# Analysis on a Stochastic Predator-Prey Model with Modified Leslie-Gower Response

Authors: Jingliang Lv; Ke Wang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper presents an investigation of asymptotic properties of a stochastic predator-prey model with modified Leslie-Gower response. We obtain the global existence of positive unique solution of the stochastic model. That is, the solution of the system is positive and not to explode to infinity in a finite time. And we show some asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained. Finally we work out some figures to illustrate our main results.


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## Body

## 1. Introduction

The dynamic interaction between predators and their prey has been one of the dominant themes in mathematical biology due to its universal existence and importance. Much literature exists on the general problem of food chains in the classical Lotka-Volterra model. In [1, 2], Leslie introduced a predator-prey model where the capacity of the predators environment is proportional to the number of preys. Leslie stresses the fact that there are upper limits to the rates of increase of both prey and predator, which are not recognized in the Lotka-Volterra model. Broer et al. [3] studied the dynamical properties of a predator-prey model with nonmonotonic response function. Reference [4] considered two-species autonomous system which incorporated a modified Leslie-Gower functional response as well as that of the Holling II as follows:
(1.1)dxdt=x[a-bx-cyx+k1],dydt=y[r-fyx+k2],
where a,b,c,r,f,k1, and k2 are all positive constants and x(t), y(t) represent the population densities at time t.Hsu and Huang [5] studied the global stability property of the following predator-prey system: (1.2)dxdt=rx(1-xk)-yp(x),dydt=y(s(1-hyx)),x0>0,y0>0,r,s,k,h>0.Recently, [6] discussed the following model with modified Leslie-Gower response: (1.3)dxdt=rx(1-xk)-nλxyλx+Ay,dydt=y(s(1-hyλx+b)),x0>0,y0>0,
where r,k,n,A,s,h,b,andλ are all positive constants and r, s are the growth rates of prey x and predator y, respectively. Here, we change the form of the predator-prey model above which reads (1.4)dxdt=x(a-bx-cyλx+Ay),dydt=y(f-gyλx+h),x0>0,y0>0.As a matter of fact, population systems are often subject to environmental noise. Recently, more and more interest is focused on stochastic systems. Reference [7] investigated the predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation: (1.5)dx=x[a-bx-cyx+m]dt+σ1xdB1(t),dy=y[r-fyx+m]dt-σ2ydB2(t).
By virtue of comparison theorem, [7] obtained some interesting results, including globally positive solutions, persistence in mean and extinction. Moreover, [8] continued to consider the stochastic ratio-dependent predator-prey system:(1.6)dx=x[a-bx-cyx+my]dt+σ1xdB1(t),dy=y[-g+fxx+my]dt-σ2ydB2(t),
where Bi(t), i=1,2, are independent standard Brownian motions. And [8] also obtained some nice conclusions on the stochastic model.According to (1.4), taking into account the effect of randomly fluctuating environment, we will consider the corresponding autonomous stochastic system described by the Itô equation (1.7)dx(t)=x(t)[a-bx(t)-cy(t)λx(t)+Ay(t)]dt+σ1x(t)dB1(t),dy(t)=y(t)[f-gy(t)λx(t)+h]dt+σ2y(t)dB2(t),
where Bi(t), i=1,2, are independent standard Brownian motions and a,b,c,f,g,λ,A,σ1, and σ2 are all positive.When white noise is taken into account in our model (1.7), we obtain the global existence of positive unique solution of the stochastic model, that is, the solution of the system is positive and not to explode to infinity in a finite time in Section 2. Section 3 shows some fundamental asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained in Section 3. The main contributions of this paper are therefore clear.Throughout the paper, we useK to denote a positive constant whose exact value may be different in different appearances.


