A discrete almost periodic competitive system with delay is proposed and analyzed. The system admits a unique almost periodic solution, which is shown to be uniformly asymptotically stable by using the method of Lyapunov function. Some specific numerical examples are provided to verify our analytical results.

1. Introduction

As we know, a discrete time model governed by difference equations is more suitable than a corresponding continuous version when the populations have a short-life expectancy and nonoverlapping generations and can also provide efficient computation for numerical simulation (see [1–7]). On the other hand, the coefficients of model are changed owing to environmental variation. The assumption of almost periodicity of the coefficients is a way of incorporating the time-dependent variability of the environment if the various components of the environment are with incommensurable periods. One of the important ecological problems associated with the investigation of populations interaction in an almost periodic environment is the positive almost periodic solution which plays the significant role played by the equilibrium of the autonomous model (see [8–14]). In this contribution, we discuss the positive almost periodic solutions of a delay almost periodic competitive system in discrete time, and our motivation comes from the works of [8, 9, 15].

Firstly, let us introduce the following autonomous differential model which was proposed by Ayala et al. [15]:
(1)x1′(t)=x1(t)[r1-x1(t)-a1x2(t)-c1x22(t)],x2′(t)=x2(t)[r2-x2(t)-a2x1(t)-c2x12(t)],
where x1 and x2 stand for the population densities of two competing species. r1 and r2 are the intrinsic growth rates. ai and ci represent the interspecific competitive effects, i=1,2. Assume that a species needs some time to mature and the competition occurs after some time lag required for maturity of the species; a revised version was introduced by Gopalsamy [16]
(2)x1′(t)=x1(t)[r1-x1(t)-a1x2(t-τ2)-c1x22(t-τ2)],x2′(t)=x2(t)[r2-x2(t)-a2x1(t-τ1)-c2x12(t-τ1)].
Furthermore, considering the biological parameters naturally being subject to almost periodic fluctuation in time and the influence of many generations on the density of species population, we establish the following model:
(3)x1n+1=x1nexpr1(n)-x1na1(n)x2(n-τ2)-c1(n)x22(n-τ2)-a1(n)x2(n-τ2)-c1(n)x22(n-τ2),x2n+1=x2(n)expr2n-x2n-a2(n)x1(n-τ1)-c2(n)x12(n-τ1)-a2(n)x1(n-τ1)-c2(n)x12(n-τ1),n=0,1,2,…, and the initial conditions satisfy
(4)x1(θ)=ϕ1(θ)≥0,x2(θ)=ϕ2(θ)≥0,ϕ10>0,ϕ20>0,θ∈-τ,-τ+1,…,0,τ=max{τi,i=1,2}.
Here xi(n) and ri(n) are, respectively, the densities and intrinsic growth rates of species xi at the nth generation. ai(n) and ci(n) measure the interspecific influence of the (n-τj)th generation of species xj on species xi(i,j=1,2;i≠j). The delays τ1 and τ2 are positive integers and the coefficients {ri(n)}, {ai(n)}, and {ci(n)} are bounded positive almost periodic sequences, i=1,2.

The rest part of this paper is organized as follows. In Section 2, some preliminaries are given. Sufficient conditions for the uniformly asymptotic stability of a unique positive almost periodic solution for the system are established in Section 3. In final section, some specific numerical examples are carried out to illustrate the feasibilities of our theoretical results.

2. Preliminaries

This section is concerned with some notations, definitions, and lemmas which will be used for our main results.

Let R, R+, Z, and Z+ be, respectively, the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers. Assign R2 and Rk which denote the cone of 2-dimensional and k-dimensional real Euclidean space, respectively. Denote [a,b]Z=[a,b]⋂Z, a,b∈Z, and then K=[-τ,+∞)Z, where τ is defined in (4). For simplicity in the following discussion, we use the notations AU=supn∈Z+{A(n)}, AL=infn∈Z+{A(n)}, where {A(n)} is an almost periodic sequence.

Definition 1 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

A sequence x:Z→Rk is called an almost periodic sequence if the ɛ-translation set of x(5)Eɛ,x:=τ∈Z:xn+τ-xn<ɛ,∀n∈Z
is a relatively dense set in Z for all ɛ>0, that is, for any given ɛ>0, there exists an integer l(ɛ)>0 such that each discrete interval of length l(ɛ) contains an integer τ=τ(ɛ)∈E{ɛ,x} such that |x(n+τ)-x(n)|<ɛ, ∀n∈Z. τ is called theɛ-translation number of x(n).

