# Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

Authors: Li Zu; Daqing Jiang; Fuquan Jiang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a predator-prey model in which the preys disperse amongn patches (n≥2) with stochastic perturbation. We show that there is a unique positive solution
and find out the sufficient conditions for the extinction to the system with any given positive initial
value. In addition, we investigate that there exists a stationary distribution for the system and it has
ergodic property. Finally, we illustrate the dynamic behavior of the system with n=2 via numerical
simulation.


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## Body

## 1. Introduction

Interest has been growing in the study of the dynamic relationship between predators and their preys due to its universal existence and importance. However, due to the spatial heterogeneity and the increasing spread of human activities, the habitats of many biological species have been separated into isolated patches. In some of these patches, without the contribution from other patches, the species will go to extinction. Recently, the effect of dispersion on the species survival has been an important subject in population biology (see [1–10] and the references cited therein). Particularly, two species predator-prey systems with dispersal have received great attention from both theoretical and mathematical biologists and many good results have been achieved (see [1, 2, 7, 9–11]). The analysis of these papers has been centered around the coexistence of populations, stability (local and global), and permanence of equilibria. Zhang and Teng [11] established the sufficient conditions on the boundedness, permanence, and existence of positive periodic solution for two species predator-prey model. Kuang and Takeuchi [1] studied a predator-prey system with prey dispersal in a two-patch environment; they obtained the existence,  local and global stability of the positive steady state and analyzed both the stabilizing and destabilizing effects of dispersion by introducing examples.Li and Shuai [2] considered the model
(1.1)x˙i=xi(ri-bixi-eiyi)+∑j=1n‍dij(xj-αijxi),y˙i=yi(-γi-δiyi+εixi),i=1,2,…,n,
where xi, yi denote the densities of preys and predators on the patch i, respectively. The parameters in the model are nonnegative constants and ei, εi are positive. The constants dij are the dispersal rate from patch j to i, and the constants αij can be selected to represent different boundary conditions in the continuous diffusion case [12]. Let (dij) denote n×n dispersal matrix. By constructing a Lyapunov function and using graph theory, Li and Shuai proved the uniqueness and globally asymptotically stability of a positive equilibrium, whenever it exists, if (dij) is irreducible and there exists i such that bi>0 or δi>0.The model mentioned above is a deterministic model which assumes  that the parameters in the model are all deterministic irrespective environmental fluctuations. In fact, population dynamics is inevitably affected by environmental white noise, such as weather and epidemic disease. Therefore, the deterministic models are often subject to stochastic perturbation, and it is useful to reveal how the noise affects the population system. There are some authors who have studied the dynamics of predator-prey models with stochastic perturbations (see [13–15]). Ji et al. [13] studied a predator-prey with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation; they got some good results about existence, uniqueness, and extinction of positive solution. Cai and Lin [15] investigated a predator-prey stochastic system with competition among predators and obtained the probability distribution of the system state variables at the state of statistical stationarity. But, until now, few people study the dynamical behavior of the predator-prey system with diffusion under the influence of white noise. However, the diffusion phenomenon and environmental white noise are universal existence in nature. Therefore, we want to study the effect of random perturbations on the predator-prey system on the basis of the existing diffusion model and the contents of this paper are of great significance.In this paper, we take into account the effect of randomly fluctuating and stochastically perturb intrinsic growth rate in each equations of (1.1):
(1.2)ri→ri+σ1iB˙1i(t),-γi→-γi+σ2iB˙2i(t),i=1,2,…,n,
where B1i(t),B2i(t) are mutually independent Brownian motions and σ1i,σ2i are positive constants representing the intensity of the white noises, respectively. Then the stochastic system takes the following form:
(1.3)dxi=[xi(ri-bixi-eiyi)+∑j=1n‍dij(xj-αijxi)]dt+σ1ixidB1i(t),dyi=yi(-γi-δiyi+εixi)dt+σ2iyidB2i(t),i=1,2,…,n.
