# Almost Periodic Solution of a Discrete Schoener’s Competition Model with Delays

Authors: Hui Zhang; Yingqi Li; Bin Jing; Xiaofeng Fang; Jing Wang
Journal: Journal of Difference Equations
 (2014)

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## Abstract

We consider an almost periodic discrete Schoener’s competition model with delays. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result.


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## Body

## 1. Introduction

In 2009, Wu et al. [1] had studied a discrete Schoener’s competition mode with delays:
(1)x1(n+1)=x1(n)exp⁡⁡[a10(n)x1(n-τ10)+k1(n)-a11(n)x1(n-τ11)-a12(n)x2(n-τ12)-c1(n)a10(n)x1(n-τ10)+k1(n)],x2(n+1)=x2(n)exp⁡⁡[a20(n)x2(n-τ20)+k2(n)-a21(n)x1(n-τ21)-a22(n)x2(n-τ22)-c2(n)a20(n)x2(n-τ20)+k2(n)],
where {ki(n)},{aij(n)}, and {ci(n)} are real positive bounded sequences and τij are positive integers, i=1,2;j=0,1,2. Sufficient conditions which guarantee the permanence and the global attractivity of positive solutions for system (1) are obtained.By the biological meaning, the system (1) is considered together with the following initial condition:
(2)xi(s)=φi(s)≥0,φi(0)>0,s∈[-τ,0]∩Z={-τ,-τ+1,…,0},i=1,2,
where τ=max⁡1≤i≤2,0≤j≤2{τij}. Let (x1(n),x2(n)) be any solution of system (1) with the initial condition (2). One could easily see that xi(n)>0, i=1,2 for all n∈N.Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [1–6] and the references cited therein). Recently, a few papers investigate the global stability of the pure-delay model (see [7–13]).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 [14–21] and the references cited therein). But to the best of the author’s knowledge, to this day, still no scholars have studied the almost periodic version which is corresponding to system (1). Therefore, with stimulation from the works of [12, 19], the main purpose of this paper is to derive a set of sufficient conditions ensuring the existence of a unique strictly positive almost periodic solution of system (1) which is globally attractive.Denote byZ and Z+ the set of integers and the set of nonnegative integers, respectively. For any bounded sequence {f(n)} defined on Z, define fu=sup⁡n∈Zf(n),fl=inf⁡n∈Zf(n).Throughout this paper, we assume the following.(H1)
ki(n),aij(n), and ci(n) are bounded positive almost periodic sequences such that
(3)0<kil≤ki(n)≤kiu,0<aijl≤aij(n)≤aiju,0<cil≤ci(n)≤ciu,i=1,2,j=0,1,2.The remaining part of this paper is organized as follows. In Section2, we will introduce some definitions and several useful lemmas. In Section 3, by applying the theory of difference inequality, we present the permanence results for system (1). In Section 4, we establish the sufficient conditions for the existence of a unique globally attractive almost periodic solution of system (1). The main result is illustrated by an example with a numerical simulation in the last section.


## 2. Preliminaries

In this section, we give the definitions and lemmas of the terminologies involved.Definition 1 (see [22]).
A sequencex:Z→R is called an almost periodic sequence if the ε-translation set of x(4)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 interval of length l(ε) contains an integer τ∈E{ε,x} with
(5)|x(n+τ)-x(n)|<ε,∀n∈Z.τ is called an ε-translation number of x(n).Definition 2 (see [23]).
LetD be an open subset of Rm,f:Z×D→Rm. f(n,x) is said to be almost periodic in n uniformly for x∈D if for any ε>0, and any compact set S⊂D, there exists a positive integer l=l(ε,S) such that any interval of length l contains an integer τ for which
(6)|f(n+τ,x)-f(n,x)|<ε,∀(n,x)∈Z×S.τ is called an ε-translation number of f(n,x).Definition 3 (see [24]).
The hull off, denoted by H(f), is defined by
(7)H(f)={g(n,x):lim⁡k→∞f(n+τk,x)=g(n,x)uniformlyon⁡Z×Slim⁡k→∞f},
for some sequence {τk}, where S is any compact set in D.Definition 4.
Suppose thatX(n)=(x1(n),x2(n)) is any solution of system (1). X(n) is said to be a strictly positive solution in Z if
(8)0<inf⁡n∈Zxi(n)≤sup⁡n∈Zxi(n)<∞,i=1,2,
for n∈Z.Lemma 5 (see [25]).
Assume that{x(n)} satisfies x(n)>0 and
(9)x(n+1)≤x(n)exp⁡⁡{a(n)-b(n)x(n)}
for n∈N, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants. Then
(10)limsup⁡n→+∞x(n)≤1blexp⁡⁡{au-1}.Lemma 6 (see [25]).
Assume that{x(n)} satisfies
(11)x(n+1)≥x(n)exp⁡⁡{a(n)-b(n)x(n)},n≥N0,limsup⁡n→+∞x(n)≤x*,
and x(N0)>0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and N0∈N. Then
(12)liminf⁡n→+∞x(n)≥min⁡⁡{albuexp⁡{al-bux*},albu}.