Definition 2 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let f:Z×D→Rk, where D is an open set in C:{φ:[-τ,0]Z→Rk}. f(n,φ) is said to be almost periodic in n uniformly for φ∈D, if for any ɛ>0 and any compact set S in D, there exists a positive integer l(ɛ,S) such that any interval of length l(ɛ,S) contains an integer τ for which
(6)fn+τ,φ-fn,φ<ɛ,∀n∈Z,φ∈S,
and τ is called the ɛ-translation number of f(n,φ).

Lemma 3 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

{x(n)} is an almost periodic sequence if and only if for any sequence {tk′}⊂Z there exists a subsequence {tk}⊂{tk′} such that x(n+tk) converges uniformly on n∈Z as k→+∞.

Zhang and Zheng [12] considered the following almost periodic delay difference system:
(7)x(n+1)=f(n,xn),n∈Z+,
where f:Z+×CB→Rk, CB={φ∈C:∥φ∥<B}, C={φ:[-τ,0]Z→Rk} with φ=sups∈[-τ,0]Z|φ(s)|, f(n,φ) is almost periodic in n uniformly for φ∈CB and is continuous in φ, while xn∈CB is defined as xn(s)=x(n+s) for all s∈[-τ,0]Z. The product system of (7) is as follows:
(8)xn+1=fn,xn,y(n+1)=f(n,yn).
A discrete Lyapunov function of (8) is a function V:Z+×CB×CB→R+ which is continuous in its second and third variables. Define the difference of V along the solution of system (8) by
(9)ΔV(8)(n,φ,ψ)=V(n+1,xn+1(n,φ),yn+1(n,ψ))-V(n,φ,ψ),
where (x(n,φ),y(n,ψ)) is a solution of system (8) through (n,(φ,ψ)),φ,ψ∈CB.

Lemma 4 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Suppose that there exists a Lyapunov function V(n,φ,ψ) satisfying the following conditions.

a(|φ(0)-ψ(0)|)≤V(n,φ,ψ)≤b(∥φ-ψ∥), where a,b∈P with P={a:[0,+∞)→[0,+∞)∣a(0)=0 and a(u) is continuous, increasing in u}.

|V(n,φ1,ψ1)-V(n,φ2,ψ2)|≤L(φ1-φ2+ψ1-ψ2), where L>0 is a constant.

ΔV(n,φ,ψ)≤-αV(n,φ,ψ), where 0<α<1 is a constant.

Moreover, if there exists a solution x(n) of system (7) such that xn≤B*≤B for all n∈Z+, then there exists a unique uniformly asymptotically stable almost periodic solution q(n) of system (7) which satisfies |q(n)|≤B* for all n∈K. In particular, if f(n,φ) is periodic of period ω, then system (7) has a unique uniformly asymptotically stable periodic solution of period ω.
Remark 5 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Condition (III) of Lemma 4 can be replaced by

(III)′ΔV(n,φ,ψ)≤-β(|φ(0)-ψ(0)|), where β∈{c:[0,+∞)→[0,+∞)∣c is continuous, c(0)=0 and c(s)>0 for s>0}.

Lemma 6 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Assume that x(n)>0 and
(10)xn+1≤xnexprn1-axnforn∈n1,+∞,
where r(n) is a bounded positive sequence and a is a positive constant. Then
(11)limsupn→+∞x(n)≤1arUexp(rU-1).

Lemma 7 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Assume that {x(n)} satisfies
(12)x(n+1)≥x(n)exp(r(n)(1-ax(n))),n≥N0,limsupn→+∞x(n)≤M and x(N0)>0, where r(n) is a bounded positive sequence and a is a positive constant such that aM>1 and N0∈Z+. Then
(13)liminfn→+∞x(n)≥1aexp(rU(1-aM)).

If Δi>0, then any positive solution (x1(n),x2(n)) of system (3) satisfies hi≤liminfn→+∞xi(n)≤limsupn→+∞xi(n)≤Hi, i=1,2.

Proof.

