Global asymptotical stability of almost periodic solutions for a non-autonomous competing model with time-varying delays and feedback controls

In this paper, we are concerned with a non-autonomous competing model with time delays and feedback controls. Applying the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. An example with its numerical simulations is given to illustrate the feasibility of our results.


Introduction
In recent years, the competitive prey-predator systems have been investigated by many scholars. The dynamical behavior such as permanence, global attractivity and global stability of continuous differential competitive prey-predator systems was extensively studied and many excellent results were reported. For example, Sarwardi et al. [27] focused on the local and the global stability and the bifurcations of a competitive preypredator system with a prey refuge, Ko and Ahn [16] discussed the global attractor, persistence and the stability of all non-negative equilibria of a diffusive one-prey and two-competing-predator system with a ratio-dependent functional response, Pan et al. [25] considered Gause , s principle in interspecific competition of the cyclic predatorprey system, Qun Liu et al. [23] addressed the global stability of a stochastic predatorprey system with infinite delays. For more related works on this topic, one can see [2,3,6,[9][10][11][12]14,15,18,20,24,26,28,29,31,34,35,38]. In 2001, Zhang et al. [36] investigated the permanence of the following non-autonomous competing model where x 1 (t), x 2 (t) represent the densities of two competing populations, respectively, r i , a i , b i , c i , k i : (0, +∞) → +∞, i = 1, 2, are continuous functions. In real world, the situation of competitive populations is often distributed by unpredictable force which can result in changes in parameters of systems such as the intrinsic growth rates, thus it is important to investigate models with control variables which are so-called disturbance functions [13,21,22,30,37]. Moreover, the unpredictable forces are seldom happened immediately, there is a time delay. Inspired by the above analysis, we modify system (1) as the following non-autonomous competing model with multiple delays and feedback controls where u 1 , u 2 are the control variables. Many scholars [7,30] argue that periodic phenomenon and almost periodic phenomena are widespread in nature, and almost periodic phenomenon is more frequent than periodic one. Hence, they have been the object of intensive analysis by numerous authors. In particular, there have been extensive results on existence of almost periodic solutions of differential equations in the literatures (see [1,8,17,19,33]). The main object of this paper is to investigate the almost periodic solutions of model (2). By the analysis on the almost periodic solutions of model (2), we can find the coexistence conditions of two competing populations, which can help human beings to control ecological balance. Also the analysis results enrich and develop the ecological theory. We believe that this investigation on the global stability of two competitive populations has important theoretical value and tremendous potential for application in administering process for both ecology and mathematical ecology. Let R and R + denote the set of all real numbers and nonnegative real numbers, respectively. Let f be a continuous bounded function defined on R and we set f u = sup t∈R f (t) and f l = inf t∈R f (t). In order to obtain our main results, throughout this paper, we assume that for i = 1,2, whereġ(t) expresses the derivative of g with respect to time t for any differentiable g.
We consider (2) together with the following initial condition The innovativeness of this article is listed as follows: • A non-autonomous competing model has been generalized to a non-autonomous competing model with time-varying delays and feedback controls. • Some new sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are presented. The research shows that the delay and feedback control term have an important effect on global asymptotical stability of almost periodic solutions of considered system. • Constructing an appropriate Lyapunov functional to handle the global asymptotical stability of almost periodic solutions is a challenging work. The analytic results of this article will enrich and develop the stability theory of delayed differential equations and also improve the earlier works.
The remainder of the paper is organized as follows: in Section 2, several useful lemmas are introduced. In Section 3, by applying the comparison theorem of the differential equation and constructing a suitable Lyapunov functional, a set of sufficient conditions which ensure the existence of a unique globally asymptotically stable nonnegative almost periodic solution for system (2) are established. In Section 4, a suitable example with its simulations is given to illustrate the feasibility of the main results. Brief conclusions are drawn in Section 5.

Preliminaries
In order to obtain the main result of this paper, we shall first state several lemmas which will be useful in the proving the main result.
where f is an m-vector, t is a real scalar and x is an n-vector, is said to be almost periodic in t uniformly with respect to x ∈ R n , if f (t, x) is continuous in t ∈ R and x ∈ R, and if for any ε > 0, it is possible to find a constant l(ε) > 0 such that in any interval of length l(ε) there exists a σ such that the inequality is satisfied for all t ∈ R and x ∈ R n . The number σ is called the ε-translation number or ε-almost period of f (t, x).

Definition 2.2 ([32]):
A function f : R → R is said to be asymptotically almost periodic function if there exists an almost-periodic function h(t) and a continuous function

Lemma 2.3: Any solution
, u 2 (t)) T be any positive solution of system (2). It follows from system (2) that It follows from (2) that for any ε > 0, there exists a T 1 > 0 such that for all t > T 1 , Letting ε → 0 in the above inequality leads to lim sup Proof: By (H1) and (H2), there exists a sequence {t n } with t n → +∞ as n → +∞ such that is uniformly bounded and equicontinuous on each bounded subset of R. Therefore by Ascoli , s theorem we can conclude that there exists a subsequence z(t + t k ) which converges to a continuous functionz(t) = (x 1 (t),ū 1 (t),x 2 (t),ū 2 (t)) T as k → +∞ uniformly on each bounded subset of R. Let T 0 ∈ R be given. We may assume that t k + T 0 ≥ T for all k. For t ≥ 0, we have According to Lebesgue' dominated convergence theorem, and letting k → +∞ in (6), we havex Since T 0 ∈ R is arbitrarily given, Thus z(t) ∈ . This completes the proof.

Stability of almost periodic solution
In this section, we will consider the stability of the almost periodic solution of system (2).

Remark 3.1:
Generally, it is difficult to construct a suitable Lyapunov functional to obtain the results we need. In this paper, we skillfully construct a suitable Lyapunov functional which does not appear in previous papers to achieve our goal. In this sense, the paper has novelty of techniques.

An example and its computer simulations
In this section, we present an example with its numerical simulations to demonstrate the analytical predictions obtained in the previous section.

Conclusions
In the present paper, a non-autonomous competing model with time-varying delays and feedback controls has been investigated. With the aid of the comparison theorem of differential equations and constructing a suitable Lyapunov functional, we establish some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the model. We find that under some suitable conditions, the two competitive species can come to a dynamical balance. The time delays and feedback control effect play an important role in affecting the fate of two competitive species. Our results are new and complement of the existed results in [36]. However, in real complex situation, considering that the discrete competing models are more appropriate to describe the dynamics relationship between two competitive species, thus the investigation on the discrete competitive species models has some theoretical and practical meanings and value to a certain extent. We will leave this topic for our future work.

Disclosure statement
No potential conflict of interest was reported by the authors.