Throughout this paper, we assume dij are nonnegative constants, (dij) is irreducible, and the parameters ri, γi, bi, ei, δi, εi are positive constants.In order to obtain better dynamic properties of the SDE (1.3), we will show that there exists a unique positive global solution with any initial positive value, and its pth moment is bounded in Section 2. In the study of a population dynamics, permanence is a very important and interesting topic regarding the survival of populations in ecological system. In a deterministic system, it is usually solved by showing the global attractivity of the positive equilibrium of the system. But, as mentioned above, it is impossible to expect stochastic system (1.3) to tend to a steady state. So we attempt to investigate the stationary distribution of this system by Lyapunov functional technique. The stationary can be considered a weak stability, which appears as the solution is fluctuating in a neighborhood of the equilibrium point of the corresponding deterministic model. In Section 3, we will show if the white noise is small, there is a stationary distribution of SDE (1.3) and it has ergodic property. Existing results on dynamics in a patchy environment have largely been restricted to extinction analysis which means that the population system will survive or die out in the future due to the increased complexity of global analysis. In Section 4, we give the sufficient conditions for extinction. In Section 5, we make numerical simulation to conform our analytical results. Finally, for the completeness of the paper, we give an appendix containing some theories which will be used in previous sections.The key method used in this paper is the analysis of Lyapunov functions [6, 13, 14, 16]. We will also use the graph theory in Section 3 and some graph definitions can be found in the appendix.Throughout this paper, unless otherwise specified, let(Ω,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). Let R+2n denote the positive cone of R2n, namely, R+2n={(x1,y1,…,xn,yn)∈R2n:xi>0,yi>0,i=1,2,…,n}. For convenience and simplicity in the following discussion, denote X(t)=(x1(t),y1(t),x2(t),y2(t),…,xn(t),yn(t)). If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by |A|=trace(ATA) whilst its operator norm is denoted by ∥A∥=sup{|Ax|:|x|=1}.


## 2. Positive and Global Solutions

In order for a stochastic differential equation to have a unique global (i.e., no explosion at any finite time) solution, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (see [17]). However, the coefficients of SDE (1.3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of SDE (1.3) may explode at a finite time. In this section, we will prove the solution of stochastic system (1.3) with any positive initial value is not only positive but also not exploive in infinity at any finite time.Theorem 2.1.
For any given initial valueX(0)∈R+2n, there is a unique positive solution X(t) of SDE (1.3), and the solution will remain in R+2n with probability 1.Proof.
We define aC2-function V:R+2n→R+:
(2.1)V(X(t))=∑i=1n‍[εi(xi-1-logxi)+ei(yi-1-logyi)].
Applying Itô’s formula, we have
(2.2)LV=∑i=1n‍(εi{(xi-1)[ri-bixi-eiyi+∑j=1n‍dij(xjxi-αij)]+σ1i22}+ei[(yi-1)(-γi-δiyi+εixi)+σ2i22]dt[ri-bixi-eiyi+∑j=1n‍dij(xjxi-αij)])=∑i=1n‍[-εibixi2-eiδiyi2+εi(ri-∑j=1n‍dijαij+bi-ei)xi+εi∑j=1n‍dijxj-εi∑j=1n‍dijxjxi+ei(-γi+δi+εi)yi+εi(∑j=1n‍dijαij-ri+σ1i22)+ei(γi+σ2i22)]dt≤∑i=1n‍[-εibixi2-eiδiyi2+εi(ri-∑j=1n‍dijαij+bi-ei)xi+εi∑j=1n‍dijxj+ei(-γi+δi+εi)yi+εi(∑j=1n‍dijαij-ri+σ1i22)+ei(γi+σ2i22)].
It is clear that the coefficient of quadratic term is negative, so we must be able to find a positive constant number K that satisfies
(2.3)LV≤K,
and K is independent of xi, yi, and t. By the similar proof of Li and Mao [18, Theorem  2.1], we can obtain the desired assertion.Theorem2.1 shows that the solution of SDE (1.3) will remain in the positive cone R+2 with probability 1. This nice property provides us with a great opportunity to discuss the pth moment and stochastically ultimately boundedness of the solution.Definition 2.2 (see [18]).
The solutionX(t) of SDE (1.3) is said to be stochastically ultimately bounded, if for any ϵ∈(0,1), there exists a positive constant χ(=χ(ϵ)), such that for any initial value X(0)∈R+2n, the solution X(t) to (1.3) has the property that
(2.4)limsupt→∞P{|x(t)|>χ}<ϵ.Lemma 2.3.
For any given initial valueX(0)∈R+2n, there exist positive constants κ(p), pi, and qi(i=1,2,…,n), such that the solution X(t) of SDE (1.3) has the following property:
(2.5)E[∑i=1n‍(pixip(t)+qiyip(t))]≤κ(p),t≥0,p>1.Proof.
