# Stochastically Ultimate Boundedness and Global Attraction of Positive Solution for a Stochastic Competitive System

Authors: Shengliang Guo; Zhijun Liu; Huili Xiang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.


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## Body

## 1. Introduction

In recent years, many researches have been done on the dynamics of many types of Lotka-Volterra competitive systems. Owing to their theoretical and practical significance, these competitive systems have been investigated extensively and there exists a large volume of literature relevant to many good results (see [1–7]). Particularly, in [8], Gopalsamy introduced the following autonomous two-species competitive system:
(1)y1′(t)=y1(t)[a1-b1y1(t)-c1y2(t)-d1y12(t)],y2′(t)=y2(t)[a2-b2y2(t)-c2y1(t)-d2y22(t)],
where y1(t) and y2(t) can be interpreted as the population size of two competing species at time t, respectively. All parameters involved with the above model are positive constants and can be interpreted in more detail; a1 and a2 are the intrinsic growth rates of two species; b1, d1, b2, and d2 represent the effects of intraspecific competition; c1 and c2 are the effects of interspecific competition. Recently, Tan et al. [9] have considered the effect of impulsive perturbations and discussed the uniformly asymptotic stability of almost periodic solutions for a corresponding nonautonomous impulsive version of (1). It has also been noticed that the ecological systems, in the real world, are often perturbed by various types of environmental noise. May [10] also pointed out that, due to environmental fluctuation, the birth rate, the death rate, and other parameters usually show random fluctuation to a certain extent. To accurately describe such systems, it is necessary to use stochastic differential equations. Recently, various stochastic dynamical models have been introduced extensively in [11–17] and many interesting and valuable results including extinction, persistence, and stability can be found in [18–20].Motivated by the above works, in this contribution, we assume that the environmental noise affects mainly the intrinsic growth rateai with
(2)ai⟶ai+σiw˙i(t),i=1,2,
where w˙i(t) are independent white noises, wi(t) are standard Brownian motions defined on the complete probability space (Ω,F,{Ft}t≥0,P) with a filtration {Ft}t≥0 satisfying the usual conditions, and σi2 represent the intensities of the white noises. Then the stochastically perturbed sytem (1) can be Itô’s equations
(3)dy1(t)=y1(t)[a1-b1y1(t)-c1y2(t)-d1y12(t)]dt+σ1y1(t)dw1(t),dy2(t)=y2(t)[a2-b2y2(t)-c2y1(t)-d2y22(t)]dt+σ2y2(t)dw2(t)
with the initial values yi(0)>0.In this paper, we will focus on the stochastically ultimate boundedness and global attraction of positive solutions of system (3). To the best of our knowledge, there are few published papers concerning system (3). The rest of this paper is organized as follows. In Section 2, we present some assumptions, definitions, and lemmas. In Section 3, we investigate the existence and uniqueness of positive solution, and then, we discuss the stochastically ultimate boundedness of positive solutions. In Section 4, we discuss the global attraction of positive solutions. Finally, we conclude and present a specific example to justify the analytical results.