## 2. Positive and Global Solution

Asx(t), y(t) of the SDE (1.7) are sizes of the species in the system at time t, it is obvious that the positive solutions are of interest. The coefficients of (1.7) are locally Lipschitz continuous and do not satisfy the linear growth condition, so the solution of (1.7) may explode at a finite time. The following Theorem shows that the solution will not explode at a finite time.Theorem 2.1.
For a given initial valueX0=(x0,y0)∈R+2, there is a unique positive solution X(t)=(x(t),y(t)) to (1.7) on t≥0, and the solution will remain in R+2 with probability one, namely, X(t)∈R+2 for all t≥0 almost surely.Proof.
The proof is similar to [9, 10]. Since the coefficients of the equation are locally Lipschitz continuous, for a given initial value X0=(x0,y0)∈R+2, there is a unique local solution X(t) on t∈[0,τe), where τe is the explosion time. To show that this solution is global, we need to show that τe=+∞ a.s. Let k0>0 be sufficiently large for every component of x(t) and y(t) all lying within the interval [1/k0,k0]. For each integer k≥k0, define the stopping time
(2.1)τm=inf⁡{t∈[0,τe):x(t)∉(1k,k)ory(t)∉(1k,k)},
where throughout this paper we set inf∅=∞. Obviously, τk is increasing as k→∞. Let τ∞=limk→∞τk, whence τ∞≤τe a.s. If we can show that τ∞=∞ a.s., then τe=∞ a.s. and X(t)∈R+2 a.s. for all t≥0. So we just prove that τ∞=∞ a.s. If not, there is ϵ∈(0,1) and T>0 such that
(2.2)P{τ∞≤T}>ϵ.
Hence, there is an integer k1≥k0 such that P{τk≤T}≥ϵ for all k≥k1. Define a function V:R+2→R+ by V(x,y)=(x-1-lnx)+(y-1-lny). The nonnegativity of this function can be seen from
(2.3)u-1-ln⁡⁡u≥0,on⁡u>0.
If X(t)=(x(t),y(t))∈R+2, by virtue of u≤2[u-1-lnu]+2 on u>0, we obtain
(2.4)LV(x,y)=(x-1)(a-bx-cyλx+Ay)+(y-1)(f-gyλx+h)+σ12+σ222=ax-bx2-cxyλx+Ay-a+bx+cyλx+Ay+fy-f-gy2λx+h+gyλx+h+σ12+σ222≤K1V(x,y)+K2,
dropping t from x(t) and y(t). Making use of the Itô formula yields
(2.5)EV(x(τk∧T),y(τk∧T))≤V(x0,y0)+K2T+K1∫0TEV(x(τk∧T),y(τk∧T)).
The Gronwall inequality yields
(2.6)EV(x(τk∧T),y(τk∧T))≤[V(x0,y0)+K2T]eK1T.
Set Ωk=τk≤T for k≥k1; then P(Ωk)≥ϵ. Note that, for every ω∈Ω, there is x(τk,ω) or y(τk,ω) equal to either k or 1/k, and hence V(x(τk,ω)) is no less than either
(2.7)k-1-ln⁡⁡k
or
(2.8)1k-1-ln⁡⁡(1k)=1k-1+ln⁡⁡k.
Therefore
(2.9)V(x(τk∧T),y(τk∧T))≥([k-1-ln⁡⁡k]∧[1k-1+ln⁡⁡k]).
So
(2.10)[V(x0,y0)+K2T]eK1T≥E[1ΩkV(x(τk∧T),y(τk∧T))]≥ϵ([k-1-ln⁡⁡k]∧[1k-1+ln⁡⁡k]),
where 1Ωk is the indicator function of Ωk. Letting k→∞ implies the contradiction
(2.11)∞>[V(x0,y0)+KT]=∞.
So we have that τ∞=∞ a.s. The proof is complete.Theorem2.1 shows that the solution of the SDE (1.7) will remain in the positive cone R+2 for any initial value (x0,y0)∈R+2. The conclusion is fundamental which will be used later.