By Itô’s formula and Young inequality, we compute(2.6)d(1pxip)=xip-1dxi+p-12xip-2(dxi)2=[-bixip+1+(ri+p-12σ1i2-∑j=1n‍dijαij)xip-eixipyi+∑j=1n‍dijxip-1xj]dt+σ1ixipdB1i(t)≤[-bixip+1+(ri+p-12σ1i2+∑j=1n‍dijαij)xip+∑j=1n‍dij(p-1pxip+1pxjp)]dt+σ1ixipdB1i(t)=[-bixip+1+(ri+p-12σ1i2+∑j=1n‍dijαij+p-1p∑j=1n‍dij)xip+1p∑j=1n‍dijxjp]dt+σ1ixipdB1i(t),d(1pyip)=yip-1dyi+p-12yip-2(dyi)2=[-δiyip+1+(-γi+p-12σ2i2)yip+εixiyip]dt+σ2iyipdB2i(t)≤[-δiyip+1+(-γi+p-12σ2i2)yip+εi(1p+1ki-pxip+1+pp+1kiyip+1)]dt+σ2iyipdB2i(t)=[(-δi+pεikip+1)yip+1+(-γi+p-12σ2i2)yip+εiki-pp+1xip+1]dt+σ2iyipdB2i(t),
where ki(i=1,2,…,n) are positive constants to be determined. Hence, for positive constants pi,qi, we have
(2.7)d[∑i=1n‍(pixip+qiyip)]≤∑i=1n‍{-ppibixip+1+pi[pri+p(p-1)2σ1i2+∑j=1n‍dij(p-1+pαij)+∑j=1n‍djipjpi]xip+pqi(-δi+pεikip+1)yip+1+pqi(-γi+p-12σ2i2)yip+pqiεiki-pp+1xip+1[pri+p(p-1)2σ1i2+∑j=1n‍dij(p-1+pαij)+∑j=1n‍djipjpi]}dt+p∑i=1n‍piσ1ixipdB1i(t)+p∑i=1n‍qiσ2iyipdB2i(t)=∑i=1n‍{-p(pibi-qiεiki-pp+1)xip+1-pqi(δi-pεikip+1)yip+1+pi[pri+p(p-1)2σ1i2+∑j=1n‍dij(p-1+pαij)+∑j=1n‍djipjpi]xip+pqi(-γi+p-12σ2i2)yip(pibi-qiεiki-pp+1)}dt+p∑i=1n‍piσ1ixipdB1i(t)+p∑i=1n‍qiσ2iyipdB2i(t).
Next, we claim that there exist pi>0,qi>0, and ki>0 such that
(2.8)pibi-qiεiki-pp+1>0,δi-pεikip+1>0.
In fact, we only need 0<ki<(p+1)δi/pεi and 0<qi=pi/m, where m is a sufficiently large positive integer. Let
(2.9)αi=:pri+p(p-1)2σ1i2+∑j=1n‍dij(p-1+pαij)+∑j=1n‍djipjpi,αi′=:p(-γi+p-12σ2i2),βi=:ppi-1/p(bi-εiki-pm(p+1)),βi′=:pqi-1/p(δi-pεikip+1),αˇ=:max{α1,α1′,…,αn,αn′},β^=:min{β1,β1′,…,βn,βn′}.
It is clear that α˘>0,β^>0. Then
(2.10)d[∑i=1n‍(pixip(t)+qiyip(t))]≤[αˇ∑i=1n‍(pixip(t)+qiyip(t))-β^∑i=1n‍(pi(p+1)/pxip+1(t)+qi(p+1)/pyip+1(t))]dt+p∑i=1n‍piσ1ixip(t)dB1i(t)+p∑i=1n‍qiσ2iyip(t)dB2i(t).
Integrating it from 0 to t and taking expectations of both sides, we obtain that
(2.11)dE[∑i=1n(pixip(t)+qiyip(t))]dt≤{αˇE[∑i=1n‍(pixip(t)+qiyip(t))]-β^E[∑i=1n‍(pi(p+1)/pxip+1(t)+qi(p+1)/pyip+1(t))]}≤{αˇE[∑i=1n‍(pixip(t)+qiyip(t))]-(2n)-1/pβ^E[∑i=1n‍(pixip(t)+qiyip(t))](p+1)/p}=E[∑i=1n‍(pixip(t)+qiyip(t))]{αˇ-(2n)-1/pβ^E[∑i=1n‍(pixip(t)+qiyip(t))]1/p}.
