JDE Journal of Difference Equations 2356-7848 2356-7856 Hindawi Publishing Corporation 10.1155/2014/256094 256094 Research Article Almost Periodic Solution of a Discrete Schoener’s Competition Model with Delays http://orcid.org/0000-0003-4458-5517 Zhang Hui http://orcid.org/0000-0002-4423-7830 Li Yingqi http://orcid.org/0000-0003-0201-314X Jing Bin http://orcid.org/0000-0001-8961-3150 Fang Xiaofeng Wang Jing Çinar Cengiz 1 Mathematics and OR Section Xi’an Research Institute of High-Tech Hongqing Town Xi’an Shaanxi 710025 China 2014 2472014 2014 08 06 2014 12 07 2014 24 7 2014 2014 Copyright © 2014 Hui Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In 2009, Wu et al.  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=max1i2,0j2{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 nN.

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  and the references cited therein). Recently, a few papers investigate the global stability of the pure-delay model (see ).

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  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 by Z and Z+ the set of integers and the set of nonnegative integers, respectively. For any bounded sequence {f(n)} defined on Z, define fu=supnZf(n),fl=infnZf(n).

Throughout this paper, we assume the following.

ki(n),aij(n), and ci(n) are bounded positive almost periodic sequences such that (3)0<kilki(n)kiu,0<aijlaij(n)aiju,0<cilci(n)ciu,i=1,2,j=0,1,2.

The remaining part of this paper is organized as follows. In Section 2, 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 [<xref ref-type="bibr" rid="B22">22</xref>]).

A sequence x:ZR is called an almost periodic sequence if the ε-translation set of x(4)E{ε,x}={τZ:|x(n+τ)-x(n)|<ε,nZ} 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 τE{ε,x} with (5)|x(n+τ)-x(n)|<ε,nZ.τ is called an ε-translation number of x(n).

Definition 2 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let D be an open subset of Rm,f:Z×DRm. f(n,x) is said to be almost periodic in n uniformly for xD if for any ε>0, and any compact set SD, 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 [<xref ref-type="bibr" rid="B24">24</xref>]).

The hull of f, denoted by H(f), is defined by (7)H(f)={g(n,x):limkf(n+τk,x)=g(n,x)  uniformly  on  Z×Slimkf}, for some sequence {τk}, where S is any compact set in D.

Definition 4.

Suppose that X(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<infnZxi(n)supnZxi(n)<,i=1,2, for nZ.

Lemma 5 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Assume that {x(n)} satisfies x(n)>0 and (9)x(n+1)x(n)exp{a(n)-b(n)x(n)} for nN, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants. Then (10)limsupn+x(n)1blexp{au-1}.

Lemma 6 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Assume that {x(n)} satisfies (11)x(n+1)x(n)exp{a(n)-b(n)x(n)},nN0,limsupn+x(n)x*, and x(N0)>0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and N0N. Then (12)liminfn+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 [<xref ref-type="bibr" rid="B1">1</xref>]).

Any solution (x1(n),x2(n)) of system (1) with the initial condition (2) is positive and ultimately bounded; that is, (13)limsupn+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)liminfn+xi(n)mi,i=1,2, where m1 and m2 are defined by (24) and (26), respectively.

Proof.

For any small positive constant ε>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 nN1+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 nN1+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].

When ε 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)liminfn+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}.

Letting ε0, it follows that (24)liminfn+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)liminfn+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 Proposition 8 is weakened compared to condition (H) in .

Theorem 9 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Assume that (H2) holds; system (1) is permanent.

Proposition 10 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Assume that (H2) holds and further that there exist positive constants α1,α2, and η>0 such that (28)(H3)αiEii-(αiFij+αjGji)η,i,j=1,2,ij, 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;ij,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, ij. 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)limn+(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 nZ. 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 , 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 Lemma 11, 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 nZ. 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)miliminfn+xi(n)limsupn+xi(n)Mi,i=1,2. And also (33)0<infnZ+xi(n)supnZ+xi(n)<,i=1,2.