## 3. Asymptotic Behavior

### 3.1. Limit Results

To begin our discussion, we impose the following assumption:(H)
a-c/A-σ12/2>0,f-σ22/2>0.And we list the interesting lemma as follows.Lemma 3.1 (see [7, 8]).
Consider one-dimensional stochastic differential equation(3.1)dx=x[a-bx]dt+σxdB(t),
where a,b,andσ are positive and B(t) is standard Brownian motion. Under condition a>σ2/2, for any initial value x0>0, the solution x(t) has the properties
(3.2)lim⁡t→∞ln⁡⁡x(t)t=0,a.s,lim⁡t→∞1t∫0tx(s)ds=a-σ2/2b,a.s.To demonstrate asymptotic properties of the stochastic system (1.7), we firstly discuss the long time behavior of lnx(t)/t and lny(t)/t.On the one hand, by the comparison theorem of stochastic equations, it is obvious that(3.3)dx≤x[a-bx]dt+σ1xdB1(t).Denote byX2(t)  the solution of the following stochastic equation: (3.4)dX2=X2[a-bX2]dt+σ1X2dB1(t),X2(0)=x0.
We have that (3.5)x(t)≤X2(t),t∈[0,+∞),a.s.
On the other hand, by the comparison theorem of stochastic equations, it is obvious that we denote by X1 the solution of stochastic differential equation (3.6)dX1=X1[a-cA-bX1]dt+σ1X1dB1(t),X1(0)=x0.
Consequently (3.7)x(t)≥X1(t),t∈[0,+∞),a.s.
To sum up, we have that(3.8)X1(t)≤x(t)≤X2(t),t∈[0,+∞),a.s.
So we have the explicit solutions of X1(t) and X2(t) as follows: (3.9)X1(t)=e[(a-c/A-σ12/2)t+σ1B1(t)]1/x0+b∫0te[(a-c/A-σ12/2)s+σ1B1(s)]ds,(3.10)X2(t)=e[(a-σ12/2)t+σ1B1(t)]1/x0+b∫0te[(a-σ12/2)s+σ1B1(s)]ds.Theorem 3.2.
Under assumption (H), for any initial valuex0>0, the solutions X1(t) and X2(t) satisfy
(3.11)lim⁡t→∞ln⁡⁡X1(t)t=0,a.s.,lim⁡t→∞ln⁡⁡X2(t)t=0,a.s.Proof.
By assumption (H) and Lemma3.1, the assertion is straightforward.Theorem 3.3.
Under assumption (H), for any initial valuex0>0, the solution x(t) satisfies
(3.12)lim⁡t→∞ln⁡⁡x(t)t=0,a.s.Proof.
By virtue of (3.8) and Theorem 3.2, we can imply the desired assertion.Now let us continue to consider the asymptotic behavior of the speciesy(t). By the comparison theorem of stochastic equations, we have that (3.13)dy(t)≤y(t)(f-gy(t)λX2(t)+h)dt+σ2y(t)dB2(t).Denote byY2(t) the solution of the stochastic equation as follows: (3.14)dY2=Y2(f-gY2λX2(t)+h)dt+σ2Y2dB2(t),Y2(0)=y0.
We have that(3.15)y(t)≤Y2(t),t∈[0,+∞),a.s.
On the other hand, applying the comparison theorem again, denote by Y1 the solution of stochastic equation (3.16)dY1=Y1(f-ghY1)dt+σ2Y1dB2(t),Y1(0)=y0.
Consequently,   (3.17)y(t)≥Y1(t),t∈[0,+∞),a.s.
To sum up, we have that  (3.18)Y1(t)≤y(t)≤Y2(t),t∈[0,+∞),a.s.
Moreover, Y1(t) and Y2(t) have the explicit solutions, respectively,  (3.19)Y1(t)=e[(f-σ22/2)t+σ2B2(t)]1/y0+(g/h)∫0te[(f-σ22/2)s+σ2B2(s)]ds,(3.20)Y2(t)=e[(f-σ22/2)t+σ2B2(t)]1/y0+∫0t(g/(λX2(s)+h))e[(f-σ22/2)s+σ2B2(s)]ds.Lemma 3.4.
Under assumption (H), for any initial valuey0>0, the solutions Y1(t) and Y2(t) satisfy
(3.21)lim⁡t→∞ln⁡⁡Y1(t)t=0,a.s.,limsup⁡t→∞ln⁡⁡Y2(t)t≤0,a.s.Proof.
The proof is motivated by [7]. Obviously, Lemma 3.1 and assumption (H) yield