## 2. Preliminaries

Throughout this paper, we give the notationR+2={y1>0,y2>0} and assumptions.(S1)
b1>d1, b2>d2.(S2)
For any initial value (y1(0),y2(0))∈R+2, there exists p>1 such that
(4)yi(0)≤ai+(1/2)(p-1)σi2bi,i=1,2.(S3)
b1>c2, b2>c1.In the following, let us briefly review several basic definitions and lemmas which will be useful for establishing our main results.Definition 1.
The solution(y1(t),y2(t)) of system (3) is stochastically ultimately bounded a.s. if for arbitrary εi∈(0,1), there exists a positive constant φi=φ(εi) such that
(5)limsup⁡t→+∞P{|yi(t)|>φi}<εi,i=1,2.Definition 2.
Let(y1(t),y2(t)) be a positive solution of system (3). Then (y1(t),y2(t)) is said to be globally attractive provided that any other solution (y1*(t),y2*(t)) of system (3) satisfies
(6)lim⁡t→+∞|y1(t)-y1*(t)|=0,lim⁡t→+∞|y2(t)-y2*(t)|=0a.s.Lemma 3 (see [21]).
Letp>2,g∈Mp([0,T];Rm) such that
(7)E∫0T‍|g(s)|pds<∞,
where Mp([0,T];Rm) is the family of processes {h(t)}0≤t≤T such that
(8)E∫0T‍|h(t)|pdt<∞.
Then
(9)E|∫0T‍g(s)dw(s)|p≤(p(p-1)2)p/2T(p-2)/2E∫0T‍|g(s)|pds.Lemma 4 (see [22]).
Suppose thata1,a2,…,an are real numbers; then
(10)|a1+a2+⋯+an|p≤Cp(|a1|p+|a2|p+⋯+|an|p),
where p>0 and
(11)Cp={1,0<p≤1;np-1,p>1.Lemma 5 (see [23]).
Assume that an n-dimensional stochastic processX(t) on t≥0 satisfies the condition
(12)E|X(t)-X(s)|α≤λ|t-s|1+β,0≤s,t<∞,
for positive constants α, β, and λ. Then there exists a continuous version X~(t) of X(t) which has the property that, for every ϑ∈(0,β/α), there is a positive random variable ψ(ω) such that
(13)P{ω:sup⁡0<|t-s|<ψ(ω),0≤s,t<∞|X~(t,ω)-X(t,ω)||t-s|ϑ≤21-2-ϑ}=1.
In other words, almost every sample path of X~(t) is locally but uniformlyHölder continuous with exponent ϑ.Lemma 6 (see [24]).
Letf(t) be a nonnegative function on t≥0 such that f(t) is integrable on t≥0 and is uniformly continuous on t≥0. Then lim⁡t→+∞f(t)=0.


