.
In fact, for any nite subset A = {l , l , · · ·, lm} ⊆ Z + , where m is a nite number, tp + l j > N (j = , , · · ·, m), when p is large enough. Therefore m i − ε ≤ x i (k + tp) ≤ M i + ε(i = , , · · ·, n); that is, x i (k + tp) are uniformly bounded when p is su ciently large. Next, for l ∈ A, we choose a subsequence {t ( ) p } of {tp} such that x i (k + t ( ) p ) uniformly converge on Z + for p su ciently large. Similar to the arguments of l , for l ∈ A, one can select a subsequence {t ( ) p } of {t ( ) p } such that x i (k + t ( ) p ) uniformly converge on Z + for p su ciently large. Repeating above-mentioned process, for lm ∈ A, one obtains a subsequence {t (m) p ) uniformly converge on Z + for p su ciently large. Based on the above, one selects the sequence {t (m) p } which is a subsequence of {tp}, still denoted by {tp}, then one gets x i (k + tp) → x * i uniformly in k ∈ A as k → ∞. So the conclusion holds truly due to the arbitrariness of A. Thus we have a sequence {y i (k)} such that for k ∈ Z + , ( ) which, together with (36), yields that ( )
( ) are two solutions of system (42) de ned on D, . Consider the product system of (42)

( ) Denoting
On the other side, Thus condition (i) in Lemma 9 is satis ed if we take a(x) = x and b(x) = ρx, where a, b ∈ C(R + , R + ). By using (3.2) in Ref. [16], for any W k , Z k ,W k ,Z k ∈ D, we have so, condition (ii) in Lemma 9 is also satis ed. By the mean value theorem, it derives that where θ i (k) lie between ω i (k) and z i (k), and Calculating the V(k) along with the solution of system (47), we have Denoting c(x) = λx ∈ C(R + , R + ), also, the condition in Remark 10 is satis ed. Therefore, system (42) has a unique uniformly asymptotically stable almost periodic solution denoted by (ω * (k), · · ·, ω * n (k), u * (k), · · ·, u * n (k)), which is equivalent to saying that the system (2) has a unique uniformly asymptotically stable almost periodic solution denoted by (x * (k), · · ·, x * n (k), u * (k), · · ·, u * n (k)). If the coe cients are bounded positive periodic sequences, we have the following corollary. Corollary 14. System (2) shows a unique positive periodic solution which is uniformly asymptotically stable under the same assumptions of Theorem 13.

Numerical Simulations
We give an example to check the feasibility of our results. Example 15. Consider the following system: ( ) Clearly, the assumption of Theorem 13 is satis ed, i.e., system (56) admits a unique uniformly asymptotically stable positive almost periodic solution. The numerical simulations support our results (see Figure 1).

Discussion
In this paper we consider a discrete competitive system with delays and feedback controls. By constructing Lyapunov functional and using mean value theorem, the conditions on the asymptotical stability of the positive almost periodic solution are established. Compared with the Theorem 10 in [3], it is easy to see that the feedback controls, to some extent, destroy the stability of the system.