Therefore, letting z(t)=E[∑i=1n‍(pixip(t)+qiyip(t))], we have
(2.12)dz(t)dt≤z(t)(αˇ-(2n)-1/pβ^z1/p(t)).
Notice that the solution of equation
(2.13)dz¯(t)dt=z¯(t)(αˇ-(2n)-1/pβ^z¯1/p(t))
obeys
(2.14)z¯(t)→2n·(αˇβ^)p,ast→∞.
By comparison theorem, we can get
(2.15)limsupt→∞E[∑i=1n‍(pixip(t)+qiyip(t))]≤2n·(αˇβ^)p:=L(p),
which implies that there is a T>0, such that
(2.16)E[∑i=1n‍(pixip(t)+qiyip(t))]≤2L(p),t>T.
In addition E[∑i=1n‍(pixip(t)+qiyip(t))] is continuous, so we have
(2.17)E[∑i=1n‍(pixip(t)+qiyip(t))]≤C(p),t∈[0,T].
Let κ(p)=max{2L(p),C(p)}, then
(2.18)E[∑i=1n‍(pixip(t)+qiyip(t))]≤κ(p),t≥0,p>1.
This completes the proof.Theorem 2.4.
The solutions of system (1.3) are stochastically ultimately bounded for any initial value X(0)∈R+2n.Proof.
From Theorem2.1, the solution X(t) will remain in R+2n with probability 1. Let Q=min{p1,q1,p2,q2,…,pn,qn}. Note that |X(t)|=[∑i=1n‍(xi2+yi2)]1/2 and |X(t)|p≤(2n)p/2[∑i=1n‍(xip+yip)]. Therefore, we get
(2.19)E[∑i=1n‍(pixip(t)+qiyip(t))]≥QE[∑i=1n‍(xip(t)+yip(t))]≥(2n)-p/2QE|X(t)|p,
and by (2.5) we have
(2.20)E|X(t)|p≤(2n)p/2Qκ(p)=:κ~(p)<+∞.
Applying the Chebyshev inequality yields the required assertion.


## 3. Stationary Distribution

In this section, we investigate there is a stationary distribution for SDE (1.3) instead of asymptotically stable equilibria. Before giving the main theorem, we first give a lemma (see [2]).Lemma 3.1 (see [2]).
Assume(dij) is irreducible. If there exists i such that bi>0 or δi>0, then, whenever a positive equilibrium E* exists for system (1.1), it is unique and globally asymptotically stable in the positive cone R+2n.In the section, we assume system (1.1) exists and the positive equilibrium E*=(x1*,y1*,…,xn*,yn*) satisfies the equation
(3.1)xi*(ri-bixi*-eiyi*)+∑j=1n‍dij(xj*-αijxi*)=0,yi*(-γi-δiyi*+εixi*)=0,i=1,2,…,n,
where xi*>0,yi*>0.We now state a theorem in which the graph theory will be used. Assumeci(i=1,2,…,n) defined as in Lemma A.1 are nonnegative constants and A=(dijεixj*)n×n(i=1,2,…,n).Theorem 3.2.
Assumeδ<min{min1≤i≤n{ciεibi(xi*)2},min1≤i≤n{cieiδi(yi*)2}}. Then there is a stationary distribution μ(·) for SDE (1.3) and it has ergodic property. Here (x1*,y1*,…,xn*,yn*) is the solution of (3.1), and δ=(1/2)∑i=1n‍(ciεibixi*σ1i2+cieiδiyi*σ2i2).Proof.
DefineV:El=R+2n→R+:
(3.2)V(X(t))=∑i=1n‍[εi(xi-xi*-xi*logxixi*)+ei(yi-yi*-yi*logyiyi*)].
Then
(3.3)LV=∑i=1n‍ci{εi(xi-xi*)[(ri-bixi-eiyi)+∑j=1n‍dij(xjxi-αij)]+ei(yi-yi*)(-γi-δiyi+εixi)+12εixi*σ1i2+12eiyi*σ2i2[(ri-bixi-eiyi)+∑j=1n‍dij(xjxi-αij)]}.
By (3.1), we have
(3.4)ri=bixi*+eiyi*-∑j=1n‍dij(xj*xi*-αij),-γi=δiyi*-εixi*,i=1,2,…,n.
