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

Authors: Heping Ma; Jianguo Gao; Lingling Xie
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

A discrete two-species
competitive model is investigated. By using some preliminary
lemmas and constructing a Lyapunov function, the existence and uniformly
asymptotic stability of positive almost periodic solutions of the system are
derived. In addition, an example and numerical simulations are
presented to illustrate and substantiate the results of this paper.


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## Body

## 1. Introduction

Gopalsamy has presented the following Lotka-Volterra competitive system with continuous time version in 1992 (see [1]):
(1)x1′t=x1tr1t-b1tx1t-a2tx2t1+d1tx1t,x2′t=x2tr2t-b2tx2t-a1tx1t1+d2tx2t,hhhhhhhhhhhhhhhhhhhihhhhhhhhhhhn=0,1,2,…,
where x1(t),x2(t) represent the population densities of two competing species; r1(t),r2(t) are the intrinsic growth rates of species; b1(t),b2(t) stand for the rates of intraspecific competition of the first and second species, respectively; a2tx2t/(1+d1(t)x1(t)),a1(t)x1(t)/(1+d2tx2t) 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]):
(2)x1n+1=x1nexp⁡r1n-b1nx1n-a2nx2n1+d1nx1n,x2n+1=x2nexp⁡r2n-b2nx2n-a1nx1n1+d2nx2n,hhhhhhhhhhhhhhhhhhhhhhihhhhhhhhhhhn=0,1,2,…,
where xi(n) (i=1,2; i≠j) denote the densities of species xi at the nth generation, ri(n) stand for the natural growth rates of species xi at the nth generation, bi(n) represent the self-inhibition rate, respectively, and aj(n) and di(n) are the interspecific effects of the nth generation of species xi on species xj.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 [11–16]). 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 [14–16]).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 time-dependent variability of the environment (see [6, 8–10, 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.(i)
Assume that all the coefficients{ri(n)}, {bi(n)}, {ai(n)}, and {di(n)} are bounded nonnegative periodic sequences. We explore the existence and global stability of positive periodic solutions of system (2) with positive periodic coefficients.(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 Section2, 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.


## 2. Notations and Preliminaries Lemmas

Throughout this paper, the notations below will be used:(3)hu=sup⁡n∈Z+hn,hl=inf⁡n∈Z+hn,
where {h(n)} is a bounded sequence and Z+={0,1,2,3,…}.Denote byR, R+, Z, and Z+ the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. R2 and Rk are the cones of 2-dimensional and k-dimensional real Euclidean spaces, respectively.Definition 1 (see [10]).
A sequencey:Z→Rk is called an almost periodic sequence provided that the following ɛ-translation set of y(4)Iɛ,y≔τ∈Z:yn+τ-yn<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 a τ=τ(ɛ)∈I{ɛ,y} such that
(5)yn+τ-yn<ɛ,∀n∈Z;τ is referred to as the ɛ-translation number of y(n).Definition 2 (see [10]).
Supposeh:Z×A→Rk, where A is an open set in Rk. h(n,x) is said to be almost periodic in n uniformly for x∈A or uniformly almost periodic for short, if for any ɛ>0 and any compact set S in A there exists a positive integer l(ɛ,S) such that any interval of length l(ɛ,S) contains an integer τ for which
(6)hn+τ,x-hn,x<ɛ,∀n∈Z,x∈S;τ is called the ɛ-translation number of h(n,x).Lemma 3 (see [10]).
{
x
(
n
)
} is an almost periodic sequence if and only if for any sequence {pt′}⊂Z there exists a subsequence {pt}⊂{pt′} such that x(n+pt) converges uniformly on n∈Z as t→∞. Thus, the limit sequence is also an almost periodic sequence.Furthermore, we consider the following almost periodic difference system:(7)xn+1=hn,xn,n∈Z+,
where h:Z+×CB→Rk, CB={x∈C:x<B}, and h(n,x) is almost periodic in n uniformly for x∈CB and is continuous in x.The product system of (7) is in the following form:
(8)xn+1=hn,xn,yn+1=hn,yn,
and [20] obtained the following lemma, where (x(n,ϕ),y(n,ψ)) is a solution of (8).Lemma 4 (see [20]).
