# Asymptotic Behavior of Positive Solutions of a Competitive System Subject to Environmental Noise

Authors: Huili Xiang; Zhuang Fang; Zuxiong Li; Zhijun Liu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

A competitive system subject to environmental noise is established. By using the theory of stochastic differential equations
and Lyapunov function, sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solution of the above system
are established, respectively. An example together with its corresponding numerical simulations is presented to confirm our analytical results.


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## Body

## 1. Introduction

Mathematical modelling plays an important role in the mathematical ecology. In the past several years, ecological models based on determinate systems emerged in large numbers (see [1–9]). While the disturbance of environmental noise is unavoidable in the real world, more and more researchers start to pay attention to the study on nonlinear dynamic systems with environmental noise and many valuable results have been obtained (see [10–24]).In [25], Gopalsamy introduced the following competitive system:
(1)dx1t=x1tr1-a1x1t-c2x2t1+x2tdt,dx2t=x2tr2-a2x2t-c1x1t1+x1tdt,
where xi(t) may represent the densities of species. The coefficients ri, ai, bi, and ci are all positive constants. In the absence of interspecific interactions, each species is governed by the logistic equation; however, in the presence of interspecific interactions, each species retains the average growth rate of the other. In this contribution, we consider the influence of environmental noise and obtain the following form:
(2)dx1t=x1tr1-a1x1t-c2x2t1+x2tdt+σ1x1tdw1t,dx2t=x2tr2-a2x2t-c1x1t1+x1tdt+σ2x2tdw2t,
where xi(0)>0, dwi(t) is independent white noise with wi(0)=0, t≥0, and σi2 represents the intensity of the noise, i=1,2. wi(t) is standard Brownian motion defined on the complete probability space (Ω,F,{Ft}t≥0,P) with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets).In this paper, we focus on the asymptotic behavior of positive solution of system (2). To the best of our knowledge, there are few published papers concerning system (2). The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. The existence, uniqueness, and stochastic boundedness of positive solution of system (2) are discussed in Section 3. The global attraction of system (2) is studied in Section 4. As an application of our main results, we present an example and its numerical simulations to support our theoretical results in Section 5.


## 2. Preliminaries

In this section, we introduce some definitions and lemmas which are useful for establishing our main results.Definition 1.
The solution(x1(t),x2(t)) of system (2) is stochastically bounded if, for any εi∈(0,1), there exist positive constants Hi=H(εi) such that
(3)lim⁡t→+∞sup⁡Pxit>Hi<εi,i=1,2.Definition 2.
Let(x1(t),x2(t)) be a positive solution of system (2). If another positive solution (x1*(t),x2*(t)) of system (2) satisfies
(4)lim⁡t→+∞Ex1t,x2t-x1*t,x2*t=0,
then (x1(t),x2(t)) is global attractive.Definition 3.
Let(x1(t),x2(t)) be a positive solution of system (2). The solution x(t)=(x1(t),x2(t)) to system (2) is said to be exponentially extinct with probability one if
(5)lim⁡sup⁡t→∞ln⁡xi(t)t<0,a.s.i=1,2.Lemma 4 (Cp inequality).
Suppose thata1,a2,…,an are all real numbers; then for any positive real number p we have
(6)a1+a2+⋯+anp≤Cpa1p+a2p+⋯+anp,
where
(7)Cp=1,0<p≤1,np-1,p>1.Lemma 5 (see [26]).
Letf(t) be a nonnegative integrable and uniformly continuous function defined on [0,+∞) such that f(t) is integrable and uniformly continuous on [0,+∞). Then lim⁡t→+∞f(t)=0.Lemma 6 (see [27, 28]).
Suppose that a stochastic processX(t) on t≥0 satisfies the condition
(8)EXt-Xs|α≤ct-s|1+β,0≤s,t<+∞,
for some positive constants α, β, and c. Then there exists a continuous modification X~(t) of X(t), which has the property that, for every γ∈(0,β/α), there is a positive random variable h(ω) such that
(9)Pω:sup⁡0<t-s<hω,0≤s,t<+∞X~t,ω-X~s,ωt-sγ≥21-2-γ=1.
In other words, almost every sample path of X~ is locally but uniformlyHölder continuous with exponent γ.