Substituting this into (3.3) gives
(3.5)LV=∑i=1n‍ci[-εibi(xi-xi*)2-eiδi(yi-yi*)2+εi(xi-xi*)∑j=1n‍dij(xjxi-xj*xi*)+12εixi*σ1i2+12eiyi*σ2i2∑j=1n‍dij(xjxi-xj*xi*)]=∑i=1n‍ci[∑j=1n‍dijxj*(xjxj*-xixi*-xi*xjxixj*+1)-εibi(xi-xi*)2-eiδi(yi-yi*)2+12εixi*σ1i2+12eiyi*σ2i2+εi∑j=1n‍dijxj*(xjxj*-xixi*-xi*xjxixj*+1)]=∑i=1n‍ci{∑j=1n‍dijxj*[(-xixi*+logxixi*)-(-xjxj*+logxjxj*)+(1-xi*xjxixj*+logxi*xjxixj*)]-εibi(xi-xi*)2-eiδi(yi-yi*)2+12εixi*σ1i2+12eiyi*σ2i2+εi∑j=1n‍dijxj*[(-xixi*+logxixi*)-(-xjxj*+logxjxj*)+(1-xi*xjxixj*+logxi*xjxixj*)]}≤∑i=1n‍ci{∑j=1n‍dijxj*[(-xixi*+logxixi*)-(-xjxj*+logxjxj*)]-εibi(xi-xi*)2-eiδi(yi-yi*)2+12εixi*σ1i2+12eiyi*σ2i2+εi∑j=1n‍dijxj*[(-xixi*+logxixi*)-(-xjxj*+logxjxj*)]},
Here we use the fact: 1-a+loga≤0, for a>0 with equality holding if and only if a=1. Since (dij) is irreducible, εi>0, xj*>0, we know matrix A=(dijεixj*) is irreducible. Let Gi(xi)=-xi/xi*+log(xi/xi*),Gj(xj)=-xj/xj*+log(xj/xj*), and by Lemma A.1, we have
(3.6)∑i,j=1n‍ci(dijεixj*)(-xixi*+logxixi*)=∑i,j=1n‍ci(dijεixj*)(-xjxj*+logxjxj*)
which implies that
(3.7)LV≤∑i=1n‍[-ciεibi(xi-xi*)2-cieiδi(yi-yi*)2]+12∑i=1n‍(ciεixi*σ1i2+cieiyi*σ2i2).
The following proof of ergodicity is similar to Theorem 3.2 in [19]. Assume δ=(1/2)∑i=1n‍(ciεixi*σ1i2+cieiyi*σ2i2), then
(3.8)LV≤∑i=1n‍[-ciεibi(xi-xi*)2-cieiδi(yi-yi*)2]+δ.
Note that δ<min{min1≤i≤n{ciεibi(xi*)2},min1≤i≤n{cieiδi(yi*)2}}, then the ellipsoid
(3.9)∑i=1n‍[-ciεibi(xi-xi*)2-cieiδi(yi-yi*)2]+δ=0
lies entirely in R+2n. We can take U to be a neighborhood of the ellipsoid with U-⊂El=R+2n, so for X∈El∖U, LV≤-N (N is a positive constant), which implies the condition (B.2) in Lemma A.2 is satisfied. Therefore, the solution X(t) is recurrent in the domain U, which together with Remark A.3 and Lemma A.4 implies X(t) is recurrent in any bounded domain D⊂R+2n. Thus, for any D, there is
(3.10)M=min{min1≤i≤n{σ1i2xi2},min1≤i≤n{σ2i2yi2},(x1,y1,…,xn,yn)∈D-}>0
such that
(3.11)∑i,j=1n‍aij(x)ξiξj=∑i=1n‍(σ1i2xi2ξi2+σ2i2yi2ξi2)≥M∥ξ∥2,
for all (x1,y1,…,xn,yn)∈U-, ξ∈Rn, which implies that condition (B.1) is also satisfied. Therefore, the stochastic system (1.3) has a stable stationary distribution μ(·) and it is ergodic.


## 4. Extinction

In this section, we will show that if the noise is sufficiently large, the solution to the associated SDE (1.3) will become extinct with probability 1, although the solution to the original equation (1.1) may be persistent. For example, recall a simple case, namely, the scalar logistic equation
(4.1)dN(t)=N(t)(a-bN(t))dt,t≥0,
with initial value N0>0. It is well known that, when a>0,b>0, the solution N(t) is persistent, because
(4.2)limt→∞N(t)=ba.