(3.22)lim⁡t→∞ln⁡⁡Y1(t)t=0,a.s.
On the other hand, it follows from (3.20) that
(3.23)1Y2(t)=e[-(f-(σ22/2))t-σ2B2(t)][1y0+∫0tgλX2(s)+he[(f-σ22/2)s+σ2B2(s)]ds].
Choose T satisfying e(a-(σ12/2))t≥2 for t≥T. Thus we have that e(a-σ12/2)t/2≤e(a-σ12/2)t-1 for t≥T. Then for s≥T, from (3.10), we obtain
(3.24)X2(t)=e[(a-σ12/2)t+σ1B1(t)]1/x0+b∫0te[(a-σ12/2)s+σ1B1(s)]ds≤e[(a-σ12/2)t+σ1B1(t)]b∫0te[(a-σ12/2)s+σ1B1(s)]ds≤e[(a-σ12/2)t+σ1B1(t)]be(σ1min⁡0≤s≤tB1(s))∫0te(a-σ12/2)sds=a-σ12/2be[(a-σ12/2)t+σ1B1(t)]e(σ1min⁡0≤s≤tB1(s))[e(a-σ12/2)t-1]≤2(a-σ12/2)be[(a-σ12/2)t+σ1B1(t)]e(σ1min⁡0≤s≤tB1(s))e(a-σ12/2)t=2a-σ12beσ1(B1(t)-min⁡0≤s≤tB1(s)),∫TtgλX2(s)+he[(f-σ22/2)s+σ2B2(s)]ds≥∫Ttgλ((2a-σ12)/b)eσ1(B1(s)-min⁡0≤u≤sB1(u))+h×e[(f-σ22/2)s+σ2B2(s)]ds≥bgλ(2a-σ12)+bh∫Tte-σ1(B1(s)-min⁡0≤u≤sB1(u))×e[(f-σ22/2)s+σ2B2(s)]ds≥bgλ(2a-σ12)+bheσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2min⁡0≤s≤tB2(s)×∫Tte(f-σ22/2)sds≥2bg(λ(2a-σ12)+bh)(2f-σ22)×(e(f-σ22/2)t-e(f-σ22/2)T)×eσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2min⁡0≤s≤tB2(s).
Thus,
(3.25)1Y2(t)≥e[-(f-σ22/2)(t-T)-σ2(B2(t)-B2(T))]×[1y(T)+2bg(e(f-σ22/2)t-e(f-σ22/2)T)(λ(2a-σ12)+bh)(2f-σ22)eσ1[min⁡0≤s≤t⁡B1(s)-max⁡0≤s≤t⁡B1(s)]+σ2min⁡0≤s≤tB2(s)]≥2bge(f-σ22/2)Tσ2B2(T)(1-e-(f-σ22/2)(t-T))(λ(2a-σ12)+bh)(2f-σ22)×eσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2[min⁡0≤s≤tB2(s)-max⁡0≤s≤tB2(s)]:=K(t)eσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2[min⁡0≤s≤tB2(s)-max⁡0≤s≤tB2(s)],
where K(t)=2bge(f-σ22/2)Tσ2B2(T)(1-e-(f-σ22/2)(t-T))/(λ(2a-σ12)+bh)(2f-σ22). So we derive
(3.26)-ln⁡⁡Y2(t)≥ln⁡⁡K(t)+σ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2[min⁡0≤s≤tB2(s)-max⁡0≤s≤tB2(s)].
Dividing t on both sides yields
(3.27)ln⁡⁡Y2(t)t≤-ln⁡⁡K(t)t-σ1[min⁡0≤s≤tB1(s)t-max⁡0≤s≤tB1(s)t]-σ2[min⁡0≤s≤tB2(s)t-max⁡0≤s≤tB2(s)t].
The distributions of max0≤s≤tB1(s) and max0≤s≤tB2(s) are that same  as |B1(t)| and |B2(t)|, respectively, and min0≤s≤tB1(s) and min0≤s≤tB2(s) have the same distributions as -max0≤s≤tB1(s) and -max0≤s≤tB1(s), respectively.
From the representation ofK(t), we can simplify it as follows:
(3.28)K(t)=A1eA2B2(T)(1-A3eA4t).
By assumption (H), constants Ai(i=1,2,3,4) satisfy A1>0,A2>0,A3>0,andA4<0. Then,
(3.29)ln⁡⁡K(t)=ln⁡⁡A1+A2B2(T)+ln⁡⁡(1-A3eA4t).
It follows from (lnB2(t)/t)→0,t→∞, that
(3.30)ln⁡⁡K(t)t⟶0,t⟶∞.
Hence, letting t→∞ and by the strong law of large numbers, we have that
(3.31)min⁡0≤s≤tB1(s)t⟶0,max⁡0≤s≤tB1(s)t⟶0,t⟶∞,min⁡0≤s≤tB2(s)t⟶0,max⁡0≤s≤tB2(s)t⟶0,t⟶∞.
Then,
(3.32)limsup⁡t→∞ln⁡⁡Y2(t)t≤0,a.s.,
as desired.Theorem 3.5.
Under assumption (H), for any initial valuey0>0, the solution y(t) of (1.7) has the property
(3.33)lim⁡t→∞ln⁡⁡y(t)t=0,a.s.Proof.
It follows from (3.18) and Lemma 3.4 that
(3.34)0≤liminf⁡t→∞ln⁡⁡Y1(t)t≤liminf⁡t→∞ln⁡⁡y(t)t≤limsup⁡t→∞ln⁡⁡y(t)t≤limsup⁡t→∞ln⁡⁡Y2(t)t≤0,a.s.
Consequently,
(3.35)lim⁡t→∞ln⁡⁡y(t)t=0,a.s.
The proof is complete.


