AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 720283 10.1155/2014/720283 720283 Research Article Persistence and Nonpersistence of a Predator Prey System with Stochastic Perturbation Li Haihong 1,2 Jiang Daqing 1 Cong Fuzhong 2,3 Li Haixia 4 Chu Jifeng 1 College of Science China University of Petroleum (East China) Qingdao 266580 China cmu.edu.cn 2 Department of Basic Courses Air Force Aviation University Changchun Jilin 130022 China chinamil.com.cn 3 School of Mathematics Jilin University Changchun Jilin 130024 China jlu.edu.cn 4 School of Business Northeast Normal University Changchun Jilin 130024 China nenu.edu.cn 2014 3 4 2014 2014 13 12 2013 25 02 2014 3 4 2014 2014 Copyright © 2014 Haihong Li 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 analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.

1. Introduction

Recently, the dynamic relationship between predator and prey has been one of the dominant themes in both ecology and mathematical ecology due to its universal importance. Especially, the predator prey model is the typical representative. Thereby it significantly changed the biology and the understanding of the existence and development of the basic law and has made the model become the research hot spot. One of the most famous models for population dynamics is the Lotka-Volterra predator prey system which has received plenty of attention and has been studied extensively; we refer the reader to  for details. Specially persistence and extinction of this model are interesting topics.

The predator prey model is described as follows: (1)x˙(t)=rx(t)(1-x(t)K)-cx(t)y(t),y˙(t)=-μy(t)+mcx(t)y(t), where x(t), y(t) denote the population densities of the species at time t. The parameters r,K,c,μ,m, are positive constants that stand for prey intrinsic growth rate, carrying capacity, the maximum ingestion rate, predator death rate, and the conversion factor, respectively. From a biological viewpoint, we not only require the positive solution of the system but also require its unexploded property in any finite time and stability. We know that system (1) has a unique positive equilibrium (x*,y*) which is a stable node or focus if the following condition holds, mcK>μ: (2)x*=μmc,y*=rmcK-rμmc2K and the system (1) has a unique limit cycle which is stable (see ).

However, population dynamics in the real world is inevitably affected by environmental noise (see, e.g., ). Parameters involved in the system are not absolute constants; they always fluctuate around some average values. The deterministic models assume that parameters in the systems are deterministic irrespective of environmental fluctuations which impose some limitations in mathematical modeling of ecological systems. So we cannot omit the influence of the noise on the system. Recently many authors have discussed population systems subject to white noise (see, e.g., ). May (see, e.g., ) pointed out that due to continuous fluctuation in the environment, the birth rates, death rates, saturated rate, competition coefficients, and all other parameters involved in the model exhibit random fluctuation to some extent, and as a result the equilibrium population distribution never attains a steady value but fluctuates randomly around some average value. Sometimes, large amplitude fluctuation in population will lead to the extinction of certain species, which does not happen in deterministic models.

Therefore, Lotka-Volterra predator prey models in random environments are becoming more and more popular. Ji et al. [14, 15] investigated the asymptotic behavior of the stochastic predator prey system with perturbation. Liu and Chen  introduced periodic constant impulsive immigration of predator into predator prey system and gave conditions for the system to be extinct and permanent.

In this paper, we introduce the white noise into the intrinsic growth rate and predator death rate of system (1); that is, rr+σ1B˙1(t), μμ+σ2B˙2(t); then, we obtain the following stochastic system: (3)x˙(t)=rx(t)(1-x(t)K)-cx(t)y(t)+σ1Kx(t)(K-x(t))B˙1(t),y˙(t)=-μy(t)+mcx(t)y(t)-σ2y(t)B˙2(t), where Bi(t)  (i=1,2) are independent white noises with Bi(0)=0, σi2>0  (i=1,2) representing the intensities of the noise.

The aim of this paper is to discuss the long time behavior of system (3). We have mentioned that (x*,y*) is the positive equilibrium of system (1). But when it suffers stochastic perturbations, there is no positive equilibrium. Hence, it is impossible that the solution of system (3) will tend to a fixed point. In this paper, we show that system (3) is persistent in time average. Furthermore, under certain conditions, we prove that the population of system (3) will die out in probability which will not happen in deterministic system and could reveal that large white noise may lead to extinction.

