# Asymptotic Behavior of a Stochastic Two-Species Competition Model under the Effect of Disease

Authors: Rong Liu; Guirong Liu
Journal: Complexity
 (2018)

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## Abstract

This paper is concerned with a stochastic two-species competition model under the effect of disease. It is assumed that one of the competing populations is vulnerable to an infections disease. By the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model. Then, the asymptotic pathwise behavior of the model is given via the exponential martingale inequality and Borel-Cantelli lemma. Next, we find a new method to prove the boundedness of thepth moment of the global positive solution. Then, sufficient conditions for extinction and persistence in mean are obtained. Furthermore, by constructing a suitable Lyapunov function, we investigate the asymptotic behavior of the stochastic model around the interior equilibrium of the deterministic model. At last, some numerical simulations are introduced to justify the analytical results. The results in this paper extend the previous related results.


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## Body

## 1. Introduction

Populations that compete for common resources are well known among ecologists. They are classically modeled by observing their interactions that hinder the growth of both populations and are thus described by negative bilinear terms in all the relevant differential equations. The classic two-species Lotka-Volterra competition model takes the form(1)dx1tdt=x1tr1-a11xt-a12x2t,dx2tdt=x2tr2-a21xt-a22x2t.There is an extensive literature concerned with model (1) and related deterministic models (see [1–3] and the references therein) and we here do not mention them in detail.As mentioned in [4], another major problem in today’s modern society is the spread of infectious diseases. A detailed account of modeling and the study of epidemic diseases can be found in the literature [5, 6]. The population biology of infectious diseases has also been presented in [7]. In [8], the authors studied the dynamics of two competing species when one of them is subject to a disease. In [4], the authors assumed that x(t) and y(t) are competing for the same resource and assumed that the disease spreads only in one of the competing species, denoted by y(t). They specified the healthy individuals x(t), the healthy individuals y1(t), and the infected individuals of the latter species denoted by y2(t). Moreover, they studied the following two-competing-species model under the effect of infectious disease(2)dxt=xta-bxt-cy1t-ηy2tdt,dy1t=y1td-e^xt-fy1t+y2t-δy2tdt,dy2t=y2tδy1t-gxt-fy1t+y2tdtwith initial value x(0)=x0, y1(0)=y10, and y2(0)=y20. The parameters in (2) are defined as follows: a and d are the intrinsic growth rates of the populations x(t) and y1(t), respectively. c is the loss rate in population x(t) due to the competitor y1(t) and η is the loss rate in population x(t) due to the competitor y2(t). e^ is the loss rate in population y1(t) due to the competitor x(t) and g is the loss rate in population y2(t) due to the competitor x(t). b, f are intraspecific coefficients of x(t), y1(t), and y2(t). δ is the transmission rate of the infection. Denote R+3={(x,y.z)∈R3:x>0,y>0,z>0}. From [4], we know that all solutions of model (2) will lie in the region(3)B1=x,y1,y2∈R+3:0≤x≤ab,0≤y1≤df,0≤y2≤δdf2as t→∞, for all positive initial values (x0,y10,y20)∈R+3. Moreover, the interior equilibrium E^=(x^,y^1,y^2) of model (2) is feasible when δ>f, g(f+δ)>e^f, a>(ca1+ηa3), and (δ-f)(gf+gδ-e^f)>gδ2. Here(4)x^=a-ca1-ηa3b+ca2+ηa4,y^1=a1+a2x^,y^2=a3+a4x^,where a1=df/δ2, a2=gf+gδ-e^f/δ2, a3=(δ-f)/δ2, a4=(δ-f)(gf+gδ-e^f)/fδ2-g/f.However, the population dynamics in the real world are often disturbed by some uncertain factors while the stochastic population model is more in line with actual situation. During the past decades, a great deal of attention has been paid to the study of stochastic biological model (see [9–20]). In [12], the authors discussed a two-species stochastic nonautonomous Lotka-Volterra competition model. Some sufficient conditions on the boundedness, extinction, nonpersistence in the mean, and weak persistence of solutions are established. In [13], the authors investigated the optimal harvesting problem for a stochastic delay competitive Lotka-Volterra model with Le´vy jumps. In [14], the authors studied the permanence and asymptotical behavior of stochastic prey-predator model with Markovian switching. In [15], the authors investigated the stochastic competitive models in a polluted environment. In [16], the authors explored an impulsive stochastic infected predator-prey system with Le´vy jumps and delays. In [20], the authors considered a stochastic susceptible-infective epidemic model in a polluted environment, which incorporates both environmental fluctuations and pollution.Parameter perturbation induced by white noise is an important and common form to describe the effect of stochasticity. In this paper, we perturb the intrinsic growth ratesa and d in model (2) with white noise; that is,(5)a→a+σ1w˙1t,d→d+σ2w˙2t,where w1(t), w2(t) are mutually independent Brownian motions and σ12, σ22 denote the intensities of the white noise. Then corresponding to the deterministic model (2), the stochastic model takes the following form(6)dxt=xta-bxt-cy1t-ηy2tdt+σ1xtdw1t,dy1t=y1td-e^xt-fy1t+y2t-δy2tdt+σ2y1tdw2t,dy2t=y2tδy1t-gxt-fy1t+y2tdtwith initial value x(0)=x0, y1(0)=y10, and y2(0)=y20. Here w={w1(t),w2(t),t≥0} represents the two-dimensional standard Brownian motion defined on a compete probability space (Ω,F,P) with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). All meanings of the parameters are exact to or similar to those for (2).The remaining part of this paper is organized as follows. The proof of the existence and the uniqueness for the global positive solution of model (6) for any positive initial value is given in Section 2. An important asymptotic property of the model is obtained by using the exponential martingale inequality and Borel-Cantelli lemma in Section 3. In Section 4, the stochastically ultimate boundedness of the positive solution is examined. In Section 5, sufficient conditions for extinction and persistence in mean of model (6) are established. In Section 6, by constructing a suitable Lyapunov function, we investigate the asymptotic behaviors of the stochastic model (6) around the interior equilibrium of the deterministic model. Numerical simulations under certain parameters are presented to illustrate our main results in Section 7. Finally, a few comments will conclude the paper.