## 3. Permanence and Global Attractivity

Now we state several lemmas which will be useful in proving our main result.Proposition 7 (see [1]).
Any solution(x1(n),x2(n)) of system (1) with the initial condition (2) is positive and ultimately bounded; that is,
(13)limsup⁡n→+∞xi(n)≤Mi,
where
(14)Mi=1aiilexp⁡⁡{ai0ukil(τii+1)-1},i=1,2.Proposition 8.
Assume that(15)(H2)a10lM1+k1u-a12uM2-c1u>0,2222222a20lM2+k2u-a21uM1-c2u>0
holds; then for any solution (x1(n),x2(n)) of system (1) with the initial condition (2), one has
(16)liminf⁡n→+∞xi(n)≥mi,i=1,2,
where m1 and m2 are defined by (24) and (26), respectively.Proof.
For any small positive constantε>0, according to Proposition 7, there exists a N1>0 such that, for all n>N1 and i=1,2,
(17)xi(n)≤Mi+ε.
It follows from (17) and the first equation of system (1), for n≥N1+τ,
(18)x1(n+1)≥x1(n)exp⁡⁡[a10l(M1+ε)+k1u-a11u(M1+ε)-a12u(M2+ε)-c1ua10l(M1+ε)+k1u]≜x1(n)exp⁡⁡{A1ε},⁡⁡⁡
where A1ε=(a10l/((M1+ε)+k1u))-a11u(M1+ε)-a12u(M2+ε)-c1u.
Thus, by using (18) we obtain
(19)x1(n-τ11)≤x1(n)exp⁡⁡{-τ11A1ε}.
Substituting (19) into the first equation of system (1), for n≥N1+τ11, it follows that
(20)x1(n+1)≥x1(n)exp⁡⁡[a10l(M1+ε)+k1u-a12u(M2+ε)-c1u-a11(n)exp⁡⁡{-τ11A1ε}x1(n)a10l(M1+ε)+k1u].
Whenε is an arbitrary small positive constant, it follows from condition (H2)
(21)a10l(M1+ε)+k1u-a12u(M2+ε)-c1u>0.
Thus, as a direct corollary of Lemma 6, according to (14) and (20), one has
(22)liminf⁡n→+∞x1(n)≥min⁡⁡{B1ε,B2ε},
where
(23)B1ε=1a11u[a10l(M1+ε)+k1u-a12u(M2+ε)-c1u]exp⁡⁡{τ11A1ε},B2ε=B1εexp⁡{a10l(M1+ε)+k1u-a12u(M2+ε)-c1u-M1a11uexp⁡⁡{-τ11A1ε}a10l(M1+ε)+k1u}.
Lettingε→0, it follows that
(24)liminf⁡n→+∞x1(n)≥12min⁡⁡{B1,B2}≜m1>0,
where
(25)B1=1a11u[a10lM1+k1u-a12uM2-c1u]exp⁡{τ11A1},B2=B1exp⁡⁡{a10lM1+k1u-a12uM2-c1u-M1a11uexp⁡⁡{-τ11A1}a10lM1+k1u},A1=a10lM1+k1u-a11uM1-a12uM2-c1u.
Similar to the analysis of (18)–(24), by applying (17), from the second equation of system (1), we also have that
(26)liminf⁡n→+∞x2(n)≥12min⁡⁡{D1,D2}≜m2>0,
where
(27)D1=1a22u[a20lM2+k2u-a21uM1-c2u]exp⁡{τ22C1},D2=D1exp⁡⁡{a20lM2+k2u-a21uM1-c2u-M2a22uexp⁡⁡{-τ22C1}a20lM2+k2u},C1=a20lM2+k2u-a22uM2-a21uM1-c2u.
The proof is completed.Note that condition (H2) of Proposition8 is weakened compared to condition (H) in [1].Theorem 9 (see [1]).
Assume that (H2) holds; system (1) is permanent.Proposition 10 (see [1]).
Assume that (H2) holds and further that there exist positive constantsα1,α2, and η>0 such that
(28)(H3)αiEii-(αiFij+αjGji)≥η,i,j=1,2,i≠j,
where
(29)Eii=min⁡⁡(aiil,2Mi-aiiu)-ai0u(kil)2(1+τiiaiiuMiBiu)-τii(aiiu)2MiBiu,Fij=τiiaiiuAiu(ai0ukil+aiiuMi+aijuMj+ciu),Gji=ajiu+τjjajjuMjBjuajiu,i,j=1,2;i≠j,Ai(s)=exp⁡⁡{θi(s)(ai0(s)xi*(s-τi0)+ki(s)-aiixi*(s-τii)-aijxj*(s-τij)-ci(s)ai0(s)xi*(s-τi0)+ki(s))},Bi(s)=exp⁡⁡{φi(s)(ai0(s)xi(s-τi0)+ki(s)-aiixi(s-τii)-aijxj(s-τij)-ci(s)ai0(s)xi(s-τi0)+ki(s))+(1-φi(s))(ai0(s)xi*(s-τi0)+ki(s)-aiixi*(s-τii)-aijxj*(s-τij)-ci(s)ai0(s)xi*(s-τi0)+ki(s))},θi(s),φi(s)∈(0,1), i,j=1,2, i≠j. Then system (1) is globally attractive; that is, for any two positive solutions (x1(n),x2(n)) and (x1*(n),x2*(n)) of system (1), we have
(30)lim⁡n→+∞(xi*(n)-xi(n))=0,i=1,2.


