# Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative Systems with Time Delay

Authors: Kaihong Zhao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper considers the existence of positive almost-periodic solutions for almost-periodic Lotka-Volterra cooperative system with time delay. By using Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost-periodic solutions are obtained. An example and its simulation figure are given to illustrate the effectiveness of our results.


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## Body

## 1. Introduction

Lotka-Volterra system is one of the most celebrated models in mathematical biology and population dynamics. In recent years, it has also been found with successful and interesting applications in epidemiology, physics, chemistry, economics, biological science, and other areas (see [1–4]). Moreover, in [5], it was shown that the continuous-time recurrent neural networks can be embedded into Lotka-Volterra models by changing coordinates, which suggests that the existing techniques in the analysis of Lotak-Volterra systems can also be applied to recurrent neural networks.Owing to its theoretical and practical significance, Lotka-Volterra system have been studied extensively (see [6–16] and the cites therein). Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment (e.g., seasonal effects of weather, food supplies, mating habits, etc.) are considered as important selective forces on systems in a fluctuating environment. Therefore, on the one hand, models should take into account both the periodically changing environment and the effects of time delays. However, on the other hand, in fact, it is more realistic and reasonable to study almost-periodic system than periodic system. Recently, there are two main approaches to obtain sufficient conditions for the existence and stability of the almost-periodic solutions of biological models: one is using the fixed point theorem, Lyapunov functional method, and differential inequality techniques (see [17–19]); the other is using functional hull theory and Lyapunov functional method (see [14–16]). However, to the best of our knowledge, there are very few published letters considering the almost-periodic solutions for nonautonomous Lotka-Volterra cooperative system with time delay by applying the method of coincidence degree theory. Motivated by this, in this letter, we apply the coincidence theory to study the existence of positive almost-periodic solutions for Lotka-Volterra cooperative system with time delay as follows:(1.1)u̇i(t)=ui(t)(ri(t)-bi(t)ui(t-τi(t))+∑j=1,j≠incij(t)uj(t-τij(t))),i=1,2,…,n,
where ui(t) stands for the ith species population density at time t∈ℝ, ri(t) is the natural reproduction rate, bi(t) represents the inner-specific competition, cij(t)(i≠j) stands for the interspecific cooperation, τi(t)>0 and τij(t)>0 are all continuous almost-periodic functions on ℝ. Throughout this paper, we always assume that ri(t), bi(t), and cij(t) are all nonegative almost periodic functions with respect to t∈ℝ.The initial condition of (1.1) is of the form(1.2)ui(s)=ϕi(s),i=1,2,…,n,
where ϕi(s) is positive bounded continuous function on [-τ,0] and τ=max1≤i,j≤nsupt∈R{|τij(t)|}.The organization of the rest of this paper is as follows. In Section2, we introduce some preliminary results which are needed in later sections. In Section 3, we establish our main results for the existence of almost-periodic solutions of (1.1). Finally, an example and its simulation figure are given to illustrate the effectiveness of our results in Section 4.