However, consider its associated stochastic equation
(4.3)dN(t)=N(t)[(a-bN(t))dt+σdB(t)],t≥0,
where σ>0, then the solution to this stochastic equation will become extinct with probability 1, that is to say, if σ2>2a,
(4.4)limt→∞N(t)=0a.s.
The following theorem reveals the important fact that the environmental noise may make the population extinct.Theorem 4.1.
For any given initial valueX(0)∈R+2n, the solution of the SDE (1.3) has the property that
(4.5)limsupt→∞log(∑i=1nxi)t≤lˇ1-σ^122a.s.
Here
(4.6)lˇ1=max{max1≤i≤n{ri+∑j=1n‍dji-∑j=1n‍dijαij},0},σ^122=12∑i=1n(1/σ1i2).
Particularly, if lˇ1-σ^12/2<0, then limt→∞(x1(t),x2(t),…,xn(t))=0a.s.Proof.
Define(4.7)V=∑i=1n‍xi.
By Itô’s formula, we have
(4.8)dV=∑i=1n‍[rixi-bixi2-eixiyi+∑j=1n‍dij(xj-αijxi)]dt+∑i=1n‍σ1ixidB1i(t)=∑i=1n‍[-bixi2-eixiyi+(ri+∑j=1n‍dji-∑j=1n‍dijαij)xi]dt+∑i=1n‍σ1ixidB1i(t).
Thus we compute
(4.9)dlogV=1VdV-12V2(dV)2=1V∑i=1n‍[-bixi2-eixiyi+(ri+∑j=1n‍dji-∑j=1n‍dijαij)xi]dt+1V∑i=1n‍σ1ixidB1i(t)-12V2∑j=1n‍σ1i2xi2dt,
Letting lˇ1=max{max1≤i≤n{ri+∑j=1n‍dji-∑j=1n‍dijαij},0}, we compute
(4.10)1V∑i=1n‍[-bixi2-eixiyi+(ri+∑j=1n‍dji-∑j=1n‍dijαij)xi]≤1V∑i=1n‍(ri+∑j=1n‍dji-∑j=1n‍dijαij)xi≤lˇ1.
By Cauchy inequality, we compute also
(4.11)-12V2∑i=1n‍σ1i2xi2=-∑i=1n(σ1i2xi2)2(∑i=1nxi)2≤-(∑i=1nxi)22∑i=1n(1/σ1i2)·(∑i=1nxi)2=-12∑i=1n(1/σ1i2)=:-σ^122,
where σ^12/2=1/2∑i=1n‍(1/σ1i2). Substituting these two inequalities into (4.9) yields
(4.12)dlogV≤(lˇ1-σ^122)dt+1V∑i=1n‍σ1ixidB1i(t).
This implies
(4.13)logV(X(t))≤logV(X(0))+∫0t‍(lˇ1-σ^122)dt+M(t),
where M(t) is a martingale defined by
(4.14)M(t)=∫0t‍1V∑i=1n‍σ1ixi(s)dB1i(s)=∫0t‍∑i=1nσ1ixi(s)dB1i(s)∑i=1nxi(s)
with M(0)=0. The quadratic variation of this martingale is
(4.15)〈M,M〉t=∫0t‍∑i=1nσ1i2xi2(s)(∑i=1nxi(s))2ds≤σˇ12∫0t‍∑i=1nxi2(s)∑i=1nxi2(s)ds=σˇ12t,
where σˇ1=max1≤i≤n{σ1i}. By the strong law of large numbers for martingale (see [17, 20]), we have
(4.16)limt→∞M(t)t=0a.s.
It finally follows from (4.13) by dividing t on the both sides and then letting t→∞ that
(4.17)limsupt→∞logVt≤limsupt→∞1t∫0t‍(lˇ1-σ^122)dt=lˇ1-σ^122a.s.
which implies the required assertion.Remark 4.2.
Theorem4.1 shows that if the condition lˇ1-σ^12/2<0 holds, that is, when the prey population is disturbed by large white noise, the species of prey will extinct.Now we give the following theorem which describes the entire extinction.Theorem 4.3.
For any given initial valueX(0)∈R+2n, the solution of the SDE (1.3) has the property that
(4.18)limsupt→∞log[∑i=1n(εixi+eiyi)]t≤lˇ-σ^22a.s.
Here
(4.19)l˘=max{max1≤i≤n{ri+∑j=1n‍(djiεjεi-dijαij)},0},σ^22=12∑i=1n(1/σ1i2+1/σ2i2).