## 2. Existence and Uniqueness of Positive Solution

Sincex(t), y1(t), and y2(t) in model (6) are the size of the populations at time t, we are interested only in the positive solutions of model (6). However, the coefficients of (6) do not satisfy the linear growth condition; the classical theory of stochastic differential equations is not applicable directly. Next, by using comparison theorem of stochastic differential equations, we show that model (6) has unique positive global solution with positive initial value.Theorem 1.
For any initial value(x0,y10,y20)∈R+3, model (6) has unique global solution (x(t),y1(t),y2(t)) defined on t≥0 and the solution will remain in R+3 with probability one.Proof.
For(x0,y10,y20)∈R+3, we first consider the following stochastic differential system(7)dut=a-σ122-beut-cev1t-ηev2tdt+σ1dw1t,dv1t=d-σ222-e^eut-fev1t+ev2t-δev2tdt+σ2dw2t,dv2t=δev1t-geut-fev1t-fev2tdtwith the initial value u(0)=ln⁡x0, v1(0)=ln⁡y10, v2(0)=ln⁡y20. Since the coefficients of (7) obey the local Lipschitz condition, and, hence, for given initial value (u(0),v1(0),v2(0)), (7) has a unique maximal local solution (u(t),v1(t),v2(t)) on [0,τe), where τe is the explosion time. Therefore, by using Ito^ formula, it follows that (x(t),y1(t),y2(t))=(eu(t),ev1(t),ev2(t)) is the unique positive local solution of (6) with the initial value (x0,y10,y20).
Now, by using the comparison theorem of stochastic differential equations, we show that(u(t),v1(t),v2(t)) is a global solution to system (7); that is, τe=∞ a.s. Let us consider the following two stochastic differential systems(8)dΦt=Φta-bΦtdt+σ1Φtdw1t,dΨ1t=Ψ1td-fΨ1tdt+σ2Ψ1tdw2t,dΨ2t=Ψ2tδΨ1t-fΨ2tdtwith initial value (Φ(0),Ψ1(0),Ψ2(0))=(x0,y10,y20)∈R+3 and(9)dϕt=ϕta-bϕt-cΨ1t-ηΨ2tdt+σ1ϕtdw1t,dψ1t=ψ1td-e^Φt-f+δΨ2t-fψ1tdt+σ2ψ1tdw2t,dψ2t=ψ2t-gΦt-fΨ1t-fψ2tdtwith initial value (ϕ(0),ψ1(0),ψ2(0))=(x0,y10,y20)∈R+3.
Thanks to [21, 22], systems (8) and (9) can be explicitly solved as follows:(10)Φt=exp⁡a-σ12/2t+σ1w1t1/x0+b∫0texpa-σ12/2s+σ1w1sds,Ψ1t=exp⁡d-σ22/2t+σ2w2t1/y10+f∫0texp⁡d-σ22/2s+σ2w2sds,Ψ2t=exp⁡δ∫0tΨ1sds1/y20+f∫0texp⁡δ∫0sΨ1rdrdsand(11)ϕt=exp⁡a-σ12/2t-∫0tcΨ1r+ηΨ2rdr+σ1w1t1/x0+b∫0texp⁡a-σ12/2s-∫0scΨ1r+ηΨ2rdr+σ1w1sds,ψ1t=exp⁡d-σ22/2t-∫0te^Φr+f+δΨ2rdr+σ2w2t1/y10+f∫0texp⁡d-σ22/2s-∫0se^Φr+f+δΨ2rdr+σ2w2sds,ψ2t=exp⁡-∫0tgΦr+fΨ1rdr1/y20+f∫0texp⁡-∫0sgΦr+fΨ1rdrds.
Note that the local solution(x(t),y1(t),y2(t)) is positive for t∈[0,τe). Then, by the comparison theorem of stochastic differential equations (see [23]) we have(12)0<ϕt≤xt≤Φta.s.,0<ψit≤yit≤Ψita.s.for t∈[0,τe), i=1,2. Thus,(13)ln⁡ϕt≤ut≤ln⁡Φta.s.,ln⁡ψit≤vit≤ln⁡Ψita.s.for t∈[0,τe), i=1,2.
Note thatln⁡ϕ(t), ln⁡Φ(t), ln⁡ψi(t), and ln⁡Ψi(t) exist on [0,∞). Hence τe=∞. Thus, for any initial value (u(0),v1(0),v2(0))=(ln⁡x0,ln⁡y10,ln⁡y20)∈R3, system (7) has a unique global solution (u(t),v1(t),v2(t)) on [0,∞) a.s. Therefore, for any initial value (x0,y10,y20)∈R+3, model (6) has a unique global positive solution (x(t),y1(t),y2(t))=(eu(t),ev1(t),ev2(t)) on [0,∞) a.s. The proof is therefore complete.