According to the first equation of system (3), one has
(15)x1n+1≤x1(n)exp[r1(n)-x1(n)]≤x1(n)[r1(n)(1-1r1Ux1(n))].
It follows from Lemma 6 that
(16)limsupn→+∞x1(n)≤exp(r1U-1)=H1.
Analogously, from the second equation of system (3) we obtain that
(17)limsupn→+∞x2(n)≤exp(r2U-1)=H2.
Assigning a positive constant ɛ arbitrarily small, it follows from (16) and (17) that there exists a large enough n1>0 such that for all n≥n1,
(18)xi(n)≤Hi+ɛ,i=1,2.
From the first equation of system (3), for n≥n1+τ,
(19)x1n+1≥x1nexpr1L-x1n-a1UH2+ɛ-c1UH2+ɛ2=x1nexpr1L-a1UH2+ɛ-c1UH2+ɛ21-x1nr1L-a1UH2+ɛ-c1UH2+ɛ2×1-x1nr1L-a1UH2+ɛ-c1UH2+ɛ2.
Using Lemma 7, for n≥n1+τ, one has
(20)liminfn→+∞x1n≥r1L-a1UH2+ɛ-c1UH2+ɛ2×exp[r1L-a1U(H2+ɛ)-c1UH2+ɛ2-H1].
Setting ɛ→0, it follows that
(21)liminfn→+∞x1n≥(r1L-a1UH2-c1UH22)×exp[r1L-a1UH2-c1UH22-H1]=h1.
Similar to the above argument, from the second equation of system (3) we can obtain that
(22)liminfn→+∞x2n≥(r2L-a2UH1-c2UH12)×exp[r2L-a2UH1-c2UH12-H2]=h2.
This completes the proof.

Denote Θ by
(23)Θ=everysolutionofsystem(3)satisfyinghi≤xin≤Hi,i=1,2∀n∈K.

Lemma 9.

If Δi>0, then Θ≠∅, where Δi are defined in (14).

Proof.

Since {ri(n)}, {ai(n)}, and {ci(n)}, i=1,2 are almost periodic sequences, there exists a positive integer sequence {tk} with tk→+∞ as k→+∞ such that
(24)rin+tk⟶rin,ain+tk⟶ain,cin+tk⟶cin,i=1,2,
as k→+∞ for n∈Z+. By (16), (17), (21), and (22), for any sufficiently small ɛ>0, there exists a positive integer n0 such that for all n>n0,
(25)hi-ɛ≤xi(n)≤Hi+ɛ.
Assign
(26)xikn=xin+tkforn≥n0+τ-tk,k=1,2,….
For any positive integer q, it is obvious that there exists a sequence {xik(n):k≥q} such that the sequence {xik(n)} has a subsequence, also denoted by {xik(n)}, converging on any definite interval of K as k→+∞. Thus, there is a sequence {wi(n)} satisfying
(27)xik(n)⟶wi(n),forn∈Kask⟶+∞,
which, together with (24), yields that
(28)x1kn+1=x1k(n)expx2k2r1(n+tk)-x1k(n)-a1n+tkx2kn-τ2-c1(n+tk)x2k2(n-τ2),x2kn+1=x2k(n)expx1k2r2(n+tk)-x2k(n)-a2(n+tk)x1k(n-τ1)-c2(n+tk)x1k2(n-τ1),
we have from (24), (27), and (28) that
(29)w1n+1=w1(n)expw22r1n-w1n-a1nw2n-τ2-c1(n)w22(n-τ2),w2n+1=w2(n)expr2(n)-w2(n)w12-a2(n)w1(n-τ1)-c2(n)w12(n-τ1).
It is easy to see that (w1(n),w2(n)) is a solution of system (3) and hi-ɛ≤wi(n)≤Hi+ɛ for n∈K. Since ɛ is small enough, we derive that hi≤wi(n)≤Hi, i=1,2 for n∈K. This completes the proof.

3. Main Result

In this section, we focus on the result of the uniformly asymptotic stability of positive almost periodic solutions of system (3).

If Δi>0 and λi>0, system (3) has a unique positive almost periodic solution which is uniformly asymptotically stable, where Δi are defined in (14).

Proof.

