# Periodic Measures of Mean-Field Stochastic Predator-Prey System

Authors: Zaitang Huang; Junfei Cao
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper discusses the dynamics of the mean-field stochastic predator-prey system. We prove the existence and pathwise uniqueness of the solution for stochastic predator-prey systems in the mean-field limit. Then we show that the solution of the mean-field equation is a periodic measure. Finally, we study the fluctuations of the periodic in distribution processes when the white noise converges to zero.


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## Body

## 1. Introduction

The predator-prey equations is one of the most famous population models(1)x˙t=xtr1-xtK-xtyta+x2t,y˙t=ytμxta+x2t-D,where x(t) denotes the prey population density and y(t) denotes the predator population density. The parameters r,K,a,μ, and D are positive real numbers. By the results in Ruan and Xiao [1], they discuss all kinds of bifurcation phenomena. Recently, system (1) was studied extensively that it exhibits complex dynamical phenomena, including bifurcation, stability, and attractive [2–8].However, population systems in the real world are very often subject to environmental noise [9–15]. According to the Markov jump approach, a classical stochastic predator-prey model can be described by(2)dxt=xtr1-xtK-ytra+x2tdt+g1xtdWt,dyt=ytμxta+x2t-Ddt+g2ytdWt,where Wt is independent Brownian motions. Biologic i in population satisfies the following equation, predator-prey model:(3)dxi,Nt=xi,Ntr1-xi,NtK-yi,Ntra+xi,N2tdt+g1xi,NtdWit+D11N-1∑j=1,j≠iNxj,Nt-xi,Ntdt,dyi,Ntt=yi,Nttμxi,Nta+xi,N2t-D+g2yi,NtdWit+D21N-1∑j=1,j≠iNyj,Nt-yi,Ntdt,where xi,N and yi,N denote the population density of x and y in the ith out of N population, D1 and D2 are nonnegative real number modelling the diffusion between the prey population density and the predator population density, and Wi(i=1,2,…,n) are independent Brownian motions.Under regularity conditions, for any fixedk, ((x1,N,y1,N),(x2,N,y2,N),…,(xk,N,yk,N)) converge in law when N→∞, and then system (3) becomes(4)dxt=xtr1-xtK-ytra+x2tdt+g1xtdWt+D1Ext-xtdt,dyt=ytμxta+x2t-Ddt+g2ytdWt+D2Eyt-ytdt,Lx0,y0=μ.where g1(x(t)) and g2(x(t)) are Noise intensity functions and D1 and D1 are real number. According to the mathematical approach [16–20], these systems can appear very standardized. However, many real world problems process the nature of mixing randomness and periodicity, e.g., due to change of temperatures on earth, harvesting seasons, seasonal economic data, individual lifecycle, and seasonal effects of weather [15]. Biological populations are very often subject to random perturbations that come in a more-or-less periodic way. A number of random periodic results have been studied in the literature [15, 16, 21–24], but none of them covers (4). And my method can also be extended to other noise, for example, the telephone noise, Markovian switching, and Lévy jumps [25–27].In the paper, we investigate the dynamics of mean-field stochastic predator-prey system. First, based on martingale approach and Vlassov-Limits, we prove the existence and uniqueness of the solution for mean-field stochastic predator-prey systems, then, by Tihonov’s fixed point theorem and martingale techniques, we prove that the solution of the stochastic predator-prey systems in the mean-field limit is a strictly periodic law under some suitable assumptions. Finally, we study the fluctuations of the periodic in distribution processes when the white noise converges to zero.


## 2. Preliminaries

Throughout this paper, let(Ω,F,Ftt≥0,P) be a complete probability space. Suppose that r,K,a,μ, and D are positive constants, g1 and g2 satisfy a Lipschitz condition with constant L, and g1 and g2 are bounded,(5)g10=0,g20=0,D1,D2≥0.Definition 1 (see [28]).
Letτ>0 be fixed. A random H-valued process {x(t):t∈R} is called a d-periodic (in distribution) with period τ if (6)∀n∈N,∀t1,t2,…,tn∈R,and∀B1,B2,…,Bn∈BH,P⋂k=1nωk:xω,tk+τ∈Bk=P⋂k=1nωk:xω,tk∈Bk,where B(H) is the Borel a-algebra in H.Letw≔{w(t):t∈R} be an H-valued Wiener process with P{w(0)=0¯}=1. Note that, for any {z1,z2}∈H and 0<s<t,(7)Ewt,z1,ws,z2¯=sz2,Wz1with a nuclear operator W and(8)Ewt-ws2=t-strW,t,s∈R,where Ft:σ(w(v)-w(u):u≤v≤t),t∈R.Theorem 2 (see [1]).
Ifμ2>16/3aD2 and u+μ2-4aD2/2D>K>2u-μ2-4aD2/2D, then system (1) has three equilibria: two hyperbolic saddles (0,0) and (K,0) and an unstable focus (or node) u-μ2-4aD2/2D,r1-u-μ2-4aD2/2KDa+u-μ2-4aD2/2D2 in the interior of the first quadrant. Moreover, system (1) has a unique limit cycle, which is stable.


## 3. Existence and Uniqueness

In the section, under some suitable assumptions, we prove the existence and uniqueness of the solution for mean-field stochastic predator-prey systems.Theorem 3.
For everyN≥2,N∈N, let ν denote a probability measure on [0,∞)×[0,∞) such that (9)∬y4dνx,y<∞and∬exp⁡γx2dνx,y<∞for  someγ>0.(i) Then there exists a unique global strong solution of system (3).(ii) Then there exists a unique nonnegative solution of system (3) satisfying L(x(0),y(0))=μ, r≤1,μ≤1, and ∫0tExs+Eysds<∞ for all t≥0.Proof.
(i) To show that this solution is global, we prove that it does not explode in finite time.
Let(10)zi,N=xi,N+yi,N.From (3), we have (11)dzi,N=xi,Ntr1-xi,NtK-yi,Ntra+xi,N2tdt+g1xi,NtdWit+D11N-1∑j=1,j≠iNxj,Nt-xi,Ntdt+yi,Nttμxi,Nta+xi,N2t-D+g2yi,NtdWit+D21N-1∑j=1,j≠iNyj,Nt-yi,Ntdt.
Form∈N, define Hm1,2:R2→[0,∞) by (12)Hm1x,y=xya+x2+rx2ifx,y≤mLipschitz,  bounded,  nonnegativeotherwiseHm2x,y=μxya+x2ifx,y≤mLipschitz,  bounded,  nonnegativeotherwiseand gi,m(i=1,2) by (13)gi,mx=gixif0≤x≤mLipschitz,  bounded,otherwise.