Particularly, if lˇ-σ^2/2<0, then limt→∞X(t)=0a.s.Proof.
The proof of the theorem is similar to Theorem4.1, we only give the main proof procedure. Define
(4.20)V(X(t))=∑i=1n‍(εixi+eiyi).
Let lˇ=:max{max1≤i≤n{ri+∑j=1ndji(εj/εi)-∑j=1ndijαij},0} and σ^2/2=1/2∑i=1n(1/σ1i2+1/σ2i2). By Itô’s formula, we compute
(4.21)dlogV≤1V∑i=1n‍εi(ri+∑j=1n‍djiεjεi-∑j=1n‍dijαij)xidt-12V2∑i=1n‍(εi2σ1i2xi2+ei2σ2i2yi2)dt+1V(∑i=1n‍εiσ1ixidB1i(t)+∑i=1n‍eiσ2iyidB2i(t))≤(lˇ-σ^22)dt+1V∑i=1n‍(εiσ1ixidB1i(t)+eiσ2iyidB2i(t)).
The rest of the proof is similar to Theorem4.1.Remark 4.4.
Theorem4.3 states that when the prey and predator population are all disturbed by large white noise and the condition lˇ-σ^2/2<0 holds, the two species will be extinct.


## 5. Numerical Simulation

In this section, in order to better study the effect of white noise in diffusion system, we assumeαij=1, dij are nonnegative constants, (dij) is irreducible (i,j=1,2), and d11=d22=1. Consider the predator-prey system with n=2, that is,
(5.1)dx1=[x1(r1-b1x1-e1y1)+d12(x2-x1)]dt+σ11x1dB11(t),dy1=y1(-γ1-δ1y1+ε1x1)dt+σ21y1dB21(t),dx2=[x2(r2-b2x2-e2y2)+d21(x1-x2)]dt+σ12x2dB12(t),dy2=y2(-γ2-δ2y2+ε2x2)dt+σ22y2dB22(t).In order to better study the previous results, we will numerically simulate the solution of (5.1). By the method mentioned in [21], we consider the following discretized equation:
(5.2)x1,k+1=x1,k+[x1,k(r1-b1x1,k-e1y1,k)+d12(x2,k-x1,k)]h+σ11x1,khξ1,k+12σ112x1,k(hξ1,k2-h),y1,k+1=y1,k+y1,k(-γ1-δ1y1,k+ε1x1,k)h+σ21y1,khη1,k+12σ212y1,k(hη1,k2-h),x2,k+1=x2,k+[x2,k(r2-b2x2,k-e2y2,k)+d21(x1,k-x2,k)]h+σ12x2,khξ2,k+12σ122x2,k(hξ2,k2-h),y2,k+1=y2,k+y2,k(-γ2-δ2y2,k+ε2x2,k)h+σ22y2,khη2,k+12σ222y2,k(hη2,k2-h).Using the discretized equation and the help of Matlab software, we choose the appropriate parametersr1=0.4,r2=0.3,b1=0.5,b2=0.4,e1=e2=0.2,d11=d22=1,d12=1.1,d21=1.2,γ1=γ2=0.2,δ1=δ2=0.1,ε1=ε2=0.3, the initial value (x1(0),y1(0),x2(0),y2(0))=(0.8,0.2,0.7,0.18), and time step h=0.01; then E*=(x1*,y1*,x2*,y2*)≐(0.7174,0.1545,0.7107,0.1356). In order to better investigate the white noise, we divide its intensity into small, medium, and large three cases to study.In Figures1, 2, and 3, we choose σ11=σ12=σ21=σ22=0.01 and c1≐0.86088,c2≐0.78177 (c1,c2 satisfy Lemma A.1 and Theorem 3.2); then δ=(1/2)∑i=12‍(ciεibixi*σ1i2+cieiδiyi*σ2i2)≐9.9×10-6, and so the condition δ<min{min1≤i≤2{ciεibi(xi*)2},min1≤i≤2{cieiδi(yi*)2}}≐4.11×10-4 is also satisfied. Therefore, by Theorem 3.2, there is a stationary distribution (see the histogram on the right in Figure 1). From the left picture in Figure 1, we can see that the solution of system (5.1) is fluctuating in a small neighborhood. Moreover, from Figure 2, we find that almost all population distribution lies in the neighborhood, which can be imagined by a circular or elliptic region centered at (x1*,y1*),(x2*,y2*) (see the scatter picture in Figure 2). Figure 3 shows that when the white noise is small, stochastic system imitates deterministic system and their curves are almost coincident. Hence, the solution of (5.1) is ergodicity, although there is no equilibrium of the stochastic system as the deterministic system. All of these imply system (5.1) is a stochastic stability.Figure 1
The solution of stochastic system (5.1) with small white noise and its histogram: σ11=σ12=σ21=σ22=0.01.Population distribution scatter corresponding to Figure1. The scatter points around (x1*,y1*)≐(0.7174,0.1545) and (x2*,y2*)≐(0.7107,0.1356), respectively. The system has a stationary distribution.