The rest of this paper is organized as follows. In Section 2, we show that there is a unique nonnegative solution of system (3). In Section 3, we show that system (3) is persistent in time average, while in Section 4 we consider three situations when the population of the system will be extinct. In Section 5, numerical simulations are carried out to support our results.

Throughout this paper, unless otherwise specified, let (Ω,{t}t0,P) be a complete probability space with a filtration {t}t0 satisfying the usual conditions (i.e., it is right continuous and 0 contains all P-null sets). Let R+2 denote the positive cone of R2; namely, R+2={x=(x1,x2)R2:xi>0,i=1,2}, R-+2={x=(x1,x2)R2:xi0,i=1,2}.

2. Existence and Uniqueness of the Nonnegative Solution

To investigate the dynamical behavior, first, we should concern whether the solution is global existence. Moreover, for a population model, we should also consider whether the solution is nonnegative. Hence, in this section we show that the solution of system (3) is global and nonnegative. As we have known, in order for a stochastic differential equation to have a unique global (i.e., no explosion at a finite time) solution with any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (see, e.g., ). It is easy to see that the coefficients of system (3) are locally Lipschitz continuous, so system (3) has a local solution. By Lyapunov analysis method, we show the global existence of this solution.

By the classical comparison theorem of stochastic differential equations, we could get the following.

Lemma 1.

Let (x(t),y(t)) be a positive solution of system (3) with (x(0),y(0))R+2. Then, we have (4)x(t)X(t),y(t)Y(t),a.s., where (X(t),Y(t)) are solutions of the following stochastic differential equations: (5)X˙(t)  =rX(t)(1-X(t)K)+σ1KX(t)(K-X(t))B˙1(t),X˙(t)  =rX(t)(1-X(t)K)X(0)=x(0),Y˙(t)=-μY(t)+mcX(t)Y(t)-σ2Y(t)B˙2(t),Y˙(t)=-μY(t)+mcXY(0)=y(0).

Consider the stochastic logistic equation (6)dN(t)=N(t)[(r-rKN(t))dt+αK(K-N(t))dB(t)],dN(t)=N(t)mmmmmmmmvvvvvvvr,K>0. Jiang et al.  studied system (6) and obtained the following result.

Lemma 2.

There exists a unique continuous positive solution 0<N(t)<K to system (6) for any initial value N(0)=N0 with 0<N0<K. If r>α2/2, then (7)limtN(t)=K,a.s.

From Lemmas 1 and 2, it is easy to get the following result.

Lemma 3.

Let (x(t),y(t)) be a positive solution of system (3) with 0<x(0)<K. Then, we have (8)0<x(t)<K.a.s.

Theorem 4.

For any initial value {(x(0),y(0))R+2,x(0)(0,K)}, there is a unique solution (x(t),y(t)) of system (3) on t0, and the solution will remain in R+2 with probability 1.

Proof.

It is clear that the coefficients of system (3) are locally Lipschitz continuous for the given initial value {(x(0),y(0))R+2,x(0)(0,K)}. So there is a unique local solution (x(t),y(t)) on t[0,τe), where τe is the explosion time (see, e.g., ). To show this solution is global, we need to show that τe= a.s. Let k01 be sufficiently large so that x(0) and y(0) all lie within the interval [1/k0,k0]. For each integer kk0, define the stopping time (9)τm=inf{1kt[0,τe):min{x(t),y(t)}sssssssssssss1k  or  max{x(t),y(t)}k}. Throughout this paper, we set inf= (as usual denotes the empty set). Clearly, τk is increasing as k. Set τ=limkτk; then, ττe a.s. If we can show that τ= a.s., then τe= and (x(t),y(t))R+2 a.s. for all t0. In other words, to complete the proof all we need to show is that τ= a.s. If this statement is false, then there is a pair of constants T>0 and ϵ(0,1) such that (10)P{τT}>ϵ. Hence, there is an integer k1k0 such that (11)P{τkT}ϵ  kk1.

