Global Stability of Positive Periodic Solutions and Almost Periodic Solutions for a Discrete Competitive System

a 2 (t)x 2 (t)/(1 + d 1 (t)x 1 (t)), a 1 (t)x 1 (t)/(1 + d 2 (t)x 2 (t)) denote the competitive response function, respectively. All the coefficients above are continuous and bounded above and below by positive constants. The discrete-time systems governed by difference equations recently have been won wide-spread attention and applied in studying population growth, the transmission of tuberculosis and HIV/AIDS and influenza prevention and control (see [2, 3]), just because discrete-time models conform better to the reality than the continuous ones, especially for the populations with a short life expectancy or non-overlapping generations. In addition, some works about the bifurcation, chaos, and complex dynamical behaviors of the discrete specie systems have been done (see [4, 5]). In practice, according to the discrete data measured, the discrete-time models commonly provide efficient computational models of continuous models for numerical simulations (see [2, 3, 6–11]). Therefore, we derive the discrete analogue of system (1) by using the same discretization method (see [11]):

From an evolutionary perspective, because of the selectivity of species evolution, the periodically varying environments are of vital importance for survival of the fittest.For instance, any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes (see [11][12][13][14][15][16]). Therefore, the coefficients of many systems constructed in ecology are usually considered as periodic functions (see [12,13]).Not long ago, Wang (see [12]) studied a delayed predator-prey model with Hassell-Varley type functional responses and obtained the sufficient conditions for the existence of positive periodic solutions by applying the coincidence degree theorem.Many excellent results concerned with the discrete periodic systems are obtained (see [14][15][16]).
In nature, however, there hardly exists necessarily commensurate periods in the various environment components like seasonal weather change, food supplies, mating habits and harvesting, and so forth.Compared with the periodic systems, we can thus incorporate the assumption of almost periodicity of the coefficients of (1) to reflect the timedependent variability of the environment (see [6,[8][9][10]17]).Recently, Li et al. (see [18]) have proposed an almost periodic discrete predator-prey models with time delays and investigated permanence of the system and the existence of a unique uniformly asymptotically stable positive almost periodic sequence solution.Afterwards, by using Mawhins continuation theorem of the coincidence degree theory, reference [19] achieved some sufficient conditions for the existence of positive almost periodic solutions for a class of delay discrete models with Allee-effect.
Notice that the investigation of periodic solutions and almost periodic solutions is one of the most important topics in the qualitative theory of the difference equations.In this paper, based on the ideas mentioned above, for system (2), one carries out two main works.
(ii) Furtherly, one discusses the almost periodic solutions of system (2) with positive almost periodic coefficients.
The organization of this paper is as follows.In Section 2, we present some notations and preliminary lemmas.In Section 3, we seek sufficient conditions which ensure the existence and global stability of positive periodic solutions of system (2).In Section 4, we further investigate the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solutions for system (2) above.In Section 5, we present an example and its numerical simulations are carried out to illustrate the feasibility of our main results.In Section 6, a conclusion is given to conclude this work.

Notations and Preliminaries Lemmas
Throughout this paper, the notations below will be used: where {ℎ()} is a bounded sequence and Z + = {0, 1, 2, 3, . ..}. Denote by R, R + , Z, and Z + the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively.R 2 and R  are the cones of 2-dimensional and -dimensional real Euclidean spaces, respectively.Definition 1 (see [10] is referred to as the -translation number of ().
Lemma 3 (see [10]).{()} is an almost periodic sequence if and only if for any sequence {   } ⊂ Z there exists a subsequence {  } ⊂ {   } such that (+  ) converges uniformly on  ∈ Z as  → ∞.Thus, the limit sequence is also an almost periodic sequence.Furthermore, we consider the following almost periodic difference system: where ℎ : Z + × C  → R  , C  = { ∈ C : ‖‖ < }, and ℎ(, ) is almost periodic in  uniformly for  ∈ C  and is continuous in .
The product system of ( 7) is in the following form: and [20] obtained the following lemma, where ((, ), (, )) is a solution of (8).
Moreover, suppose that there exists a solution () of system (7) such that ‖  ‖ ≤  * <  for all  ∈ Z + ; then there exists a unique uniformly asymptotically stable almost periodic solution () of system (7) which satisfies |()| ≤  * .In particular, if ℎ(, ) is periodic of period , then system (7) has a unique uniformly asymptotically stable periodic solution with period .

Existence and Stability of Positive Periodic Solutions
Apparently, the permanence of system (2) can be obtained according to Lemmas 5 and 6.In the following, we will show the existence and stability of positive periodic solutions of system (2).To this end, let us assume that all the coefficients of system (2) are -periodic; namely, Lemma 7 (see [16]).If the assumption (10) holds, then system (2) has at least one strictly positive -periodic solution and is denoted by 2) is globally stable if each other solution ( 1 (),  2 ()) with positive initial value defined for all  > 0 satisfies lim Now, we present the main results.
Theorem 9. Let the following assumption and ( 10) hold; then the positive periodic solution of system ( 2) is globally stable.
Denote exp  1 () =  1 ()/ * 1 () and exp  2 () =  2 ()/  * 2 (); then we have which, according to the mean value, yields where all the constants  1 ,  2 ,  3 ,  4 ∈ (0, 1).Obviously, together with (14) we can find a sufficiently small  such that It follows from Lemmas 5 and 6 that there exists an  0 such that  >  0 ; we have Then one obtains the fact that both  * 1 () exp( Similar to the arguments as above, we must have We denote

Existence and Stability of Positive Almost Periodic Solutions
In this section, we discuss the existence of positive almost periodic solutions of system (2).
For any  ∈ Z + , assume that   +  ≥  0 when  is large enough.By an inductive argument of system (2) from   +  to  +   + , where  ∈ Z + , one obtains Hence, (25) yields Let  → +∞; one has It is easy to see that ( * 1 (),  * 2 ()) is a solution of system (2) on Z + for arbitrary , and Then we get (29) due to  which is an arbitrarily small positive constant: This completes the proof.
Finally, we are ready to state our main result in this section.

Example and Simulations
In this section, we only give the following example about almost periodic solutions to check feasibility of the assumptions of Theorem 11 considering that the simulation about periodic model is similar.(42) Clearly, the assumptions of Theorem 11 are satisfied and all the coefficients are appropriate.Hence, system (40) admits a unique uniformly asymptotically stable positive almost periodic solution.From Figure 1, we easily see that there exists a positive almost periodic solution ( * 1 (),  * 2 ()), and the 2-dimensional and 3-dimensional phase portraits of almost periodic system (40) are revealed in Figure 2, respectively.Figure 3 shows that any positive solution ( 1 (),  2 ()) tends to the almost periodic solution ( * 1 (),  * 2 ()).

Conclusions
In this paper, we consider a discrete two-species competitive model whose periodic solutions and almost periodic solutions are discussed, respectively.By the scale law and meanvalue theorem, a good understanding of the existence and stability of positive periodic solutions is gained.Furthermore, by constructing Lyapunov functions, the conditions on the asymptotic stability of the positive almost periodic solution are established.The assumption in (10) implies that the () should be suitably large.