## 4. Almost Periodic Solution

In this section, by means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, we will study the existence of a globally attractive almost periodic solution of system (1) with initial condition (2) and obtain the sufficient conditions.Let{δk} be any integer valued sequence such that δk→∞ as k→∞. According to Lemma 5, taking a subsequence if necessary, we have ki(n+δk)→ki*(n),aij(n+δk)→aij*(n),ci(n+δk)→ci*(n),i=1,2,j=0,1,2, as k→∞ for n∈Z. Then we get a hull equation of system (1) as follows:
(31)x1(n+1)=x1(n)exp⁡⁡[a10*(n)x1(n-τ10)+k1*(n)-a11*(n)x1(n-τ11)-a12*(n)x2(n-τ12)-c1*(n)a10*(n)x1(n-τ10)+k1*(n)],x2(n+1)=x2(n)exp⁡⁡[a20*(n)x2(n-τ20)+k2*(n)-a21*(n)x1(n-τ21)-a22*(n)x2(n-τ22)-c2*(n)a20*(n)x2(n-τ20)+k2*(n)].By the almost periodic theory, we can conclude that if system (1) satisfies (H2) and (H3), then the hull equation (31) of system (1) also satisfies (H2) and (H3).By Theorem 3.4 in [26], we can easily obtain the lemma as follows.Lemma 11.
If each hull equation of system (1) has a unique strictly positive solution, then the almost periodic difference system (1) has a unique strictly positive almost periodic solution.
Now we investigate a globally attractive almost periodic solution of system (1).Theorem 12.
If the almost periodic difference system (1) satisfies (H1), (H2), and (H3), then the almost periodic difference system (1) admits a unique strictly positive almost periodic solution, which is globally attractive.Proof.
By Lemma11, we only need to prove that each hull equation of system (1) has a unique globally attractive almost periodic sequence solution; hence we firstly prove that each hull equation of system (1) has at least one strictly positive solution (the existence) and then we prove that each hull equation of system (1) has a unique strictly positive solution (the uniqueness).
Now we prove the existence of a strictly positive solution of any hull equation (31). By the almost periodicity of {ki(n)},{aij(n)}, and {ci(n)}, i=1,2, j=0,1,2, there exists an integer valued sequence {τk} with τk→∞ as k→∞ such that ki*(n+τk)→ki*(n),aij*(n+τk)→aij*(n),ci*(n+τk)→ci*(n), i=1,2, j=0,1,2, as k→∞ for n∈Z. Suppose that X=(x1(n),x2(n)) is any solution of hull equation (31). By the proof of Propositions 7 and 8, we have
(32)mi≤liminf⁡n→+∞xi(n)≤limsup⁡n→+∞xi(n)≤Mi,i=1,2.
And also
(33)0<inf⁡n∈Z+xi(n)≤sup⁡n∈Z+xi(n)<∞,i=1,2.
Letε be an arbitrary small positive number. It follows from (32) that there exists a positive integer n0 such that mi-ε≤xi(n)≤Mi+ε, n≥n0, i=1,2. Write Xk(n)=X(n+τk)=(x1k(n),x2k(n)), for all n≥n0+τ-τk, k∈Z+. We claim that there exists a sequence {yi(n)}, and a subsequence of {τk}, which we still denote by {τk} such that xik(n)→yi(n) uniformly in n on any finite subset B of Z as k→∞, where B={a1,a2,…,am}, ah∈Z(h=1,2,…,m), and m is a finite number.