Applying Itô’s formula, we have(14)dzi,Nmt2≤2zi,Nmxi,Nmtrdt+2zi,Nmg1xi,NmtdWit+2D1zi,Nm1N-1∑j=1,j≠iNxj,Nmtdt+g12xi,Nmtdt+2zi,Nmg2yi,NtdWit+2D2zi,Nm1N-1∑j=1,j≠iNyj,Nmtdt+g22yi,Nmtdt≤2rzi,Nm2dt+2zi,Nmg1xi,NmtdWit+2D11N-1∑j=1,j≠iNzj,Nmtzi,Nmtdt+g12xi,Nmtdtr+2zi,Nmg2yi,NtdWit+2D21N-1∑j=1,j≠iNzj,Nmzi,Nmtdt+g22yi,Nmtdt.Then, we get(15)Ezi,Nmt2≤Ezi,Nm02+2r∫0tEzi,Nms2ds+∫0tEg12xi,Nms+g22yi,Nmsds+2D1+D21N-1∫0t∑j=1,j≠iNEzj,Nmszi,Nmsds.Since the law of (xi,N(m)(t),yi,N(m)(t)) is symmetric (i=1,2,…,N), we get(16)Ez1,Nmt2≤Ez1,Nm02+2r∫0tEz1,Nms2ds+2L∫0tEz1,Nms2ds+2D1+D2∫0tEz1,Nms2ds=Ez1,Nm02+2r+L+D1+D2∫0tEz1,Nms2dsApplying the Bellman inequality, we have(17)Ez1,Nmt2≤Ez1,Nm02×1+2r+L+D1+D2∫0te2r+L+D1+D2t-sds.Due to Ez1,N(m)(0)2=Ez1,N(0)2 and from [29], then (x1,N(m)(t),y1,N(m)(t)) converge weakly to (x1,N(t),y1,N(t)) when m→∞; by Fatou’s lemma, we obtain that(18)supN≥2⁡sup0≤t≤T⁡Ez1,Nt2<∞for  allT>0.Then we have(19)Pxi,N≥0,yi,N≥0for  allt≥0=1.Therefore, we prove the first assertion of the theorem.
(ii) Next, we will prove that there exists a unique nonnegative solution of system (3).
Let∫0TEx(s)+Ey(s)ds<∞ for all T>0; (x(t),y(t)) denotes nonnegative real solution of (4). Set a(t)=Ex(t),b(t)=Ey(t), and hα(g1(x(t))) where (20)hαx=xifx≤α,x∈Rαifx≥α,x∈R.Then (xα(t),yα(t)) denote the real solution of (4) with hα(g1(x(t))) instead of g1(x(t)).
Let(21)zt=xt+ytandzαt=xαt+yαt.
Applying Itô’s formula, we get(22)zαt=zα0+∫0txαsr1-xαsK-yαsra+xα2sds+∫0thαg1xtdWs+D1∫0tas-xαsds+D2∫0tbs-yαsds+∫0tyαsμxαsa+xα2s-Dds+∫0tg2yαsdWsand(23)Ezαt≤Ezα0+D1∫0tasds+D2∫0tbsds.Then, we get(24)supα>0⁡sup0≤t≤T⁡Ezαt<∞and(25)Ezαt+D1-r∫0tExαsds+D2∫0tEyαsds≤Ezα0+D1∫0tasds+D2∫0tbsds.When α→∞, then (xα(·),yα(·)) converge in law to (x(·),y(·)) in C([0,+∞),R+2); by dominated convergence, we have Ez(t)≤Ez(0).
To prove the uniqueness of solution, it will prove that there are someε>0 and some γ~>0 such that(26)sup0≤t≤1⁡Eexp⁡δx2<∞andsup0≤t≤1⁡Ez4t<∞By iteration method, next, we prove the pathwise uniqueness on [0,∞).
Firstly. Applying Itô’s formula to x, we get(27)xnt=xn0+∫0tnrxns1-xsK-ysra+x2sds+∫0tnxn-1sg1xsdWs+nD1∫0txn-1sExs-xsds+nn-12∫0txn-2sg22xsds.Let τα≔inf⁡{t≥0:z(t)≥α}, B≔supx≥0⁡g21(x), and(28)xα,nt=xnt∧τα.Then, we have(29)Exα,nt≤Exn0+E∫0t∧ταnrxnsds+nn-1B2E∫0t∧ταxn-2sds+nD1E∫0t∧ταxn-1sEz0ds.For all 1≤k≤n-1, by sup0≤t≤1⁡Exk<∞ and Fatou’s theorem, when α→∞, we have sup0≤t≤1⁡Exn<∞, and(30)Ex2nt≤Ex2n0+2nr∫0tx2nsds+n2n-1B∫0tEx2n-2sds+2nD1∫0tx2n-1sEz0ds.Since x2n-1≤x2n+1, it shows that (forn≥1)(31)Ex2nt≤Ex2n0+2nD1z0T+2nr+D1Ez0∫0tEx2nsds+n2n-1B∫0tEx2n-2sds.Let r>0 so that(32)Ex02+rn≥Ex2n0+2nD1z0T.Then from (31), we obtain that(33)Ex2nt≤Ex02+rn+2nr+D1Ez0∫0tEx2nsds+n2n-1B∫0tEx2n-2sds.Now, we consider the equation(34)dX=r+D1Ez0Xdt+B1/2dwt.By the Itô formula, we have(35)EX2nt=EX02+rn+2nr+D1Ez0∫0tEX2nsds+n2n-1B∫0tEX2n-2sds.By (34) and (35), as n=1, we infer that(36)Ex2t≤Ex02+rn+Bt+2r+D1Ez0∫0tEx2sds.(37)EX2t≤EX02+rn+Bt+2r+D1Ez0∫0tEX2sds.By Gronwall’s inequality, we get(38)Ex2t≤Ex02+rn+Bt+2r+D1Ez0∫0te2r+D1Ez0t-sEx02+rn+Bsds=EX2t.From (33) and (35), by Gronwall’s inequality, we get(39)Ext2n≤EXt2n,n≥1,0≤t≤T.Hence, if for some δ>0,EeδXt2<∞, then Eeδxt2≤EeδXt2<∞.
Note that ifEeεx02<∞ for some ε, then we have(40)EeεX02=eεrEeεx02<∞,and therefore there exists δ=δ(T)>0 such that(41)sup0≤t≤1⁡EeδX2<∞.
Secondly. To prove sup0≤t≤1⁡Ez4<∞, defining τα=inf⁡{t≥0:z(t)≥α}, then, for i=2,3,4 and 0≤t≤1,(42)zit=zi0+i∫0tzi-1xsr1-xsK-ysra+x2sds+i∫0tzi-1g1xtdWs+D1i∫0tzi-1Exs-xsds+D2i∫0tzi-1Eys-ysds+i∫0tzi-1ysμxαsa+x2s-Dds+∫0tg2ysdWs+12ii-1∫0tzi-2g12xs+g22ysds,and for z(α,i)=zi(t∧τα) and for all k≤i-1, let sup0≤t≤1⁡Ezk, then we have(43)Ezα,it=Ezα,i0+4∫0tzα,i-1sxsr+D1+D2Ez0ds+6L2∫0tzα,isds≤Ezα,i0+4∫0tzα,isr+D1+D2Ez0ds+6L2∫0tzα,isds=Ezα,i0+∫0tzα,is4r+6L+4D1+D2Ez0ds.By Gronwall’s inequality and Fatou’s theorem, when α→∞, we get(44)sup0≤t≤1⁡Ez4<∞.
