# Uniformly Asymptotic Stability of Positive Almost Periodic Solutions for a Discrete Competitive System

Authors: Qinglong Wang; Zhijun Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is devoted to the study of almost periodic solutions of a discrete two-species competitive system. With the help of the methods of the Lyapunov function, some analysis techniques, and preliminary lemmas, we establish a criterion for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the system. Numerical simulations are presented to illustrate
the analytical results.


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## Body

## 1. Introduction

In recent years, many works have been done for the difference system (see [1–14] and the references cited therein) since the discrete time models governed by the difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world. In particular, Qin et al. [1] introduced the following discrete Lotka-Volterra competitive system:(1)x1(n+1)=x1(n)exp[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)],x2(n+1)=x2(n)exp[r2(n)-a2(n)x2(n)-c1(n)x1(n)1+x1(n)],=p[r2(n)-a2(n)x2(n)-c1(n)x1(n)1+x1(n)].n=0,1,2…,
where xi(0)>0, xi(n) stand for the densities of species xi at the nth generation, ri(n) represent the natural growth rates of species xi at the nth generation, ai(n) are the intraspecific effects of the nth generation of species xi on own population, and ci(n) measure the interspecific effects of the nth generation of species xi on species xj(i,j=1,2;i≠j). They investigated the permanence and global asymptotic stability of positive periodic solutions of system (1).Notice that the investigation of almost periodic solutions for difference equations is one of most important topics in the qualitative theory of difference equations due to the applications in biology, ecology, neural network, and so forth (see [10–14] in detail), and few work has been done previously on an almost periodic version which is corresponding to periodic system (1). In this paper, we will further investigate the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the above almost periodic version. To this end, we assume that the coefficients of system (1) {ri(n)},{ai(n)} and {ci(n)} are bounded nonnegative almost periodic sequences.For the sake of simplicity and convenience in the following discussion, the notations below will be used throughout this paper:(2)fU=supn∈ℤ+{f(n)},fL=infn∈ℤ+{f(n)},
where {f(n)} is a bounded sequence and ℤ+={0,1,2,…}.The remaining part of this paper is organized as follows. In the next section, we introduce some notations, definitions, and lemmas which are available for our main results. In Section3, sufficient conditions for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of system (1) are given. Numerical simulations are carried out to substantiate the above analytical results in Section 4. Finally, we give some proofs of theorems in the appendices for convenience in reading this paper.


## 2. Preliminaries

In this section, we will need some preparations and give some notations, definitions, and lemmas which will be useful for our main results.Denote byℝ, ℝ+, ℤ, and ℤ+ the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. ℝ2 and ℝk denote the cone of 2-dimensional and k-dimensional real Euclidean space, respectively.Definition 1 (see [13]).
A sequencex:ℤ→ℝk is called an almost periodic sequence if the following ε-translation set of x(3)E{ε,x}≔{τ∈ℤ:|x(n+τ)-x(n)|<ε,∀n∈ℤ}
is a relatively dense set in ℤ for all ε>0; that is, for any given ε>0, there exists an integer l(ε)>0 such that each discrete interval of length l(ε) contains a τ=τ(ε)∈E{ε,x} such that
(4)|x(n+τ)-x(n)|<ε,∀n∈ℤ.τ is called the ε-translation number of x(n).Definition 2 (see [13]).
Letf:ℤ×𝔻→ℝk, where 𝔻 is an open set in ℝk. f(n,x) is said to be almost periodic in n uniformly for x∈𝔻, or uniformly almost periodic for short, if for any ε>0 and any compact set 𝕊 in 𝔻 there exists a positive integer l(ε,𝕊) such that any interval of length l(ε,𝕊) contains an integer τ for which
(5)|f(n+τ,x)-f(n,x)|<ε,
for all n∈ℤ and all x∈𝕊. τ is called the ε-translation number of f(n,x).Lemma 3 (see [13]).
{
x
(
n
)
} is an almost periodic sequence if and only if for any sequence {hk′}⊂ℤ there exists a subsequence {hk}⊂{hk′} such that x(n+hk) converges uniformly on n∈ℤ as k→∞. Furthermore, the limit sequence is also an almost periodic sequence.Consider the following almost periodic difference system:(6)x(n+1)=f(n,x(n)),n∈ℤ+,
where f:ℤ+×𝕊B→ℝk, 𝕊B={x∈ℝk:∥x∥<B}, and f(n,x) is almost periodic in n uniformly for x∈𝕊B and is continuous in x. The product system of (6) is the following system:
(7)x(n+1)=f(n,x(n)),y(n+1)=f(n,y(n)),
and Zhang [14] obtained the following lemma.Lemma 4 (see [14]).
Suppose that there exists a Lyapunov functionV(n,x,y) defined for n∈ℤ+, ∥x∥<B, ∥y∥<B satisfying the following conditions:(i)
a(∥x-y∥)≤V(n,x,y)≤b(∥x-y∥), where a,b∈K with K={a∈C(ℝ+,ℝ+):a(0)=0 and a is increasing};(ii)
|V(n,x1,y1)-V(n,x2,y2)|≤L(∥x1-x2∥+∥y1-y2∥), where L>0 is a constant;(iii)
ΔV(2.2)(n,x,y)≤-βV(n,x,y), where 0<β<1 is a constant and
(8)ΔV(2.2)(n,x,y)=V(n+1,f(n,x),f(n,y))-V(n,x,y).Moreover, if there exists a solution φ(n) of system (6) such that ∥φ(n)∥≤B*<B for n∈ℤ+, then there exists a unique uniformly asymptotically stable almost periodic solution p(n) of system (6) which satisfies ∥p(n)∥≤B*. In particular, if f(n,x) is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of system (6) of periodic ω.Lemma 5 (see [1]).
Any positive solution(x1(n),x2(n)) of system (1) satisfies
(9)limsupn→+∞xi(n)≤Mi=def[exp(riU-1)]aiL,i=1,2.Lemma 6 (see [1]).
Suppose that system (1) satisfies the following assumptions:
(10)r1L>c2U,r2L>c1U.
Then, any positive solution (x1(n),x2(n)) of system (1) satisfies
(11)liminfn→+∞xi(n)≥mi=defriL-cjUaiUexp(riL-aiUMi-cjU),i≠j;i,j=1,2.


