# Analysis of Stochastic Gilpin-Ayala Competition System

Authors: Lei Liu; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper is concerned with the asymptotic behavior for stochastic Gilpin-Ayala competition system. The sufficient conditions for existence of stationary distribution and extinction are established. And a certain asymptotic property of the solution is also obtained. A nontrivial example is provided to illustrate our results.


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## Body

## 1. Introduction

One of the most common phenomena considering ecological population is that many species which grow in the same environment compete for the limited resources or in some way inhibit others’ growth. It is therefore very important to study the competition models for multispecies. It is well known that one of the famous models is the following classical Lotka-Volterra competition system:(1)dxidt=xibi-∑j=1n‍aijxj,i=1,…,n,
where xi(t) represents the population size of species i at time t, the constant bi is the growth rate of species i, and aij represents the effect of interspecific (i≠j) or intraspecific (i=j) interaction. The Lotka-Volterra models have often been severely criticized. One disadvantage of Lotka-Volterra models is that in such a model, the rate of change in the density of each species is a linear function of densities of the interacting species. In order to yield significantly more accurate results, Gilpin and Ayala proposed the the following Gilpin-Ayala models; detailed studies related to the model may be found in [1]:
(2)dxidt=xibi-aiixiθi-∑j≠i‍aijxj,i=1,…,n,
where θi are the parameters to modify the classical Lotka-Volterra model.On the other hand, population systems are inevitably affected by environmental noise. It is therefore useful to reveal how the noise affects the population systems. Recall that the parameterbi in (2) represents the intrinsic growth rate of the population. In practice we usually estimate it by an average value plus an error which follows a normal distribution; then the intrinsic growth rate becomes
(3)bi⟶bi+σiB˙i(t),
where Bi(t)  (i=1,…,n) are Brown motions with Bi(0)=0 and σi2 represent the intensities of the noise. As a result, system (2) becomes the stochastic Gilpin-Ayala system as follows:
(4)dxi=xibi-aiixiθi-∑j≠i‍aijxjdt+σixidBit,hhhhhhhhhhhhhhhhhhhhhhhhihhi=1,…,n,
and we impose the following condition:
(5)θi>1,hhhhaii>0,hhhhaij≥0,hhhhhhh1≤i,hhj≤n,hhi≠j.The stochastic Lotka-Volterra model has been extensively studied due to its universal existence and importance; see [2–10]. More recently, the existence of stationary distribution and extinction of stochastic Lotka-Volterra system have received a lot of attention, which can give a good explanation of the recurring phenomena in population system. Under what conditions can a stochastic Lotka-Volterra system has a stationary distribution? It is an open topic until very recently Mao [11] gave a positive answer. Since then, this topic has received a lot of attention; the readers are referred to [11–14]. In addition, the asymptotic behavior of log⁡⁡xi(t)/t,  i=1,…,n for various stochastic Lotka-Volterra systems has been considered by many authors [4, 5, 10, 12], which is an important and useful property on asymptotic estimation for corresponding population systems.However, these properties for stochastic Gilpin-Ayala system (4) have not been investigated, which remain an interesting research topic. We aim to establish new results on these properties for system (4). It is well known that the stochastic Gilpin-Ayala system (4) is a highly nonlinear system; the method for classic Lotka-Volterra system cannot be directly applied to system (4). By the Lyapunov methods, and some techniques to deal with the nonquadratic item, sufficient criteria are established which ensure the existence of a stationary distribution and extinction. By using some stochastic analysis techniques, an asymptotic property for system (4) is obtained.