### 3.2. Persistent in Mean and Extinction

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now we present the definition of persistence in mean proposed in [7, 11].Definition 3.6.
System (1.7) is said to be persistent in mean if
(3.36)liminf⁡t→∞∫0tx(t)dst>0,liminf⁡t→∞∫0ty(t)dst>0,a.s.Theorem 3.7.
Assume that condition (H) holds. Then system (1.7) is persistent in mean.Proof.
Define the functionV=lnx; by the Itô formula, we get
(3.37)ln⁡⁡x(t)-ln⁡⁡x0=(a-σ122)t-b∫0tx(s)ds-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t).
That is,
(3.38)b∫0tx(s)ds=-ln⁡⁡x(t)+ln⁡⁡x0+(a-σ122)t-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t)≥-ln⁡⁡x(t)+ln⁡⁡x0+(a-σ122)t-ctA+σ1B1(t).
Dividing t on both sides and using the strong law of large numbers, it follows from Theorem 3.3 that
(3.39)liminf⁡t→∞∫0tx(t)dst≥a-c/A-σ12/2b>0,a.s.
Moreover, define the functionV=lny; using the Itô formula again, we have that
(3.40)ln⁡⁡y(t)-ln⁡⁡y0=(f-σ222)t-∫0tgy(s)λx(s)+hds+σ2B2(t).
Thus,
(3.41)gh∫0ty(s)ds≥∫0tgy(s)λx(s)+hds=-ln⁡⁡y(t)+ln⁡⁡y0+(f-σ222)t+σ2B2(t).
Dividing both sides by t and letting t→∞ and also by the strong law of large numbers and Theorem 3.5, we have that
(3.42)liminf⁡t→∞1t∫0ty(s)ds≥h(f-σ22/2)g>0,a.s.
So the system is persistent in mean and we complete the proof.Under condition (H), we show that the system is persistent in mean. To a large extent, (H) is the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.Theorem 3.8.
Assume that conditiona-σ12/2<0,f-σ22/2<0 holds. Then system (1.7) will become extinct exponentially with probability one.Proof.
Define the functionV=lnx; by the Itô formula, we get
(3.43)ln⁡x(t)-ln⁡⁡x0=(a-σ122)t-b∫0tx(s)ds-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t).
Then,
(3.44)ln⁡⁡x(t)-ln⁡⁡x0≤(a-σ122)t+σ1B1(t).
By the strong law of large numbers of martingales, we have that
(3.45)lim⁡t→∞B1(t)t=0,a.s.
Therefore,
(3.46)limsup⁡t→∞ln⁡⁡x(t)t≤a-σ122<0,a.s.
On the other hand, by the Itô formula again, we derive(3.47)ln⁡⁡y(t)=ln⁡⁡y0+(f-σ222)t-∫0tgy(s)λx(s)+hds+σ2B2(t)≤ln⁡⁡y0+(f-σ222)t+σ2B2(t).
Applying the strong law of large numbers of martingales, we obtain
(3.48)limsup⁡t→∞ln⁡⁡y(t)t≤f-σ222<0,a.s.
The proof is complete.We continue to discuss the asymptotic behaviors of the stochastic system (1.7).Theorem 3.9.
Assume that conditiona-c/A-σ12/2>0,f-σ22/2<0 holds. Then the prey x(t) of system (1.7) is persistent in mean; however, the predator y(t) will become extinct exponentially with probability one.Proof.
Define the functionV=lnx; by the Itô formula, we get
(3.49)ln⁡⁡x(t)-ln⁡⁡x0=(a-σ122)t-b∫0tx(s)ds-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t).
Thus,
(3.50)b∫0tx(s)ds≥-ln⁡⁡x(t)+ln⁡⁡x0+(a-σ122-cA)t+σ1B1(t).
Under condition a-c/A-σ12/2>0, it follows from the proof of Theorem 3.3 that
(3.51)lim⁡t→∞ln⁡⁡x(t)t=0,a.s.
So
(3.52)liminf⁡t→∞∫0tx(t)dst≥a-c/A-σ12/2b>0,a.s.
That is, the prey x(t) is persistent in mean. However, under condition f-σ22/2<0, from the proof of Theorem 3.8, we have that
(3.53)limsup⁡t→∞ln⁡⁡y(t)t≤f-σ222<0,a.s.
That is, the predator y(t) will become extinct exponentially with probability one.