Define a C2-function V:R+2R-+ by (12)V(x,y)=(x-a-alogxa)+1m(y-1-logy), where a is a positive constant to be determined later. The nonnegativity of this function can be seen from u-1-logu0, for all  u>0. Using Itô’s formula, we get (13)dV=LVdt+σ1K(x-a)(K-x)dB1(t)+σ2m(y-1)dB2(t), where (14)LV=(x-a)(r-rKx-cy)+aσ122K2(K-x)2+1m(y-1)(-μ+mcx)+σ222m=-ar+aσ122+μm+σ222m+(r+ar-aσ12K-c)x-(rK-aσ122K2)x2-(μm-ac)y. Choose a=μ/mc such that μ/m-ac=0, together with Lemma 3; then, (15)LV-ar+aσ122+μm+σ222m+(r+ar-aσ12K-c)x-(rK-aσ122K2)x2M, where M is a positive constant. Therefore, (16)0τkTdV(x(t),y(t))0τkTMdt+0τkTσ1K(x-μmc)(K-x)dB1(s)+σ2m(y(s)-1)dB2(s),E[V(x(τkT),y(τkT))]V(x(0),y(0))+E0τkTKdtV(x(0),y(0))+MT. Set Ωk={τkT} for kk1; then, by (11), we know that P(Ωk)ϵ. Note that for every ωΩk, there is at least one of x(τk,ω) and y(τk,ω) equals either k or 1/k; then, (17)V(x(τk),y(τk))(k-a-alogka)(1k-a+alog(ak))1k(k-1-logk)1k(1k-1+logk). It then follows from (11) and (16) that (18)V(x(0),y(0))+KTE[1Ωk(ω)V(x(τk),y(τk))]ϵ(k-a-alogka)(1k-a+alog(ak))1k(k-1-logk)1k(1k-1+logk), where 1Ωk(ω) is the indicator function of Ωk. Letting k leads to the contradiction that >V(x(0),y(0))+MT=. So we must, therefore, have τ= a.s.

3. Permanence

There is no equilibrium of system (3). Hence, we cannot show the permanence of the system by proving the stability of the positive equilibrium as the deterministic system. In this section we first show that this system is persistent in mean.

3.1. Persistent in Time Average

L. S. Chen and J. Chen in  proposed the definition of persistence in mean for the deterministic system. Here, we also use this definition for the stochastic system.

Definition 5.

System (3) is said to be persistent in mean if (19)liminft1t0ty(s)ds>0,a.s.

Lemma 6 (Xia et al. [<xref ref-type="bibr" rid="B15">20</xref>, Lemma 17]).

Let fC([0,+)×Ω,(0,+)), FC([0,+)×Ω,R). If there exist positive constants λ0,λ, such that (20)logf(t)λt-λ00tf(s)ds+F(t),t0a.s. and limt(F(t)/t)=0  a.s., then (21)liminft1t0tf(s)dsλλ0,a.s.

Assumption 7.

We have (22)(r-σ122)Kmc-r(μ+σ222)>0.

Theorem 8.

If Assumption 7 is satisfied, then the solution (x(t),y(t)) of system (3) with any initial value {(x(0),y(0))R+2,x(0)(0,K)} has the following property: (23)liminft1t0ty(s)ds(r-(σ12/2))Kmc-r(μ+(σ12/2))Kmc2>0,a.s.

Proof.

According to Ito’s formula, the system (3) is changed into (24)dlogx(t)=r-rKx(t)-cy(t)-σ122K2(K-x(t))2+σ1K(K-x(t))dB1(t),dlogy(t)=-μ-σ222+mcx(t)-σ2dB2(t); then, (25)d(logx(t)+rKmclogy(t))=r-rKmc(μ+σ222)-cy(t)-σ122K2(K-x(t))2+σ1K(K-x(t))dB1(t)-rσ2KmcdB2(t)(r-rKmc(μ+σ222)-σ122)-cy(t)+σ1K(K-x(t))dB1(t)-rσ2KmcdB2(t). After that (26)logx(t)-logx(0)t+rKmclogy(t)-logy(0)t(r-rKmc(μ+σ222)-σ122)-c0ty(s)dst+σ1Kt0t(K-x(s))dB1(s)-rσ2Kmct0tdB2(s); besides, from Lemma 3, it is clear that (27)limsuptlogx(t)t0, where M1(t)=0t(K-x(s))dB1(s) and M2(t)=0tdB2(s) are martingale with Mi(0)=0  (i=1,2), and from Lemma 3 we get (28)limsuptM1,M1t,=limsupt1t0t(K-x(s))2dsK2; then, by strong law of large numbers, we know that limt(Mi/t)=0  (i=1,2).