## 2. Preliminaries

To obtain the existence of an almost-periodic solution of system (1.1), we first make the following preparations.Definition 2.1 (see [20]).
Letu(t):ℝ→ℝ be continuous in t.u(t) is said to be almost-periodic on ℝ, if, for any ϵ>0, the set K(u,ϵ)={δ:|u(t+δ)-u(t)|<ϵ,foranyt∈ℝ} is relatively dense, that is, for any ϵ>0, it is possible to find a real number l(ϵ)>0, for any interval with length l(ϵ), there exists a number δ=δ(ϵ) in this interval such that |u(t+δ)-u(t)|<ϵ, for any t∈ℝ.Definition 2.2.
A solutionu(t)=(u1(t),u2(t),…,un(t))T of (1.1) is called an almost periodic solution if and only if for each i=1,2,…,n, ui(t) is almost periodic.For convenience, we denoteAP(ℝ) the set of all real-valued, almost-periodic functions on ℝ and for each j=1,2,…,n, let(2.1)∧(fj)={λ̃∈R:lim⁡T→∞1T∫0Tfj(s)e-iλ̃sds≠0},mod⁡(fj)={∑i=1Nniλĩ:ni∈Z,N∈N+,λĩ∈∧(fj)}
be the set of Fourier exponents and the module of fj, respectively, where fj(·) is almost periodic. Suppose fj(t,ϕj) is almost periodic in t, uniformly with respect to ϕj∈C([-τ,0],ℝ). Kj(fj,ϵ,ϕj) denote the set of ϵ-almost periods for fj uniformly with respect to ϕj∈C([-τ,0],ℝ). lj(ϵ) denote the length of inclusion interval. Let m(fj)=(1/T)∫0Tfj(s)ds be the mean value of fj on interval [0,T], where T>0 is a constant. clearly, m(fj) depends on T.m[fj]=limT→∞(1/T)∫0Tfj(s)ds.Lemma 2.3 (see [20]).
Suppose thatf and g are almost periodic. Then the following statements are equivalent.(i)
mod(f)⊃mod(g),(ii)
for any sequence{tn*}, if limn→∞f(t+tn*)=f(t) for each t∈ℝ, then there exists a subsequence {tn}⊆{tn*} such that limn→∞g(t+tn)=f(t) for each t∈ℝ.Lemma 2.4 (see [21]).
Letu∈AP(ℝ). Then ∫t-τtu(s)ds is almost periodic.LetX and Z be Banach spaces. A linear mapping L:dom(L)⊂X→Z is called Fredholm if its kernel, denoted by ker(L)={X∈dom(L):Lx=0}, has finite dimension and its range, denoted by Im(L)={Lx:x∈dom(L)}, is closed and has finite codimension. The index of L is defined by the integer dimK(L)-codimdom(L). If L is a Fredholm mapping with index 0, then there exist continuous projections P:X→X and Q:Z→Z such that Im(P)=ker(L) and ker(Q)=Im(L). Then L|dom(L)∩ker(P):Im(L)∩ker(P)→Im(L) is bijective, and its inverse mapping is denoted by KP:Im(L)→dom(L)∩ker(P). Since ker(L) is isomorphic to Im(Q), there exists a bijection J:ker(L)→Im(Q). Let Ω be a bounded open subset of X and let N:X→Z be a continuous mapping. If QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact, then N is called L-compact on Ω, where I is the identity.LetL be a Fredholm linear mapping with index 0 and let N be a L-compact mapping on Ω¯. Define mapping F:dom(L)∩Ω¯→Z by F=L-N. If Lx≠Nx for all x∈dom(L)∩∂Ω, then by using P, Q, KP, J defined above, the coincidence degree of F in Ω with respect to L is defined by(2.2)degL(F,Ω)=deg⁡(I-P-(J-1Q+KP(I-Q))N,Ω,0),
where deg(g,Γ,p) is the Leray-Schauder degree of g at p relative to Γ.Then The Mawhin’s continuous theorem is given as follows.Lemma 2.5 (see [22]).
LetΩ⊂X be an open bounded set and let N:X→Z be a continuous operator which is L-compact on Ω¯. Assume(a)
for eachλ∈(0,1), x∈∂Ω∩dom(L),Lx≠λNx;(b)
for eachx∈∂Ω∩L, QNx≠0;(c)
deg(JNQ,Ω∩ker(L),0)≠0.Then Lx=Nx has at least one solution in Ω¯∩dom(L).In this paper, since we need some related properties ofM-matrix we introduce them as follows. In addition, A matrix A=(aij)≥0 means that each elements aij≥0.Definition 2.6 (see [23]).
If a real matrixA=(aij)n×n satisfies the following conditions (1) and (2):(1)
aii>0, i=1,2,…,n, aij≤0, i≠j, i,j=1,2,…,n,(2)
A is a positive-definite matrix,then A is called a M-matrix.Lemma 2.7 (see [23]).
If matrixA=(aij)n×n is a M-matrix, then A-1 exists and its every element is nonnegative.Lemma 2.8.
Suppose that matrixA=(aij)n×n is a M-matrix, then AX≤B implies X≤A-1B.Proof.
In fact, there exists a nonnegative positive vectorɛ0=(ɛ1,ɛ2,…,ɛn)T∈Rn such that AX-B+ɛ0=(0,0,…,0)T which imply that X-A-1B+A-1ɛ0=(0,0,…,0)T. According to Lemma 2.4, there exists at least one positive element in the every row of A-1, which imply A-1ɛ0≥(0,0,…,0)T. Thus, we obtain X≤A-1B.