## 3. Stochastically Ultimate Boundedness

In this section, we first show, under the assumption(S1), that the positive solution of system (3) will not explode to infinity at any finite time.Lemma 7.
Let(S1) hold and the initial value (y1(0),y2(0))∈R+2. Then system (3) has a unique solution (y1(t),y2(t)) on t≥0, which will remain in R+2 with probability one.Proof.
It is easy to see that the coefficients of system (3) satisfy the local Lipschitz condition. Then for any given initial value (y1(0),y2(0))∈R+2, there exists a unique local solution (y1(t),y2(t)) on [0,τe), where τe is the explosion time. To show that the positive solution is global, we only need to show that τe=+∞a.s. Let n0 be sufficiently large for every component of (y1(0),y2(0)) which remains in the interval [1/n0,n0]. For each integer n≥n0, one can define the stopping time
(14)τn=inf⁡{t∈[0,τe):y1(t)∉(1n,n)ory2(t)∉(1n,n)}.
Clearly, τn is increasing as n→+∞. Assign τ+∞=lim⁡n→+∞τn, whence τ+∞≤τea.s. If we can show that τ+∞=+∞a.s., then τe=+∞a.s. and (y1(t),y2(t))∈R+2a.s. for all t≥0. In other words, to complete the proof, we just need to show that τ+∞=+∞a.s.
By reduction to absurdity, we suppose thatτ+∞≠+∞; then there exists a pair of constants T>0 and ɛ∈(0,1) such that
(15)P{τ+∞≤T}>ɛ.
As a result, there exists an integer n1≥n0 such that for all n≥n1(16)P{τn≤T}≥ɛ.
Define a C2-function V:R+2→R+ as
(17)V(y1,y2)=y1-1-ln⁡y1+y2-1-ln⁡y2.
Obviously, V(y1,y2) is a nonnegative function. If (y1(t),y2(t))∈R+2, then using Itô’s formula, one can derive that
(18)dV(y1,y2)=(1-1y1)dy1+0.51y12(dy1)2+(1-1y2)dy2+0.51y22(dy2)2={-d1y13(t)-b1y12(t)+d1y12(t)hhhhh+a1y1(t)+b1y1(t)+c2y1(t)hhhhh-c1y1(t)y2(t)-c2y1(t)y2(t)-d2y23(t)hhhhh-b2y22(t)+d2y22(t)+a2y2(t)hhhhh+b2y2(t)+c1y2(t)-a1-a2hhhhh+0.5σ12+0.5σ22}dt+(y1(t)-1)σ1dw1(t)+(y2(t)-1)σ2dw2(t)=F(y1,y2)dt+(y1(t)-1)σ1dw1(t)+(y2(t)-1)σ2dw2(t),
where
(19)F(y1,y2)=-d1y13(t)-b1y12(t)+d1y12(t)+a1y1(t)+b1y1(t)+c2y1(t)-c1y1(t)y2(t)-c2y1(t)y2(t)-d2y23(t)-b2y22(t)+d2y22(t)+a2y2(t)+b2y2(t)+c1y2(t)-a1-a2+0.5σ12+0.5σ22.
A simple calculation shows that
(20)F(y1,y2)≤-(b1-d1)y12(t)+(a1+b1+c2)y1(t)-(b2-d2)y22(t)+(a2+b2+c1)y2(t)-a1-a2+0.5σ12+0.5σ22.
It then follows from (S1) that the upper bound of F(y1,y2), noted by K, exists. We therefore have
(21)dV(y1,y2)≤Kdt+(y1(t)-1)σ1dw1(t)+(y2(t)-1)σ2dw2(t).
Integrating both sides from 0 to τk∧T yields that
(22)∫0τk∧T‍dV(y1,y2)≤∫0τk∧T‍Kdt+∫0τk∧T‍(y1(t)-1)σ1dw1(t)+∫0τk∧T‍(y2(t)-1)σ2dw2(t),
whence taking expectations leads to
(23)EV(y1(τk∧T),y2(τk∧T))≤V(y1(0),y2(0))+KE(τk∧T)≤V(y1(0),y2(0))+KT.
Set Ωn={τn≤T} for n≥n1, and then P(Ωn)≥ɛ. Note that arbitrary ω∈Ωn, there exist some i such that yi(τn,ω) equals either n or 1/n. Then V(y1(τn,ω),y2(τn,ω)) is not less than
(24)min⁡{(n-1-ln⁡n),(1n-1+ln⁡n)}.
As a consequence,
(25)V(y1(0),y2(0))+KT≥E[1Ωn(ω)V(y1(τn,ω),y2(τn,ω))]≥ɛmin⁡{(n-1-ln⁡n),(1n-1+ln⁡n)},
where 1Ωn is the indicator function of Ωn. Let n→+∞ lead to the contradiction
(26)+∞>V(y1(0),y2(0))+KT=+∞.
So we must have τ+∞=+∞a.s. This completes the proof of Lemma 7.Lemma7 is fundamental in this paper. In the following, we will show that the pth moment of the positive solution of system (3) is upper bounded and then discuss the stochastically ultimate boundedness.Theorem 8.
Let(S1) and (S2) hold; then the positive solution (y1(t),y2(t)) of system (3) with initial value (y1(0),y2(0))∈R+2 is stochastically ultimately bounded.Proof.
From Lemma7, we can see that system (3) has a unique positive solution under assumption (S1). Assign p>1 arbitrarily; then by Itô’s formula we can show that
(27)dyip(t)=pyip-1(t)dyi(t)+12p(p-1)yip-2(dyi(t))2=pyip(t)(12ai-biyi(t)-ciyj(t)-diyi2(t)hhhhhhhhh+12(p-1)σi2)dt+pσiyipdwi(t).
Integrating and taking expectations on both sides yield that
(28)E(yip(t))-E(yip(0))=∫0t‍pE{yip(s)(p-12ai-biyi(s)-ciyj(s)hhhhhhhhhhhhhihh-diyi2(s)+p-12σi2)}ds.
We then derive that
(29)dE(yip(t))dt=pE{yip(t)(p-12ai-biyi(t)-ciyj(t)hhhhhhhhhhhi-diyi2(t)+p-12σi2)}≤paiE(yip(t))-pbiE(yip+1(t))+12p(p-1)σi2E(yip(t))=pE(yip(t))[ai+p-12σi2]-pbiE(yip+1(t)).
By Hölder inequality one has
(30)E(yip+1(t))≥(E(yip(t)))(p+1)/p,
and moreover
(31)dE(yip(t))dt≤pE(yip(t)){ai+p-12σi2-bi[E(yip(t))]1/p}.
Denote Xi(t)=E(yip(t)), Xi(0)=yip(0); then (31) can be rewritten as
(32)dXi(t)dt≤pXi(t)[ai+p-12σi2-biXi1/p(t)].
It follows from (S2) that
(33)0<biXi1/p(0)=byi(0)≤ai+12(p-1)σi2;
that is,
(34)0<Xi(0)≤[ai+(1/2)(p-1)σi2bi]p≔Hi(p).
Meanwhile, it is easy to see that
(35)dXi(t)dt|Xi(t)=Hi(p)≤0.
By the standard comparison theorem, we therefore derive that
(36)E(yip(t))≤Hi(p),
which implies that the pth moment of positive solution is upper bounded.
Let us now proceed to discuss the stochastically ultimate boundedness of system (3). Setting φi=[Hi(p)/εi]1/p, then by the Chebyshev inequality, we obtain that
(37)P{|yi(t)|>φi}<E(yip(t))φip≤Hi(p)Hi(p)/εi=εi.
This gives that
(38)limsup⁡t→+∞P{|yi(t)|>φi}<εi,i=1,2.
The proof of Theorem 8 is complete.