Hence, (29)limt0((r-(r/Kmc)(μ+(σ22/2))-(σ12/2))t+(σ1/K)M1(t)-(rσ2/Kmc)M2(t)(r-(r/Kmc)(μ+(σ12/2))-(σ12/2)(r-(r/Kmc)(μ+(σ12/2))-(σ12/2)))(r-(r/Kmc)(μ+(σ12/2))-(σ12/2))(r-(r/Kmc)(μ+(σ12/2))-(σ12/2))(r-(r/Kmc)(μ+(σ12/2))-(σ12/2)))×(t)-1=r-rKmc(μ+σ222)-σ122. With Lemma 6 and Assumption 7 we could get (30)limt1t0ty(s)dsr-(σ12/2)-(r/Kmc)(μ+(σ22/2))c=(r-(σ12/2))Kmc-r(μ+(σ22/2))Kmc2>0.

3.2. Stationary Distribution and Ergodicity for System (<xref ref-type="disp-formula" rid="EEq1.2">3</xref>)

In this section we show there is a stationary distribution of system (3).

Theorem 9.

Let (x(t),y(t)) be the solution of system (3) with any initial value {(x(0),y(0))R+2,x(0)(0,K)}. If σ22<μ<min{mcK,rmcK/σ12} and σ1>0, σ2>0, such that σ12<Kr/x* and (31)(12+lx*)x*σ12+(12+ly*m)y*σ22m<min{12(rK-x*σ12K2)(x*)2,sssssssssssl2(μ-σ22)(y*)2,(K-x*)2(rK-x*σ12K2)12}, where (x*,y*) is the positive equilibrium of system (1) and l is defined as in the proof, then system (3) exists as a stationary distribution and it is ergodic.

Proof.

Since μ<mcK, then there is a positive equilibrium (x*,y*) of system (1), and (32)r=rKx*+cy*,μ=mcx*. Define (33)V1(x,y)=(x-x*-x*logxx*)+1m(y-y*-y*logyy*), and let L be the generating operator of system (3). Then, (34)LV1=(x-x*)(r-rKx-cy)+x*σ122K2(K-x)2+1m(y-y*)(-μ+mcx)+y*σ222m=(x-x*)[-rK(x-x*)-c(y-y*)]+x*σ122K2(K-x*-(x-x*))2+c(x-x*)(y-y*)+y*σ222m-rK(x-x*)2+x*σ12K2×((K-x*)2+(x-x*)2)+y*σ222m=-(rK-x*σ12K2)(x-x*)2+x*σ122K2(K-x*)2+y*σ222m.