Finally. Let (x(t),y(t)) and (x~(t),y~(t)) denote two solutions of (4) on the same space. Ones already showed that(45)sup0≤t≤1⁡Ex4t+Ey4t+Ex~4t+Ey~4t<∞.Set(46)cs=Eys-Ey~s,ds=Exs-Ex~s,x¯s=xs-x~t,y¯s=ys-y~t,z¯s=zs-z~t.By truncation technique, we get(47)dx¯2t=2x¯tdx¯t+g1xt-g1x~t2dt=2x¯trx¯t-rx2tK-ytxta+x2t+rx~2tK+x~ty~ta+x~2tdt+2x¯tg1xt-g1x~tdWt+2x¯tD1dt-D1x¯tdt+g1xt-g1x~t2dt,and(48)Ex¯2t≤E2x¯trx¯t-rx2tK-ytxta+x2t+rx~2tK+x~ty~ta+x~2tdt+2D1∫0td2sds-2D1∫0tEx¯2sds+L2∫0tEx¯2sds.Since(49)d2s≤Ex¯2tthen we have(50)Ex¯2t≤∫0t2Ex¯srx~2sK-rx2sK+x~sy~sa+x~2s-ytxsa+x2sds+L2+2r∫0tEx¯2sds≤∫0t2Ex¯sM2sads+L2+2r∫0tEx¯2sds,where M(s)=max⁡{x~(s),y~(s)}. Furthermore, we have(51)dz¯2t≤2rz¯x¯s+D1dt+D2ct-D1x¯t-D2y¯tdt+2z¯g1x-g1x~dWt+2z¯g2y-g2y~dWt+g1x-g1x~2+g2y-g2y~2dt.Therefore, we have(52)Ez¯2t≤2r+1+2D1+2D2+K2∫0tEx¯2sds+1+2D1+2D2+K2∫0tEy¯2sds.Let(53)Hs=Ex¯2s+Ey¯2s,we get(54)Ht=Ez¯t-x¯t2+Ex¯2s≤2Ez¯2t+3Ex¯2s≤k1∫0tHsds+6a∫0tEy~sM2sds,where(55)k1=10r+2+4D1+4D2+2K2+3L2.For n>0,s≤1, let q=1+1/n and p=n+1. Then 1/p+1/q=1. By Gronwall’s inequality, we have(56)EM2sx¯s=EM2sx¯q-1/qsx¯1/qs≤EM2psx¯s1/pEx¯s1/q≤EM4ps1/2pEx¯2s1/2pEx¯s1/q≤k2αpEx¯2s1/2p,where(57)αp=sup0≤s≤1⁡EM4p1/2p,k2=sup0≤s≤1⁡Ex¯s1/q+1<∞.
By Gronwall’s inequality, fort∈[0,1], we showed that(58)Ht≤k1∫0tHsds+6k2αpa∫0tHsn/n+1ds.D(·) is continuous, and by Hölder’s inequality, we get(59)0≤Ht≤k1+6k2αn+1a1n+1tn+1for0≤t≤ε1.For some ε1≥ε2>0, it is easily seen that there is a sequence (nk)k∈N→∞(k→∞) such that sup⁡H(t)→0 when nk→∞ uniformly in t∈[0,ε2] iff(60)liminfn→∞⁡k1+6k2αn+1a1n+1<∞and(61)liminfn→∞⁡1nsup0≤s≤ε2⁡EM2ns1/n<∞.By Chebychev’s inequality, we have, for s∈[0,ε2],(62)PMs≥σ≤Pxs≥σ+Px~s≥σ≤k3e-δσ2,where(63)k3=sups∈0,ε2⁡Eexp⁡δx2t+sups∈0,ε2⁡Eexp⁡δx~2t<∞.Hence, we have, for n∈N,(64)1nEM2ns1/n=1n∫0∞PM2ns≥σdσ1/n≤1nk3∫0∞e-δσ1/ndσ1/n=1nk3∫0∞e-δunun-1du1/n=1nk3n!δ-n1/n.Since n!<nn for n∈N, we get (65)1nEM2ns1/n≤k31/nδ-1.In summary, we proved the theorem on [0,∞).Theorem 4.
Suppose that Theorem3 is satisfied. When N→∞, then(66)x1,N,y1,N,x2,N,y2,N,…,xk,N,yk,Nconverge to k independent copies of solutions of (4) in C([0,∞),R2k).Proof.
LetPN be the law of 1/N∑i=1Nεxi,N(·),xi,N(·), where νa is the measure (67)νa=1a∈A0otherwise.Then {PN,N∈N,N≥2} is relatively compact and (P~N=L(x1,N,y1,N))N=2,3,… is a tight family. So, we have(68)limϵ↓0⁡supN≥2⁡P~Nsup0≤s<t≤T,t-s<ϵ⁡xt-xs+yt-ys>ρ=0.Let(69)x¯N=1N∑i=1Nxi,Nt,y¯N=1N∑i=1Nyi,Nt,andz¯t=x¯t+y¯t.Next, we will prove that(70)z¯t+∫0tz¯sdsis a submartingale. It is easy to see that(71)z¯t≤z¯0+∫0trK4-Dy¯Nsds+∫0t1N∑i=1Ng1xi,NtdWs+∫0t1N∑i=1Ng2yi,NsdWs.By martingale theory, we get(72)z¯t-z¯0-∫0trK4-Dy¯Nsdsis a martingale. Therefore, we have(73)z¯t+D∫0tz¯Nsds≥0is a submartingale. By martingale inequality [29], we get(74)PNsup0≤t≤T⁡z¯t≥α>0≤PNsup0≤t≤T⁡z¯t+D∫0tz¯Ns≥α>0≤1αEz¯T+D∫0Tz¯Ns≤1αDT+1z¯0+rK4T+T22.Furthermore, we have that(75)z1,Nt-z1,N0-∫0trK4-Dy1,Nsds+D11N-1∑i=2Nxi,N-x1,N+D21N-1∑i=2Nyi,N-y1,Nis a martingale, then we get that(76)z1,Nt+D1+D1+D1∫0tz1,Nsds≥0is a submartingale. Furthermore, we have (77)PNsup0≤t≤T⁡z¯1,nt≥α≤1αTD1+D1+D1+1z¯1,N0+rK4T+T22+D1+D1+D1.So, for every T>0, it is easy to see that(78)limα↑∞⁡supN≤2⁡PNsup0≤t≤T⁡z¯t≥α=0and(79)limα↑∞⁡supN≤2⁡PNsup0≤t≤T⁡z¯1,Nt≥α=0.Let(80)τα,N=inf⁡t≥0∣z¯t≥αorz¯1,Nt≥αand τα,N=∞ if there does not exist such t. For any T>0,ρ,>0,σ>0 and α>0, it is easy to see that (81)P¯Nsup0≤s<t≤T,t-s<σ⁡xt-xs+yt-ys>ρ≤PNτα,N≤T+PNsup0≤s<t≤T,t-s<σ⁡x1,Nt∧τα,N-x1,Ns∧τα,N+y1,Nt∧τα,N-y1,Ns∧τα,N>ρ.
Letf∈C0∞(R2),p∈N; it is easy to see that g¯1,g¯2,…,g¯p are continuous bounded function from R2 to R and set 0≤sp<sp-1<…<s1≤s<t. Let(82)M0=ν∈M:sup0≤u<t⁡∫xω,u+yω,udνω<∞and for ν∈M0(83)Fν=ν,fxt,yt-fxs,ys-∫st∫Afx·,u,y·,u,xω,u,yω,uνdωdu·∏j=1pg¯jx·,sj,y·,sjwhere (84)Afx,y,x′,y′=∂f∂xx,yrx1-xK-yra+x2+D1x′-x+∂f∂yx,yyμxa+x2-D+D2y′-y+12g12x∂2f∂x2x,y+12g22y∂2f∂y2x,y.