## 2. Notation

Throughout this paper, unless otherwise specified, let(Ω,F,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). Let B(t)=(Bt1,…,Btn) be an-dimensional Brownian motion defined on the probability space. If a and b are real numbers, then a∨b denotes the maximum of a and b, and a∧b stands for the minimum of a and b. If A∈Rn×n is symmetric, its largest and smallest eigenvalues are denoted by λmax⁡(A) and λmin⁡(A). Let x*=(x1*,…,xn*) be the positive equilibrium of the deterministic Gilpin-Ayala competition system (2), that is, the solution of the following equation:
(6)bi=aiixi*θi+∑j≠i‍aijxj*,i=1,…,n.
In the same way as Mao et al. [8] did, we can also show the following result on the existence of global positive solution.Lemma 1.
Assume that condition (5) holds. Then, for any given initial value x0∈R+n, there is a unique solution x(t,x0) to system (4) and the solution will remain in R+n with probability 1; namely,
(7)Pxt,x0∈R+n,∀t≥0=1,
for any x0∈R+n.Lemma 2.
Let condition (5) hold. Then, for any p>0 and any given initial value x0∈R+n, there exists a constant Kp such that
(8)sup⁡0≤t≤∞E∑i=1n‍xipt,x0<Kp.The proof of the lemma is rather standard so it is omitted.