## 3. Main Result

In this section, we state and prove our main results of our this paper. By making the substitution(3.1)ui(t)=eyi(t),i=1,2,…,n,
Equation (1.1) can be reformulated as(3.2)ẏi(t)=ri(t)-bi(t)eyi(t-τii(t))+∑j=1,i≠jncij(t)eyj(t-τij(t)),i=1,2,…,n.
The initial condition (1.2) can be rewritten as follows: (3.3)yi(s)=ln⁡ϕi(s)=:ψi(s),i=1,2,…,n.
Set X=Z=V1⊕V2, where(3.4)V1={y(t)=(y1(t),y2(t),…,yn(t))T∈C(R,Rn):yi(t)∈AP(R),mod⁡(yi(t))⊂mod⁡(Hi(t)),∀λĩ∈∧(yi(t))satisfies|λ̃i|>β,i=1,2,…,n(y1(t),y2(t),…,yn(t))T},V2={y(t)≡(h1,h2,…,hn)T∈Rn},Hi(t)=ri(t)-bi(t)eψi(-τii(0))+∑j=1,i≠jncij(t)eψj(-τij(0))
and ψ(·) is defined as (3.3), i=1,2,…,n,β>0is a given constant. For y=(y1,y2,…,yn)T∈Z, define ∥y∥=max1≤i≤nsupt∈ℝ|yi(t)|.Lemma 3.1.
Z is a Banach space equipped with the norm ∥·∥.Proof.
Ify{k}⊂V1 and y{k}=(y1{k},y2{k},…,yn{k})T converges to y¯=(y¯1,y¯2,…,y¯n)T, that is, yj{k}→y¯j, as k→∞,j=1,2,…,n. Then it is easy to show that y¯j∈AP(ℝ) and mod(y¯j)∈mod(Hj). For any |λ̃j|≤β, we have that
(3.5)lim⁡T→∞1T∫0Tyj{k}(t)e-iλ̃jtdt=0,j=1,2,…,n;
therefore,
(3.6)lim⁡T→∞1T∫0Ty¯j(t)e-iλ̃jtdt=0,j=1,2,…,n,
which implies y¯∈V1. Then it is not difficult to see that V1 is a Banach space equipped with the norm ∥·∥. Thus, we can easily verify that X and Z are Banach spaces equipped with the norm ∥·∥. The proof of Lemma 3.1 is complete.Lemma 3.2.
LetL:X→Z, Ly=ẏ, then L is a Fredholm mapping of index 0.Proof.
Clearly,L is a linear operator and ker(L)=V2. We claim that Im(L)=V1. Firstly, we suppose that z(t)=(z1(t),z2(t),…,zn(t))T∈Im(L)⊂Z. Then there exist z{1}(t)=(z1{1}(t),z2{1}(t),…,zn{1}(t))T∈V1 and constant vector z{2}=(z1{2},z2{2},…,zn{2})T∈V2 such that
(3.7)z(t)=z{1}(t)+z{2},
that is,
(3.8)zi(t)=zi{1}(t)+zi{2},i=1,2,…,n.
From the definition of zi(t) and zi{1}(t), we can easily see that ∫t-τtzi(s)ds and ∫t-τtzi{1}(s)ds are almost-periodic functions. So we have zi{2}≡0,i=1,2,…,n, then z{2}≡(0,0,…,0)T, which implies z(t)∈V1, that is Im(L)⊂V1.
On the other hand, ifu(t)=(u1(t),u2(t),…,un(t))T∈V1∖{0}, then we have ∫0tuj(s)ds∈AP(ℝ), j=1,2,…,n. If λ̃j≠0, then we obtain
(3.9)lim⁡T→∞1T∫0T(∫0tuj(s)ds)e-iλ̃jtdt=1iλ̃jlim⁡T→∞1T∫0Tuj(t)e-iλ̃jtdt,j=1,2,…,n.
It follows that
(3.10)∧[∫0tuj(s)ds-m(∫0tuj(s)ds)]=∧(uj(t)),j=1,2,…,n,
hence
(3.11)∫0tu(s)ds-m(∫0tu(s)ds)∈V1⊂X.
Note that ∫0tu(s)ds-m(∫0tu(s)ds) is the primitive of u(t) in X, we have u(t)∈Im(L), that is, V1⊂Im(L). Therefore, Im(L)=V1.
Furthermore, one can easily show thatIm(L) is closed in Z and
(3.12)dim⁡ker⁡(L)=n=co⁡dim⁡Im⁡(L);
therefore, L is a Fredholm mapping of index 0. The proof of Lemma 3.2 is complete.Lemma 3.3.
LetN:X→Z, Ny=(G1y,G2y,…,Gny)T, where
(3.13)Giy=ri(t)-bi(t)yi(t-τi(t))+∑j=1,i≠jncij(t)eyj(t-τij(t)),i=1,2,…,n.
Set
(3.14)P:X⟶Z,Py=(m(y1),m(y2),…,m(yn))T,Q:Z⟶Z,Qz=(m[z1],m[z2],…,m[zn])T.
Then N is L-compact on Ω¯, where Ω is an open bounded subset of X.Proof.
Obviously,P and Q are continuous projectors such that
(3.15)Im⁡P=ker⁡(L),Im⁡(L)=ker⁡(Q).
It is clear that (I-Q)V2={(0,0,…,0)}, (I-Q)V1=V1. Hence
(3.16)Im⁡(I-Q)=V1=Im⁡(L).
Then in view of
(3.17)Im⁡(P)=ker⁡(L),Im⁡(L)=ker⁡(Q)=Im⁡(I-Q),