Let ε be an arbitrary small positive number. It follows from (32) that there exists a positive integer n0 such that mi-εxi(n)Mi+ε, nn0, i=1,2. Write Xk(n)=X(n+τk)=(x1k(n),x2k(n)), for all nn0+τ-τk, kZ+. 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}, ahZ(h=1,2,,m), and m is a finite number.

In fact, for any finite subset BZ, when k is large enough, τk+ah-τ>n0, h=1,2,,m. So (34)mi-εxi(n+τk)Mi+ε,i=1,2; that is, {xi(n+τk)} are uniformly bounded for k large enough.

Now, for a1B, we can choose a subsequence {τk(1)} of {τk} such that {xi(a1+τk(1))} uniformly converges on Z+ for k large enough.

Similarly, for a2B, 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, for amB, 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 nB, we have xi(n+τk)yi(n) uniformly in nB, as k.

By the arbitrary of B, 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-εyi(n)Mi+ε, i=1,2, for nZ. Since ε is an arbitrary small positive number, it follows that miyi(n)Mi, i=1,2, for nZ; that is, (37)0<infnZyi(n)supnZyi(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)),nZ, where (39)Vi1*(n)=|lnxi*(n)-lnyi*(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)=Mil=n-τi0n-1ai0(l+τi0)ki2(l+τi0)Bi(l+τi0)×|xj*(l)-yj*(l)|s=l+τi0+1l+τi0+τiiaii(s)+Mil=n-τiin-1Bi(l+τii)aii(l+τii)×|xi*(l)-yi*(l)|s=l+τii+1l+2τiiaii(s)+Mil=n-τijn-1Bi(l+τij)aij(l+τij)×|xj*(l)-yj*(l)|s=l+τij+1l+τij+τiiaii(s),i,j=1,2,ij.

Calculating the difference of V*(n) along the solution of the hull equation (31), one has (40)ΔV*(n)-ηi=12|xi*(n)-yi*(n)|,nZ. 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)ηk=n0i=12|xi*(k)-yi*(k)|V*(0)-V*(n+1),n<0. Note that V*(n) is bounded. Hence we have (42)k=-0i=12|xi*(k)-yi*(k)|<+, which implies that (43)limn-|xi*(n)-yi*(n)|=0,i=1,2.

Define Q=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,ij.

Let ε 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,ij, (45)Vi1*(n)1mi|xi*(n)-yi*(n)|1miεQ,Vi2*(n)τi0ai0u(kil)2maxpn|xi*(p)-yi*(p)|+τijaijumaxpn|xj*(p)-yj*(p)|+τii2aiiu[Aiu(ai0ukil+aiiuMi+aijuMj+ciu)×maxpn|xi*(p)-yi*(p)|+MiBiuai0u(kii)2maxpn|xi*(p)-yi*(p)|+MiBiuaiiumaxpn|xi*(p)-yi*(p)|+MiBiuaijumaxpn|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)2aiiumaxpn|xj*(p)-yj*(p)|+Miτii2Biu(aiiu)2maxpn|xi*(p)-yi*(p)|+MiτijτiiBiuaijuaiiumaxpn|xj*(p)-yj*(p)|{Miτi0τiiai0uBiu(kil)2aiiu+Miτii2Biu(aiiu)2+MiτijτiiBiuaijuaiiuai0uBiu(kil)2}εQ,i,j=1,2,ij.

It follows from (38) and above inequalities that (46)V*(n)i=12βiQiεQ=ε,n<-n1, so limn-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 nZ. 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 710 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.

Let τij=0, i=1,2, j=0,1,2. Like in the proof of Theorem 12, we have the following corollary.

Corollary 13.

Let τij=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,ij, 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)M10.2033,M20.1626,a10lM1+k1u-a12uM2-c1u0.0237,a20lM2+k2u-a21uM1-c2u0.0018,E111.1047,E221.377,F121.0075,F211.3366,G120.0548,G210.0217. 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).

Dynamic behavior of the first component x1(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.

Dynamic behavior of the second component x2(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.

Dynamic behavior of the first component x1(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.

Dynamic behavior of the second component x2(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.

Conflict of Interests

There are no financial interest conflicts between the authors and the commercial identity.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 61132008) and Scientific Research Program Funded by Shaanxi Provincial Education Department of China (no. 2013JK1098).

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