## 3. An Asymptotic Property

The main aim of this section is to consider the large time behavior oflog⁡xi(t)/t,  i=1,…,n. To this end, we consider two auxiliary stochastic differential equations as follows:
(9)dφi=φibi-aiiφiθidt+σiφidBi(t),φi0=xi0,i=1,…,n,(10)dyi=yibi-aiiyiθi-∑j≠i‍aijφjdt+σiyidBit,yi0=xi0,i=1,…,n.
Then it follows from comparison principle (see [15]) that
(11)yit≤xit≤φit,i=1,…,n.Lemma 3.
Let condition (5) hold. Then the solution to system (9) has the following property:
(12)log⁡φitt=0,i=1,…,n,a.s.The proof is similar to Li et al. [5] and is omitted here.Theorem 4.
Let condition (5) hold and x(t,x0) be the global solution to system (4) with any positive initial value x0. Assume moreover that
(13)bi-σi22>0,bi-σi22-∑j≠i‍aijbj-σj2/2ajjj1/θj>0,hhhihhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,…,n.
Then the solution x(t,x0) of system (4) has the following property:
(14)lim⁡t→∞log⁡⁡xit,x0t=0,i=1,…,n,a.s.Proof.
Letx(t) be x(t,x0) for simplicity. By virtue of Lemma 3 and (11), we have limsup⁡t→∞log⁡xi(t)/t≤0,  i=1,…,n,  a.s. Thus it remains to show that liminf⁡t→∞log⁡xi(t)/t≥0,  i=1,…,n,a.s. It is sufficient to show
(15)liminf⁡t→∞log⁡⁡yi(t)t≥0,i=1,…,n,a.s.
By Ito’s formula, yi(t) satisfies
(16)1yiθi(t)=1xiθi(0)exp⁡⁡∑j≠i‍∫0t‍aijφj(s)dsσi22-biθit-θiσiBi(t)+θi∑j≠i‍∫0t‍aijφj(s)ds+aiiθi∫0t‍exp⁡∑j≠i‍∫st‍aijφjτdτσi22-biθit-s-θiσiBit-Bis+θi∑j≠i‍∫st‍aijφjτdτds=:Ji1+Ji2,i=1,…,n.
A simple computation shows that
(17)∫st‍φθiτdτ=1aiilog⁡φis-log⁡φit+bi-σi2/2aii(t-s)+σ1aiiBit-Bis≔ki(t-s)+miBit-Bis+dilog⁡φis-log⁡φit,i=1,…,n.
The well-known Hölder inequality yields
(18)∫st‍φiτdτ≤t-s1-1/θi×kit-s+miBit-Bis+dilog⁡φis-log⁡φit1/θi,i=1,…,n.
For i=1,…,n, it follows from the inequality (a+b+c)p≤3(p-1)∨0(ap+bp+cp) that
(19)∫st‍φiτdτ≤t-s1-1/θikiθit-sθimax⁡0≤s≤tlog⁡φis-log⁡φitθi+miθiBit-min⁡0≤s≤tBisθi+diθimax⁡0≤s≤tlog⁡φis-log⁡φitθi1/θi.
Fori=1,…,n, set
(20)Bi*t≔Bit-min⁡0≤s≤tBis,ξit≔max⁡0≤s≤tlog⁡φis-log⁡φit.
Substituting these inequalities into (16) yields
(21)Ji1≤1xiθi0exp⁡⁡∑j≠i‍σi22-biθit-θiσiBit+θi∑j≠i‍aijt1-1/θj×kit+miBi(t)+dilog⁡φi0-log⁡φit1/θj∑j≠i‍≤1xiθi(0)exp⁡∑j≠i‍σi22-biθit+θiσimax⁡0≤s≤tBis-Bit+θi∑j≠i‍aijt1-1/θj×ξjt1/θjξjt1/θjkj1/θjt-s1/θj+mj1/θjBj*t1/θj+dj1/θjξjt1/θj∑j≠i‍≤1xiθi0exp⁡θiσi22-bi+∑j≠i‍aijkj1/θjt+θiσimax⁡0≤s≤tBis-Bit+θi∑j≠i‍t1-1/θjaij×mj1/θjBj*t1/θj+dj1/θjξjt1/θjσi22-bi+∑j≠i‍aijkj1/θjds,i=1,…,n.
Similarly, we get
(22)Ji2≤aiiθi∫0t‍exp⁡⁡∑j≠i‍σi22-biθit-s-θiσiBit-Bis+θi∑j≠i‍t-s1-1/θj×kj(t-s)+mjBjt-Bjs+djlog⁡φjs-log⁡φjt1/θj∑j≠i‍ds≤aiiθi∫0t‍exp⁡∑j≠i‍σi22-biθit-s+θiσimax⁡0≤s≤tBis-Bit+θi∑j≠i‍t-s1-1/θjaij×ξjt1/θjkj1/θjt-s1/θj+mj1/θjBj*t1/θj+dj1/θjξjt1/θj∑j≠iσi22-bi‍ds≤aiiθi∫0t‍exp⁡θiσi22-bi+∑j≠i‍aijkj1/θj×t-s+θiσimax⁡0≤s≤tBis-Bit+θi∑j≠i‍t1-1/θjaij×mj1/θjBj*t1/θj+dj1/θjξjt1/θjσi22-bi+∑j≠i‍aijkj1/θjds,i=1,…,n.
Substituting (21) and (22) into (16) yields
(23)1yiθit≤1xiθi0exp⁡⁡θiσi22-bi+∑j≠i‍aijkj1/θjt+θiaii∫0t‍exp⁡⁡θiσi22-bi+∑j≠i‍aijkj1/θj×t-sσi22-bi+∑j≠i‍aijkj1/θjds×exp⁡θi∑j≠i‍θiσimax⁡0≤s≤tBis-Bit+θi∑j≠i‍t1-1/θjmj1/θjBj*t1/θj+dj1/θjξjt1/θjθi∑j≠i‍:=Zi-1texp⁡∑j≠it1-1/θjθiσimax⁡0≤s≤tBis-Bit+θi∑j≠it1-1/θjmj1/θjBj*t1/θj+dj1/θjξjt1/θj∑j≠it1-1/θj,i=1,…,n,
where Zi(t) is the solution of the following system:
(24)Z˙i(t)=θiZibi-σi22-∑j≠i‍aijkj1/θj-aiiZit,Zi0=xi0,i=1,…,n.
A simple computation shows that
(25)log⁡⁡yiθitt≥-log⁡⁡Zitt-θiσimax⁡0≤s≤t⁡Bis-Bitt-∑n≠j‍aijmjBj*tt1/θj-∑n≠j‍aijdjξjtt1/θj,i=1,…,n.
Using the property of Brownian motion, we conclude that
(26)lim⁡t→∞max⁡0≤s≤t⁡Bis-Bitt=0,lim⁡t→∞Bjt-min⁡0≤s≤t⁡Bjst=0,i=1,…,n,a.s.
It is easy to see that if bi-σi2/2-∑j≠iaijkj1/θj>0, then we have
(27)lim⁡t→∞log⁡⁡Zi(t)t=0,i=1,…,n,a.s.
Besides, it follows from Lemma 3 that
(28)lim⁡t→∞⁡max⁡0≤s≤tlog⁡⁡φjs-log⁡⁡φjtt=0,hhhhhhhhlhhhhhhhhhhi=1,…,n,a.s.
The required assertion (15) follows by letting t→∞ on both sides of (25) and using conditions (26)–(28). The proof is therefore completed.