Suppose there exists a Lyapunov functionV(n,ϕ,ψ) defined for n∈Z+, ϕ<B, and ψ<B satisfying the following conditions:(1)
α(|ϕ-ψ|)≤V(n,ϕ,ψ)≤β(ϕ-ψ), where α,β∈p with p={η:[0,∞)→[0,∞)∣η(0)=0 and η(ν) is continuous, increasing in ν};(2)
|V(n,ϕ1,ψ1)-V(n,ϕ2,ψ2)|≤L(ϕ1-ϕ2+ψ1-ψ2), where L>0 is a constant;(3)
ΔV(8)(n,ϕ,ψ)≤-γV(n,ϕ,ψ), where 0<γ<1 is a constant and ΔV8(n,ϕ,ψ)=V(n+1,xn+1(n,ϕ),yn+1(n,ψ))-V(n,ϕ,ψ).Moreover, suppose that 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*. In particular, if h(n,ϕ) is periodic of period ω, then system (7) has a unique uniformly asymptotically stable periodic solution with period ω.Lemma 5 (see [21]).
Any positive solution(x1(n),x2(n)) of system (2) satisfies
(9)limsup⁡n→+∞xin≤exp⁡riu-1bil≡Mi,i=1,2.Lemma 6 (see [21]).
Let system (2) satisfy the following assumptions:
(10)min⁡r1l-a2uM2,r2l-a1uM1>0.
Then, any positive solution (x1(n),x2(n)) of system (2) satisfies
(11)xin≥ril-ajuMjbiuexp⁡ril-ajuMj-biuMi≡mi,i≠j;i,j=1,2.


## 3. 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 δ-periodic; namely,
(12)rin+δ=rin,ain+δ=ain,bin+δ=bin,din+δ=din,i=1,2.Lemma 7 (see [16]).
If the assumption (10) holds, then system (2) has at least one strictly positive δ-periodic solution and is denoted by (x1*(n),x2*(n)).Definition 8.
A positive periodic solution(x1*(n),x2*(n)) of system (2) is globally stable if each other solution (x1(n),x2(n)) with positive initial value defined for all n>0 satisfies
(13)limn→+∞x1n-x1*n=0,limn→+∞x2n-x2*n=0.
Now, we present the main results.Theorem 9.
Let the following assumption(14)ξ1=max⁡1+a2ud1uM1M2-b1lm1,hhhhhhhh1+a2ld1lm1m2-b1uM1+a2uM2+a2ud1uM1M2<1,ξ2=max⁡1+a1ud2uM1M2-b2lm2,hhhhhhhh1+a1ld2lm1m2-b2uM2+a1uM1+a1ud2uM1M2<1
and (10) hold; then the positive periodic solution of system (2) is globally stable.Proof.
Let(x1*(n),x2*(n)) be a positive periodic solution of system (2).