We make the change of variables
(31)y1(n)=lnx1(n),y2(n)=lnx2(n),
and then system (3) can be rewritten as
(32)y1n+1=y1(n)+r1(n)-ey1(n)-a1(n)ey2(n-τ2)-c1(n)e2y2(n-τ2),y2n+1=y2(n)+r2(n)-ey2(n)-a2(n)ey1(n-τ1)-c2(n)e2y1(n-τ1).
It follows from Lemma 9 that there exists a bounded solution (y1(n),y2(n)) of system (32) satisfying
(33)lnh1≤y1n≤lnH1,lnh2≤y2n≤lnH2,∀n∈K,
which implies that |y1(n)|≤B1, |y2(n)|≤B2, where Bi=max{|lnhi|,|lnHi|}, i=1,2. Assign
(34)Yns=y1n+s,y2n+s,Zns=z1n+s,z2n+s,n∈Z+,s∈-τ,0Z
are two solutions of system (32) defined on D, where
(35)D=(y1(n),y2(n))∣D=5(y1(n),y2(n))∣lnhi≤yin≤lnHi,i=1,2,n∈K.
Definning
(36)Yn(s)=y1n+s,y2n+s=sups∈[-τ,0]Zy1n+s+y2n+s,
where (y1(n+s),y2(n+s))∈R2, we have
(37)Yn≤A,Zn≤A,
where A=B1+B2.

Let us consider the associate product system of system (32)(38)y1n+1=y1(n)+r1(n)-ey1(n)-a1(n)ey2(n-τ2)-c1(n)e2y2(n-τ2),y2n+1=y2(n)+r2(n)-ey2(n)-a2(n)ey1(n-τ1)-c2(n)e2y1(n-τ1),z1n+1=z1(n)+r1(n)-ez1(n)-a1(n)ez2(n-τ2)-c1(n)e2z2(n-τ2),z2n+1=z2(n)+r2(n)-ez2(n)-a2(n)ez1(n-τ1)-c2(n)e2z1(n-τ1).
Construct the following Lyapunov function V(n)=V(n,Yn,Zn) defined on Z+×D×D(39)Vn=V(n,Yn,Zn)=y1n-z1n+y2n-z2n+∑ζ=n-τ1n-1a2UH1+2c2UH12y1ζ-z1ζ+∑ζ=n-τ2n-1a1UH2+2c1UH22y2ζ-z2ζ.
Apparently,
(40)Yn0-Zn0≤V(n)≤y1n-z1n+y2n-z2n+∑ζ=n-τn-1a2UH1+2c2UH12y1ζ-z1ζ+∑ζ=n-τn-1a1UH2+2c1UH22y2ζ-z2ζ≤y1n-z1n+y2n-z2n+∑ζ=n-τn-1σy1ζ-z1ζ+y2ζ-z2ζ≤(1+τσ)sups∈[-τ,0]Zy1n+s-z1n+s+y2n+s-z2n+s=ρYn-Zn,
where
(41)Yn0-Zn0=y1n-z1n2+y2n-z2n21/2≤y1n-z1n+y2n-z2n,(42)ρ=1+τσ,σ=max{a2UH1+2c2UH12,a1UH2+2c1UH22}.
Denote
(43)ax=x,b=ρx,a,b∈C(R+,R+),
and thus condition (I) in Lemma 4 is satisfied.