## 4. Stationary Distribution

The main aim of this section is to study the existence of a unique stationary distribution of the system (4). Let us prepare a known lemma (see Hasminskii [16, pp. 106–125]). Let X(t) be a homogeneous Markov process in En⊂Rn described by the following stochastic differential equation:
(29)dXt=bXdt+∑m=1d‍σmXdBmt.
The diffusion matrix is
(30)A(x)=aijx,aij(x)=∑m=1d‍σmi(x)σmj(x).
To be more precise, let Px0,t denote the probability measure induced by X(t,x0), that is
(31)Px0,tA=PXt,x0∈A,A∈BEn,
where B(En) is the σ-algebra of all the Borel sets A⊂En.Lemma 5 (see [16]).
We assume that there is a bounded open subsetG⊂En with a regular (i.e., smooth) boundary such that its closure G¯⊂En, and consider the following:(i)
in the domainG and some neighborhood, therefore, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero;(ii)
ifx∈En∖G, the mean time τ at which a path issuing from x reaches the set G is finite, and sup⁡x∈KExτ<+∞ for every compact subset K∈En. And throughout this paper one sets inf⁡∅=∞.
We then have the following assertions.(1)
The Markov processX(t) has a stationary distribution μ(·) with density in En, such that, for any borel set B⊂En,
(32)lim⁡t→∞∫B‍f(y)Px0,t(dy)=∫B‍f(y)μ(dy).(2)
(ergodic property) Letf(x) be a function integrable with respect to the measure μ(·). Then
(33)Plim⁡t→∞1t∫0t‍fxsds=∫En‍fyμdylim⁡t→∞1t∫0t‍=1.Remark 6.
The proof is given by [16] in detail. Exactly, the existence of stationary distribution with density is referred to Theorem 4.3 on page 117 while ergodic property (33) is referred to Theorem 4.2, page 110.Theorem 7.
Let condition (5) hold and x(t,x0) be the global solution to system (4) with any positive initial value x0. Assume that there exists c=(c1,…,cn)≫0 such that
(34)cix*θi-1aii-12∑i≠j‍ciaij+cjaji>0,(35)12∑k=1n‍cixi*σi2<min⁡1≤i≤nxi*2cix*θi-1aii-∑i≠j‍ciaij+cjaji,i=1,…,n.
Then there is a stationary distribution for system (4) and it has the ergodic property.Proof.
By Lemma5, it suffices to prove that there exists some neighborhood U and a nonnegative C2-function V(x) such that the diffusion matrix H(x)=diag⁡(σ1x1,…,σnxn) is uniformly elliptical in U and, for any x∈R+n∖U, LV(x) is negative (for details refer to [11]).
Applying Itô’s formula toV(x)=∑i=1nci(xi-xi*-xi*log⁡xi/xi*) yields
(36)LVx=∑i=1n‍∑i≠jn‍ciaiixi-xi*xiθi-xi*θi+∑i≠jn‍ciaijxj*-xjxi*-xi+12∑i=1n‍ciσi2xi*.
If xi>xi*, since (xi/xi*)θi≥(xi/xi*), for θi≥1, then
(37)xi-xi*xiθi-xi*θi=xi*θi+1xixi*-1xixi*θi-1≥xi*θi+1xixi*-1xixi*-1=xi*θi-1xi-xi*2.
If xi<xi*, then
(38)xi-xi*xiθi-xi*θi=xi*θi+11-xixi*1-xixi*θ≥xi*θi+11-xixi*1-xixi*=xi*θi-1xi-xi*2.
Substituting (37) and (38) into (36) yields
(39)LVx≤-∑i=1n‍xi*θi-1ciaiixi-xi*2+∑i=1n∑i≠jn‍ciaij(xj*-xj)(xi*-xi)+12∑i=1n‍ciσi2xi*.
By the inequality ab≤1/2(a2+b2), we have
(40)LVx≤-∑i=1n‍xi*θi-1ciaiixi-xi*2+∑i=1n∑i≠jn‍ciaij(xj*-xj)(xi*-xi)+12∑i=1n‍ciσi2xi*≤-∑i=1n‍xi*θi-1ciaiixi-xi*2+12∑i=1n∑i≠jn‍ciaijxi*-xi2+12∑i=1n∑i≠jn‍ciaijxj*-xj2+12∑i=1n‍ciσi2xi*=-∑i=1n‍xi*θi-1ciaiixi-xi*2+12∑i=1n∑i≠jn‍ciaijxi*-xi2+12∑i=1n∑i≠jn‍cjajixi*-xi2+12∑i=1n‍ciσi2xi*=-∑i=1n‍cix*θi-1aii-12∑i≠jn‍ciaij+cjaji×xi*-xi2+12∑i=1n‍ciσi2xi*.