## 3. Main Result

From (9) and (11), we denote by Ω the set of all solutions (x1(n),x2(n)) of system (1) satisfying mi≤xi(n)≤Mi, i=1,2 for all n∈ℤ+. According to Lemma 4, we first prove that there is a bounded solution of system (1), and then structure a suitable Lyapunov function for system (1).Theorem 7.
If the assumptions in (10) hold, then Ω≠Φ.The proof of Theorem7 is given in Appendix A.Theorem 8.
If the assumptions in (10) are satisfied, furthermore, 0<β<1, where β=min{s1,s2}, and
(12)s1=2a1Lm1-a1U2M12-(1+a1UM1)c2UM2(1+m2)2-(1+a2UM2)c1UM1(1+m1)2-c1U2M12(1+m1)4,s2=2a2Lm2-a2U2M22-(1+a2UM2)c1UM1(1+m1)2-(1+a1UM1)c2UM2(1+m2)2-c2U2M22(1+m2)4,
then there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by Ω for all n∈ℤ+.The proof of Theorem8 is given in Appendix B.


## 4.  Numerical Simulations

In this section, we give the following example to check the feasibility of the assumptions of Theorem8.Example 9.
Consider the following discrete system:(13)x1(n+1)=x1(n)exp[(0.025+0.002cos(2nπ))x2(n)1+x2(n)1.20-0.02sin(2nπ)=x1(n)exp-(1.05+0.01sin(2nπ))x1(n)x1(n)exp-(0.025+0.002cos(2nπ))x2(n)1+x2(n)],x2(n+1)=x2(n)exp[(0.035+0.005sin(3nπ))x1(n)1+x1(n)]1.15-0.02cos(2nπ)=x2(n)exp-(1.02+0.02cos(2nπ))x2(n)x2(n)exp-(0.035+0.005sin(3nπ))x1(n)1+x1(n)].
A computation shows that
(14)r1L-c2U=1.1530>0,r2L-c1U=1.0900>0,
and moreover, we have
(15)s1=0.3515,s2=0.2502,
that is, 0<β=min{s1,s2}=0.2502<1. It is easy to see that the assumptions of Theorem 8 are satisfied. Hence, in system (13) there exists a unique uniformly asymptotically stable positive almost periodic solution. From Figure 1, it is easy to see that there exists a positive almost periodic solution (x1*(t),x2*(t)), and the 2-dimensional and 3-dimensional phase portraits of almost periodic system (13) 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 (13). (a), (c) Time-series x1*(n) and x2*(n) with initial values x1*(0)=1.19, x2*(0)=1.18 for n∈[0,100], respectively. (b), (d) Time-series x1*(n) and x2*(n) with the above initial values for n∈[700,800], respectively.
(a)
(b)
(c)
(d)Phase portrait. (a), (b) 2-dimensional phase portrait of almost periodic system (13). Time-series x1*(n) and x2*(n) with initial values x1*(0)=1.19, x2*(0)=1.18 for n∈[0,100] and n∈[700,800], respectively. (c), (d) 3-dimensional phase portrait of almost periodic system (13). Time-series x1*(n) and x2*(n) with the above initial values for n∈[0,100] and n∈[700,800], respectively.
(a)
(b)
(c)
(d)Uniformly asymptotic stability. (a), (c) Time-seriesx1*(n) and x2*(n) with initial values x1*(0)=1.19, x2*(0)=1.18 and x1(n) and x2(n) with initial values x1(0)=1.06, x2(0)=1.03 for n∈[0,100], respectively. (b), (d) Time-series x1*(n), x2*(n), x1(n), and x2(n) with the above initial values for n∈[700,800], respectively.
(a)
(b)
(c)
(d)



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