Defining(85)x¯u=1Nk∑i=1Nkxi,Nkuandy¯u=1Nk∑i=1Nkyi,Nkuwhere (Nk)k=1,2,… are a sequence such that PNk→k→∞P∞. By (78), we have P∞(M0)=1. Furthermore(86)∫M0FνdPNkν=ENk1Nk∑i=1Nkfxit,yit-fxis,yis-∫stAfxiu,yiu,x¯u,y¯udu∏j=1pg¯jxisj,yisj2=ENkNk-1Nk∏i=1Nkfxit,yit-fxis,yis-∫stAfxiu,yiu,x¯u,y¯udu∏j=1pg¯jxisj,yisj2+1NkENkfx1t,y1t-fx1s,y1s-∫stAfx1u,y1u,x¯u,y¯udu∏j=1pg¯jx1sj,y1sj2.When k→∞, the last term is o(1). Then, for 1≤i≤Nk, we have that(87)Θiτ=fxit,yit-fxis,yis-∫0τAfxiu,yiu,x¯u,y¯udu+∫0τ∂f∂xxiu,yiuD1x¯u-1Nk-1∑j=1j≠iNkxjuare PNk-martingales and Θi,Θj for j≠i.
Using(88)x¯u-1Nk-1∑j=2Nkxju=1Nk-1x1u-x¯u,for all N≥2, we get(89)ENkx¯u+y¯u2<∞,ENkx1u+y2u2<∞,then we have(90)limk→∞⁡∫M0FνdPNkν=0.Next, we prove that(91)limk→∞⁡∫MFνdP∞ν=0.For α>0, define hα:[0,∞]→∞ by (92)hα=xx≤ααx≥α.We will show that (93)alimk→∞⁡∫M0F2νdPNkν=0blimα→∞⁡supk⁡∫M0Fα2νdPNkν-∫M0F2νdPNkν=0blimk→∞⁡∫MFα2νdPNkν=∫MFα2νdP∞νdlimα→∞⁡∫MFα2νdP∞ν≥∫MF2νdP∞ν.It is easy to see that we already proved (a). By the definition of weak convergence, (c) is proven directly.
By Fatou’s theory, forν∈M0, then(94)Fαν=Fν+ν,∫st∫∂f∂xx·,u,x·,uD1xω,u-hαxω,u+∂f∂yx·,u,x·,uD2yω,u-hαyω,udνωdu∏j=1Pg¯jx·,sj,x·,sj,and it is easy to see that (d) is proved. By similar way, we can prove (b).
Then(a)-(c) give(95)limα→∞⁡∫MFα2νdP∞ν=0.So (d) implies(96)∫MF2νdP∞ν=0and therefore F(ν)=0P∞-a.s.
By the law of large numbers, we have proved thatP∞-a.s. the projection of ν at t=0 is equal to μ. By Lemma 3.1 in [30], we proved the assertion of the Theorem 4.


## 4. Periodic Distribution

In the section, under some suitable assumptions, we prove that the solution of mean-field stochastic predator-prey systems possesses a strictly periodic distribution.Lemma 5.
Suppose thatT>0,b>0,b¯>0,κ1>0,κ2>0. Let(97)z¯t=gyt+xtwhere(98)A=b¯+r4KT+1and(99)gx=A0≤x≤A-1xx≥A+1arbitraryotherwise,  but  such  thatg∈C20,∞and0≤dgxdx<1.Then(100)An=supEz0≤b¯⁡supEz¯n0≤b⁡supD2≤0⁡supκ2D2+κ1≥D1≤0⁡sup0≤t≤T⁡Ez¯nt<∞.for all n∈N.Proof.
Applying Itô’s formula toz¯n(t), we get(101)dz¯nt=nz¯n-1tdxt+g′ydy+12g′′yg22ytdt+12nn-1z¯n-2tg12xt+g′2ytg22ytdt.Suppose that Ez¯n(0)≤b and Ezn(0)≤b~.
Definingτ(α)=inf⁡{t≥0:z¯≥α} and (102)1αt=0ifsup0≤s≤t⁡z¯s≥α1otherwiseand(103)z¯αt=z¯αt∧τα.Then, we have(104)E1αtz¯nt≤z¯αnt=Ez¯n0+n∫0tE1αsz¯n-1sxsr1-xsK-ysra+x2s+g′yysμxsa+x2s-D+D2Eys-ys+D1Exs-xs+12g′′ysg22ysds+12nn-1∫0tE1αsz¯n-2tg12xt+g′2ytg22ytds.
Case 1 (D2≥D1). Substituting D1Ex(s)-x(s) by -D1z¯(s)+D2g(y(s))+D2Ex(s), we have(105)E1αtz¯nt=Ez¯n0-nD1∫0tE1αsz¯nsds+∫0tRsds+R0t.Solving the above equation, we have(106)E1αtz¯nt≤e-D1ntEz¯n0+∫0teD1nsRsds≤Ez¯n0+∫0tb1D1+b2e-D1t-sds+nn-1L2∫0tE1αtz¯nse-D1nt-sds≤b+b1n+b2T+nn-1L2∫0tE1αtz¯nsds,where b1 and b2 are real number independent of D1,D2, but N possibly based on b,b¯ and T, then we have(107)0≥D1g′ysEys-ys≥D2g′ysEys-ys,since g′(y(s))≠0, it implies(108)Eys-y0≤b¯+r4KT-A-1=0,for0≤s≤T.Using Gronwall’s Inequality and Fatou’s theory, when α→∞, we obtain the result.
Case 2 (D1≥D1>0 and D1≤κ2D2+κ1≥D1). By (104), (109)0=-nD2∫0tE1αsz¯nsds+nD2∫0tE1αsz¯n-1sxs+gysds.Then we get(110)E1αtz¯nt=Ez¯n0-nD2∫0tE1αsz¯nsds+∫0tR¯sds+R0t.As before, we have (111)E1αtz¯nt≤Ez¯n0+∫0tb¯2D2+b1e-D2nt-sds+nn-1L2∫0tE1αtz¯nsds,where b¯1 and b¯2 are real numbers not relying on D1,D2, and N but possibly relying on b¯,b,n,T,κ1, and κ2, then we get(112)D1Exs-xs≤κ2D2+κ1Exs-D2xs.Similarly the proof is as in Case 1, so we prove Case 2 directly.Lemma 6.
Suppose that the condition of Lemma5 is satisfied. Let n∈N∖{1}, αn-1>0, and N¯>L2(n-1). The following conditions hold:
(1) For allD1,D2 satisfying N¯≤D1≤κ2D2+κ1, D2≥N.
(2) There exists a constantA¯n such that if ∞>Ez¯n(0)≥A¯n,Ez¯n-1(0)≤αn-1 and Ez(0)≤b¯.
ThenEz¯n(t)≤Ez¯n(0) for all t∈[0,T].Proof.
LetEz¯n(0)<∞ and Ez¯n-1(0)≤αn-1. By (105) and (106), we can estimate R(s) and R¯(s) by(113)Rs≤nn-1L2E1αsz¯ns+b2D1+b1and(114)R¯s≤nn-1L2E1αsz¯ns+b¯2D2+b¯2,then, for any α>0, we have(115)E1αtz¯nt≤Ez¯n0+∫0tγD¯+λ-nD¯-L2n-1E1αsz¯nsds,where D¯=min⁡{D1,D2}≥N¯ and γ>0 and λ>0 are real number independent of D1,D2 and Ez¯n(0) but possibly based on κ1,κ2,T,n, and αn-1. When α↑∞, we have(116)Ez¯nt≤Ez¯n0+∫0tγD¯+λ-nD¯-L2n-1Ez¯nsds.Furthermore(117)Ez¯nt≤Ez¯nu+∫utγD¯+λ-nD¯-L2n-1Ez¯nsds.where 0≤u≤t≤T.