Note that (ci(x*)θi-1aii-∑i≠j‍(ciaij+cjaji))>0,  i=1,…,n, and
(41)12∑k=1n‍cixi*σi2<min⁡1≤i≤nxi*2cix*θi-1aii-∑i≠j‍ciaij+cjaji.
Then the ellipsoid
(42)∑i=1n‍cix*θi-1aii-12∑i≠jn‍ciaij+cjajixi*-xi2=12∑i=1n‍ciσi2xi*,
lies entirely in R+n. Let U⊂U¯⊂R+n be a neighborhood of the ellipsoid such that, for any x∈R+n∖U, LV<0. We therefore have verified condition (ii) in Lemma 5.
Now we begin to verify condition (i) in Lemma5. It is easy to see that λmin⁡(HT(x)H(x))≥0. If λmin⁡(HT(x)H(x))=0, then there exists ξ≠0 such that ξTHT(x)H(x)ξ=0. This implies that (diag⁡(σ1x1,…,σnxn)ξ=0. Then we have ξ=0, which contradicts the fact that ξ≠0. Noting that λmin⁡(HT(x)H(x)) is a continuous function of x∈U¯, we therefore have
(43)min⁡x∈U¯λmin⁡(HT(x)H(x))>0.
This immediately implies condition (i) in Lemma 5. The proof is completed.Now we denote byμ(·) the stationary distribution. The mean vector of μ(·) is important and useful information on population systems, from which we can infer asymptotically the mean of xi(t) and the size of each species. If we can show that ∫R+n‍|z|μ(dz)<∞, then the mean vector μ¯=(μ¯1,…,μ¯n)T is well defined. In this case, the ergodic theory stated above implies that
(44)μ¯i:=lim⁡t→∞1t∫0t‍xisds,i=1,…,n,a.s.Theorem 8.
Let assumptions in Theorems4 and 7 hold. Then
(45)aiiμ¯iθi+∑j≠i‍aijμ¯j≤bi-σi22,i=1,…,n.Proof.
The proof is composed of two parts. The first part is to show the well-definition ofμ by dominated control convergence theorem. The second part is to prove assertion (45). Let x(t)=x(t,x0) for simplicity.
By the ergodic property of stationary distribution, form>0,  p>0, we have
(46)lim⁡t→∞1t∫0t‍xips∧mds=∫0∞‍zip∧mμdy,hhhhhhhhhhhhhhhhhhhhhhlhhi=1,…,n,a.s.
The dominated convergence theorem yields that
(47)Elim⁡t→∞1t∫0t‍xips∧mds=1t∫0t‍Exip∧mds,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhli=1,…,n.
It follows from Lemma 2 that
(48)∫0∞‍zip∧mμdz≤Kp,i=1,…,n.
Letting m→∞ yields
(49)∫0∞‍zipμdy≤Kp,i=1,…,n.
That is to say, for any p>0, the functions yp are integrable with respect to the measure μ(·). The well-definition of μ follows by letting p=1 in (49) straightforward.
Now we process to show assertion (45). For i=1,…,n, simple computation shows that
(50)log⁡xitt=log⁡xi0t+1t∫0t‍bi-σi22-aiixiθi-∑j=1,j≠in‍aijxjsds+1t∫0t‍σidBi(s).
The well-known Hölder inequality yields
(51)∫0t‍xi(s)ds≤∫0t‍xiθisds1/θi∫0t‍1ds1/θi′i=1,…,n,
where 1/θi′=1-1/θi. This implies
(52)∫0t‍xi(s)ds≤∫0t‍xiθisds1/θit1-1/θi,i=1,…,n.
The well-known Hölder inequality yields
(53)1t∫0t‍xisds≤1t∫0t‍xiθisds1/θi,i=1,…,n.
This implies
(54)log⁡xitt≤log⁡x0t+bi-σi22-aii1t∫0t‍xisdsθi-1t∫0t‍∑j=1,j≠in‍aijxj(s)ds+1t∫0t‍σidBi(s),i=1,…,n.
By the law of strong large numbers for martingales and Theorem 4, letting t→∞ on both sides of (54) yields
(55)aiiμ¯iθi+∑j≠i‍aijμ¯j≤bi-σi22,i=1,…,n.
which is the required assertion (45).