## 4. Global Attraction

In this section, we will establish sufficient conditions for global attraction of system (3).Lemma 9.
Let(S2) hold and let (y1(t),y2(t)) be a solution of (3) with initial value (y1(0),y2(0))∈R+2; then almost every sample path of (y1(t),y2(t)) is uniformly continuous for t≥0.Proof.
We first provey1(t). Let us consider the following integral equation:
(39)y1(t)=y1(0)+∫0t‍f1(s,y1(s),y2(s))ds+∫0t‍g1(s,y1(s),y2(s))dw1(s),
where
(40)f1(s,y1(s),y2(s))=y1(s)[a1-b1y1(s)-c1y2(s)-d1y12(s)],g1(s,y1(s),y2(s))=σ1y1(s).
Recalling (S2), (32), and the standard comparison theorem, we can know that
(41)E(yip(t))≤Hi(p),i=1,2.
Then from Lemma 4 and (41) one derives that
(42)E(|f1(s,y1(s),y2(s))|p)=E(y1p(s)|a1-b1y1(s)-c1y2(s)-d1y12(s)|p)≤0.5E(y12p(s))+0.5E(-d1y12(s)|2p|y12a1-b1y1(s)-c1y2(s)hhhhhhhhhhhhhhhhhhhhh-d1y12(s)|2p)≤0.5E(y12p(s))+0.5E(42p-1(|d1y12(s)|2p|a1|2p+|b1y1(s)|2phhhhhhhhhhhhhhh+|c1y2(s)|2p+|d1y12(s)|2p))≤0.5E(y12p(s))+0.5·42p-1|a1|2p+0.5·42p-1|b1|2pE(y12p(s))+0.5·42p-1|c1|2pE(y22p(s))+0.5·42p-1|d1|2pE(y14p(s))≤0.5H1(2p)+0.5·42p-1|a1|2p+0.5·42p-1|b1|2pH1(2p)+0.5·42p-1|c1|2pH2(2p)+0.5·42p-1|d1|2pH1(4p)≔L(p),E(|g1(s,y1(s),y2(s))|p)=E(σ1py1p(s))≤σ1pE(y1p(s))≤σ1pH1(p)≔R(p).
Meanwhile, by Lemma 3, we obtain that, for 0≤t1<t2<+∞ and p>2,
(43)E|∫t1t2‍g1(s,y1(s),y2(s))dw1(s)|p≤[p(p-1)2]p/2(t2-t1)(p-2)/2×∫t1t2E|g1(s,y1(s),y2(s))|pds.
Let
(44)t2-t1≤1,1p+1q=1,
and then from (42), (43), and Lemma 4, one can derive that
(45)E|y1(t2)-y1(t1)|p=E|∫t1t2‍f1(s,y1(s),y2(s))dshhhhhh+∫t1t2‍g1(s,y1(s),y2(s))dw1(s)|p≤2p-1E(∫t1t2‍|f1(s,y1(s),y2(s))|ds)p+2p-1E|∫t1t2‍g1(s,y1(s),y2(s))dw1(s)|p≤2p-1(∫t1t2‍1qds)p/qE(∫t1t2|f1(s,y1(s),y2(s))|pds)+2p-1[p(p-1)2]p/2(t2-t1)(p-2)/2×∫t1t2‍E|g1(s,y1(s),y2(s))|pds≤2p-1(∫t1t2‍1qds)p/q∫t1t2‍L(p)ds+2p-1[p(p-1)2]p/2(t2-t1)(p-2)/2∫t1t2‍R(p)ds≤2p-1(t2-t1)p/q+1L(p)+2p-1[p(p-1)2]p/2(t2-t1)p/2R(p)≤2p-1(t2-t1)pL(p)+2p-1[p(p-1)2]p/2(t2-t1)p/2R(p)≤2p-1(t2-t1)p/2×{(t2-t1)p/2L(p)+[p(p-1)2]p/2R(p)}≤2p-1(t2-t1)p/2{L(p)+[p(p-1)2]p/2R(p)}.
Thus, it follows from Lemma 5 that almost every sample path of y1(t) is locally but uniformly Hölder-continuous with exponent ϑ for ϑ∈(0,(p-2)/2p) and therefore almost every sample path of y1(t) is uniformly continuous on t≥0.