we obtain that the inverse KP:Im(L)→ker(P)∩dom(L) of LP exists and is given by
(3.18)KP(z)=∫0tz(s)ds-m[∫0tz(s)ds].
Thus,
(3.19)QNy=(m[G1y],m[G2y],…,m[Gny])TKP(I-Q)Ny=(f(y1)-Q(f(y1)),f(y2)-Q(f(y2)),…,f(yn)-Q(f(yn)))T,
where
(3.20)f(yi)=∫0t(Giy-m[Giy])ds,i=1,2,…,n.
Clearly,QN and (I-Q)N are continuous. Now we will show that KP is also continuous. By assumptions, for any 0<ϵ<1 and any compact set ϕi⊂C([-τ,0],ℝ), let li(ϵi) be the length of the inclusion interval of Ki(Hi,ϵi,ϕi), i=1,2,…,n. Suppose that {zk(t)}⊂Im(L)=V1 and zk(t)=(z1k(t),z2k(t),…,znk(t))T uniformly converges to z¯(t)=(z¯1(t),z¯2(t),…,z¯n(t))T, that is zik→z¯i, as k→∞, i=1,2,…,n. Because of ∫0tzk(s)ds∈Z, k=1,2,…,n, there exists σi(0<σi<ϵi) such that Ki(Hi,σi,ϕi)⊂Ki(∫0tzik(s)ds,σi,ϕi), i=1,2,…,n. Let li(σi) be the length of the inclusion interval of Ki(Hi,σi,ϕi) and
(3.21)li=max⁡{li(ϵi),li(σi)},i=1,2,…,n.
It is easy to see that li is the length of the inclusion interval of Ki(Hi,σi,ϕi) and Ki(Hi,ϵi,ϕi), i=1,2,…,n. Hence, for any t∉[0,li], there exists ξt∈Ki(Hi,σi,ϕi)⊂Ki(∫0tzik(s)ds,σi,ϕi) such that t+ξt∈[0,li], i=1,2,…,n. Hence, by the definition of almost periodic function we have
(3.22)‖∫0tzk(s)ds‖=max⁡1≤i≤nsup⁡t∈R|∫0tzik(s)ds|≤max⁡1≤i≤nsup⁡t∈[0,li]|∫0tzik(s)ds|+max⁡1≤i≤nsup⁡t∉[0,li]|∫0tzik(s)ds-∫0t+ξtzik(s)ds+∫0t+ξtzik(s)ds|≤2max⁡1≤i≤nsup⁡t∈[0,li]|∫0tzik(s)ds|+max⁡1≤i≤nsup⁡t∉[0,li]|∫0tzik(s)ds-∫0t+ξtzik(s)ds|≤2max⁡1≤i≤n|∫0lizik(s)ds|+max⁡1≤i≤nϵi.
From this inequality, we can conclude that ∫0tz(s)ds is continuous, where z(t)=(z1(t),z2(t),…,zn(t))T∈Im(L). Consequently, KP and KP(I-Q)Ny are continuous.
From (3.22), we also have ∫0tz(s)ds and KP(I-Q)Ny also are uniformly bounded in Ω¯. Further, it is not difficult to verify that QN(Ω¯) is bounded and KP(I-Q)Ny is equicontinuous in Ω¯. By the Arzela-Ascoli theorem, we have immediately concluded that KP(I-Q)N(Ω¯) is compact. Thus N is L-compact on Ω¯. The proof of Lemma 3.3 is complete.Theorem 3.4.
Assume that the following conditions(H1) and (H2) hold:(H1)
ei∶=m[ri(t)]>0,m[bi(t)]>0,m[cij(t))]>0,i,j=1,2,…,n;(H2)
D is a positive-definite matrix, where dii=m[bi(t)], dij=m[cij(t)], i≠j, i,j=1,2,…,n,
(3.23)D=(d11-d12⋯-d1n-d21d22⋯-d2n⋮⋮⋮⋮-dn1-dn2⋯dnn). Then (1.1) has at least one positive almost periodic solution.Proof.
To use the continuation theorem of coincidence degree theorem to establish the existence of a solution of (3.2), we set Banach space X and Z the same as those in Lemma 3.1 and set mappings L, N, P,Q the same as those in Lemmas 3.2 and 3.3, respectively. Then we can obtain that L is a Fredholm mapping of index 0 and N is a continuous operator which is L-compact on Ω¯.
Now, we are in the position of searching for an appropriate open, bounded subsetΩ for the application of the continuation theorem. Corresponding to the operator equation
(3.24)Ly=λNy,λ∈(0,1),
we obtain
(3.25)ẏi(t)=λ[ri(t)-bi(t)eyi(t-τi(t))+∑j=1,i≠jncij(t)eyj(t-τij(t))],i=1,2,…,n.
Assume that y(t)=(y1(t),y2(t),…,yn(t))T∈X is a solution of (3.25) for some λ∈(0,1). Denote M¯i=supt∈R{yi(t)}, M̲i=inft∈R{yi(t)}.
On the one hand, by (3.25), we derive