## 3. Existence, Uniqueness, Stochastic Boundedness, and Extinction

We first present the existence and uniqueness of positive solution of system (2).Theorem 7.
System (2) has a unique positive solution, say (x1(t),x2(t)), on t≥0. Furthermore, the solution will remain in R+2={(x1,x2)∈R2:xi>0,i=1,2} with probability one.Proof.
The proof of this lemma is rather standard. It is obvious that the coefficients of system (2) are local Lipschitz continuous. Then, for any initial value (x1(0),x2(0)) with xi(0)>0, there exists a unique local solution (x1(t),x2(t)),t∈[0,τ*), where τ* is the explosion time (see [10, 19]). Therefore, to prove that the local solution is also global, we only need to show that τ*=+∞ a.s. Let n0>0 be sufficiently large so that every component of (x1(0),x2(0)) lies in [1/n0,n0]. For each integer n≥n0, we define the stopping time as follows:
(10)τn=inf⁡t∈0,τ*:x1t∉1n,norx2t∉1n,n.
Here we set inf⁡∅=+∞ (∅ denotes the empty set). Obviously, τn is increasing as n→+∞. Denote τ+∞=lim⁡n→+∞τn, whence τ+∞≤τ* a.s. We need to show that τ+∞=+∞ a.s. Otherwise, there exist constants T>0 and ε∈(0,1) such that P{τ+∞≤T}>ε. Then, by denoting Ωn=τn≤T, there exists an integer n1≥n0 such that for all n≥n1,
(11)PΩn≥ε.
We now define a C2-function V as
(12)Vx1,x2=x1-ln⁡x1-1+x2-ln⁡x2-1,
where x1>0 and x2>0. It is obvious that V(x1,x2) is nonnegative. ByItô’s formula, one has
(13)dVx1t,x2t=r1+a1x1-c2x2t1+x2t1-x1thhhih-a1x12t+12σ12-r1hhhih+r2+a2x2-c1x1t1+x1t1-x2thhhh-a2x22(t)+12σ22-r2r1+a1x1-c2x2t1+x2t1-x1tdt+1-x1t-1σ1x1tdw1t+1-x2t-1σ2x2tdw2t=fx1t,x2tdt+1-x1t-1σ1x1tdw1t+1-x2t-1σ2x2tdw2t,
where
(14)fx1,x2=r1+a1x1+r2+a2x2-c2x21+x21-x1-c1x11+x11-x2-a1x12-a2x22+12σ12+12σ22-r1-r2.
A calculation can show that f(x1,x2) is upper bounded, denoted by H. Thus (13) can be rewritten as
(15)dVx1t,x2t≤Hdt+1-x1t-1σ1x1tdw1t+1-x2t-1σ2x2tdw2t.
Integrating both sides from 0 to τn∧T, we acquire that
(16)∫0τn∧T‍Vx1t,x2t≤∫0τn∧T‍1-x1t-1σ1x1tdw1thhhhhhhh+1-x2t-1σ2x2tdw2t+∫0τn∧T‍Hdt.
As a consequence, one has
(17)Vx1τn∧T,x2τn∧T-Vx10,x20≤Hτn∧T+∫0τn∧T‍1-x1t-1σ1x1tdw1thhhhhhhhhhhhhhhhhh.+1-x2t-1σ2x2tdw2t.
Since τn∧T>0, taking expectations one shows that
(18)EVx1τn∧T,x2τn∧T≤Vx10,x20+HEτn∧T.
Thus
(19)EVx1τn∧T,x2τn∧T≤Vx10,x20+HT.
On the other hand, for every ω∈Ωn, either x1(τn,ω) or x2(τn,ω) equals to either n or 1/n. Then V(x1(τn,ω),x2(τn,ω)) is not less than either n-1-ln⁡n or 1/n-1+ln⁡n. Consequently, from (19) we have
(20)Vx10,x20+HT≥E1ΩnVx1τn,ω,x2τn,ω≥εn-1-ln⁡n∧1n-1+ln⁡n,
where 1Ωn is the indicator function of Ωn. Let n→+∞, one can show the following contradiction:
(21)+∞≤Vx10,x20+HT<+∞.
Hence, τ+∞=+∞ a.s. and there exists a unique positive solution (x1(t),x2(t)) of system (2) on t≥0. This completes the proof.Next, we investigate the stochastic boundedness of the positive solutions of system (2). To this end, we first give the following Lemma 8.Lemma 8.