## 3. Asymptotic Property

In this section, by using the exponential martingale inequality and Borel-Cantelli lemma, we investigate an important asymptotic property of positive solutions of model (6).Theorem 2.
Let(x(t),y1(t),y2(t)) be the solution of model (6) with initial value (x0,y10,y20)∈R+3. Then(14)lim⁡supt→∞⁡ln⁡xtln⁡t≤1,lim⁡supt→∞⁡ln⁡yitln⁡t≤1a.s.i=1,2.Proof.
For populationx, applying Ito^’s formula to etln⁡x leads to(15)etln⁡xt=ln⁡x0+∫0tesln⁡xsds+∫0tesa-bxs-cy1s-ηy2s-σ122ds+M1t,where M1(t)=∫0tσ1esdw1(s) is a continuous local martingale vanishing at time 0 and the quadratic variation of M1(t) is(16)M1,M1t=∫0tσ12e2sds.Let n=1,2,⋯, γ>0, θ>1, and 0<ε<1. Choose T=nγ, α=εe-nγ, and β=(θenγln⁡n)/ε. By the exponential martingale inequality (see Theorem 1.7.4 in [24]), we deduce that(17)Psup0≤t≤T⁡M1t-α2M1,M1t>β≤e-αβ=1nθ.Since ∑n=0∞1/nθ<∞ for θ>1, the Borel-Cantelli lemma (see Lemma 1.2.4 in [24]) implies that there exists a set Ω0∈F with P(Ω0)=1 and an integer-valued random variable n0=n0(ω) such that for every ω∈Ω0(18)M1t≤θenγln⁡nε+εe-nγ2M1,M1tholds for all 0≤t≤nγ, n≥n0. Substituting the above inequality into (15), we see that(19)etln⁡xt≤ln⁡x0+∫0tesln⁡xs+a-bxsds-12∫0tσ12esds+εe-nγ2∫0tσ12e2sds+θenγln⁡nεholds for all 0≤t≤nγ, n≥n0. Note that, for 0≤s≤t≤nγ,(20)12εe-nγσ12e2s-12σ12es=12σ12esεes-nγ-1≤12σ12esε-1<0.Therefore, for all 0≤t≤nγ, n≥n0, it follows from (19) that(21)etln⁡xt≤ln⁡x0+∫0tesln⁡xs+a-bxsds+θenγln⁡nε.Let us consider function q(x)=ln⁡x+a-bx on (0,∞). It is easy to show that q has maximum value for x=1/b>0 and maximum value of function q is qmax=ln⁡1/b+a-1. Denote K1≐(ln⁡1/b+a-1)∨1. Then(22)etln⁡xt≤ln⁡x0+K1et+θenγln⁡nεholds for all 0≤t≤nγ, n≥n0. Therefore, for all 0≤(n-1)γ≤t≤nγ, n≥n0, we have(23)ln⁡xtln⁡t≤ln⁡x0etln⁡t+K1ln⁡t+θeγln⁡nεln⁡n-1γ.Letting n→∞ (and so t→∞), we obtain(24)lim⁡supt→∞⁡ln⁡xtln⁡t≤θeγε.Moreover, letting θ↓1, γ↓0, and ε↑1, we can get(25)lim⁡supt→∞⁡ln⁡xtln⁡t≤1a.s.
For populationy, denote y(t)=y1(t)+y2(t) and y0=y10+y20. It follows from (6) that(26)dyt=dy1t-e^xty1t-gxty2t-fy2tdt+σ2y1tdw2t.Applying Ito^’s formula to etln⁡y leads to(27)etln⁡yt≤ln⁡y0+∫0tesln⁡ys+d-fysds-12∫0tσ22esy12sy2sds+M2t,where M2(t)=∫0tσ2esy1(s)/y(s)dw2(s) is a continuous local martingale with initial value M2(0)=0 and the quadratic variation of M2(t) is(28)M2,M2t=∫0tσ22e2sy12sy2sds.Similarly, we can derive that there exists a set Ω1∈F with P(Ω1)=1 and an integer-valued random variable n1=n1(ω) such that, for every ω∈Ω1 and n≥n1,(29)M2t≤θenγln⁡nε+εe-nγ2M2,M2tholds for all 0≤t≤nγ. Moreover, for 0≤s≤t≤nγ, we have(30)12εe-nγσ22e2s-12σ22es<0.Substituting these inequalities into (27), we see that(31)etln⁡yt≤ln⁡y0+∫0tesln⁡ys+d-fysds+θenγln⁡nεholds for all 0≤t≤nγ, n≥n1. Similarly, for almost all 0≤s≤nγ, there exists a positive constant K2≐(ln⁡1/f+d-1)∨1 such that ln⁡y(s)+d-fy(s)≤K2. This, together with (31), yields(32)etln⁡yt≤ln⁡y0+K2et+θenγln⁡nεfor all 0≤t≤nγ, n≥n0. Thus,(33)lim⁡supt→∞⁡ln⁡ytln⁡t≤1a.s.Since y1(t) and y2(t) are positive for all t≥0, it follows that(34)lim⁡supt→∞⁡ln⁡yitln⁡t≤lim⁡supt→∞⁡ln⁡ytln⁡t≤1a.s.,i=1,2.The proof is therefore complete.Remark 3.
It follows fromlimt→∞⁡ln⁡t/t=0 that(35)lim⁡supt→∞⁡ln⁡xtt≤0,lim⁡supt→∞⁡ln⁡yitt≤0a.s.,i=1,2.Hence, the sample Lyapunov exponents of the solutions of model (6) are less than or equal to zero.