Let(118)A¯n=γN¯+λnN¯-L2n-1=nλ1-L2n-1N¯+γnN¯-L2n-1≥γD¯+λnD-L2n-1,Fix Ez¯n(0)≥A¯n, and if there are some t≤T satisfying Ez¯n(0)≤Ez¯n(t). Define u=sup⁡{τ≤t:Ez¯n(0)≥Ez¯n(τ)}. Since(119)Ez¯ns+h-Ez¯ns≤γD¯+λhfor all 0≤s≤s+h≤T, then u<t and Ez¯n(u)≥Ez¯n(0); it implies contradiction (117).Lemma 7.
Suppose thatκ1≥0,κ2≥0,T>0, and b>0. The following conditions hold:
(i) There are constantsα,α¯≥0 such that, for any β1,β2 and β=max⁡β1,β2,(120)0≤D1≤κ2D2+κ1andD2≥max⁡0,α2β2-D,min⁡2r-α¯β1≥D1>0.
(ii) For all initial conditions satisfyingEz4(0)≤b, (121)EEx0-x02≤β1,Ey0-y02≤β2.
Then there exist constantsΛ≥0 such that(122)Ext-f1t2+Eyt-f2t2≤Λβ,where (f1,f2) is the solution of (1) with f1(0)=Ex(0),f2(0)=Ey(0).Proof.
Let(123)Φt=xt-Ext,Ψt=yt-Eyt.Applying Itô’s formula, we have(124)dΨ2t=-2D+D2Ψ2t+2Ψtμxtyta+x2t-Eμxtyta+x2tdt+g22ytdt+2Ψtg2ytdW2.Then(125)EΨ2t=EΨ20e-2D+D2t+∫0te-2D+D2t-s2EμxtytΨta+x2t+Eg22ytdt.By Lemma 5, we have(126)α=supEz40≤b⁡supD2≥0⁡sup0≤D1≤κ2D2+κ1⁡sup0≤s≤T⁡2EμxtytΨta+x2t+Eg22yt<∞.Therefore(127)EΨ2t≤EΨ20e-2D+D2t+α2D+D21-e-2D+D2t≤max⁡EΨ20,α2D+D2.
Next, applying Itô’s formula, we get(128)dΦ2t=-2r-D1Ψ2t+2ΦtEμxtyta+x2t-μxtyta+x2tdt+2ΦtrEx2t-x2tK+g12ytdt+2Φtg1ytdW2≤-2r-D1Ψ2t+2μaΦtxtyt+Extytdt+2rKΦtEx2t+x2t+g12ytdt+2Φtg1ytdW2.Then (129)EΦ2t≤EΦ20e-2r-D1t+∫0te-2r-D1t-sEΦs2μa+2rKx2s+y2s+Ex2s+Ey2s+g12ytΦsds.By Lemma 5, we also have(130)α¯=supEz40≤b⁡supD2≥0⁡sup0≤D1<r⁡sup0≤s≤T⁡EΦsx2s+y2s+Ex2s+Ey2s+Eg12yt<∞.Therefore(131)EΦ2t≤EΦ20e-2r-D1t+α¯2r-D11-e-2r-D1t≤max⁡Φ20,α¯2r-D1.
LetEΨ2(0)≤β, and we have EΨ2(t)≤β. Then (132)Ext-f1t2=∫0t2Exs-f1srExs-f1s+rEx2sK-f12tK-Exsysa+x2s-f1sf2sa+f12sds,Eyt-f2s2=∫0s2Eys-f2s-DEys-f2s+Extyta+x2t-f1tf2ta+f12tds.Let(133)Ht=Ext-f1t2+Eyt-f2t2,and we get(134)Ht≤∫0t2rHs+2rKExs-f1sEΦ2s+2EΦsExs+E2xs-f12sds+∫0t2aExs-f1s+Eys-f2sEΦ2s1/2EΨ2s1/2+EΦsEys+EΨExs+ExsEys+f1sf2sds≤∫0t2rHs+Θsds,where(135)Θt=2rKExs-f1sEΦ2s+2EΦsExs+E2xs-f12s+2aExs-f1s+Eys-f2sEΦ2s1/2EΨ2s1/2+EΦsEys+EΨExs+ExsEys+f1sf2s.By Ez2(0)≤b,f1<∞,f2<∞,Ex(t)<∞ and Ey(t)<∞, for ∀t∈[0,T], for some Λ and 0≤t≤T, we have (136)Θt=2rKExs-f1sEΦ2s+2EΦsExs+E2xs-f12s+2aExs-f1s+Eys-f2sEΦ2s1/2EΨ2s1/2+EΦsEys+EΨExs+ExsEys+f1sf2s≤Λ.Then, we get(137)Ht≤∫0t2rHs+Λds.By Gronwall’s inequality, we have(138)Ht≤∫0tΛds+2rΛ∫0tse2rt-sds≤ΛT+2reT-T-1.Then, we prove the assertion of Lemma 7.Theorem 8.
Suppose thatμ2>16/3aD2 and u+μ2-4aD2/2D>K>2u-μ2-4aD2/2D. Let κ1≥0 and κ2≥0, if the following conditions hold:
(i) There is a real numberN∗ such that for all D1≥N∗, 0≤D1≤κ2D2+κ1.
(ii) There is a probability measureν∗ on [0,∞)×[0,∞) and some τ∗>0 such that L(x(0),y(0))=ν∗.
ThenL(x(τ∗),y(τ∗))=ν∗ but L(x(s),y(s))≠ν∗ for 0<s<τ∗; i.e., (4) possesses a periodic distribution.Proof.
( f 1 ∗ , f 1 ∗ ) denote the unique periodic solution of (1), where f1∗(0)=θ,f2∗(0)>ϑ (see [1]) and(139)θ=u-μ2-4aD22D,ϑ=r1-u-μ2-4aD22KDa+u-μ2-4aD22D2.Let(140)b2>f2∗>b1>ϑand (141)T=2maxb1≤b≤b2⁡min⁡u>0:f1u=θ,f1u>ϑ,f1u=θ,f2u=b,f1,f1solve1.(f1(i),f2(i)) denote the solution of system (1) starting at f1(i)(0)=θ,f2(i)(0)=bi(i=1,2) and let (142)t¯=minb1≤b≤b2⁡min⁡u≥0:f1u=θ,f1u<ϑ,f10=θ,f20=b,f1,f1solve1b¯≔b2+θN¯=b2ϵ1≔min0≤t≤Tf11t-θ2+f21t-ϑ21/2ϵ2≔distθ,b;ϑ≤b≤b1,f11t,f21t;t¯≤t≤Tϵ3≔distθ,b;b≥b2,f12t,f22t;t¯≤t≤T.Since (f1(1)(t),f2(1)(t));0≤t≤T is a compact set but not included in (θ,ϑ), because there exists the uniqueness of the solution of system (1), it is easy to see that ϵ1>0 is proven. Next, we prove that ϵ2>0. Suppose that there exist some t¯≤t≤T such that f1(1)(t)=a and ϑ≤f2(1)(t)≤b1. Then the line segment (θ,u),c1≤u≤f1(1)(t) and the curve (f1(1)(s),f2(1)(s));0≤s≤t form a set Ω which is invariant, it is easy see that Ω contains the limit cycle (f1∗,f2∗) and Ω≠(θ,ϑ), due to the globally stable limit cycle, then we prove ϵ2>0. By the same way, we prove that ϵ3>0.