By a similar procedure as above,y2(t) can be proven. Thus, (y1(t),y2(t)) is uniformly continuous on t≥0. The proof of Lemma 9 is complete.Theorem 10.
Let(S1), (S2), and (S3) hold; then the unique positive solution of system (3) is globally attractive for initial data (y1(0),y2(0))∈R+2.Proof.
It follows from(S1) that, for initial data (y1(0),y2(0))∈R+2, system (3) has a unique solution (y1(t),y2(t))∈R+2 (see Lemma 7). Assume that (y1*(t),y2*(t)) is another solution of system (3) with initial values y1*(0),y2*(0)>0.
Define a Lyapunov functionV(t) as
(46)V(t)=|ln⁡y1(t)-ln⁡y1*(t)|+|ln⁡y2(t)-ln⁡y2*(t)|.
Using Itô’s formula, a calculation of the right differential D+V(t) along (3) shows that
(47)D+V(t)=sgn⁡(y1(t)-y1*(t))d(ln⁡y1(t)-ln⁡y1*(t))+sgn⁡(y2(t)-y2*(t))d(ln⁡y2(t)-ln⁡y2*(t))=sgn⁡(y1(t)-y1*(t))×{[dy1(t)y1(t)-(dy1(t))22y12(t)]-[dy1*(t)y1*(t)-(dy1*(t))22(y1*(t))2]}+sgn⁡(y2(t)-y2*(t))×{[dy2(t)y2(t)-(dy2(t))22y22(t)]-[dy2*(t)y2*(t)-(dy2*(t))22(y2*(t))2]}=sgn⁡(y1(t)-y1*(t))×{[(a1-b1y1(t)-c1y2(t)-d1y12(t)-σ122)dthhhhhhhh+σ1dw1(a1-b1y1(t)-c1y2(t)-d1y12(t)-σ122)]-[(a1-b1y1*(t)-c1y2*(t)-d1y1*(t)2-σ122)dthhhhhhhh+σ1dw1(a1-b1y1*(t)-c1y2*(t)-d1y1*(t)2-σ122)]}+sgn⁡(y2(t)-y2*(t))×{[(a2-b2y2(t)-c2y1(t)-d2y22(t)-σ222)dthhhhhhh+σ2dw2(a2-b2y2(t)-c2y1(t)-d2y22(t)-σ222)]-[(a2-b2y2*(t)-c2y1*(t)-d2y2*(t)2-σ222)dtHHHHHH+σ2dw2(a2-b2y2*(t)-c2y1*(t)-d2y2*(t)2-σ222)]}=sgn⁡(y1(t)-y1*(t))×{-b1(y1(t)-y1*(t))-c1(y2(t)-y2*(t))hhhhhh-d1(y1(t)+y1*(t))(y1(t)-y1*(t))}dt+sgn⁡(y2(t)-y2*(t))×{-b2(y2(t)-y2*(t))-c2(y1(t)-y1*(t))hhhhhh-d2(y2(t)+y2*(t))(y2(t)-y2*(t))}dt≤{-b1-d1(y1(t)+y1*(t))+c2}|y1(t)-y1*(t)|dthhihh+{-b2-d2(y2(t)+y2*(t))+c1}|y2(t)-y2*(t)|dt≤-(b1-c2)|y1(t)-y1*(t)|dt-(b2-c1)|y2(t)-y2*(t)|dt.
Integrating both sides yields that
(48)V(t)-V(0)≤-(b1-c2)∫0t‍|y1(s)-y1*(s)|ds-(b2-c1)∫0t‍|y2(s)-y2*(s)|ds;
that is,
(49)V(t)+(b1-c2)∫0t‍|y1(t)-y1*(t)|ds+(b2-c1)∫0t|y2(t)-y2*(t)|ds≤V(0)<∞.
In view of (S3) and V(t)≥0, then it follows from (49) that
(50)|y1(t)-y1*(t)|∈L1[0,∞),|y2(t)-y2*(t)|∈L1[0,∞).
So recalling Lemmas 9 and 6, we can show that
(51)lim⁡t→+∞|y1(t)-y1*(t)|=0,lim⁡t→+∞|y2(t)-y2*(t)|=0a.s.
This completes the proof of Theorem 10.