## 4. Stochastically Ultimate Boundedness

In this section, we continue to examine the stochastically ultimate boundedness which means that the solution is ultimately bounded with the large probability. Firstly, its definition will be given.Definition 4 (see [25]).
The solutions of model (6) are called stochastically ultimate bounded, if, for any ε∈(0,1), there exist three positive constants H1=H1(ε), H2=H2(ε), and H3=H3(ε) such that the solution (x(t),y1(t),y2(t)) of model (6) with any initial value in R+3 satisfies(36)lim⁡supt→∞⁡Pxt>H1<ε,lim⁡supt→∞⁡Py1t>H2<ε,lim⁡supt→∞⁡Py2t>H3<ε.Now, we prove that the solutions of model (6) are uniformly bounded in the pth moment by using the Bernoulli equation. Then, the stochastically ultimate boundedness follows directly by Chebyshev’s inequality.Lemma 5.
For any positive constantsp, α, and β, the Bernoulli equation(37)dφtdt=pβφt-pαφ1+1/pt,with the initial value φ(0)=x0>0, has the solution(38)φt=βα1-e-βt+β/αx0-1/pe-βtp.Proof.
Obviously, (37) is equivalent to(39)1pφ-1+1/ptdφtdt-βφ-1/pt=-α.Multiplying the above equation by integral factor -eβt, we obtain(40)-1peβtφ-1+1/ptdφtdt+βeβtφ-1/pt=αeβt,that is(41)ddteβtφ-1/pt=αeβt.Integrating both sides of the above equation from 0 to t yields(42)φ-1/pt=x0-1/pe-βt+αβ1-e-βt,which implies(43)φt=βα1-e-βt+β/αx0-1/pe-βtp.The proof is therefore complete.Lemma 6.
For any(x0,y10,y20)∈R+3, let (x(t),y1(t),y2(t)) be the solution of model (6) with initial value (x0,y10,y20). Then, for any p≥1,(44)lim⁡supt→∞⁡Expt≤a+p-1/2σ12bp,lim⁡supt→∞⁡Eyipt≤d+p-1/2σ22fp,i=1,2.That is, the solutions of model (6) are uniformly bounded in the pth moment.Proof.
For populationx, applying Ito^’s formula to xp leads to(45)xpt=x0p+∫0tpxpsa+p-12σ12-bxs-cy1s-ηy2sds+∫0tpσ1xpsdw1s.Taking expectation on both sides of the above inequality, we can derive(46)Expt=x0p+E∫0tpxpsa+p-12σ12-bxs-cy1s-ηy2sds,which implies the differentiability of E[xp(t)]. Denote β1≐a+p-1/2σ12. Then, using Ho¨lder inequality (Exp)1/p≤(Exp+1)1/p+1, we obtain that(47)dExptdt≤pβ1Expt-pbExp+1t≤pβ1Expt-pbExpt1+1/p.Then, from Lemma 5 and the comparison theorem, it follows that(48)Expt≤β1b1-e-β1t+β1/bx0-1e-β1tp.Note that p≥1. Then β1=a+(p-1)/2σ12>0. Thus,(49)lim⁡supt→∞⁡Expt≤β1bp=a+p-1/2σ12bp.
For populationy, denote y(t)=y1(t)+y2(t) and y0=y10+y20. Applying Ito^’s formula to yp leads to(50)ypt=∫0tpyp-1sdy1s-e^xsy1s-gxsy2s-fy2s+pp-12σ22y12syp-2sds+∫0tpσ2y1syp-1sdw2s+y0p.Taking expectation on both sides of the above inequality, we can derive(51)Eypt=y0p+E∫0tpyp-1sdy1s-e^xsy1s-gxsy2s-fy2s+pp-12σ22y12syp-2sds,which implies the differentiability of E[yp(t)]. Denote β2≐d+p-1/2σ22. Using the Ho¨lder inequality, we have(52)dEyptdt≤pβ2Eypt-pfEyp+1t≤pβ2Eypt-pfEypt1+1/p.Then, from Lemma 5 and the comparison theorem, it follows that(53)Eypt≤β2f1-e-β2t+β2/fy0-1e-β2tp.Note that p≥1. Then β2=d+(p-1)/2σ22>0. Thus,(54)lim⁡supt→∞⁡Eypt≤β2fp=d+p-1/2σ22fp.Since y1(t) and y2(t) are positive for all t≥0, it follows that, for i=1,2,(55)lim⁡supt→∞⁡Eyipt≤lim⁡supt→∞⁡Eypt≤d+p-1/2σ22fp.The proof is therefore complete.According to Chebyshev’s inequality and the application of Lemma6, we have the following result.Theorem 7.
Solutions of model (6) are stochastically ultimate bounded.Proof.
Let(x(t),y1(t),y2(t)) be the solution of model (6) with any initial values (x0,y10,y20)∈R+3. From Lemma 6, it follows that(56)lim⁡supt→∞⁡Ext≤ab,lim⁡supt→∞⁡Eyit≤df,i=1,2.Now, for any ε∈(0,1), let H1=a/bε+1 and H2=H3=d/fε+1. Then by Chebyshev’s inequality(57)Pxt>H1≤ExtH1,Py1t>H2≤Ey1tH2,Py2t>H3≤Ey2tH3.Hence,(58)lim⁡supt→∞⁡Pxt>H1≤lim⁡supt→∞⁡ExtH1<ε,lim⁡supt→∞⁡Py1t>H2≤lim⁡supt→∞⁡Ey1tH2<ε,lim⁡supt→∞⁡Py2t>H3≤lim⁡supt→∞⁡Ey2tH3<ε.The proof is therefore complete.