## 5. Extinction

One of the most basic questions one can ask in population dynamics is extinction, which means a species will be doomed. The interesting question is can the exponential extinction rate be estimated precisely? In many cases, we need to know the extinction rate of the species in order to have a suitable policy in investment and to have timely measures to protect them from the extinct disaster.Theorem 9.
Let condition (5) and σi2>2bi,  i=1,…,n, hold and x(t,x0) be the global solution to system (4) with any positive initial value x0. Then the solution xi(t,x0) to system (4) has the property that
(56)lim⁡t→∞log⁡xi(t,x0)t=-σi22-bi,i=1,…,n,a.s.
That is, the population will become extinct exponentially with probability one and the exponential extinction rate of the ith species is -(σi2/2-bi).Proof.
Letx(t)=x(t,x0) for simplicity. It follows from Itô’s formula that
(57)log⁡xit=log⁡xi(0)+∫0t‍bi-σi22ds-∫0t‍(aiixθi(s)+∑i≠j‍aijxj(s))ds+∫0t‍σidBis,i=1,…,n,
where Mi(t)=∫0t‍σidBi(s) is the real-valued continuous local martingale vanishing at t=0, with the quadratic variation Mi(t),Mi(t)=σi2t. Dividing both sides by t yields
(58)log⁡xitt=log⁡xi0t+1t∫0t‍r-σi22ds-1t∫0t‍aiixθis+∑i≠j‍aijxjsdsW+1t∫0t‍σi2dBi(s),i=1,…,n.
Using the law of strong large numbers for martingales (see [17]), we can claim that
(59)lim⁡t→∞1t∫0t‍σidBis=0,i=1,…,n,a.s.
Letting t→∞ yields
(60)lim⁡t→∞sup⁡log⁡xi(t)t≤-σi22-bi,i=1,…,n,a.s.
This shows that, for any 1≤i≤k and ϵ∈(0,min⁡1≤i≤k{σi/2-bi}), there is a positive random variable T(ϵ) such that, with probability one,
(61)xit≤e-σi2/2-bit+ϵt,∀t>Tϵ,i=1,…,n,a.s.
It follows that
(62)xiθit≤e-ασi2/2-bit+αϵt,∀t>Tϵ,i=1,…,n,a.s.,
which means
(63)aii∫0∞‍xiθisds+∑j≠i‍aij∫0∞‍xjθjsds<∞,i=1,…,n,a.s.
The required assertion (58) follows by letting t→∞ on both sides of (54).Remark 10.
Theorem9 showed that when the perturbation is large in the sense that σi2>2bi,  i=1,…,n, the population will be forced to expire. And the exponential extinction rate is given precisely in terms of system’s coefficients.