(3.26)-|ẏi(t)|≤λ[ri(t)-bi(t)eyi(t-τi(t))+∑j=1,i≠jncij(t)eyj(t-τij(t))],i=1,2,…,n.
On the both sides of (3.26), integrating from 0 to T and applying the mean value theorem of integral calculus, we have
(3.27)0≤λ[m(ri(t))-m(bi(t))eyi(ξi-τi(ξi))+∑j=1,i≠jnm(cij(t))eyj(ηij-τij(ηij))]+m(|ẏi(t)|),i=1,2,…,n,
where ξi∈[0,T], ηij∈[0,T], i,j=1,2,…,n. In the light of (3.27), we get for i=1,2,…,n,
(3.28)λ[m(bi(t))eyi(ξi-τi(ξi))]≤λ[m(ri(t))+∑j=1,i≠jnm(cij(t))eyj(ηij-τij(ηij))]+m(|ẏi(t)|).
On the both sides of (3.28), taking the supremum with respect to ξi,ηij and letting T→+∞, we obtain
(3.29)m[bi(t)]eM¯i≤m[ri(t)]+∑j=1,j≠in+m[cij(t)]eM¯j,
that is,
(3.30)diieM¯i-∑j=1,j≠indijeM¯j≤ei,i=1,2,…,n.
Equation (3.30) can be written by the following matrix form
(3.31)(d11-d12⋯-d1n-d21d22⋯-d2n⋮⋮⋮⋮-dn1-dn2⋯dnn)(eM¯1eM¯2⋮eM¯n)≤(e1e2⋮en).
By Lemma 2.8 and assumption, we obtain
(3.32)(eM¯1eM¯2⋮eM¯n)≤D-1(e1e2⋮en)=:(H1+H2+⋮Hn+),
which imply that
(3.33)M¯i≤ln⁡Hi+,i=1,2,…,n.
On the two sides of (3.28), taking the infimum with respect to ξi, ηij, and letting T→+∞, we obtain
(3.34)M̲i≤ln⁡Hi+,i=1,2,…,n.
On the other hand, according to (3.25), we derive
(3.35)λri(t)-ẏi(t)<λbi(t)eyi(t-τi(t)),i=1,2,…,n.
On the both sides of (3.35), integrating from 0 to T and using the mean value theorem of integral calculus, we get
(3.36)λm(ri(t))<m(ẏi(t))+λm(bi(t))eyi(ζi-τi(ζi)),i=1,2,…,n,
where ζi∈[0,T], i=1,2,…,n. On the both sides of (3.36), take the supremum and infimum with respect to ζi, respectively, and let T→+∞, then we have for i=1,2,…,n,
(3.37)m[ri(t)]<m[bii(t)]eM¯i,m[ri(t)]<m[bii(t)]eM̲i,
namely,
(3.38)eM¯i>m[ri(t)]m[bi(t)]=eidii,eM̲i>m[ri(t)]m[bi(t)]=eidii
which imply that
(3.39)M¯i>ln⁡eidii,M̲i>ln⁡eidii.
Combining with (3.33), (3.34), and (3.39), we derive for all t∈ℝ,i=1,2,…,n,
(3.40)min⁡1≤i≤n{ln⁡eidii}≤ln⁡eidii<M̲i≤yi(t)≤M¯i≤ln⁡Hi+<max⁡1≤i≤n{ln⁡Hi+}+1.
Denote M=max{|min1≤i≤n{ln(ei/dii)}|,|max1≤i≤n{lnHi+}+1|}. Clearly, M is independent of λ. Take
(3.41)Ω={y=(y1,y2,…,yn)T∈X:‖y‖<M}.
It is clear that Ω satisfies the requirement (a) in Lemma 2.5. When y∈∂Ω∩ker(L),y=(y1,y2,…,yn)T is a constant vector in ℝn with ∥y∥=M. Then
(3.42)QNy=(m[G1],m[G2],…,m[Gn])T,y∈X,
where
(3.43)Gi=ri(t)-bi(t)eyi+∑j=1,j≠incij(t)eyj,i=1,2,…,n,m[Gi]=m[ri(t)]-m[bii(t)]eyi+∑j=1,j≠inm[cij(t)]eyj=ei-diieyi+∑j=1,j≠indijeyj,i=1,2,…,n.
If QNy=(0,0,…,0])T, then we have
(3.44)(d11-d12⋯-d1n-d21d22⋯-d2n⋮⋮⋮⋮-dn1-dn2⋯dnn)(ey1ey2⋮eyn)=(e1e2⋮en),
which imply that yi=lnHi+, i=1,2,…,n. Thus, y=(y1,y2,…,yn)T∈Ω, this contradicts the fact that y∈∂Ω∩ker(L). Therefore, QNy≠(0,0,…,0)T, which implies that the requirement (b) in Lemma 2.5 is satisfied. If necessary, we can let M be greater such that yTQNy<0, for any y∈∂Ω∩ker(L). Furthermore, take the isomorphism J:Im(Q)→ker(L), Jz≡z and let Φ(γ;y)=-γy+(1-γ)JQNy, then for any y∈∂Ω∩ker(L), yTΦ(γ;y)<0, we have
(3.45)deg⁡{JQN,Ω∩ker⁡(L),0}=deg⁡{-y,Ω∩ker⁡(L),0}≠0.
So, the requirement (c) in Lemma 2.5 is satisfied. Hence, (3.2) has at least one almost-periodic solution in Ω¯, that is, (1.1) has at least one positive almost periodic solution. The proof is complete.