Denoteexp⁡z1(n)=x1(n)/x1*(n) and exp⁡z2(n)=x2(n)/x2*(n); then we have
(15)z1n+1=z1n+b1nx1*n1-exp⁡z1n+a2nx2*n-x2nhhhhh+a2nd1nx1nx2*n-x1*nx2n·1+d1nx1*n1+d1nx1n-1≤z1n+b1nx1*n1-exp⁡z1n+a2nx2*n1-exp⁡z2n+a2nd1nx1*nx2*n·1-exp⁡z2n-1-exp⁡z1n,z2n+1=z2n+b2nx2*n1-exp⁡z2n+a1nx1*n-x1nhhhh+a1nd2nx1*nx2n-x1nx2*n·1+d2nx2*n1+d2nx2n-1≤z2n+b2nx2*n1-exp⁡z2n+a1nx1*n1-exp⁡z1n+a1nd2nx1*nx2*n·1-exp⁡z1n-1-exp⁡z2n,
which, according to the mean value, yields
(16)z1n+1≤z1n1-b1nx1*nexp⁡ϑ1z1nhhhhhhh+a2nd1nx1*nx2*nexp⁡ϑ1z1n-z2na2nx2*nexp⁡ϑ2z2nhhhhhhhhh+a2nd1nx1*nx2*nexp⁡ϑ2z2n,z2n+1≤z2n1-b2nx2*nexp⁡ϑ3z2nhhhhhhh+a1nd2nx1*nx2*nexp⁡ϑ3z2n-z1na1nx1*nexp⁡ϑ4z1nhhhhhhhhh+a1nd2nx1*nx2*nexp⁡ϑ4z1n,
where all the constants ϑ1,ϑ2,ϑ3,ϑ4∈(0,1). Obviously, together with (14) we can find a sufficiently small ɛ such that
(17)ξ1*=max⁡1+a2ud1uM1+ɛM2+ɛ-b1lm1-ɛ,hhhhhhh1+a2ld1lm1-ɛm2-ɛ-b1uM1+ɛ+a2uM2+ɛ+a2ud1uM1+ɛM2+ɛ<1,ξ2*=max⁡1+a1ud2uM1+ɛM2+ɛ-b2lm2-ɛ,hhhhhhh1+a1ld2lm1-ɛm2-ɛ-b2uM2+ɛ+a1uM1+ɛ+a1ud2uM1+ɛM2+ɛ<1.
It follows from Lemmas 5 and 6 that there exists an N0 such that n>N0; we have
(18)0<m1-ɛ≤x1*n≤M1+ɛ,0<m2-ɛ≤x2*n≤M2+ɛ,0<m1-ɛ≤x1n≤M1+ɛ,0<m2-ɛ≤x2n≤M2+ɛ.
Then one obtains the fact that both x1*(n)exp⁡(ϑ1z1(n)) and x1*(n)exp⁡(ϑ4z1(n)) are between x1*(n) and x1(n). Similarly, both x2*(n)exp⁡(ϑ2z2(n)) and x2*(n)exp⁡(ϑ3z2(n)) are between x2*(n) and x2(n). From the first equation of (2), one has
(19)z1n+1≤z1n1-b1nx1*nexp⁡ϑ1z1nhhhhhhhhh+a2nd1nx1*nx2*nexp⁡ϑ1z1n+z2na2nx2*nexp⁡ϑ2z2nhhhhhhhhhh+a2nd1nx1*nx2*nexp⁡ϑ2z2n≤max⁡1+a2ud1uM1+ɛM2+ɛ-b1lm1-ɛ,hhhhhhh1+a2ld1lm1-ɛm2-ɛ-b1uM1+ɛ·z1n+a2uM2+ɛ+a2ud1uM1+ɛM2+ɛz2n≤ξ1*max⁡z1n,z2n.
Similar to the arguments as above, we must have
(20)z2n+1≤ξ2*max⁡z1n,z2n.
We denote ξ*=max⁡{x1*,x2*}; then ξ*<1. Therefore, provided that n>N0,
(21)max⁡z1n+1,z2n+1≤ξ*max⁡z1n,z2n≤ξ*n-N0.
Consequently, limn→+∞|xi(n)-xi*(n)|=0, where i=1,2. By Definition 8, it follows that the positive periodic solution {x1*(n),x2*(n)} of system (2) is globally stable. This completes the proof.


## 4. Existence and Stability of Positive Almost Periodic Solutions

In this section, we discuss the existence of positive almost periodic solutions of system (2).Lemma 10.
If assumption (10) is true, then Ω≠⌀.Proof.
According to an inductive argument, system (2) is actually described as follows:
(22)x1n=x10exp⁡∑t=0n-1r1t-b1tx1t-a2tx2t1+d1tx1t,x2n=x20exp⁡∑t=0n-1r2t-b2tx2t-a1tx1t1+d2tx2t.