## 5. Extinction and Persistence

In this section, we will investigate the extinction and persistence of the population. In order to obtain our main results, several lemmas will be given. For the sake of convenience and simplicity, we first introduce the following notation:(59)xt=1t∫0txsds.Lemma 8 (see [26]).
Supposex∈C(Ω×[0,+∞),R+) and F∈C(Ω×[0,+∞),(-∞,+∞)).
(I) If there are two positive constantsλ0 and T such that(60)ln⁡xt≤λt-λ0∫0txsds+Ft,a.s.,fort≥T,where limt→∞⁡F(t)/t=0, then(61)lim⁡supt→∞⁡xt≤λλ0,a.s.λ≥0,limt→∞⁡xt=0,a.s.λ<0.
(II) If there are three positive constantsλ0, T, and λ such that(62)ln⁡xt≥λt-λ0∫0txsds+Ft,a.s.,fort≥T,where limt→∞⁡F(t)/t=0, then(63)lim⁡inft→∞⁡xt≥λλ0a.s.The following theorem investigates how the intensity of noise affects the competition population.Theorem 9.
Let(x(t),y1(t),y2(t)) be the solution of model (6) with any initial value (x0,y10,y20)∈R+3.(i)
Ifa-0.5σ12<0, then limt→∞⁡x(t)=0 a.s.Ifa-0.5σ12=0, then limt→∞⁡〈x(t)〉=0 a.s.Ifa-0.5σ12>0, then lim⁡supt→∞⁡〈x(t)〉≤(a-0.5σ12)/b a.s.(ii)
Ifd-0.5σ22<0, then limt→∞⁡y1(t)=0 a.s.Ifd-0.5σ22=0, then limt→∞⁡〈y1(t)〉=0 a.s.Ifd-0.5σ22>0, then lim⁡supt→∞⁡〈y1(t)〉≤(d-0.5σ22)/f a.s.(iii)
Ifd-0.5σ22<0, then limt→∞⁡〈y2(t)〉=0 a.s.Proof.
(i) For populationx, using Ito^’s formula results in(64)ln⁡xt=∫0ta-0.5σ12-bxs-cy1s-ηy2sds+σ1w1t+ln⁡x0≤a-0.5σ12t-b∫0txsds+σ1w1t+ln⁡x0.Clearly, Brownian motion w1(t) is a real-valued continuous local martingale vanishing at time 0. Then, from the strong law of large numbers [24], it follows that(65)limt→∞⁡w1tt=0.Thus,(66)limt→∞⁡σ1w1t+ln⁡x0t=0.Note that x(t)>0. Then lim⁡inft→∞⁡〈x(t)〉≥0. Thus, from Lemma 8, it follows that(67)limt→∞⁡xt=0,a.s.a-0.5σ12<0,limt→∞⁡xt=0,a.s.a-0.5σ12=0,lim⁡supt→∞⁡xt≤a-0.5σ12b,a.s.a-0.5σ12>0.
(ii) For populationy1, applying Ito^ formula, we have(68)ln⁡y1t=∫0td-0.5σ22-e^xs-fy1s+y2s-δy2sds+σ2w2t+ln⁡y10≤d-0.5σ22t-f∫0ty1sds+σ2w2t+ln⁡y10.A similar discussion to that in the above for x, we also have(69)limt→∞⁡y1t=0,a.s.d-0.5σ22<0,limt→∞⁡y1t=0,a.s.d-0.5σ22=0,lim⁡supt→∞⁡y1t≤d-0.5σ22f,a.s.d-0.5σ22>0.
(iii) For populationy2, using Ito^ formula results in(70)dln⁡y2t=δ-fy1t-gxt-fy2tdt.It follows from d-0.5σ22<0 that limt→∞⁡y1(t)=0 a.s. Let Ω2=ω∈Ω:limt→∞⁡y1t,ω=0; then limt→∞⁡y1(t)=0 a.s. implies P(Ω2)=1. Hence, for any ω∈Ω2 and any constant ε>0, there exists a constant t0=t0(ω,ε)>0 such that, for any t≥t0,(71)-ε≤y1t,ω≤ε.Thus, for