In fact, for any finite subsetB⊂Z, when k is large enough, τk+ah-τ>n0, h=1,2,…,m. So
(34)mi-ε≤xi(n+τk)≤Mi+ε,i=1,2;
that is, {xi(n+τk)} are uniformly bounded for k large enough.
Now, fora1∈B, we can choose a subsequence {τk(1)} of {τk} such that {xi(a1+τk(1))} uniformly converges on Z+ for k large enough.
Similarly, fora2∈B, we can choose a subsequence {τk(2)} of {τk(1)} such that {xi(a2+τk(2))} uniformly converges on Z+ for k large enough.
Repeating this procedure, foram∈B, we can choose a subsequence {τk(m)} of {τk(m-1)} such that {xi(am+τk(m))} uniformly converges on Z+ for k large enough.
Now pick the sequence{τk(m)} which is a subsequence of {τk}, which we still denote by {τk}, then for all n∈B, we have xi(n+τk)→yi(n) uniformly in n∈B, as k→∞.
By the arbitrary ofB, the conclusion is valid.
Combining with(35)x1k(n+1)=x1k(n)exp⁡⁡[a10*(n+τk)x1k(n-τ10)+k1*(n+τk)-a11*(n+τk)x1k(n-τ11)-a12*(n+τk)x2k(n-τ12)-c1*(n+τk)a10*(n+τk)x1k(n-τ10)+k1*(n+τk)],x2k(n+1)=x2k(n)exp⁡⁡[a20*(n+τk)x2k(n-τ20)+k2*(n+τk)-a21*(n+τk)x1k(n-τ21)-a22*(n+τk)x2k(n-τ22)-c2*(n+τk)a20*(n+τk)x2k(n-τ20)+k2*(n+τk)],
gives
(36)y1(n+1)=y1(n)exp⁡⁡[a10*(n)y1(n-τ10)+k1*(n)-a11*(n)y1(n-τ11)-a12*(n)y2(n-τ12)-c1*(n)a10*(n)y1(n-τ10)+k1*(n)],y2(n+1)=y2(n)exp⁡⁡[a20*(n)y2(n-τ20)+k2*(n)-a21*(n)y1(n-τ21)-a22*(n)y2(n-τ22)-c2*(n)a20*(n)y2(n-τ20)+k2*(n)].
We can easily see that Y(n)=(y1(n),y2(n)) is a solution of hull equation (31) and mi-ε≤yi(n)≤Mi+ε, i=1,2, for n∈Z. Since ε is an arbitrary small positive number, it follows that mi≤yi(n)≤Mi, i=1,2, for n∈Z; that is,
(37)0<inf⁡n∈Zyi(n)≤sup⁡n∈Zyi(n)<∞,i=1,2.
Hence each hull equation of almost periodic difference system (1) has at least one strictly positive solution.
Now we prove the uniqueness of the strictly positive solution of each hull equation (31). Suppose that the hull equation (31) has two arbitrary strictly positive solutions (x1*(n),x2*(n)) and (y1*(n),y2*(n)). We construct a Lyapunov functional
(38)V*(n)=∑i=12βi(Vi1*(n)+Vi2*(n)+Vi3*(n)),n∈Z,
where
(39)Vi1*(n)=|ln⁡⁡xi*(n)-ln⁡⁡yi*(n)|Vi2*(n)=∑s=n-τi0n-1ai0(s+τi0)ki2(s+τi0)|xi*(s)-yi*(s)|+∑s=n-τijn-1aij(s+τij)|xj*(s)-yj*(s)|+∑s=nn-1+τii‍aii(s)×∑u=s-τiin-1‍{Ai(u)[ai0(u)ki(u)+aii(u)Mi+aij(u)Mj+ci(u)ai0(u)ki(u)]×|xi*(u)-yi*(u)|+MiBi(u)ai0(u)ki2(u)×|xi*(u-τi0)-yi*(u-τi0)|+MiBi(u)aii(u)×|xi*(u-τii)-yi*(u-τii)|+MiBi(u)aij(u)×|xj*(u-τij)-yj*(u-τij)|ai0(u)ki(u)},Vi3*(n)=Mi∑l=n-τi0n-1‍ai0(l+τi0)ki2(l+τi0)Bi(l+τi0)×|xj*(l)-yj*(l)|∑s=l+τi0+1l+τi0+τii‍aii(s)+Mi∑l=n-τiin-1‍Bi(l+τii)aii(l+τii)×|xi*(l)-yi*(l)|∑s=l+τii+1l+2τii‍aii(s)+Mi∑l=n-τijn-1‍Bi(l+τij)aij(l+τij)×|xj*(l)-yj*(l)|∑s=l+τij+1l+τij+τii‍aii(s),i,j=1,2,i≠j.