Ifxi(0)<ri/ai, i=1,2, then for any real number p≥1 the solution (x1(t),x2(t)) of system (2) satisfies
(22)Exipt≤Kip,
where
(23)Ki(p)=ri+p-1/2σi2aip,i=1,2.Proof.
ByItô’s formula, one can show that
(24)dxipt=pxip-1tdxit+pp-12xip-1tdxitdxit=pxip-1txitri-aixit-cjxjt1+xjthhi+pp-12σi2xiptdt+pσixiptdwit.
Integrating from 0 to t, we have
(25)xipt-xip0=∫0t‍pxipsri-aixis-cjxjt1+xjthhhhhhhhhhhhhi+p-12σi2ds+∫0t‍pσixiptdwit.
Taking expectations, we obtain that
(26)Exipt-Exip0=∫0t‍pExipsri-aixis-cjxjs1+xjshhhhhhhhhhhhhhii+p-12σi2ds.
So
(27)dExiptdt=pExiptri-aixit-cjxjt1+xjt+p-12σi2≤priExipt-paiExip+1t+p-12σi2Exipt=pExiptri+p-12σi2-aiExipt1/p.
Let
(28)Zit=Exipt;
we have
(29)dZitdt≤pZitri+p-12σi2-aiZit1/p.
As a consequence
(30)dZitZit≤pri+p-12σi2-aiZit1/pdt.
Noting that xi(0)<ri+1/2p-1σi2/ai, i=1,2, we have
(31)0<aiZi01/p=aixi0<ri+12p-1σi2.
Furthermore, using the standard comparison principle, one can show that
(32)Exitp1/p=Zit1/p≤ri+1/2(p-1)σi2ai.
Then we can obtain that
(33)Exipt≤Kip,
where
(34)Kip=ri+p-1/2σi2aip.
This completes the proof.Finally, we discuss the stochastic boundedness of the positive solutions of system (2).Theorem 9.
Ifxi(0)<ri/ai, i=1,2, then the solution (x1(t),x2(t)) of system (2) is stochastically bounded.Proof.
On one hand, for any positive numberδi, i=1,2, one derives that
(35)Pxit-Exit≥δi≥Pxit≥Exit+δi.
On the other hand, by theChebyshev inequality and Lemma 8, we obtain that
(36)Pxit-Exit>δi≤Var⁡xitδi2≤Exi2tδi2≤ri+1/2σi22ai2δi2.
Let δi=ri+1/2σi2/εiai. Then we have
(37)Pxi(t)≥Exit+ri+1/2σi2εiai≤εi.
It follows from Lemma 8 that
(38)Exit≤riai,
which, together with (37), leads to
(39)Pxit≥riai+ri+1/2σi2εiai≤Pxit≥Exit+ri+1/2σi2εiai≤εi.
Let Hi(εi)=εi+1ri+1/2σi2/aiεi, and noting that xi(t)>0, one shows that
(40)Pxit≥Hiεi<εi.
Therefore,
(41)lim⁡t→∞sup⁡Pxit≥Hiεi<εi,
which implies that the solution of system (2) is stochastically bounded. The proof is complete.Theorem 10.
Suppose that all coefficients of system (2) are positive and ri<σi2/2, i=1,2. Then the solution (x1(t),x2(t)) of system (2) is exponentially extinct with probability one.Proof.
Define, respectively, Lyapunov functionsln⁡x1(t) and ln⁡x2(t). Then the following conclusions can be obtained byItô’s formula
(42)dln⁡x1t=r1-12σ12-a1x1t-c2x2t1+x2tdt+σ1dw1t,dln⁡x2t=r2-12σ22-a2x1t-c1x1t1+x1tdt+σ2dw2t.
Integrating from 0 to t, one concludes that
(43)ln⁡x1t≤ln⁡x10+r1-12σ12t+∫0t‍σ1dw1s,ln⁡x2t≤ln⁡x20+r2-12σ22t+∫0t‍σ2dw2s.
Dividing t on both sides of (43), sending t→∞, and employing the strong law of large numbers for local martingales, one acquires that
(44)lim⁡sup⁡t→∞ln⁡xi(t)t<0,a.s.i=1,2.
This completes the proof.