Combining Lemmas 5 and 6, for any solution (x1(n),x2(n)) of system (2) and an arbitrarily small constant ɛ>0, there must be n0 which is sufficiently large such that
(23)m1-ɛ≤x1n≤M1+ɛ,m2-ɛ≤x2n≤M2+ɛ,hhhhhhhhhhihhh∀n≥n0.
Assuming that τk is any positive integer sequence such that {τk}→+∞ as k→+∞, we can prove that there is a subsequence of {τk} still denoted by {τk}, such that xi(n+τk)→xi*(n), i=1,2 uniformly in n on any finite subset L of Z+ as k→+∞, where L={l1,l2,…,lm}, lj∈Z+(j=1,2,…,m), and m is a finite number.
In fact, for any finite subsetL⊂Z+, τk+lj>n0, (j=1,2,…,m), when k is large enough. Therefore, mi-ɛ≤xi(n+τk)≤Mi-ɛ, i=1,2; that is, xi(n+τk) are uniformly bounded when k is sufficiently large.
Next, forl1∈L, we choose a subsequence {τk(1)} of {τk} such that x1(l1+τk(1)) and x2(l1+τk(1)) uniformly converge on Z+ for k sufficient large.
Similar to the arguments ofl1, for l2∈L, one can select a subsequence {τk(2)} of {τk(1)} such that x1(l2+τk(2)) and x2(l2+τk(2)) uniformly converge on Z+ for large enough k.
Repeating above-mentioned process, forlm∈L, one obtains a subsequence {τk(m)} of {τk(m-1)} such that x1(lm+τk(m)) and x2(lm+τk(m)) uniformly converge on Z+ for sufficiently large k.
Based on the above, one selects the sequence{τk(m)} which is a subsequence of {τk} still denoted by {τk}; then, for n∈L, one gets xi(n+τk)→xi*(n), i=1,2 uniformly in n∈L as k→+∞. So the conclusion holds truely due to the arbitrariness of L.
Different from Section3, we suppose that all the coefficients {ri(n)}, {ai(n)}, {bi(n)}, and {di(n)}, i=1,2, are bounded nonnegative almost periodic sequences; for the above sequence {τk}, τk→+∞ as k→+∞, there exists a subsequence denoted by {τk} such that
(24)rin+τk⟶rin,bin+τk⟶bin,ain+τk⟶ain,din+τk⟶din,i=1,2,
as k→+∞ uniformly on Z+.
For anyρ∈Z+, assume that τk+ρ≥N0 when k is large enough. By an inductive argument of system (2) from τk+ρ to n+τk+ρ, where n∈Z+, one obtains
(25)x1n+τk+ρ=x1τk+ρ·exp⁡∑t=τk+ρn+τk+ρ-1r1t-b1tx1t-a2tx2t1+d1tx1t,x2n+τk+ρ=x2τk+ρ·exp⁡∑t=τk+ρn+τk+ρ-1r2t-b2tx2t-a1tx1t1+d2tx2t.
Hence, (25) yields
(26)x1n+τk+ρ=x1τk+ρ·exp⁡∑t=ρn+ρ-1r1t+τk-b1t+τkx1t+τka2t+τkx2t+τk1+d1t+τkx1t+τkhhhhhhhhhhhh-a2t+τkx2t+τk1+d1t+τkx1t+τk,x2n+τk+ρ=x2τk+ρ·exp⁡∑t=ρn+ρ-1r2t+τk-b2t+τkx2t+τka1t+τkx1t+τk1+d2t+τkx2t+τkhhhhhhhhhhhh-a1t+τkx1t+τk1+d2t+τkx2t+τk.
Let k→+∞; one has
(27)x1*n+ρ=x1*ρ·exp⁡∑t=ρn+ρ-1r1t-b1tx1*t-a2tx2*t1+d1tx1*t,x2*n+ρ=x2*ρ·exp⁡∑t=ρn+ρ-1r2t-b2tx2*t-a1tx1*t1+d2tx2*t.