(a)
(b)The solution of stochastic system compared to the deterministic system:σ11=σ12=σ21=σ22=0.01. The blue and red lines represent the solutions of system (5.1), and the black and green lines represent the solutions of corresponding undisturbed system. These lines are almost coincident.
(a)
(b)
(c)
(d)Comparing with small white noise as in Figures1, 2, and 3, we select the relatively large white noise σ11=0.4,σ12=0.3,σ21=0.3,σ22=0.4,c1≐0.86088,c2≐0.78177 in Figures 4, 5, and 6. We find that δ=0.0107>min{min1≤i≤2{ciεibi(xi*)2},min1≤i≤2{cieiδi(yi*)2}}=2.875×10-4, so the conditions of Theorem 3.2 are not satisfied; therefore, there is not a stationary distribution although the deterministic system is global asymptotic stability. The condition lˇ-σ^2/2>0, that is, all extinction condition does not hold by Theorem 4.3 and the predator y2 will die. From Figure 4, we see that the fluctuations on the left figures are more intense and histogram distribution is not concentrated comparing with Figure 1, y2(t) close to the point 0.Figure 4
The solution of stochastic system with large white noise and its histogram:σ11=0.4,σ12=0.3,σ21=0.3,σ22=0.4. The fluctuations on the left figures are more intense and histogram distribution is not concentrated comparing with Figure 1.Population distribution scatter corresponding to Figure4. The scatter is not around the equilibrium points of the corresponding deterministic system. There is not a stationary distribution in system (5.1).
(a)
(b)The solution of stochastic system compared to the deterministic system:σ11=0.4,σ12=0.3,σ21=0.3,σ22=0.4. The predator population y2 will die out although its corresponding deterministic system is globally stable. The blue and red lines represent the solutions of system (5.1), and the black and green lines represent the solutions of corresponding undisturb system.
(a)
(b)
(c)
(d)In Figure7, we assume both the predator and the prey population suffered large white noise; we choose σ11=1.4,σ12=1.5,σ21=0.01,σ22=0.01, which satisfy the cases said in Theorem 4.1, that is lˇ1-σ^12/2≐-0.0239<0. As the case in Theorem 4.1 expected, the large white noise leads to the extinction of the prey which also leads to the extinction of the predator, so the solution of system (5.1) tends to zero.The solution of stochastic system compared to the deterministic system:σ11=1.4,σ12=1.5,σ21=0.01,σ22=0.01. The prey population suffer the large white noise which leads to the prey and predator extinction. The blue and red lines represent the solutions of system (5.1), and the black and green lines represent the solutions of corresponding undisturb system.
(a)
(b)
(c)
(d)In Figure8, we choose the same parameters as in Figure 1, but change the value of σ21,σ22 to σ21=0.8,σ22=0.7, which do not meet the conditions for extinction of the two species as in Theorem 4.3. The predator population suffer the large white noises and then die out, but the prey will survive.The solution of stochastic system compared to the deterministic system:σ11=σ12=0.01,σ21=0.8,σ22=0.7. The predator population suffer the large white noise and then extinction. The blue and red lines represent the solutions of system (5.1), and the black and green lines represent the solutions of corresponding undisturb system.
(a)
(b)
(c)
(d)From these figures, we can get the following conclusions: when the white noise is small, system (5.1) imitates its deterministic system (see Figures 1, 2, and 3). But when the white noise is relatively large, it will bring more big deviation (see Figures 4, 5, and 6) even the extinction of the species (see Figures 6, 7, and 8), which will not happen in the deterministic system. However, when the intensity of the white noise on prey species is large, the predator and prey population will be extinct (see Figure 7). In real world, the large white noise may be bad weather, serious epidemic, which can be considered as the decisive factors responsible for the extinction of populations. Therefore, the study of stochastic model has great practical significance.



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*Source: 547152-2012-10-17.xml*