Forn=2,3,4, we take values A¯n satisfying the condition of Lemma 6 with αn-1=A¯n-1,=2,3,4 and α1=c¯. Let c=A¯4; fix Λ and β1,β2 satisfying (Λβ)1/2<min{ϵ1,ϵ2,ϵ3} and rK/4>Λ. Now let N∗≥N¯ be large enough such that(143)α=supEz40≤b⁡supD2≥0⁡sup0≤D1≤κ2D2+κ1⁡sup0≤s≤T⁡2EμxtytΨta+x2t+Eg22yt<2D+N∗β1and (144)α¯=supEz40≤b⁡supD2≥0⁡sup0≤D1<r⁡sup0≤s≤T⁡EΦsx2s+y2s+Ex2s+Ey2s+Eg12yt<2r+N∗β2.From (127) and (131), it is shown that sup0≤t≤T⁡EΦ2(t)≤EΦ2(0) if EΦ2(0)≥β1 and EΨ2(t)≤EΨ2(0) if EΨ2(0)≥β1, Ez4(0)≤b, and D2>N∗. By Lemma 6, it is easy see that sup0≤t≤T⁡Ez¯4(t)≤b if Ez¯i(0)≤A¯n and Ez¯(0)≤b¯. It implies that there are real numbers δ>0 and σ>1 such that sup0≤t≤T⁡Eexp⁡(δx2(t))≤σ whenever Eexp⁡(δx2(0))≤σ. Let U1(0,∞2) denote the set on [0,∞)2 and (145)U=ν∈U10,∞2:Eνx0=θ,b1≤Eνx0≤c2,EνΦ20≤β1,Eνzn0≤A¯nforn=2,3,4andEexp⁡δx20≤σand for ν∈U(146)τν=inf⁡t>0:Eνxt=θ,Eνyt>ϑ,∃t≥s≥0:Eνyt≤ϑand(147)S:U→U10,∞2Sν=Lxτν,yτν.
Next, we will prove thatS maps U into U; hence, we show that there are real numbers δ>0 and σ>1 satisfying sup0≤t≤T⁡Eexp⁡(δx2(t))≤σ whenever Eexp⁡(δx2(0))≤σ.
LetC=supδ≥0⁡g12(x),δ=N¯/4C; suppose that Ez¯(0)≤b and sup⁡Eexp⁡(δx2(0))≤∞. By the proof of Theorem 3, it proves that sup0≤t≤T⁡Eexp⁡(δx2n(t))<∞ for all n≥1 and for A¯=c1/4(148)dx2nt=2nx2ntr1-xtK-ytra+x2t+n2n-1x2n-2g12xtdt+x2n-1g1xtdWt+D12nx2n-1tExt-xtdt≤2nx2ntr-D1+n2n-1x2n-2Bdt+x2n-1g1xtdWt+D12nx2n-1tExtdt.Then, we have (149)Ex2nt≤∫t2nr-D1x2ns+n2n-1Bx2n-2s+2nD1A¯Ex2n-1tds.Therefore, we get(150)E∑n=0Nδx2ntn!≤Eexp⁡δx20+∑n=1Nδnn!∫0tEx2ns2n+1Bδ+nr-D1ds+∑n=1Nδnn!∫0tnD1Ex2n-1s2A¯-xsds+δBt-δN+1N!2N+1B∫0tExNsds.Due to δ=N¯/4B≤D1/4B, we show that the first integrand is negative real number for large enough n. The second integrand is not larger than D1n2A¯2n and(151)∑n=1Nδnn!D1n2A¯2nt≤δD12A¯2texp⁡δ2A¯2.Hence, we get(152)Bδ∑n=1Nδnn!∫0tEx2nds+δ2Bδ-D1∑n=1Nδn-1n-1!∫0tEx2n-2x2sdsand using 2Bδ-D1≤-1/2D1, we have (153)Eexp⁡δxt≤Eexp⁡δx0+Bδ∫0tEexp⁡δx2sds-12δD1∫0tEx2sexp⁡δx2sds+4δD1A¯2texp⁡δ2A¯2Clearly, Ex2exp⁡(δx2)-α¯Eexp⁡(δx2)→∞ when Eexp⁡(δx2)→∞ for any -α¯>0. Therefore, we choose σ>1 satisfying(154)D1Ex2sexp⁡δx2-2BEexp⁡δx2>8D1A¯2exp⁡δ2A¯2.It shows that sup0≤t≤T⁡Eexp⁡δx2t≤σ whenever Eexp⁡δx20≥σ.
Next, we prove thatS is weakly continuous. Hence, if we have proved that τ:U→[0,T] is continuous, it is easy to see that S is weakly continuous on U. Then it implies that τ:U→[0,T] is continuous. For ν∈U, we have (155)ddtExτν=Erxτν-xτνK-xτνyτνa+x2τν=rK4-Exτν-K/22K+xτνyτνa+x2τν≥rK4-Λ=ϱ>0.Let ν∈U and take t0>0 such that d/dtEνx(t)>ϱ/2 for all t∈[τ(ν)-t0,τ(ν)+t0]. Fix 0<ϵ≤ϱ/2, let νn→n→∞ν(νn∈U) and take n0 such that sup0≤t≤T⁡Eνnx(t)-Eνx(t)<ϵ for all n≥n0(ε). Then |τ(ν)-τ(νn)|≤2/ϱϵ for n≥n0(ε); it is easy to see that τ is continuous on U and therefore the proof is completed.Corollary 9.
Suppose thatμ2>16/3aD2 and u+μ2-4aD2/2D>K>2u-μ2-4aD2/2D, κ1>0,κ2>0, and the following conditions hold:
(i) There exists a sequence(D1,n,D2,n)n∈N satisfying the conditions of Theorem 8 for every n∈N and D1,n→n→∞∞.
(ii) There is a sequenceνn∗ on [0,∞)×[0,∞) such that (4) with D1=D1,n,D2=D2,n and Lx0,y0=νn∗.
Then system (4) possesses a periodic distribution and(a)
Eνn∗xt,Eνn∗yt→n→∞f1∗(t),f2∗(t);(b)
Lνn∗x·→n→∞ϵf1∗(·).Proof.
Letb1(n) and b2=b2(n) converge to f2∗(0) with D1,n≥N∗=N∗(n). It is easy to see that β=β(n) converges to zero. Then, the first assertion of theorem is proved. Furthermore, we have (156)dΨt=-D+D2,nΨt+μxtyta+x2t-Eμxtyta+x2tdt+g2ytdWtand hence, solving for Ψ(t), we get (157)Ψt=e-D+D2,ntΨ0+∫0te-D+D2,nt-sμxtyta+x2t-Eμxtyta+x2tdt+∫0te-D+D2,nt-sg2ytdWt.Applying Chebychev’s inequality, we have(158)Psup0≤t≤T⁡Ψt≥R≤PΨ0≥R+P∫0Te-D+D2,nt-sμxtyta+x2t-Eμxtyta+x2tdt≥R+Psup0≤t≤T⁡∫0te-D+D2,nt-sg2ytdWt≥R≤1REΨ0+1RΘD+D2,n+Psup0≤t≤T⁡Qnt≥R,where(159)Qnt=∫0te-D+D2,nt-sg2ytdWtand(160)Θ=sup0≤t≤T⁡Eμxtyta+x2t-Eμxtyta+x2t.Since(161)EΨ0≤EΨ202≤β1/2n→n→∞0then we prove the second assertion of theorem that is(162)Psup0≤t≤T⁡Qnt≥R→n→∞0for  everyR>0.By [31, pp 142], we get(163)Qnt=-D+D2,nQntdt+g2ytdWt,Qn0=0.Therefore, for (164)fy=y4,y≤R4bounded,C2R,then (165)fQnt+D+D2,n∫0tf′QnsQnsds-12∫0tf′′Qnsg2ysdsis a martingale. Denoting the stopping time,(166)τ≔inf⁡t≥0:Qns=R∧T.By Chebychev’s inequality, we get (167)Psup0≤t≤T⁡Qnt≥R=PfQnτ≥R≤1R4EfQnτ=1R4E∫0τ6Qn2sg22ys-4D+D2,nQn4sds.For -R4≤y≤R4,(168)6y2g22ys-4D+D2,ny4=-4D+D2,nx2-3g22ys4D+D2,n2+9g24ys4D+D2,nand hence, we have (169)6y2g22ys-4D+D2,ny4≤9g24ys4D+D2,nifg22ys≤R8D+D2,n6R8g22ys-4D+D2,nR16ifg22ys>R8D+D2,n;since supn⁡E∫0Tg24(y(s))ds<∞, it implies (170)Psup0≤t≤T⁡Qnt≥R≤1R4E∫0T1g22ys≤R8D+D2,n9g24ys4D+D2,n+1g22ys>R8D+D2,n9g24ys4D+D2,n6g24ysD+D2,nds→n→∞0.Therefore, the proof is completed.