It is easy to see that (x1*(n),x2*(n)) is a solution of system (2) on Z+ for arbitrary ρ, and
(28)0<m1-ɛ≤x1*n≤M1-ɛ,0<m2-ɛ≤x2*n≤M2-ɛ,hhhhhhhhhihhhhhhhn∈Z+.
Then we get (29) due to ɛ which is an arbitrarily small positive constant:
(29)0<m1≤x1*n≤M1,0<m2≤x2*n≤M2,hhhhhhhhhhhn∈Z+.
This completes the proof.Finally, we are ready to state our main result in this section.Theorem 11.
Let the assumption (10) be satisfied and the following statement in which 0<γ<1 holds true, where γ=min⁡{β1,β2}:
(30)β1=2b1lm1-a2uM2+a2ub1uM1M21+d1lm1-a1uM1+a1ub2uM1M21+d2lm2-b1uM12-2a2ud1uM1M2-a2lb1ld1lm12m21+d1lm12-a1uM121+d2lm22-a2u2d1uM1M221+d1lm13-a1u2d2uM12M21+d2lm23-a2ud1uM1M221+d1lm14,β2=2b2lm2-a2uM2+a2ub1uM1M21+d1lm1-a1uM1+a1ub2uM1M21+d2lm2-b2uM22-2a1ud2uM1M2-a2lb2ld2lm1m221+d2lm22-a2uM221+d1lm12-a2u2d1uM1M221+d1lm13-a1u2d2uM12M21+d2lm23-a1ud2uM1M221+d2lm24;
then system (2) admits a unique positive almost periodic solution, which is uniformly asymptotically stable.Proof.
Denoteu1(n)=ln⁡x1(n) and u2(n)=ln⁡x2(n). It follows from (2) that
(31)u1n+1=u1n+r1n-b1neu1n-a2neu2n1+d1neu1n,u2n+1=u2n+r2n-b2neu2n-a1neu1n1+d2neu2n.
By Lemma 10, it is easy to see that for system (31) there exists a bounded solution (u1(n),u2(n)) satisfying
(32)ln⁡m1≤u1n≤ln⁡M1,ln⁡m2≤u2n≤ln⁡M2,n∈Z+.
Thus |u1(n)|≤s1 and |u2(n)|≤s2, where s1=max⁡{|ln⁡M1|,|ln⁡m1|} and s2=max⁡{|ln⁡M2|,|ln⁡m2|}. Define the norm:
(33)u1n,u2n=u1n+u2n,
where (u1(n),u2(n))∈R2.
Consider the product system of system (31) as follows:
(34)u1n+1=u1n+r1n-b1neu1n-a2neu2n1+d1neu1n,u2n+1=u2n+r2n-b2neu2n-a1neu1n1+d2neu2n,w1n+1=w1n+r1n-b1new1n-a2new2n1+d1new1n,w2n+1=w2n+r2n-b2new2n-a1new1n1+d2new2n.
Assume that H=(u1(n),u2(n)) and I=(w1(n),w2(n)) are any two solutions of system (18) defined on Γ. And then H≤S and I≤S, where S=s1+s2, and Γ={(u1(n),u2(n))∣ln⁡mi≤ui(n)≤ln⁡Mi,i=1,2,n∈Z+}.
In the following, construct a Lyapunov function which is defined onZ+×Γ×Γ:
(35)Vn,H,I=u1n-w1n2+u2n-w2n2.
Notice that the form H-I=|u1(n)-w1(n)|+|u2(n)-w2(n)| is equivalent to H-IΔ=[(u1(n)-w1(n))2+(u2(n)-w2(n))2]1/2, which implies that there exist two positive constants K1,K2 such that K1H-I≤H-I≤K2H-I. Obviously, K12(H-I)2≤V(n,H,I)≤K22(H-I)2.