Calculating the difference ofV*(n) along the solution of the hull equation (31), one has
(40)ΔV*(n)≤-η∑i=12‍|xi*(n)-yi*(n)|,n∈Z.
From (40), we can see that V*(n) is a nonincreasing function on Z. Summing both sides of the above inequalities from n to 0, we have
(41)η∑k=n0‍∑i=12‍|xi*(k)-yi*(k)|≤V*(0)-V*(n+1),n<0.
Note that V*(n) is bounded. Hence we have
(42)∑k=-∞0‍∑i=12‍|xi*(k)-yi*(k)|<+∞,
which implies that
(43)lim⁡n→-∞|xi*(n)-yi*(n)|=0,i=1,2.
DefineQ=∑i=12βiQi, where
(44)Qi=1mi+τi0ai0u(kil)2(1+MiBiuaiiuτij)+τijaij2+(τiiaiiu+τijaiju)MiBiuτiiaiiu+τii2aii2[[ai0u(Kil)2+aiiu+aiju]Aiu(Ai0uKil+AiiuMi+AijuMj+ciu)+BiuMi[ai0u(Kil)2+aiiu+aiju]],i,j=1,2,i≠j.
Letε be an arbitrary small positive number. It follows from (43) that there exists a positive integer n1>0 such that |xi*(n)-yi*(n)|<ε/Q,n<-n1,i=1,2. Therefore, for n<-n1,i,j=1,2,i≠j,
(45)Vi1*(n)≤1mi|xi*(n)-yi*(n)|≤1miεQ,Vi2*(n)≤τi0ai0u(kil)2max⁡p≤n|xi*(p)-yi*(p)|+τijaijumax⁡p≤n|xj*(p)-yj*(p)|+τii2aiiu[Aiu(ai0ukil+aiiuMi+aijuMj+ciu)×max⁡p≤n⁡|xi*(p)-yi*(p)|+MiBiuai0u(kii)2max⁡p≤n⁡|xi*(p)-yi*(p)|+MiBiuaiiumax⁡p≤n|xi*(p)-yi*(p)|+MiBiuaijumax⁡p≤n|xj*(p)-yj*(p)|ai0ukil]≤{τi0ai0u(kil)2+τijaiju+τii2aiiu[Aiu(ai0ukil+aiiuMi+aijuMj+ciu)+MiBiuai0u(kii)2+MiBiuaiiu+MiBiuaijuai0ukil]τi0ai0u(kil)2}εQ,Vi3*(n)≤Miτi0τiiai0uBiu(kil)2aiiumax⁡p≤n|xj*(p)-yj*(p)|+Miτii2Biu(aiiu)2max⁡p≤n|xi*(p)-yi*(p)|+MiτijτiiBiuaijuaiiumax⁡p≤n|xj*(p)-yj*(p)|≤{Miτi0τiiai0uBiu(kil)2aiiu+Miτii2Biu(aiiu)2+MiτijτiiBiuaijuaiiuai0uBiu(kil)2}εQ,i,j=1,2,i≠j.
It follows from (38) and above inequalities that
(46)V*(n)≤∑i=12βiQiεQ=ε,n<-n1,
so lim⁡n→-∞V*(n)=0. Note that V*(n) is a nonincreasing function on Z, and then V*(n)≡0; that is xi*(n)=yi*(n), i=1,2, for all n∈Z. Therefore, each hull equation of system (1) has a unique strictly positive solution.
In view of the above discussion, any hull equation of system (1) has a unique strictly positive solution. By Propositions 7–10 and Lemma 11, the almost periodic difference system (1) has a unique strictly positive almost periodic solution which is globally attractive. The proof is completed.Letτij=0, i=1,2, j=0,1,2. Like in the proof of Theorem 12, we have the following corollary.Corollary 13.
Letτij=0, i=1,2, j=0,1,2. Assume that
(47)a10lM1+k1u-a12uM2-c1u>0,a20lM2+k2u-a21uM1-c2u>0
hold and further that there exist two positive constants α1 and α2, such that
(48)αi[min⁡⁡{aiil,2Mi-aiiu}-ai0u(kil)2]-αjajiu>0,2222222222222222222222i,j=1,2,i≠j,
where Mi=(1/aiil)exp⁡{(ai0u/kil)-1}. Then the almost periodic difference system (1) admits a unique strictly positive almost periodic solution, which is globally attractive.