## 3.1. Limit Results

To begin our discussion, we impose the following assumption:(H)
a-c/A-σ12/2>0,f-σ22/2>0.And we list the interesting lemma as follows.Lemma 3.1 (see [7, 8]).
Consider one-dimensional stochastic differential equation(3.1)dx=x[a-bx]dt+σxdB(t),
where a,b,andσ are positive and B(t) is standard Brownian motion. Under condition a>σ2/2, for any initial value x0>0, the solution x(t) has the properties
(3.2)lim⁡t→∞ln⁡⁡x(t)t=0,a.s,lim⁡t→∞1t∫0tx(s)ds=a-σ2/2b,a.s.To demonstrate asymptotic properties of the stochastic system (1.7), we firstly discuss the long time behavior of lnx(t)/t and lny(t)/t.On the one hand, by the comparison theorem of stochastic equations, it is obvious that(3.3)dx≤x[a-bx]dt+σ1xdB1(t).Denote byX2(t)  the solution of the following stochastic equation: (3.4)dX2=X2[a-bX2]dt+σ1X2dB1(t),X2(0)=x0.
We have that (3.5)x(t)≤X2(t),t∈[0,+∞),a.s.
On the other hand, by the comparison theorem of stochastic equations, it is obvious that we denote by X1 the solution of stochastic differential equation (3.6)dX1=X1[a-cA-bX1]dt+σ1X1dB1(t),X1(0)=x0.
Consequently (3.7)x(t)≥X1(t),t∈[0,+∞),a.s.
To sum up, we have that(3.8)X1(t)≤x(t)≤X2(t),t∈[0,+∞),a.s.
So we have the explicit solutions of X1(t) and X2(t) as follows: (3.9)X1(t)=e[(a-c/A-σ12/2)t+σ1B1(t)]1/x0+b∫0te[(a-c/A-σ12/2)s+σ1B1(s)]ds,(3.10)X2(t)=e[(a-σ12/2)t+σ1B1(t)]1/x0+b∫0te[(a-σ12/2)s+σ1B1(s)]ds.Theorem 3.2.
Under assumption (H), for any initial valuex0>0, the solutions X1(t) and X2(t) satisfy
(3.11)lim⁡t→∞ln⁡⁡X1(t)t=0,a.s.,lim⁡t→∞ln⁡⁡X2(t)t=0,a.s.Proof.
By assumption (H) and Lemma3.1, the assertion is straightforward.Theorem 3.3.
Under assumption (H), for any initial valuex0>0, the solution x(t) satisfies
(3.12)lim⁡t→∞ln⁡⁡x(t)t=0,a.s.Proof.
By virtue of (3.8) and Theorem 3.2, we can imply the desired assertion.Now let us continue to consider the asymptotic behavior of the speciesy(t). By the comparison theorem of stochastic equations, we have that (3.13)dy(t)≤y(t)(f-gy(t)λX2(t)+h)dt+σ2y(t)dB2(t).Denote byY2(t) the solution of the stochastic equation as follows: (3.14)dY2=Y2(f-gY2λX2(t)+h)dt+σ2Y2dB2(t),Y2(0)=y0.
We have that(3.15)y(t)≤Y2(t),t∈[0,+∞),a.s.
On the other hand, applying the comparison theorem again, denote by Y1 the solution of stochastic equation (3.16)dY1=Y1(f-ghY1)dt+σ2Y1dB2(t),Y1(0)=y0.
Consequently,   (3.17)y(t)≥Y1(t),t∈[0,+∞),a.s.
To sum up, we have that  (3.18)Y1(t)≤y(t)≤Y2(t),t∈[0,+∞),a.s.
Moreover, Y1(t) and Y2(t) have the explicit solutions, respectively,  (3.19)Y1(t)=e[(f-σ22/2)t+σ2B2(t)]1/y0+(g/h)∫0te[(f-σ22/2)s+σ2B2(s)]ds,(3.20)Y2(t)=e[(f-σ22/2)t+σ2B2(t)]1/y0+∫0t(g/(λX2(s)+h))e[(f-σ22/2)s+σ2B2(s)]ds.Lemma 3.4.
Under assumption (H), for any initial valuey0>0, the solutions Y1(t) and Y2(t) satisfy
(3.21)lim⁡t→∞ln⁡⁡Y1(t)t=0,a.s.,limsup⁡t→∞ln⁡⁡Y2(t)t≤0,a.s.Proof.
The proof is motivated by [7]. Obviously, Lemma 3.1 and assumption (H) yield