For any y,z,y¯,z¯, we obtain that
(44)y-z-y¯-z¯=y-z-y¯-z¯,ify-z≥y¯-z¯,y¯-z¯-y-z,ify¯-z¯>y-z≤y-y¯+z¯-z,ify-z≥y¯-z¯,y¯-y+z-z¯,ify¯-z¯>y-z≤y-y¯+z¯-z,ify-z≥y¯-z¯,y¯-y+z-z¯,ify¯-z¯>y-z=y-y¯+z-z¯.
Hence, for any Yn,Zn,Y¯n,Z¯n∈D, we obtain, by (44), that
(45)Vn,Yn,Zn-Vn,Y¯n,Z¯n=y1n-z1n+y2n-z2n+∑ζ=n-τ1n-1a2UH1+2c2UH12y1ζ-z1ζ+∑ζ=n-τ2n-1a1UH2+2c1UH22y2ζ-z2ζ-y¯1n-z¯1n-y¯2n-z¯2n-∑ζ=n-τ1n-1a2UH1+2c2UH12y¯1ζ-z¯1ζ-∑ζ=n-τ2n-1a1UH2+2c1UH22y¯2ζ-z¯2ζ≤y1(n)-z1(n)+y2n-z2n-y¯1n-z¯1n-y¯2n-z¯2n+∑ζ=n-τ1n-1(a2UH1+2c2UH12)×y1ζ-z1ζ-y¯1ζ-z¯1ζ+∑ζ=n-τ2n-1(a1UH2+2c1UH22)×y2(ζ)-z2(ζ)-y¯2ζ-z¯2ζ≤y1n-y¯1n+y2n-y¯2ny1n-y¯1n+z1n-z¯1n+z2n-z¯2n+∑ζ=n-τn-1a2UH1+2c2UH12×y1ζ-y¯1ζ+z1ζ-z¯1ζ+∑ζ=n-τn-1(a1UH2+2c1UH22)×y2ζ-y¯2ζ+z2ζ-z¯2ζ≤y1n-y¯1n+y2n-y¯2ny1n-y¯1n+z1n-z¯1n+z2n-z¯2n+∑ζ=n-τn-1σy1ζ-y¯1ζ+z1ζ-z¯1ζ+y2ζ-y¯2ζ+z2ζ-z¯2ζ≤(1+τσ)sups∈[-τ,0]Zy1n+s-y¯1n+s+y2n+s-y¯1n+s+z1n+s-z¯1n+sy1n.s-y¯1n+s+z2n+s-z¯2n+s≤ρYn-Y¯n+Zn-Z¯n,
where ρ and σ are defined in (42). That is, condition (II) in Lemma 4 is also satisfied.

It follows from the mean-value theorem that we find
(46)eyi(n)-ezi(n)=eθi(n)(yi(n)-zi(n)),eyi(n-τi)-ezi(n-τi)=eηi(n-τi)(yi(n-τi)-zi(n-τi)),e2yi(n-τi)-e2zi(n-τi)=2e2ςi(n-τi)(yi(n-τi)-zi(n-τi)),i=1,2,
where θi(n) lie between yi(n) and zi(n) and ηi(n-τi), ςi(n-τi) all lie between yi(n-τi) and zi(n-τi), respectively. Then
(47)lnhi≤θin,ηin-τi,ςin-τi≤lnHi,n∈Z+.
This one together with system (38) and (46) yields that
(48)y1n+1-z1n+1+y2n+1-z2n+1=y1(n)-z1(n)-eθ1(n)(y1(n)-z1(n))-a1(n)eη2(n-τ2)(y2(n-τ2)-z2(n-τ2))-2c1(n)e2ς2(n-τ2)(y2(n-τ2)-z2(n-τ2))+y2(n)-z2(n)-eθ2(n)(y2(n)-z2(n))-a2(n)eη1(n-τ1)(y1(n-τ1)-z1(n-τ1))-2c2(n)e2ς1(n-τ1)(y1(n-τ1)-z1(n-τ1))≤1-eθ1n·y1n-z1n+(a1(n)eη2(n-τ2)+2c1(n)e2ς2(n-τ2))×y2n-τ2-z2n-τ2+1-eθ2n·y2n-z2n+(a2(n)eη1(n-τ1)+2c2(n)e2ς1(n-τ1))×y1n-τ1-z1n-τ1≤1-eθ1n·y1n-z1n+a1UH2+2c1UH22y2n-τ2-z2n-τ2+1-eθ2n·y2n-z2n+a2UH1+2c2UH12y1n-τ1-z1n-τ1.
Combining with (48) and calculating the ΔV of V along the solution of (38), one has
(49)ΔVn=V(n+1)-V(n)=y1n+1-z1n+1+y2n+1-z2n+1+∑ζ=n+1-τ1na2UH1+2c2UH12y1ζ-z1ζ+∑ζ=n+1-τ2na1UH2+2c1UH22y2ζ-z2ζ-y1n-z1n-y2n-z2n-∑ζ=n-τ1n-1a2UH1+2c2UH12y1ζ-z1ζ-∑ζ=n-τ2n-1a1UH2+2c1UH22y2ζ-z2ζ≤1-eθ1n·y1n-z1n+a1UH2+2c1UH22y2n-τ2-z2n-τ2+1-eθ2n·y2n-z2n+a2UH1+2c2UH12y1n-τ1-z1n-τ1+∑ζ=n+1-τ1na2UH1+2c2UH12y1ζ-z1ζ+∑ζ=n+1-τ2na1UH2+2c1UH22y2ζ-z2ζ-y1n-z1n-y2n-z2n-∑ζ=n-τ1n-1a2UH1+2c2UH12y1ζ-z1ζ-∑ζ=n-τ2n-1a1UH2+2c1UH22y2ζ-z2ζ=1-eθ1n-1+a2UH1+2c2UH12·y1n-z1n+{1-eθ2n-1+(a1UH2+2c1UH22)}·y2n-z2n≤-{1-max(1-h1,1-H1)-(a2UH1+2c2UH12)}·y1n-z1n-{1-max(1-h2,1-H2)-(a1UH2+2c1UH22)}·y2n-z2n=-λ1y1n-z1n-λ2y2n-z2n≤-β(y1n-z1n+y2n-z2n)≤-βy1n-z1n2+y2n-z2n21/2=-β(Yn0-Zn0),
where β=min{λ1,λ2}. Since λi>0, β>0. Denote c∈C(R+,R+), c(x)=βx; therefore, the condition in Remark 5 is satisfied. From Lemma 4 and Remark 5, system (32) has a unique uniformly asymptotically stable almost periodic solution denoted by (y1*(n),y2*(n)), which is equivalent to saying that system (3) has a unique uniformly asymptotically stable positive almost periodic solution denoted by (x1*(n),x2*(n)). This proof of Theorem 10 is completed.