## 4. Global Attraction

In this section, we first introduce Lemma11 before we show the global attraction of system (2).Lemma 11.
Ifci≤ai, xi(0)<ri/ai, i=1,2, then almost every sample path of the solution (x1(t),x2(t)) of system (2) is uniformly continuous on t≥0.Proof.
It follows from system (2) that
(45)x1t=x10+∫0t‍g1s,x1s,x2sds+∫0t‍h1s,x1s,x2sdw1s,
where
(46)g1s,x1s,x2s=x1sr1-a1x1s-c2x2s1+x2s,h1s,x1s,x2s=σ1x1s.
Applying Lemmas 4 and 8, for any p>1, one derives that
(47)Eg1s,x1s,x2sp=Ex1psr1-a1x1s-c2x2s1+x2sp≤12Ex12ps+12Er1-a1x1s-c2x2s1+x2s2p≤12Ex12ps+12E32p-1r12p+32p-1a1x1s2phhhhhhhhhhhhhhhhhhi+32p-1c2x2s2p=12Ex12ps+1232p-1r12p+1232p-1a12pEx12ps+1232p-1|c2|2p-1Ex22ps≤12K12p+1232p-1r12p+1232p-1a12pK1(2p)+1232p-1c22p-1K22p≜L1p,E(h1s,x1s,x2sp)=Eσ1px1ps≤σ1pEx1ps≜M1p.
Without loss of generality, we assume that p>2. Using the moment inequality (see [10]) to stochastic integral (45), we can obtain that
(48)E∫t1t2‍h1s,x1s,x2sdw1sp⩽pp-12p/2t2-t1p/2×∫t1t2‍Eh1s,x1s,x2spds,
where 0⩽t1<t2<+∞ and p>2. We further let
(49)t2-t1<1,1p+1q=1;
then by (47), (48), and Lemma 4, one yields that
(50)Ex1t2-x1t1p=E∫t1t2‍g1s,x1s,x2sdshhhhhhi+∫t1t2‍h1s,x1s,x2sdw1sp≤E2p-1∫t1t2‍g1s,x1s,x2sdsp∫t1t2‍h1s,x1s,x2sdw1sphhhhhi+2p-1∫t1t2‍h1s,x1s,x2sdw1sp≤2p-1E∫t1t2‍g1s,x1s,x2spds+2p-1E∫t1t2‍h1s,x1s,x2spdw1s≤2p-1∫t1t2‍1qdsp/qE∫t1t2‍g1s,x1s,x2spds+2p-1pp-12p/2t2-t1p-2/2×∫t1t2‍Eh1s,x1s,x2spds≤2p-1∫t1t2‍1qdsp/qE∫t1t2‍L1pds+2p-1pp-12p/2t2-t1p-2/2∫t1t2‍M1pds=2p-1L1pt2-t1p/2+2p-1pp-12p/2t2-t1p-2/2M1p≤2p-1t2-t1p-2/2L1p+pp-12p/2M1p.
It follows from Lemma 6 that almost every sample path of x1(t) is uniformly continuous on t≥0. Similarly, we can show that almost every sample path of x2(t) is uniformly continuous on t≥0. Therefore, (x1(t),x2(t)) is uniformly continuous on t≥0, a.s. This completes the proof.We can now present the result on global attraction of system (2).Theorem 12.
Ifci≤ai, xi(0)<ri/ai, i=1,2, then system (2) has a unique global attractive positive solution, denoted by (x1(t),x2(t)), on t≥0.Proof.
It follows from Theorem7 that system (2) has a unique positive solution (x1(t),x2(t)). Assume that (x1*(t),x2*(t)) is another positive solution of system (2). Consider a Lyapunov function V(t) defined by
(51)Vt=ln⁡x1t-ln⁡x1*t+ln⁡x2t-ln⁡x2*t,t≥0.
ApplyingItô’s formula, a calculation of the right differential D+V(t) of V(t) along the solution, one yields that
(52)D+Vt=sgn⁡x1t-x1*tdx1tx1t-dx1t22x12thhhhhhhhhhhhhhhhhhhi-dx1*tx1*t-dx1*t22x1*t2+sgn⁡x2t-x2*tdx2tx2t-dx2t22x22thhhhhhhhhhhhhhhhhhhih-dx2*tx2*t-dx2*t22x2*t2=sgn⁡(x1(t)-x1*(t))×r1-a1x1t-c2x2t1+x2t-σ122dthhhhhihii+σ1dw1tr1-a1x1t-c2x2t1+x2t-σ122hhhhhhi-r1-a1x1*t-c2x2*t1+x2*t-σ122dthhhhhhhhh+σ1dw1tr1-a1x1t-c2x2t1+x2t-σ122r1-a1x1t-c2x2t1+x2t-σ122dt+sgn⁡(x2(t)-x2*(t))×r2-a2x2t-c1x1t1+x1t-σ222dthhhhhhhi+σ2dw2tr2-a2x2t-c1x1t1+x1t-σ222hhhhhhh-r2-a2x2*t-c1x1*t1+x1*t-σ222dthhhhhhhhhi+σ2dw2tr2-a2x2*t-c1x1*t1+x1*t-σ222=sgn⁡(x1(t)-x1*(t))×x2*(t)1+x2*(t)-x2(t)1+x2(t)a1x1*t-x1thhhhhh+c2x2*t1+x2*t-x2t1+x2tdt+sgn⁡x2t-x2*t×x1*t1+x1*t-x1t1+x1ta2x2*t-x2thhhhhh+c1x1*t1+x1*t-x1t1+x1tdt.
Integrating from 0 to t and taking expectations one can show that
(53)EVt-V0=E∫0t‍sgn⁡x1s-x1*shhhhhhhh×x2*s1+x2*s-x2s1+x2sa1x1*s-x1shhhhhhhhhh+c2x2*s1+x2*s-x2s1+x2sdshhhhhh+∫0t‍sgn⁡x2s-x2*shhhhhhhhih×x1*s1+x1*s-x1s1+x1sa2x2*s-x2shhhhhhhhhhhi+c1x1*s1+x1*s-x1s1+x1sds.
Thus
(54)dE(V(t))dt=Ex1*t1+x1*t-x1t1+x1tsgn⁡x1t-x1*thhhhhh×x2*t1+x2*t-x2t1+x2ta1x1*t-x1thhhhhhhhi+c2x2*t1+x2*t-x2t1+x2thhhhhii+sgn⁡x2t-x2*thhhhhii×x1*t1+x1*t-x1t1+x1ta2(x2*(t)-x2(t))hhhhhhhiii+c1x1*t1+x1*t-x1t1+x1t≤-a1Ex1t-x1*t+c2Ex2t-x2*t-a2E(x2t-x2*t)+c1Ex1t-x1*t=c1-a1E(x1t-x1*t)+c2-a2E(x2t-x2*t),
and hence integrating from 0 to t one derives that
(55)EVt≤∫0t‍c1-a1E(x1s-x1*s)ds+∫0t‍c2-a2Ex2s-x2*sds+V0,
which implies that
(56)Ex1t,x2t-x1*t,x2*t=Ex1t-x1*t2+x2t-x2*t21/2≤Ex1t-x1*t+Ex2t-x2*t∈L10,+∞.
So
(57)Ex1t,x2t-x1*t,x2*t=Ex1t-x1*t2+x2t-x2*t21/2≤Ex1t-x1*t+Ex2t-x2*t∈L10,∞,
which, together with Lemmas 5 and 11, leads to
(58)lim⁡t→∞Ex1t,x2t-x1*t,x2*t=0
and hence (x1(t),x2(t)) is global attractive on t≥0.