every, t≥t0,(72)ln⁡y2t=∫t0tδ-fy1s-gxs-fy2sds+ln⁡y2t0.Consequently, if δ≤f, then(73)ln⁡y2t≤f-δεt-t0-f∫t0ty2sds+ln⁡y2t0,whereas if δ>f, then(74)ln⁡y2t≤δ-fεt-t0-f∫t0ty2sds+ln⁡y2t0.Making use of Lemma 8 and the arbitrariness of ε, we have lim⁡supt→∞⁡〈y2(t)〉≤0 a.s. Note that lim⁡inft→∞⁡〈y2(t)〉≥0 a.s. Thus,(75)limt→∞⁡y2t=0a.s.The proof is therefore complete.The following theorem tells us that competition coefficients play an important role in determining extinction of species for stochastic model. Denote(76)Δ=bf-ce^,Δ1=a-0.5σ12f-d-0.5σ22c,Δ2=cf+δ-fη,Δ3=ηf-δ-cf,Δ4=d-0.5σ22b-a-0.5σ12e^,Δ5=e^η-bf+δ,Δ6=fe^-f+δg.Theorem 10.
Let(x(t),y1(t),y2(t)) be the solution of model (6) with initial value (x0,y10,y20)∈R+3. Assume that a-0.5σ12>0 and d-0.5σ22>0. Suppose Δ≥0  (it is easy to see that Δ1<0 and Δ4<0 cannot simultaneously hold in this case).
(i) IfΔ1<0 and Δ2≤0, then limt→∞⁡x(t)=0 a.s. Furthermore, if Δ3≥0, then(77)limt→∞⁡y2t=0a.s.,limt→∞⁡y1t=d-0.5σ22fa.s.
(ii) IfΔ4<0 and Δ5≤0 then limt→∞⁡y1(t)=0 a.s. Moreover, if Δ6≥0, then(78)limt→∞⁡y2t=0a.s.,limt→∞⁡xt=a-0.5σ12ba.s.Proof.
It follows from Ito^ formula that(79)ln⁡xt=a-σ122t-b∫0txsds-c∫0ty1sds-η∫0ty2sds+σ1w1t+ln⁡x0,(80)ln⁡y1t=d-σ222t-e^∫0txsds-f∫0ty1sds-f+δ∫0ty2sds+σ2w2t+ln⁡y10,(81)ln⁡y2t=-g∫0txsds-f-δ∫0ty1sds-f∫0ty2sds+ln⁡y20.
(i) First, computing (79) ×  f − (80) ×  c leads to(82)fln⁡xt-cln⁡y1t=fln⁡x0-cln⁡y10+Δ1t-Δ∫0txsds+Δ2∫0ty2sds+fσ1w1t-cσ2w2t,which, together with the conditions of Theorem 10, yields(83)ln⁡xtt≤Δ1f+1ftcln⁡y1t+fσ1w1t-cσ2w2t+fln⁡x0-cln⁡y10.Then by applying the strong law of large numbers and Remark 3, we have(84)lim⁡supt→∞⁡ln⁡xtt≤Δ1f<0.Thus,(85)limt→∞⁡xt=0a.s.Next, computing (81) ×  η  − (79) ×  f leads to(86)ηln⁡y2t-fln⁡xt=-fa-σ122t-gη-bf∫0txsds-Δ3∫0ty1sds-fσ1w1t+ηln⁡y20-fln⁡x0,which, together with the conditions of Theorem 10, yields(87)ηln⁡y2tt≤-fa-σ122-gη-bft∫0txsds+fln⁡xtt-fσ1w1tt+ηln⁡y20t-fln⁡x0t.Let Ω3={ω∈Ω:limt→∞⁡x(t,ω)=0}; then (85) implies P(Ω3)=1. Hence, for any ω∈Ω3 and any constant ε>0, there exists a constant T=T(ω,ε)>0 such that, for any t≥T,(88)xt,ω≤ε.Now, we introduce two cases.
1 ∘ Suppose that gη-bf<0. From (87), it follows that, for any t≥T,(89)ηln⁡y2tt≤-fa-σ122-gη-bfεt-Tt+fln⁡xtt-gη-bft∫0Txsds-fσ1w1tt+ηln⁡y20t-fln⁡x0t.Thus, from Remark 3 and the strong law of large numbers, it follows that(90)lim⁡supt→∞⁡ln⁡y2tt≤1η-fa-σ122-gη-bfε.Then, by the arbitrariness of ε, we have(91)lim⁡supt→∞ln⁡y2tt≤-fηa-σ122<0.