Letα(x)=K12x2 and β(x)=K22x2. Then condition (1) of Lemma 4 is satisfied. Furthermore, for any (n,H,I),(n,H^,I^)∈Z+×Γ×Γ, we replace (ui(n)-wi(n)) and (u^i(n)-w^i(n)), i=1,2 with Di(n) and D^i(n), respectively; we have
(36)Vn,H,I-Vn,H^,I^=D12n+D22n-D^12n-D^22n≤D12n-D^12n+D22n-D^22n=D1n+D^1nD1n-D^1n+D2n+D^2nD2n-D^2n≤u1n+w1n+u^1n+w^1n·u1n-u^1n+w1-w^1n+u2n+w2n+u^2n+w^2n·u2n-u^2n+w2-w^2n≤Gu1n-u^1n+u2n-u^2nhhhh+w1-w^1n+w2-w^2n=GH-H^+I-I^,
where H^=(u^1(n),u^2(n)), I^=(w^1(n),w^2(n)), and G=4max⁡{s1,s2}. Consequently, condition (2) of Lemma 4 is satisfied. At last, calculating the ΔV(n) of V(n) along the solutions of system (34) yields
(37)ΔV34n=Vn+1-Vn=D12n+1+D22n+1-D12n-D22n=D12n+1-D1n2+D22n+1-D2n2=D1n-b1neu1n-ew1na2neu2n1+d1neu1n-a2new2n1+d1new1nhhh-a2neu2n1+d1neu1n-a2new2n1+d1new1n2-D1n2+D2n-b2neu2n-ew2na1neu1n1+d2neu2n-a1new1n1+d2new2nhhhhh-a1neu1n1+d2neu2n-a1new1n1+d2new2n2-D2n2=-2b1nD1neu1n-ew1n-2a2nD1n·eu2n1+d1neu1n-ew2n1+d1new1n+2a2nb1neu1n-ew1n·eu2n1+d1neu1n-ew2n1+d1new1n+b12neu1n-ew1n2+a22neu2n1+d1neu1n-ew2n1+d1new1n2-2b2nD2neu2n-ew2n-2a1nD2n·eu1n1+d2neu2n-ew1n1+d2new2n+2a1nb2neu2n-ew2n·eu1n1+d2neu2n-ew1n1+d2new2n+b22neu2n-ew2n2+a12neu1n1+d2neu2n-ew1n1+d2new2n2.
Applying the one-dimensional and two-dimensional mean value theorem, we arrive at a simple result as follows:
(38)euin-ewin=ζinuin-win,euin1+djneujn-ewin1+djnewjn=θin1+djnθjnuin-win-djnθinθjn1+djnθjn2ujn-wjn,
where i,j=1,2, i≠j, and ζi(n) and θi(n) lie between eui(n) and ewi(n), respectively. Substituting (38) into (37), one obtains
(39)ΔV34n=-2b1nζ1nD12n+2a2nd1nθ1nθ2n1+d1nθ1n2D12n-2a2nθ2n1+d1nθ1nD1nD2n+a22nd1nθ1nθ2n21+d1nθ1n4D12nhhhhhhhhhh+θ22n1+d1nθ1n2D22nhhhhhhhhhh-d1nθ1nθ2n21+d1nθ1n42d1nθ1nθ22n1+d1nθ1n3D1nD2n-2a2nb1nd1nζ1nθ1nθ2n1+d1nθ1n2D12n+2a2nb1nζ1nθ2n1+d1nθ1nD1nD2n+b12nζ12nD12n-2b2nζ2nD22n+2a1nd2nθ1nθ2n1+d2nθ2n2D22n-2a1nθ1n1+d2nθ2nD1nD2n+a12nd2nθ1nθ2n21+d2nθ2n4D22nhhhhhhhhhh+θ12n1+d2nθ2n2D12nhhhhhhhhhh-d2nθ1nθ2n21+d2nθ2n42d2nθ1n2nθ2n1+d2nθ2n3D1nD2n-2a1nb2nd2nζ2nθ1nθ2n1+d2nθ2n2D22n+2a1nb2nζ2nθ1n1+d2nθ2nD1nD2n+b22nζ22nD22n=-2b1nζ1n+2a2nd1nθ1nθ2n1+d1nθ1n2a1nθ1n21+d2nθ2n2hhhh+b12nζ12nhhhh+a2nd1nθ1nθ2n21+d1nθ1n4+a1nθ1n21+d2nθ2n2hhhh-a1nθ1n21+d2nθ2n22a2nb1nd1nζ1nθ1nθ2n1+d1nθ1n2D12n+-2b2nζ2n+2a1