## 5. An Example

In this section, we first give an example to verify the feasibilities of Theorems9 and 12. Using the Milsten method mentioned in Higham [29], we can derive the following discrete version of system (2):
(59)x1k+1-x1k=x1kr1-a1x1k-c2x2k1+x2kΔt+σ1x1(k)ΔtN1(k)+12σ12x1kN12k-1Δt,x2k+1-x2k=x2kr2-a2x2k-c1x1k1+x1kΔt+σ2x2kΔtN2k+12σ22x2kN22k-1Δt,
where N1(k) and N2(k) are Gaussian random variables which follow N(0,1). Let us choose r1=0.5, a1=0.4, c1=0.1, σ1=0.1, r2=0.6, a2=0.5, c2=0.3, σ2=0.1, Δt=0.001, and (x1(0),x2(0))=(0.8,1.1), (x1*(0),x2*(0))=(0.3,0.4). A calculation shows that the conditions of Theorems 9 and 12 are satisfied. Figures 1 and 2 show that the positive solution of system (59) is stochastically bounded and global attractive on t≥0.Figure 1
The sample paths ofx1(t) and x1*(t).Figure 2
The sample paths ofx2(t) and x2*(t).Recalling the whole paper, we have derived sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solutions of system (2). However, there are still some limitations in our work which need to be improved. We only especially consider the white noise which is an idealized situation. In fact, the effect of colorful noise on system (2) is more general in line with the actual situation, and we leave it for our future work.



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