If the coefficients {ri(n)}, {ai(n)}, and {ci(n)} are bounded positive periodic sequences, then system (3) becomes a periodic version. Applying Lemma 4 and Theorem 10, Corollary 11 is obtained directly.

Corollary 11.

Periodic system (3) shows a unique positive periodic solution which is uniformly asymptotically stable under the same assumptions of Theorem 10.

4. Numerical Simulations

In this section, we give two specific numerical examples to verify our analytical results, that is, Theorem 10 and Corollary 11.

Example 12.

Consider the following delay discrete almost periodic competitive system:
(50)x1n+1=x1(n)expx22(0.87+0.02sin(2nπ))-x1(n)-(0.025-0.001cos(2nπ))x2(n-2)-(0.015+0.002sin(2nπ))x22(n-2),x2n+1=x2(n)expx12(0.95+0.03cos(2nπ))-x2(n)-(0.020+0.001sin(3nπ))x1(n-1)-(0.027+0.002sin(2nπ))x12(n-1).
A computation shows that
(51)r1U=0.89,r1L=0.85,r2U=0.98,r2L=0.92,a1U=0.026,a1L=0.024,a2U=0.021,a2L=0.019,c1U=0.017,c1L=0.013,c2U=0.029,c2L=0.025,
and then we have
(52)H1≈0.8958,H2≈0.9802,Δ1≈0.8082,Δ2≈0.8779,h1≈0.7404,h2≈0.7925,Δ1>0,Δ2>0.
Moreover,
(53)λ1=1-max(1-h1,1-H1)-(a2UH1+2c2UH12)≈0.6750>0,λ2=1-max(1-h2,1-H2)-(a1UH2+2c1UH22)≈0.7343>0.
It is easy to see that the assumptions of Theorem 10 are satisfied; that is to say, system (50) has a unique positive almost periodic solution denoted by (x1*(t),x2*(t)) which is uniformly asymptotically stable (see Figure 1), and the two-dimensional and three-dimensional phase portraits are displayed in Figure 2, respectively. In Figure 3, any positive solution denoted by (x1(n),x2(n)) tends to the above almost periodic solution (x1*(n),x2*(n)).

Positive almost periodic solution of system (50). (a) Time-series of x1*(n) with initial values x1*(-1)=0.79 and x1*(0)=0.80 for n∈[0,100]. (b) Time-series of x2*(n) with initial values x2*(-2)=0.85, x2*(-1)=0.80, and x2*(0)=0.79 for n∈[0,100].

Phase portrait. (a) Two-dimensional phase portrait of x1*(n) and x2*(n) with initial values x1*(-1)=0.79 and x1*(0)=0.80 and x2*(-2)=0.85, x2*(-1)=0.80, and x2*(0)=0.79 for n∈[0,100]. (b) Three-dimensional phase portrait of n,x1*(n) and x2*(n) with the above initial values for n∈[0,100].