nd2nθ1nθ2n1+d2nθ2n2a2nθ2n21+d1nθ1n2hhhhhh+b22nζ22n+a1nd2nθ1nθ2n21+d2nθ2n4hhhhhh-2a1nb2nd2nζ2nθ1nθ2n1+d2nθ2n2hhhhhh+a2nθ2n21+d1nθ1n2D22n+-2a2nθ2n1+d1nθ1n-2a22nd1nθ1nθ22n1+d1nθ1n3hhhhhh+2a2nb1nζ1nθ2n1+d1nθ1n-2a1nθ1n1+d2nθ2nhhhhhh-2a12nd2nθ2nθ12n1+d2nθ2n3hhhhhh+2a22nd1nθ1nθ22n1+d1nθ1n32a1nb2nζ2nθ1n1+d2nθ2nD1nD2n≤-2b1nζ1n+2a2nd1nθ1nθ2n1+d1nθ1n2a2nd1nθ1nθ2n21+d1nθ1n4hhh+b12nζ12n+a2nd1nθ1nθ2n21+d1nθ1n4hhh-2a2nb1nd1nζ1nθ1nθ2n1+d1nθ1n2hhh+a1nθ1n21+d2nθ2n2+a2nθ2n1+d1nθ1nhhh+a22nd1nθ1nθ22n1+d1nθ1n3hhh+a2nb1nζ1nθ2n1+d1nθ1n+a1nθ1n1+d2nθ2nhhh+a12nd2nθ2nθ12n1+d2nθ2n3hhh+a2nd1nθ1nθ2n21+d1nθ1n4a1nb2nζ2nθ1n1+d2nθ2nD12n+-2b2nζ2n+2a1nd2nθ1nθ2n1+d2nθ2n2a1nd2nθ1nθ2n21+d2nθ2n4hhhh+b22nζ22n+a1nd2nθ1nθ2n21+d2nθ2n4hhhh-2a1nb2nd2nζ2nθ1nθ2n1+d2nθ2n2hhhh+a2nθ2n21+d1nθ1n2+a2nθ2n1+d1nθ1nhhhh+a22nd1nθ1nθ22n1+d1nθ1n3+a2nb1nζ1nθ2n1+d1nθ1nhhhh+a1nθ1n1+d2nθ2n+a12nd2nθ2nθ12n1+d2nθ2n3hhhh+a1nd2nθ1nθ2n21+d2nθ2n4a1nb2nζ2nθ1n1+d2nθ2nD22n≤-2b1lm1+a2uM2+a2ub1uM1M21+d1lm1a2ud1uM1M221+d1lm14hhh+a1uM1+a1ub2uM1M21+d2lm2+b1uM12hhh+2a2ud1uM1M2-a2lb1ld1lm12m21+d1lm12hhh+a1uM121+d2lm22+a2u2d1uM1M221+d1lm13hhh+a1u2d2uM12M21+d2lm23+a2ud1uM1M221+d1lm14D12n+-2b2lm2+a2uM2+a2ub1uM1M21+d1lm1a1ud2uM1M221+d2lm24hhhh+a1uM1+a1ub2uM1M21+d2lm2+b2uM22hhhh+2a1ud2uM1M2-a2lb2ld2lm1m221+d2lm22hhhh+a2uM221+d1lm12+a2u2d1uM1M221+d1lm13hhhh+a1u2d2uM12M21+d2lm23+a1ud2uM1M221+d2lm24D22n=-2b1lm1-a2uM2+a2ub1uM1M21+d1lm1a2ud1uM1M221+d1lm14hhhhh-a1uM1+a1ub2uM1M21+d2lm2-b1uM12hhhhh-2a2ud1uM1M2-a2lb1ld1lm12m21+d1lm12hhhhh-a1uM121+d2lm22-a2u2d1uM1M221+d1lm13hhhhh-a1u2d2uM12M21+d2lm23-a2ud1uM1M221+d1lm14D12n-2b2lm2-a2uM2+a2ub1uM1M21+d1lm1a1ud2uM1M221+d2lm24hhhhh-a1uM1+a1ub2uM1M21+d2lm2-b2uM22hhhhh-2a1ud2uM1M2-a2lb2ld2lm1m221+d2lm22hhhhh-a2uM221+d1lm12-a2u2d1uM1M221+d1lm13hhhhh-a1u2d2uM12M21+d2lm23-a1ud2uM1M221+d2lm24D22n=-β1D12n+β2D22n=-γD12n+D22n=-γVn,
where γ=min⁡{β1,β2} and 0<γ<1 which has been pointed out in Theorem 11. In addition, condition (3) of Lemma 4 is also satisfied. According to Lemma 4, there exists a uniformly asymptotically stable almost periodic solution (u1*(n),u2*(n)) of system (31) which is bounded by Γ for all n∈Z+. Namely, there exists a uniformly asymptotically stable almost periodic solution (x1*(n),x2*(n)) of system (2) which is bounded by Ω for all n∈Z+. This completed the proof.