## 5. Numerical Simulations

In this paper, we derived the sufficient conditions for the existence, uniqueness, stochastically ultimate boundness, and global attraction of positive solutions of system (3). In order to illustrate the above theoretical results, we will perform a specific example. Motivated by the Milsten method mentioned in Higham [25], we can obtain the following discrete version of system (3):
(52)y1(k+1)=y1(k)+y1(k)[d1y12a1-b1y1(k)-c1y2(k)hhhhhhhhhhhh-d1y12(k)]Δt+σ1y1(k)Δtξ1(k)+0.5σ12y1(k)(ξ12(k)-1)Δt,y2(k+1)=y2(k)+y2(k)[d2y22a2-b2y2(k)-c2y1(k)hhhhhhhhhhhh-d2y22(k)]Δt+σ2y2(k)Δtξ2(k)+0.5σ22y2(k)(ξ22(k)-1)Δt,
where ξ1(k) and ξ2(k) are Gaussian random variables that follow N(0,1). Let a1=0.6, b1=0.5, c1=0.2, d1=0.3, and σ1=0.1; a2=0.7, b2=0.4, c2=0.3, d2=0.2, and σ2=0.1; Δt=0.001; and the initial value (y1(0),y2(0))=(0.3,0.2), (y1*(0),y2*(0))=(0.6,0.5). After a calculation, we can see that the assumptions of Theorems 8 and 10 hold. Figures 1 and 2 show that the positive solution of system (52) is stochastically ultimately bounded and globally attractive.Figure 1
The sample path of(y1(t);y1*(t)) in the same coordinate system.Figure 2
The sample path of(y2(t);y2*(t)) in the same coordinate system.It follows from Theorem8 that a preliminary result on the stochastically ultimate boundness of system (3) is obtained. We would like to mention here that an interesting but challenging problem associated with the investigation of system (3) should be the stochastic permanence; we leave this for future work.



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