(3.22)lim⁡t→∞ln⁡⁡Y1(t)t=0,a.s.
On the other hand, it follows from (3.20) that
(3.23)1Y2(t)=e[-(f-(σ22/2))t-σ2B2(t)][1y0+∫0tgλX2(s)+he[(f-σ22/2)s+σ2B2(s)]ds].
Choose T satisfying e(a-(σ12/2))t≥2 for t≥T. Thus we have that e(a-σ12/2)t/2≤e(a-σ12/2)t-1 for t≥T. Then for s≥T, from (3.10), we obtain
(3.24)X2(t)=e[(a-σ12/2)t+σ1B1(t)]1/x0+b∫0te[(a-σ12/2)s+σ1B1(s)]ds≤e[(a-σ12/2)t+σ1B1(t)]b∫0te[(a-σ12/2)s+σ1B1(s)]ds≤e[(a-σ12/2)t+σ1B1(t)]be(σ1min⁡0≤s≤tB1(s))∫0te(a-σ12/2)sds=a-σ12/2be[(a-σ12/2)t+σ1B1(t)]e(σ1min⁡0≤s≤tB1(s))[e(a-σ12/2)t-1]≤2(a-σ12/2)be[(a-σ12/2)t+σ1B1(t)]e(σ1min⁡0≤s≤tB1(s))e(a-σ12/2)t=2a-σ12beσ1(B1(t)-min⁡0≤s≤tB1(s)),∫TtgλX2(s)+he[(f-σ22/2)s+σ2B2(s)]ds≥∫Ttgλ((2a-σ12)/b)eσ1(B1(s)-min⁡0≤u≤sB1(u))+h×e[(f-σ22/2)s+σ2B2(s)]ds≥bgλ(2a-σ12)+bh∫Tte-σ1(B1(s)-min⁡0≤u≤sB1(u))×e[(f-σ22/2)s+σ2B2(s)]ds≥bgλ(2a-σ12)+bheσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2min⁡0≤s≤tB2(s)×∫Tte(f-σ22/2)sds≥2bg(λ(2a-σ12)+bh)(2f-σ22)×(e(f-σ22/2)t-e(f-σ22/2)T)×eσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2min⁡0≤s≤tB2(s).
Thus,
(3.25)1Y2(t)≥e[-(f-σ22/2)(t-T)-σ2(B2(t)-B2(T))]×[1y(T)+2bg(e(f-σ22/2)t-e(f-σ22/2)T)(λ(2a-σ12)+bh)(2f-σ22)eσ1[min⁡0≤s≤t⁡B1(s)-max⁡0≤s≤t⁡B1(s)]+σ2min⁡0≤s≤tB2(s)]≥2bge(f-σ22/2)Tσ2B2(T)(1-e-(f-σ22/2)(t-T))(λ(2a-σ12)+bh)(2f-σ22)×eσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2[min⁡0≤s≤tB2(s)-max⁡0≤s≤tB2(s)]:=K(t)eσ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2[min⁡0≤s≤tB2(s)-max⁡0≤s≤tB2(s)],
where K(t)=2bge(f-σ22/2)Tσ2B2(T)(1-e-(f-σ22/2)(t-T))/(λ(2a-σ12)+bh)(2f-σ22). So we derive
(3.26)-ln⁡⁡Y2(t)≥ln⁡⁡K(t)+σ1[min⁡0≤s≤tB1(s)-max⁡0≤s≤tB1(s)]+σ2[min⁡0≤s≤tB2(s)-max⁡0≤s≤tB2(s)].
Dividing t on both sides yields
(3.27)ln⁡⁡Y2(t)t≤-ln⁡⁡K(t)t-σ1[min⁡0≤s≤tB1(s)t-max⁡0≤s≤tB1(s)t]-σ2[min⁡0≤s≤tB2(s)t-max⁡0≤s≤tB2(s)t].
The distributions of max0≤s≤tB1(s) and max0≤s≤tB2(s) are that same  as |B1(t)| and |B2(t)|, respectively, and min0≤s≤tB1(s) and min0≤s≤tB2(s) have the same distributions as -max0≤s≤tB1(s) and -max0≤s≤tB1(s), respectively.
From the representation ofK(t), we can simplify it as follows:
(3.28)K(t)=A1eA2B2(T)(1-A3eA4t).
By assumption (H), constants Ai(i=1,2,3,4) satisfy A1>0,A2>0,A3>0,andA4<0. Then,
(3.29)ln⁡⁡K(t)=ln⁡⁡A1+A2B2(T)+ln⁡⁡(1-A3eA4t).
It follows from (lnB2(t)/t)→0,t→∞, that
(3.30)ln⁡⁡K(t)t⟶0,t⟶∞.
Hence, letting t→∞ and by the strong law of large numbers, we have that
(3.31)min⁡0≤s≤tB1(s)t⟶0,max⁡0≤s≤tB1(s)t⟶0,t⟶∞,min⁡0≤s≤tB2(s)t⟶0,max⁡0≤s≤tB2(s)t⟶0,t⟶∞.
Then,
(3.32)limsup⁡t→∞ln⁡⁡Y2(t)t≤0,a.s.,
as desired.Theorem 3.5.
Under assumption (H), for any initial valuey0>0, the solution y(t) of (1.7) has the property
(3.33)lim⁡t→∞ln⁡⁡y(t)t=0,a.s.Proof.
It follows from (3.18) and Lemma 3.4 that
(3.34)0≤liminf⁡t→∞ln⁡⁡Y1(t)t≤liminf⁡t→∞ln⁡⁡y(t)t≤limsup⁡t→∞ln⁡⁡y(t)t≤limsup⁡t→∞ln⁡⁡Y2(t)t≤0,a.s.
Consequently,
(3.35)lim⁡t→∞ln⁡⁡y(t)t=0,a.s.
The proof is complete.