2 ∘ Suppose that gη-bf≥0. Then(92)ηln⁡y2tt≤-fa-σ122+fln⁡xtt-fσ1w1tt+ηln⁡y20t-fln⁡x0t.Thus, from Remark 3 and the strong law of large numbers, it follows that(93)lim⁡supt→∞⁡ln⁡y2tt≤-fηa-σ122<0.Therefore,(94)limt→∞⁡y2t=0a.s.Finally, it follows from (80) that(95)ln⁡y1t≤d-σ222t-f∫0ty1sds+σ2w2t+ln⁡y10.Note that limt→∞⁡σ2w2(t)+ln⁡y10/t=0 and d-σ22/2>0. Thus, from (I) in Lemma 8, it follows that(96)lim⁡supt→∞⁡y1t≤d-0.5σ22f.On the other hand, it follows from (85) and (94) that, for any constant ε>0, there is a positive constant t1 such that t≥t1(97)e^xt≤ε2,f+δy2t≤ε2.Substituting this into (80) yields(98)ln⁡y1t≥d-σ222t-εt-t1-f∫0ty1sds-∫0t1e^xs+f+δy2sds+σ2w2t+ln⁡y10≥d-σ222-εt-f∫0ty1sds-∫0t1e^xs+f+δy2sds+σ2w2t+ln⁡y10,for any t≥t1. Note that(99)limt→∞⁡1t∫0t1e^x+f+δy2ds-σ2w2t-ln⁡y10=0.Then by (II) in Lemma 8 and the arbitrariness of ε, one can observe that(100)lim⁡inft→∞⁡y1t≥d-0.5σ22f.Therefore, we can derive that(101)limt→∞⁡y1t=d-0.5σ22f.
(ii) A similar discussion to that in the above for (i), we also have the desired assertion (ii). This completes the proof.If we do not consider the effect of disease, then model (6) can be degraded into the following model(102)dxt=xta-bxt-cy1tdt+σ1xtdw1t,dy1t=y1td-e^xt-fy1tdt+σ2y1tdw2t,with initial value x(0)=x0 and y1(0)=y10. Model (102) is the same as the stochastic competitive population model (SM0) discussed in [15].From Theorems9 and 10, we can get the following corollaries with the proof being omitted.Corollary 11.
Let(x(t),y1(t)) be the solution of degradation model (102) with any initial value (x0,y10)∈R+2.(i)
Ifa<0.5σ12, then the population x will go to extinction almost surely;(ii)
Ifd<0.5σ22, then the population y1 will go to extinction almost surely.Corollary 12.
Let(x(t),y1(t)) be the solution of degradation model (102) with any initial value (x0,y10)∈R+2. Suppose that a>0.5σ12, d>0.5σ22, and Δ≥0.(i)
IfΔ1<0, then the population x will go to extinction almost surely; if Δ4<0, then the population y1 will go to extinction almost surely.(ii)
IfΔ1>0 and Δ4<0, then limt→∞⁡〈x(t)〉=a-0.5σ12/b a.s.; if Δ1<0 and Δ4>0, then limt→∞⁡〈y1(t)〉=d-0.5σ22/f a.s.Remark 13.
Corollaries11 and 12 are consistent with Theorems 9 and 10 in [15]. Moreover, if one considers the effects of the disease, from Theorem 10 we know that the conditions for population extinction and persistence will be more complicated. Therefore, our work can be seen as the extension of [15].