## 6. Numerical Simulations

In this section, to illustrate the usefulness and flexibility of the theorem developed in previous section, we present a numerical example.Example 11.
Consider a 2-dimensional stochastic Gilpin-Ayala system as follows:(64)dx1=x11-0.8x11.5-0.3x2dt+σx1dB1t,dx2=x2(1-x21.2-0.2x1)dt+σx2dB2(t).
System (64) is exactly system (4) with a11=0.8>0, a12=0.3>0, a21=0.2>0, a22=1>0, b1=1>0, b2=1.2>0, and θ1=1.2,  θ2=1.5. We compute that x1*=0.9109 and x2*=1.0148. The existence and uniqueness of the solution follows from Lemma 1. We consider the solution x(t,x0) with initial data x1(0)=0.5 and x2(0)=0.5. Let x(t)=x(t,x0) for simplicity.
(i)σ=0.4: simple computation shows that
(65)b1-σ22=0.92>0,b2-σ22=1.12>0,b1-σ22-a12b2-σ221/θ2=0.5903>0,b2-σ22-a21b1-σ221/θ1=0.9308>0.
By Theorem 4, the solution to system (64) has the following property
(66)lim⁡t→∞log⁡x1(t)t=0;lim⁡t→∞log⁡x2(t)t=0,a.s.
Figures 1 and 2 show the stochastic trajectories of log⁡x1(t)/t and log⁡x2(t)/t generated by the Heun scheme for time step Δ=10-3 for system (64) on [0,50], respectively.
Choosingc1=1 and  c2=0.5, we further compute that
(67)c1a11x1*0.5-12(c1a12+c2a21)=0.5635>0,c2a22x2*0.2-12(c1a12+c2a21)=0.3015>0,c1σx1*2+c2σx2*2=0.2837<min⁡{0.5635,0.3015}.
By virtue of Theorem 7, system (64) has a unique stationary distribution. Figures 3 and 4 show the stochastic trajectories of x1(t) and x2(t) generated by the Heun scheme for time step Δ=10-3 for system (64) on [0,500], respectively.
(ii) Considerσ=2.
Note that1<22/2, 1.2<22/2, by virtue of Theorem 9, system (64) is exponentially extinctive. Figures 5 and 6 show the stochastic trajectories of log⁡x1(t)/t and log⁡x2(t)/t generated by the Heun scheme for time step Δ=10-3 for system (64) on [100,500], respectively.Figure 1Figure 2Figure 3Figure 4Figure 5Figure 6


## 7. Conclusion

In this paper, we have investigated the asymptotic behavior for the stochastic Gilpin-Ayala competition system. Firstly, by utilizing stochastic analysis techniques and the stochastic comparison principle, the larger time behaviorlog⁡xi(t)/t,  i=1,…,n. has been researched. Secondly, by applying some techniques to deal with the nonquadratic item, sufficient conditions are obtained under which there is a stationary distribution to the system. Based on the condition, the estimation on the mean of the stationary distribution is presented. Finally, the sufficient criteria for extinction are established.



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*Source: 637862-2014-11-05.xml*