Define (35)V2(x,y)=12[(x-x*)+1k(y-y*)]2; Note that (36)d[(x-x*)+1m(y-y*)]=(rx-rKx2-μmy)dt+σ1Kx(K-x)dB1(t)-σ2mydB2(t)=[r(x-x*)-rK(x2-(x*)2)-μm(y-y*)]dt+σ1Kx(K-x)dB1(t)-σ2mydB2(t)=[r(x-x*)-rK((x-x*)2+2x*(x-x*))fff-μm(y-y*)dt]+σ1Kx(K-x)dB1(t)-σ2mydB2(t)=[(r-2rx*K)(x-x*)-rK(x-x*)2-μm(y-y*)2rx*K]dt+σ1Kx(K-x)dB1(t)-σ2mydB2(t); Then, (37)LV2=[(x-x*)+1m(y-y*)]×[(r-2rx*K)(x-x*)ssssss-rK(x-x*)2-μm(y-y*)(r-2rx*K)]+σ122K2x2(K-x)2+σ222m2y2(r-2rx*K)(x-x*)2+(rm-2rKmx*-μm)×(x-x*)(y-y*)-μm2(y-y*)2+σ122K2x2(K-x)2+σ222m2y2r(x-x*)2+(rm-2rKmx*-μm)×(x-x*)(y-y*)-μm2(y-y*)2+σ12[(x-x*)2+(x*)2]+σ22m2[(y-y*)2+(y*)2]=(r+σ12)(x-x*)2-μ-σ22m2(y-y*)2+(rm-2rKmx*-μm)(x-x*)(y-y*)+σ12(x*)2+σ22m2(y*)2, where L is also the generating operator of system (3). Note that (38)(rm-2rKmx*-μm)(x-x*)(y-y*)(r/m-(2r/Km)x*-μ/m)22((μ-σ22)/m2)(x-x*)2+12(μ-σ22m2)(y-y*)2=δ(x-x*)2+(μ-σ222m2)(y-y*)2; Then, (39)LV2(r+σ12+δ)(x-x*)2-(μ-σ222m2)(y-y*)2+σ12(x*)2+σ22m2(y*)2. Now define (40)V(x,y)=V1(x,y)+lV2(x,y), where l is a positive constant to be determined later. Then, (41)LV-(rK-x*σ12K2-l(r+σ12+δ))(x-x*)2-l2(μ-σ22m2)(y-y*)2+(12+lx*)x*σ12+(12+ly*m)y*σ22m. Choose l>0 such that ((r/K)-(x*σ12/K2)  -l(r+σ12+δ))=(1/2)((r/K)-(x*σ12/K2)). Then, it follows from (47) that (42)LV-12(rK-x*σ12K2)(x-x*)2-l2(μ-σ22m2)(y-y*)2+(12+lx*)x*σ12+(12+ly*m)y*σ22m. Note that (43)(12+lx*)x*σ12+(12+ly*m)y*σ22m<min{12(rK-x*σ12K2)(x*)2,ssssssssssl2(μ-σ22m2)(y*)2,(K-x*)2}. Then, the ellipsoid (44)-12(rK-x*σ12K2)(x-x*)2-l2(μ-σ22m2)(y-y*)2+(12+lx*)x*σ12+(12+ly*m)y*σ22m=0 lies entirely in D0={(x,y)R+20<x<K}. We can take U to be a neighborhood of the ellipsoid with U-El=D0, so that for xUEl,LV-C(C is a positive constant), which implies that condition (B.2) in Lemma 3.2 of  is satisfied. Hence, the solution (x(t),y(t)) is recurrent in the domain U, which together with Lemma 3.3 and Remark 3.3 of  imply that (x(t),y(t)) is recurrent in any bounded domain DD0. Besides, for D, there is a M=min{(x2σ12/K2)(K-x)2,σ22y2,(x,y)D¯  }>0, such that (45)i,j=12λijξiξj=x2σ12K2(K-x)2ξ12+σ22y2ξ22M|ξ2|xD-,ξR2, which implies that condition (B.1) in Lemma 3.2 of  is also satisfied. Therefore, system (3) has a stationary distribution μ(·) and it is ergodic.

From Lemma 3, with the initial value 0<x(0)<K, we have the property (46)0<x(t)<Ka.s. Therefore, by ergodicity property, we know that function f(z)=zp is integrable with respect to the measure μ, and (47)limt1t0txp(s)ds=R+2zpμ(dz1,dz2),a.s.

Hence, from these arguments, we get the following result.

Theorem 10.

Assume the same conditions as in Theorem 9. Then, we have (48)limt1t0txp(s)ds=R+2zpμ(dz1,dz2),a.s.

4. Extinction

In this section, we show the situation when the population of system (3) will be extinct. Before we give the result, we should do some prepare work. We first introduce a result on the Feller’s test (see, e.g., ).

Let I=(l,r),  -r+. Consider the following one-dimensional time-homogeneous stochastic differential equation: (49)dXt=μ(Xt)dt+σ(Xt)dBt,X0=x. Assume that the coefficients σ:IR,  μ:IR satisfy the following conditions: (50)(1)σ2(x)>0;xI,(2)xI,ε>0,x-εx+ε1+μ(y)σ2(y)ds<. Fixing some cI, the scale function is defined by (51)p(x)=cxe-cv(2μ(y)/σ2(y))dydv,xR.