## 5. Fluctuations

In the section, under some suitable assumptions, we study the fluctuations of the periodic in distribution processes for mean-field stochastic predator-prey systems when the white noise converges to zero.Theorem 10.
Suppose that condition of Theorem8 is satisfied. The following conditions hold:(i)
There exist real numbersN1 and N2 satisfying D1≥N1 and D2≥N2.(ii)
There is a probability measureμε on [0,∞)2 such that(171)dxt=xtr1-xtK-ytra+x2tdt+εg1xtdWt+D1Ext-xtdt,dyt=ytμxta+x2t-Ddt+εg2ytdWt+D2Eyt-ytdt,Lx0,y0=με.Then system (171) exists a periodic distribution for all 1≥ε≥0.Furthermore, there exists a periodic solution of (171) with (xε,yε)→ε→0(f1∗,f2∗) weakly on C[0,∞),R2.Proof.
By the proof of Theorem8, we prove only supt≥0⁡Ez4<∞ but, using Lemma 6, it is easy to show that there is a periodic solution such that supt≥0⁡Ez8<∞. Then the first assertion of the theorem is proven.
Next, letεn↓0. Due to Exεn(0)=θ and Eyεn(0)≤b2, {νεn}n∈N is family of compact. By the same proof of Theorem 8, it is easy to see that there is a weakly convergent subsequence of (xεn,xεn) approaching to a solution (x,y) of(172)x˙t=xtr1-xtK-ytra+x2t+D1Ext-xt,Ex0=θy˙t=ytμxta+x2t-D+D2Eyt-yt,b1≤Ey0≤b2.Let τε denote the period of (xε,yε); we choose a subsequence εnk(k=1,2,3,…) satisfying L(xεnk,yεnk) approaches to a solution of (172), and τ=limk→∞⁡τεnk exists. Based on Theorem 11.1.4 in [18, pp 264], it is easy to see that that the map (ν,ε,t)↦Lν,ε(x(t),y(t)) from U×[0,1]×[0,T] to U1C0,T,R+2 is continuous, where Lν,ε(x(t),y(t)) is the solution of (171) with Lx0,y0=ν at t.
Letν≔limk∞⁡νεnk; it will show that the periodic distribution solution of (172) is (f1∗,f2∗). Denoting z¯=g(y(t))+x(t)(0≤t≤T), where g is defined as in Lemma 5 with b¯=b2, then we have(173)dz¯tdt=xtr1-xtK-xtyta+x2t1-μg′yt+D1Ext-xt-Dg′ytyt+D2g′ytEyt-yt≤rK4-rx-K22-Dg′ytyt+D1Ext-xt,for all y≤sup0≤t≤T⁡y(t)≤b, then g′(y)=0. The right hand side of (173) is negative constant if either x or y is large enough; i.e., there has some δ>0 satisfying dz¯(t)/dt<0 whenever z¯(t)≥δ; it easily shows that the support of Lx(t),y(t) which is periodic is included in(174)x,y∣x≥0,y≥0,x+y≤δfor every t≥0.
LetΦ=x-Ex and Ψ=y-Ey; we get(175)xtyta+x2t≤12ayt=12aΨ+Eyt.Hence, (w.p.1) 0≤x+y≤δ, and therefore 0≤Ex,Ey≤δ and -δ≤Φ,Ψ≤δ(176)EΦxtyta+x2t≤12aEΦΨ+Eyt≤14aEΦ2+EΨ2EΨxtyta+x2t≤δaEΨΦ+Ext≤δ2aEΦ2+EΨ2EΦx2≤δEΦΦ+Ex≤δEΦ2and(177)ddtEΦ2+Ψ2≤-D1-r-δrK-14a-μδ2aEΦ2-D+D2-14a-μδ2aEΨ2.Then, we have(178)ddtEΦ2+Ψ2<0whenever EΦ2+Ψ2>0 if D1 and D2 are sufficiently large. Because EΦ2 and EΨ2 are periodic, it easily shows that EΦ2≡EΨ2≡0; i.e., (x(t),y(t)) is deterministic. There exists only one periodic solution of system (1) with x(0)=θ and y(0)≥b1. Therefore, the proof is completed.Theorem 11.
Suppose thatμ2>16/3aD2 and u+μ2-4aD2/2D>K>2u-μ2-4aD2/2D. Let κ1,κ2,κ3,κ4 are nonnegative constants. For D1≤κ1D2+κ2 and D2≤κ3D1+κ4, D1 and D2 are large enough,(179)sup0<ε≤1⁡δεε<∞,where (180)δε=εsupt≥0⁡EΦε4t1/4,EΨε4t1/4,Φεt=ε-1xε-Exε,Ψεt=ε-1yε-Eyε.Then there exists a periodic solution (xε,yε) of system (4).Proof.
According tosup0<ε≤1⁡δε<∞, limε↓0⁡δε=0 and Theorem 10. It will prove δε=O(ε) for ε↓0.
Define Lyapunov functions(181)VεΦ,Ψ=λεΦ+ΨφΦ,Ψ+1-λεΦ+ΨψΦ,Ψ,Φ,Ψ≥Mεwhere(182)M=sup1≥ε>0,t≥0⁡Exε+Eyε,φΦ,Ψ=Φ4+Ψ4,ψΦ,Ψ=17Φ+Ψ4,λ∈C20,∞,Rsatisfiesdλdυ≤0andλυ=1υ≤10υ≥2and (183)λευ=λα-1ευ,andα=M·max⁡κ1,κ3,1.Next, we prove that(184)supt≥0⁡sup1≥ε>0⁡supD1,D2⁡EVΦεt,Ψεt<∞Let Lt,ε denote the generator of the diffusion (Φε(t),Ψε(t)). Then, we have (185)Lt,εVΦεt,Ψεt=λεLt,εφ+1+λεLt,εψ+α-1ελε′φ-ψΦ′+Ψ′+12α-2ε2λε′′φ-ψg12+g22+α-1ελε′φΦ-ψΦg12+ψΨ-ψΨg22.