Uniformly asymptotic stability. (a) Time-series of x1*(n) and x1(n) with initial values x1*(-1)=0.79 and x1*(0)=0.80 and x1(-1)=0.83 and x1(0)=0.92 for n∈[0,100], respectively. (b) Time-series of x2*(n) and x2(n) with initial values x2*(-2)=0.85, x2*(-1)=0.80, and x2*(0)=0.79 and x2(-2)=0.81, x2(-1)=0.79, and x2(0)=0.97 for n∈[0,100], respectively.

Example 13.

Consider the following delay discrete periodic competitive system:
(54)x1n+1=x1(n)expx22(0.87+0.02sin(nπ))-x1(n)-(0.025-0.001cos(nπ))x2(n-2)-(0.015+0.002sin(nπ))x22(n-2),x2n+1=x2(n)expx12(0.95+0.03cos(nπ))-x2(n)-(0.020+0.001sin(nπ))x1(n-1)-(0.027+0.002sin(nπ))x12(n-1).
Analogously, we can see that system (54) satisfies the assumptions of Theorem 10. From Corollary 11, there exists a unique positive periodic solution of system (54) which is uniformly asymptotically stable. From Figure 4, system (54) shows a positive periodic solution denoted by (x1+(t),x2+(t)), and the two-dimensional and three-dimensional phase portraits are displayed in Figure 5, respectively. Figure 6 shows that any positive solution denoted by (x1(n),x2(n)) tends to the above periodic solution (x1+(n),x2+(n)).

Positive periodic solution of system (54). (a) Time-series of x1+(n) with initial values x1+(-1)=0.79 and x1+(0)=0.80 for n∈[0,100]. (b) Time-series of x2+(n) with initial values x2+(-2)=0.85, x2+(-1)=0.80, and x2+(0)=0.79 for n∈[0,100].

Phase portrait. (a) Two-dimensional phase portrait of x1+(n) and x2+(n) with initial values x1+(-1)=0.79 and x1+(0)=0.80 and x2+(-2)=0.85, x2+(-1)=0.80, and x2+(0)=0.79 for n∈[0,100]. (b) Three-dimensional phase portrait of n,x1+(n) and x2+(n) with the above initial values for n∈[0,100].

Uniformly asymptotic stability. (a) Time-series of x1+(n) and x1(n) with initial values x1+(-1)=0.79 and x1+(0)=0.80 and x1(-1)=0.83 and x1(0)=0.92 for n∈[0,100], respectively. (b) Time-series of x2+(n) and x2(n) with initial values x2+(-2)=0.85, x2+(-1)=0.80, and x2+(0)=0.79 and x2(-2)=0.81, x2(-1)=0.79, and x2(0)=0.97 for n∈[0,100], respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their valuable comments that greatly improved the presentation of this paper. They are also grateful to Professor Zhijun Liu for his suggestions and doctoral student Qinglong Wang for his help in numerical simulations.

LiuZ.ChenL.Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delaysAgarwalR. P.MurrayJ. D.ZhouZ.ZouX.Stable periodic solutions in a discrete periodic logistic equationFanM.WangK.Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey systemWendiW. D.MuloneG.SalemiF.SaloneV.Global stability of discrete population models with time delays and fluctuating environmentXuR.ChaplainM. A.DavidsonF. A.Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delaysWangQ. L.LiuZ. J.Uniformly asymptotic stability of almost periodic solutions for a delay difference system of plankton allelopathyLiY.ZhangT.Almost periodic solution for a discrete hematopoiesis model with time delayLiZ.ChenF.HeM.Almost periodic solutions of a discrete Lotka-Volterra competition system with delaysSongY.Positive almost periodic solutions of nonlinear discrete systems with finite delayZhangS.ZhengG.Almost periodic solutions of delay difference systemsNiuC.ChenX.Almost periodic sequence solutions of a discrete Lotka-Volterra competitive system with feedback controlZhangT.LiY.YeY.Persistence and almost periodic solutions for a discrete fishing model with feedback controlAyalaF. J.GilpinM. E.EhrenfeldJ. G.Competition between species: theoretical models and experimental testsGopalsamyK.RongY.JialinH.The existence of almost periodic solutions for a class of differential equations with piecewise constant argumentYangX.Uniform persistence and periodic solutions for a discrete predator-prey system with delays