Now, we present a useful Lemma.

Lemma 11.

Assume that conditions (1) and (2) hold in (50), and let X be a nonexplosive solution of system (49) in I, with X0=xI; we distinguish four cases:

p(l+)=-,p(r-)=+,  then  P{supt0Xt=r}=

P{inft0Xt=l}=1, and for any yI,  we have  p{t(0,),Xt=y}=1.

p(l+)>-,p(r-)=+,  then  P{supt0Xt=l}=P{inft0Xt<r}=1.

p(l+)=-,p(r-)<+,  then  P{supt0Xt=r}=P{inft0Xt>l}=1.

p(l+)>-,p(r-)<+,  then  P{supt0Xt=l}=1-P{inft0Xt=r}=(p(r-)-p(x))/(p(r-)-p(l+)).

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M187"><mml:mi>r</mml:mi><mml:mo><</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:mrow></mml:math></inline-formula>).

Consider the first equation of system (5). Let (52)X(t)=KeZ(t)1+eZ(t). Then, (53)Z(t)=logX(t)K-X(t), and so the first equation of system (5) is reformed as (54)dZ(t)=(r-σ122+eZ(t)1+eZ(t)σ12)dt+σ1dB1(t),t0, with an initial value Z(0)=log(x(0)/(K-x(0)))  . Let (55)μ(x)=r-σ122+ex1+exσ12,σ(x)=σ1. Then, (56)0v-2μ(x)σ2(x)dx=-2σ120v(r-σ122+ex1+exσ12)dx=-2σ12[(r-σ122)v+σ12log(1+ev)]+2log2. So if r1<σ12/2, by Lemma 11, we get p(-)>-,p(+)<+; then, (57)P{supt0Z(t)=-}=1-P{inft0Z(t)=+}=p()-p(x)p()-p(-). Hence, (58)P{limtX(t)=0}=1-P{limtX(t)=K}=p()-p(x)p()-p(-)=Z(0)+e(-2/σ12)(r-σ12/2)v(1+ev)-2dv-+e(-2/σ12)(r-σ12/2)v(1+ev)-2dv. Furthermore, by the classical comparison theorem of stochastic differential equations, we have (59)x(t)X(t),y(t)Y(t), where (x(t),y(t)) is the solution of system (3). We could get (60)P{limtx(t)=0}Z(0)+e(-2/σ12)(r-σ12/2)v(1+ev)-2dv-+e(-2/σ12)(r-σ12/2)v(1+ev)-2dv. So, if limkx(t)=0,a.s.,ωΩ0={ω:limkx(t)=0}, and from (24), then we know (61)dlogy(t)=-μ-σ222+mcx(t)-σ2dB2(t). Then, (62)limsuptlogy(t)t=-μ-σ222<0,ωΩ0.

Therefore, with the condition r-σ12/2<0, we obtain the fact that system (3) will be extinct in probability.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M205"><mml:mi>r</mml:mi><mml:mo>></mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:mrow><mml:mo>></mml:mo><mml:mi>m</mml:mi><mml:mi>c</mml:mi><mml:mi>K</mml:mi></mml:math></inline-formula>).

According to Ito’s formula and comparison principle, the second population of system (3) is changed into (63)dlogy(t)=-μ-σ222+mcx(t)-σ2dB2(t)-μ-σ222+mcX(t)-σ2dB2(t). Notice that X(t)<K and then let (63) be divided by t, t; we could get (64)limsuptlogy(t)t-μ-σ222+mcKa.s. If μ+σ22/2>mcK, it follows (65)limsuptlogy(t)t<0a.s.; hence, (66)limty(t)=0,a.s. That is, for ϵ>0, there are constants T0 and Ωϵ; then, if tT0 and ωΩϵ, we have P(Ωϵ)>1-ϵ and y(t)ϵ. So, (67)x(t)(r-rKx(t)-ϵ)dt+rσ1K(K-x(t))dB1(t)dx(t)x(t)(r-rKx(t))dt+rσ1K(K-x(t))dB1(t), if r>σ12/2; from the arbitrariness of ϵ>0, Lemma 2, and  (see Theorems 6.2 and 6.3), we could know that (68)limtx(t)=K,a.s. Concluding these arguments, we have the following theorem.