## 3.2. Persistent in Mean and Extinction

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now we present the definition of persistence in mean proposed in [7, 11].Definition 3.6.
System (1.7) is said to be persistent in mean if
(3.36)liminf⁡t→∞∫0tx(t)dst>0,liminf⁡t→∞∫0ty(t)dst>0,a.s.Theorem 3.7.
Assume that condition (H) holds. Then system (1.7) is persistent in mean.Proof.
Define the functionV=lnx; by the Itô formula, we get
(3.37)ln⁡⁡x(t)-ln⁡⁡x0=(a-σ122)t-b∫0tx(s)ds-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t).
That is,
(3.38)b∫0tx(s)ds=-ln⁡⁡x(t)+ln⁡⁡x0+(a-σ122)t-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t)≥-ln⁡⁡x(t)+ln⁡⁡x0+(a-σ122)t-ctA+σ1B1(t).
Dividing t on both sides and using the strong law of large numbers, it follows from Theorem 3.3 that
(3.39)liminf⁡t→∞∫0tx(t)dst≥a-c/A-σ12/2b>0,a.s.
Moreover, define the functionV=lny; using the Itô formula again, we have that
(3.40)ln⁡⁡y(t)-ln⁡⁡y0=(f-σ222)t-∫0tgy(s)λx(s)+hds+σ2B2(t).
Thus,
(3.41)gh∫0ty(s)ds≥∫0tgy(s)λx(s)+hds=-ln⁡⁡y(t)+ln⁡⁡y0+(f-σ222)t+σ2B2(t).
Dividing both sides by t and letting t→∞ and also by the strong law of large numbers and Theorem 3.5, we have that
(3.42)liminf⁡t→∞1t∫0ty(s)ds≥h(f-σ22/2)g>0,a.s.
So the system is persistent in mean and we complete the proof.Under condition (H), we show that the system is persistent in mean. To a large extent, (H) is the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.Theorem 3.8.
Assume that conditiona-σ12/2<0,f-σ22/2<0 holds. Then system (1.7) will become extinct exponentially with probability one.Proof.
Define the functionV=lnx; by the Itô formula, we get
(3.43)ln⁡x(t)-ln⁡⁡x0=(a-σ122)t-b∫0tx(s)ds-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t).
Then,
(3.44)ln⁡⁡x(t)-ln⁡⁡x0≤(a-σ122)t+σ1B1(t).
By the strong law of large numbers of martingales, we have that
(3.45)lim⁡t→∞B1(t)t=0,a.s.
Therefore,
(3.46)limsup⁡t→∞ln⁡⁡x(t)t≤a-σ122<0,a.s.
On the other hand, by the Itô formula again, we derive(3.47)ln⁡⁡y(t)=ln⁡⁡y0+(f-σ222)t-∫0tgy(s)λx(s)+hds+σ2B2(t)≤ln⁡⁡y0+(f-σ222)t+σ2B2(t).
Applying the strong law of large numbers of martingales, we obtain
(3.48)limsup⁡t→∞ln⁡⁡y(t)t≤f-σ222<0,a.s.
The proof is complete.We continue to discuss the asymptotic behaviors of the stochastic system (1.7).Theorem 3.9.
Assume that conditiona-c/A-σ12/2>0,f-σ22/2<0 holds. Then the prey x(t) of system (1.7) is persistent in mean; however, the predator y(t) will become extinct exponentially with probability one.Proof.
Define the functionV=lnx; by the Itô formula, we get
(3.49)ln⁡⁡x(t)-ln⁡⁡x0=(a-σ122)t-b∫0tx(s)ds-∫0tcy(s)λx(s)+Ay(s)ds+σ1B1(t).
Thus,
(3.50)b∫0tx(s)ds≥-ln⁡⁡x(t)+ln⁡⁡x0+(a-σ122-cA)t+σ1B1(t).
Under condition a-c/A-σ12/2>0, it follows from the proof of Theorem 3.3 that
(3.51)lim⁡t→∞ln⁡⁡x(t)t=0,a.s.
So
(3.52)liminf⁡t→∞∫0tx(t)dst≥a-c/A-σ12/2b>0,a.s.
That is, the prey x(t) is persistent in mean. However, under condition f-σ22/2<0, from the proof of Theorem 3.8, we have that
(3.53)limsup⁡t→∞ln⁡⁡y(t)t≤f-σ222<0,a.s.
That is, the predator y(t) will become extinct exponentially with probability one.


## 4. Numerical Simulations

In this section, some simulation figures are introduced to support the main results in our paper.For model (1.7), we consider the discretization equations (4.1)xk+1=xk+xk[a-bxk-cykλxk+Ayk]Δt+σ1xkΔtξk+σ122xk(ξk2-1)Δt,yk+1=yk+yk[f-gykλxk+h]Δt+σ2ykΔtηk+σ222yk(ηk2-1)Δt,
where ξk and ηk are Gaussian random variables that follow N(0,1).In Figure1, we choose a=0.4, f=0.3, c=0.1, A=0.5, σ1(t)2/2=σ22(t)/2=0.01, and (x0,y0)=(0.5,0.2). By virtue of Theorem 3.7, the system will be persistent in mean. What we mentioned above can be seen from Figure 1. The difference between the conditions of Figure 1 and Figure 2 is that the values of σ1 and σ2 are different. In Figure 1, we choose σ12/2=σ22/2=0.01. In Figure 2, we choose σ12/2=σ22/2=1. In view of Theorem 3.8, both species x and y will go to extinction. Figure 2 confirms this.Figure 1Figure 2In Figure3, we choose a=0.4, f=0.3, c=0.1, A=0.5, σ1(t)2/2=0.01,σ22/2=1, and (x0,y0)=(0.5,0.2). Then the prey x(t) is persistent in mean; however, the predator y(t) will become extinct. Figure 3 confirms the assertion (Theorem 3.9).Figure 3By comparing Figures1 and 2, with Figure 3, we can observe that small environmental noise can retain the stochastic system permanent; however, sufficiently large environmental noise makes the stochastic system extinct.Remark 4.1.
White noise is taken into account in our model in this paper. It tells us that, when the intensities of environmental noises are not too big, some nice properties such as nonexplosion and permanence are desired. However, Theorem3.8 reveals that a large white noise will force the population to become extinct while the population may be persistent under a relatively small white noise. To some extent, Theorem 3.9 shows that, though the predator y(t) has plenty of food x(t), they may be extinct because of large environmental noise.



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*Source: 518719-2011-05-23.xml*