## 4. An Example and Simulation

Consider the following two species cooperative system with time delay:(4.1)ẋ(t)=x(t)(r1(t)-b1(t)x(t-τ1(t))+c12(t)y(t-τ12(t))),ẏ(t)=y(t)(r2(t)-b2(t)y(t-τ2(t))+c21(t)x(t-τ21(t))),
where r1(t)=2+sin2t+sin3t, b1(t)=2+sin3t+sin5t, c12(t)=(2+cos2t+cos3t)/2, τ1(t)=esin2t+sin5t, τ12(t)=esint+cos2t, r2(t)=2-sin2t-sin3t, b2(t)=2+sin3t-sin5t, c21(t)=(2-cos2t+cos3t)/2, τ2(t)=esin2t+cos5t, τ21(t)=esint-cos2t. Since(4.2)e1=m[r1(t)]=2,d11=m[b1(t)]=2,d12=m[c12(t)]=1,e2=m[r1(t)]=2,d22=m[b1(t)]=2,d21=m[c12(t)]=1,D=(2-1-12),det⁡(2-1-12)=3>0,D-1=13(2112),
then, the matrix D is positive definite, and(4.3)ln⁡e1d11=ln⁡e2d22=0,(H1+H2+)=D-1(e1e2)=(22),(ln⁡H1+ln⁡H2+)=(ln⁡2ln⁡2),M=max⁡{|min⁡1≤i≤n{ln⁡eidii}|,|max⁡1≤i≤n{ln⁡Hi+}+1|}=1+ln⁡2,Ω={y=(y1,y2,…,yn)T∈X:‖y‖<1+In⁡2}.
Therefore, all conditions of Theorem 3.4 are satisfied. By Theorem 3.4, system (4.1) has one positive almost-periodic solution. The resulting numerical simulation is depicted in Figure 1.Figure 1



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*Source: 270104-2012-05-15.xml*