## 5. Example and Numerical Simulations

In this section, we only give the following example about almost periodic solutions to check the feasibility of the assumptions of Theorem11 considering that the simulation about periodic model is similar.Example 1.
Consider the following discrete system:(40)x1n+1=x1nexp⁡1.09-0.03sin2nπ0.035+0.005cos⁡2nπx2n1+2.10+0.02cos⁡5nπx1nhhhhhhhhhh-1.15-0.01cos⁡2nπx1nhhhhhhhhhh-0.035+0.005cos⁡2nπx2n1+2.10+0.02cos⁡5nπx1n,x2n+1=x2nexp⁡1.06+0.03cos⁡2nπ0.025+0.005cos⁡2nπx1n1+2.07+0.03sin5nπx2nhhhhhhhhhhh-1.11+0.01sin2nπx2nhhhhhhhhhhh-0.025+0.005cos⁡2nπx1n1+2.07+0.03sin5nπx2n,
with the following intial conditions:
(41)x1n*0=0.98,x2n*0=1.01.
By a computation, we get(42)M1≈0.9890,M2≈0.9947,m1≈0.7745,m2≈0.7971,β1≈0.3781,β2≈0.4438,r1l-a2uM2≈1.0202>0,r2l-a1uM1≈1.0003>0,b1uM1r1l-a2uM2≈1.1245>1,b2uM2r2l-a2uM2≈1.1251>1.
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 (x1*(n),x2*(n)), 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 (x1(n),x2(n)) tends to the almost periodic solution (x1*(n),x2*(n)).Positive almost periodic solution of system (22). (a) and (c) Time-series x1*(n) and x2*(n) with initial values x1*(0)=0.98 and x2*(0)=1.01, for n∈[0,100], and (b) and (d) show that x1*(n) and x2*(n) have the same initial data for n∈[900,1000], respectively.
(a)
(b)
(c)
(d)Phase portrait; 2-dimensional and 3-dimensional phase portraits of almost periodic solution of system (22). Time-series x1*(n) and x2*(n) with initial values x1*(0)=0.98 and x2*(0)=1.01; (a) and (c) indicate n∈[0,100] and (b) and (d) indicate n∈[900,1000], respectively.
(a)
(b)
(c)
(d)Uniformly asymptotic stability. Time-seriesx1*(n) and x2*(n) with initial values x1*(0)=0.98 and x2*(0)=1.01 and x1(n) and x2(n) with initial values x1(0)=0.92 and x2(0)=0.91. (a) and (c) indicate n∈[0,100] and (b) and (d) indicate n∈[900,1000], respectively.
(a)
(b)
(c)
(d)


## 6. 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 mean-value 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 r(t) should be suitably large.



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*Source: 658758-2015-02-17.xml*