Firstly, we consider the caseΦ+Ψ≥2α/ε which shows λε=0 and hence(186)Lt,εVΦεt,Ψεt=Lt,εψ.In case Φ+Ψ≥α/ε(187)117Lt,εψΦεt,Ψεt≤Φ+Ψ3r-D1+x+ExΦ-D+D2Ψ+6Φ+Ψ2g12εΦ+Ex+g22εΨ+Ey≤Φ+Ψ3-D1-r-2MΦ-D+D2Ψ+6Φ+Ψ2K2εΦ+M2+εΨ+M2.Now, for Φ,Ψ≥-M/ε,Φ+Ψ≥α/ε, and D1≥r+2M+1, we get(188)D1-r-2M-1Φ+D+D2-1Ψ≥min⁡D1-r-2M-1-Mε+D+D2-1Φ+Ψ+MεD1-r-2M-1Φ+Ψ+Mε+D+D2-1-Mε≥ε-1min⁡r+2M-1-κ1D2-κ2M+D+D2+1α+MD1-r-2M-1α+M-D+1+κ3D1+κ4M=ε-1min⁡D2α+M-κ1M+r+2M-κ2-1M+D+1α+MD1α+M-κ3M-1+r+2Mα+M-D+κ4-1M≥0,if D1 and D2 are large enough. Then, we have(189)-D1-r-2MΦ-D+D2Ψ≤-x-y.Furthermore, since ϵΦ+M>0 and εΨ+M>0, we have(190)6Φ+Ψ2K2εΦ+M2+εΨ+M2≤6Φ+Ψ2K2εΦ+M+εΨ+M2≤6Φ+Ψ2K2εΦ+Ψ+2α2≤6Φ+Ψ4K29ε2≤Φ+Ψ4for all Φ+Ψ≥α/ε and ε≤(54K2)-1/2. Then(191)117Lt,εψΦεt,Ψεt≤-3Φ+Ψ4=-317ψΦεt,Ψεtif ε is small enough.
IfΦ+Ψ≤2α/ε, then we have(192)Lt,εφΦεt,Ψεt≤4Φ3-D1-rΦ+rεKx2-Ex2+hϵ,x,y,t+4Ψ3-D+D2Ψ+μhϵ,x,y,t+6Φ2g12εΦ+Ex+6Ψ2g22εΨ+Ey≤-4D1-rΦ4-4D+D2Ψ4+4rK2α+MΦ4+4Φ3+Ψ3hϵ,x,y,t+6Φ2+Ψ2K22α+2M2,where(193)hϵ,x,y,t=ε-1Extyta+x2t-xtyta+x2t≤ε-1aExtyt+ε-1axtyt≤5ε-12aEx2+Ey2+12aεΦ2+2MΦ+εΨ2+2MΨ≤5ε-12aEεΦ+Ex2+EϵΨ+Ey2+12aεΦ2+2MΦ+εΨ2+2MΨ.Hence, there is a constant b3 satisfying(194)Φ3+Ψ3εΦ2+2MΦ+εΨ2+2MΨ≤b3Φ4+Ψ4.Furthermore, let(195)δεt≔εmax⁡EΦε4t1/4,EΨε4t1/4,then(196)εEΦε2t≤δε2tε,εEΨε2t≤δε2tε,and it shows(197)Φ3+Ψ35ε-12aEεΦ+Ex2+EϵΨ+Ey2≤Φ4+Ψ4+220δε2tεa4provided ε is small enough such that δε≤1.
Furthermore, we have(198)-4D1-r-4rK2α+MΦ4-4D+D2Ψ4+6Φ2+Ψ2K22α+2M2≤-min⁡D1-r-4rK2α+M,4D+D2Φ4+Ψ4+b4as D1 and D2 are large enough and b4 is a suitable constant.
By (197) and (198), it has proven that there are constants b4 and b5 satisfying(199)Lt,εφΦεt,Ψεt≤-2Φ4+Ψ4+b5δε8tε-4+b4if D1 and D2 are large enough and ε is small enough.
Next, our aim is to consider the remaining terms inLt,εVΦε(t),Ψε(t); it contains derivatives of λε.
Sinceλε′≤0,(200)-D1-r-2MΦ-D+D2Ψ≤0,where λε′≠0 and (201)φΦ,Ψ=Φ4+Ψ4≤-Mε4+Φ+Ψ+Mε4≤Φ+Ψ4+2Φ+Ψ4=17Φ+Ψ4=ψΦ,Ψforαε≤Φ+Ψ≤2αε.Then, we have α-1ελε′φ-ψ-D1-r-2MΦ-D+D2≤0.
Furthermore, ifε is small enough. Then we have (202)Lt,εVΦεt,Ψεt≤-2λεΦ+ΨφΦ,Ψ-31-λεΦ+ΨψΦ,Ψ+min⁡φΦ,Ψ,ψΦ,Ψ+b7+b5δε8tε-4≤-VΦεt,Ψεt+b7+b5δε8tε-4for b7=b4+b6, Φ,Ψ≥-M/ε. Hence, we have (203)ddtEVΦεt,Ψεt≤ELt,εVΦεt,Ψεt≤-EVΦεt,Ψεt+b7+b5δε4max⁡EΦε4t,EΨε4t≤-EVΦεt,Ψεt+b7+b5δε4EVΦεt,Ψεt.Because EVΦε(t),Ψε(t) are periodic and limε↓0⁡δε4=0, hence it shows that(204)supt≥0⁡sup1≥ε>0⁡supD1,D2⁡EVΦεt,Ψεt<∞if D1 and D2 satisfy the condition above and are large enough. Therefore, the proof is completed.Theorem 12.
Suppose that condition of Theorem11 is satisfied. If D1 and D2 are large enough, as ε↓0, then Φε,Ψε converge weakly to Φ0,Ψ0 which possesses the unique time-invariant distribution solution of(205)dΦ0tdΨ0t=r-D1-2rf1∗tK-af2∗t-f1∗t2f2∗ta+f1∗t22-f1∗ta+f1∗t2-μaf2∗t-f1∗t2f2∗ta+f1∗t22-μf1∗ta+f1∗t2-D2Φ0tΨ0tdt+g1f1∗tdWtg2f2∗tdWt.Proof.
By Theorem11, it can easily see that the sequence LΦεn0,Ψεn0 is tight for any sequence εn↓0. Furthermore (206)dΦεt=r-D1Φεt-rKx2-Ex2-hε,Φεt,Ψεtdt+g1εΦεt+ExεdWtdΨεt=-D+D2Ψεt+μhε,Φεt,Ψεtdt+g2εΨεt+EyεdWt,where h defined in (194).
By Theorem11.1.4 in [29], if we can prove that (205) is no larger than one solution such that L(Φ0(τ¯),Ψ0(τ¯))=L(Φ0(0),Ψ0(0)) for some τ¯>0, (Φε(t),Ψε(t)) converge weakly to the solution of (205). For any solution (Φ0,Ψ0) of (205), it can be described by(207)Φ0tΨ0t=QtΦ00Ψ00+∫0tQ-1sΓsdWtdWtwhere (208)Qt=g1f1∗s00g2f1∗s.
LetΦ0(n)(t)=Φ0(n)(t+nτ),Ψ0(n)(t)=Ψ0(n)(t+nτ),t≥0, then (Φ0(n),Ψ0(n)) converge in U1(C[0,∞),R2) to the Gaussian process(209)Kt=Qt∫-∞tQ-1sΓsdW^t=Qt+nτ∫-∞t+nτQ-1sΓsdW^s-nτwhere K(t) is a periodic. For all n∈N, because the law of Φ00,Ψ00 is periodic, it shows that LΦ0τ,Ψ0τ=LΦ00,Ψ00; therefore the laws of Φ0n,Ψ0n are identical with (Φ0,Ψ0); it is easy to see that the laws of Φ0,Ψ0 are identical with K(t) if system (205) shows the uniqueness periodic solution. Therefore, the proof is completed.



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*Source: 3584354-2018-11-06.xml*