Theorem 12.

Let (x(t),y(t)) be the solution of system (3) with any initial value {(x(0),y(0))R+2,x(0)(0,K)}; then,

if r<σ12/2, then (69)P{limtx(t)=0}Z(0)+e(-2/σ12)(r-σ12/2)v(1+ev)-2dv-+e(-2/σ12)(r-σ12/2)v(1+ev)-2dv, where Z(0)=log(x(0)/(K-x(0))) and (70)limsuptlogy(t)t=-μ-σ222<0,ωΩ0={ω:limtx(t)=0}. That is to say, system (3) will be extinct in probability.

If μ+σ22/2>mcK,r>σ22/2, then (71)limty(t)=0a.s.,limtx(t)=K,a.s.

5. Numerical Simulation

In this section, we give out the numerical experiment to support our results. Consider the equation (72)dx(t)=rx(t)(1-x(t)K)-cx(t)y(t)+σ1Kx(t)(K-x(t))dB1(t),dy(t)=-μy(t)+mcx(t)y(t)-σ2y(t)dB2(t). By the Milstein method in , we have the difference equation (73)xk+1=xk+xk[σ122K2(r-rKxk-cyk)Δt+σ1K(K-xk)ϵ1,kΔt555555555+σ122K2(K-xk)2(ϵ1,k2Δt-Δt)],yk+1=yk+yk[σ222(-μ+mcxk)Δtssssssssssss-σ2ϵ2,kΔt+σ222(ϵ2,k2Δt-Δt)], where ϵ1,k and ϵ2,k are the Gaussian random variables N(0,1). By choosing (x(0),y(0))R+2 and suitable parameters, by Matlab, we get Figures 1 and 2.

The solution of system (1) and system (3) with (x(0),y(0))=(2,1), r=1.2, μ=0.3,K=4, c=0.2,mc=0.1. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with σ1=0.01, σ2=0.01 in (a) and σ1=0.06, σ2=0.08 in (b), respectively.

The solution of system (1) and system (3) with (x(0),y(0))=(2,1),   r=1.2, μ=0.3,K=4, c=0.2, mc=0.1. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with σ1=1, σ2=1 in (a) and σ1=6, σ2=1 in (b), respectively.

In Figure 1, choose parameters satisfying the condition of Theorem 9; system (3) is ergodic and the solution will persist in time average. Between picture (a) and (b), we only change the intensity parameters σ1 and σ2 and keep other parameters unchangeable. We observe that the amplitude of fluctuation is becoming large as the intensity of white noise is increasing. And we can see that the sample path is deviating from the corresponding deterministic system as the intensity of the white noise is becoming larger.

In Figure 2, we observe two cases. We observe case (1) in Theorem 12 and choose parameters such as r>σ12/2,μ+σ22/2>mcK in (a); as Theorem 12 indicated, the prey will die out in probability and the predators will go to their carrying capacity. We also observe case (2) in Theorem 12 and choose parameters such as r<σ12/2 in (b); as Theorem 12 indicated, not only preys but also predators will die out in probability when the noise of the predators is large, and it does not happen in the deterministic system. This tells us strong environmental noise may cause species to become extinct. The larger the intensity environmental noise is, the bigger the probability of dying out is.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the Program for Changjiang Scholars and Innovative Research Team in University, NSFC of China, (no. 11371085), the Ph.D. Programs Foundation of Ministry of China (no. 200918), and the Natural Science Foundation of Jilin Province of China (no. 201115133).