## 5. Example and Numerical Simulation

In this section, we give the following example to check the feasibility of our result.Example 1.
Consider the following almost periodic discrete Schoener’s competition model with delays(49)x1(n+1)=x1(n)exp⁡⁡[0.4+0.1sin⁡(2n)x1(n-3)+10.2+0.2cos⁡⁡(2n)-(2.02+0.02sin⁡(3n))x1(n-1)-(0.015+0.005sin⁡(2n))x2(n-4)-(0.002+0.0005sin⁡(5n))sin⁡(2n)x1(n-3)(2n)⁡],x2(n+1)=x2(n)exp⁡⁡[0.3+0.2sin⁡(3n)x2(n-4)+10.5+0.5cos⁡⁡(3n)-(0.009+0.001cos⁡⁡(5n))x1(n-2)-(2.53+0.03cos⁡⁡(3n))x2(n-1)-(0.005+0.001sin⁡(2n))sin⁡(3n)0.5cos⁡⁡(3n)].
By simple computation, we derive(50)M1≈0.2033,M2≈0.1626,a10lM1+k1u-a12uM2-c1u≈0.0237,a20lM2+k2u-a21uM1-c2u≈0.0018,E11≈1.1047,E22≈1.377,F12≈1.0075,F21≈1.3366,G12≈0.0548,G21≈0.0217.
Then
(51)23E11-(23F12+G21)≈0.01,E22-(F21+23G12)≈0.026.
Also it is easy to see that conditions (H2) and (H3) are verified. Therefore, system (49) has a unique strictly positive almost periodic solution which is globally attractive. Our numerical simulations support our results (see Figures 1, 2, 3, and 4).Figure 1
Dynamic behavior of the first componentx1(n) of the solution (x1(n),x2(n)) to system (49) with the initial conditions (x1(n),x2(n))=(0.013,0.018) and (0.032,0.025),n=1,2,3,4,5 for n∈[1,100], respectively.Figure 2
Dynamic behavior of the second componentx2(n) of the solution (x1(n),x2(n)) to system (49) with the initial conditions (x1(n),x2(n))=(0.013,0.018) and (0.032,0.025),n=1,2,3,4,5 for n∈[1,100], respectively.Figure 3
Dynamic behavior of the first componentx1(n) of the solution (x1(n),x2(n)) to system (49) with the initial conditions (x1(n),x2(n))=(0.013,0.018) and (0.032,0.025),n=1,2,3,4,5 for n∈[500,550], respectively.Figure 4
Dynamic behavior of the second componentx2(n) of the solution (x1(n),x2(n)) to system (49) with the initial conditions (x1(n),x2(n))=(0.013,0.018) and (0.032,0.025),n=1,2,3,4,5 for n∈[500,550], respectively.



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*Source: 256094-2014-07-24.xml*