Gard T. C. Stability for multispecies population models in random environments Nonlinear Analysis: Theory, Methods & Applications 1986 10 12 1411 1419 10.1016/0362-546X(86)90111-2 MR869549 ZBL0598.92017 Goh B. S. Global stability in many species system American Naturalist 1997 111 135 143 Lv J. L. Wang K. Asymptotic properties of a stochastic predator-prey system with Holling II functional response Communications in Nonlinear Science and Numerical Simulation 2011 16 10 4037 4048 10.1016/j.cnsns.2011.01.015 MR2802710 ZBL1218.92072 Chen L. S. Jing Z. J. The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions Chinese Science Bulletin 1984 9 521 523 MR764720 Mao X. R. Marion G. Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics Stochastic Processes and Their Applications 2002 97 1 95 110 10.1016/S0304-4149(01)00126-0 MR1870962 ZBL1058.60046 Mao X. Delay population dynamics and environmental noise Stochastics and Dynamics 2005 5 2, article 149 10.1142/S021949370500133X MR2147279 ZBL1093.60033 Polansky P. Invariant distributions for multipopulation models in random environments Theoretical Population Biology 1979 16 1 25 34 10.1016/0040-5809(79)90004-2 MR545478 Gard T. C. Introduction to Stochastic Differential Equations 1988 New York, NY, USA Marcel Dekker Kuang Y. Smith H. L. Global stability for infinite delay Lotka-Volterra type systems Journal of Differential Equations 1993 103 2 221 246 10.1006/jdeq.1993.1048 MR1221904 ZBL0786.34077 Hu Y. Z. Wu F. K. Huang C. Stochastic Lotka-Volterra models with multiple delays Journal of Mathematical Analysis and Applications 2011 375 1 42 57 10.1016/j.jmaa.2010.08.017 MR2735693 ZBL1245.92063 Wantanabe N. I. Stochastic Differential Equations and Diffusion Processes 1981 Amsterdam, The Netherlands North-Holland Ji C. Y. Jiang D. Q. Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response Journal of Mathematical Analysis and Applications 2011 381 1 441 453 10.1016/j.jmaa.2011.02.037 MR2796222 ZBL1232.34072 May R. M. Stability and Complexity in Model Ecosystem 2001 Princeton, NJ, USA Princeton University Press Ji C. Y. Jiang D. Q. Shi N. Z. Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation Journal of Mathematical Analysis and Applications 2009 359 2 482 498 10.1016/j.jmaa.2009.05.039 MR2546763 ZBL1190.34064 Ji C. Y. Jiang D. Q. Li X. Y. Qualitative analysis of a stochastic ratio-dependent predator-prey system Journal of Computational and Applied Mathematics 2011 235 5 1326 1341 10.1016/j.cam.2010.08.021 MR2728069 ZBL1229.92076 Liu X. N. Chen L. S. Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator Chaos, Solitons & Fractals 2003 16 2 311 320 10.1016/S0960-0779(02)00408-3 MR1949478 ZBL1085.34529 Mao X. R. Stochastic Differential Equations and Applications 1997 Chichester, UK Horwood Jiang D.-Q. Zhang B.-X. Wang D.-H. Shi N.-Z. Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation Science in China A: Mathematics 2007 50 7 977 986 10.1007/s11425-007-0071-y MR2355869 ZBL1136.34324 Chen L. S. Chen J. Nonlinear Biological Dynamical System 1993 Beijing, China Science Press Xia P. Y. Zheng X. K. Jiang D. Q. Persistence and nonpersistence of a nonautonomous stochastic mutualism system Abstract and Applied Analysis 2013 2013 13 256249 10.1155/2013/256249 MR3034979 ZBL06161315 Jiang D. Q. Ji C. Y. Li X. Y. O'Regan D. Analysis of autonomous Lotka-Volterra competition systems with random perturbation Journal of Mathematical Analysis and Applications 2012 390 2 582 595 10.1016/j.jmaa.2011.12.049 MR2890539 ZBL1258.34099 Feller W. An Introduction to Probability Theory and Its Application 1971 2 New York, NY, USA John Wiley & Sons MR0270403 Gray A. Greenhalgh D. Hu L. Mao X. Pan J. A stochastic differential equation SIS epidemic model SIAM Journal on Applied Mathematics 2011 71 3 876 902 10.1137/10081856X MR2821582 ZBL1263.34068 Higham D. J. An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Review 2001 43 3 525 546 10.1137/S0036144500378302 MR1872387 ZBL0979.65007