## 6. Asymptotic Behavior around the Interior EquilibriumE^

From [4], it follows that the interior equilibrium E^=(x^,y^1,y^2) of deterministic model (2) is feasible when δ>f, g(f+δ)>e^f, a>(ca1+ηa3), and (δ-f)(gf+gδ-e^f)>gδ2. Moreover, from Theorem 3.2 in [4], if there exist positive constants k1 and k2 such that(103)c+k1e^2<bk1f,η+k2g2<bk2f,fk1+k2+δk1-k22<k1k2f,then the interior equilibrium E^ of deterministic model (2) is globally asymptotically stable. Obviously, the interior equilibrium E^ is not the solution of stochastic model (6); the following theorem demonstrates the asymptotic behavior of the solution of model (6) around the equilibrium E^. It follows from Theorem 9 that if a-σ12/2<0 and d-σ22/2<0, then limt→∞⁡x(t)=0, limt→∞⁡yi(t)=0 a.s. Therefore, in this section, we assume that a-σ12/2>0 and d-σ22/2>0. Moreover, we have the following result.Theorem 14.
Assume thatδ>f, g(f+δ)>e^f, a>(ca1+ηa3), and (δ-f)(gf+gδ-e^f)>gδ2. Let (x(t),y1(t),y2(t)) be the solution of model (6) with initial value (x0,y10,y20)∈R+3. If a-σ12/2>0 and d-σ22/2>0, then there exist positive constants li>0 and mi>0(i=1,2) such that(104)lim⁡supt→∞⁡1tE∫0tx-x^2+y1-y^12+y2-y^22ds≤1/2σ12x^+1/2σ22y^1+Dm1+2m2/l1∧l2b∧f,where a1=df/δ2, a3=(δ-f)/δ2, and D=max⁡c+e^y^1+η+gy^2,c+e^x^+2fy^2,η+gx^+2fy^1.Proof.
From [4], we know that the interior equilibrium E^ of deterministic model (2) is feasible when δ>f, g(f+δ)>e^f, a>(ca1+ηa3), and (δ-f)(gf+gδ-e^f)>gδ2. And if (103) holds, the interior equilibrium E^ of model (2) is globally asymptotically stable. The coordinates of E^ satisfy(105)bx^+cy^1+ηy^2=a,e^x^+fy^1+f+δy^2=d,δ-fy^1-gx^-fy^2=0.Define a function(106)V1x,y1,y2=x-x^-x^ln⁡xx^+y1-y^1-y^1ln⁡y1y^1+y2-y^2-y^2ln⁡y2y^2,and then using Ito^’s formula and (105), we have(107)dV1=x-x^a-bx-cy1-ηy2+y1-y^1d-e^x-fy1-f+δy2+y2-y^2δ-fy1-gx-fy2+12σ12x^+12σ22y^1dt+σ1x-x^dw1t+σ2y1-y^1dw2t=x-x^-bx-x^-cy1-y^1-ηy2-y^2+y1-y^1-e^x-x^-fy1-y^1-f+δy2-y^2+y2-y^2δ-fy1-y^1-gx-x^-fy2-y^2+12σ12x^+12σ22y^1dt+σ1x-x^dw1t+σ2y1-y^1dw2t≤-bx-x^2-fy1-y^12-fy2-y^22+c+e^y^1+η+gy^2x+c+e^x^+2fy^2y1+η+gx^+2fy^1y2+12σ12x^+12σ22y^1dt+σ1x-x^dw1t+σ2y1-y^1dw2t≤-bx-x^2-fy1-y^12-fy2-y^22+Dx+y1+y2+12σ12x^+12σ22y^1dt+σ1x-x^dw1t+σ2y1-y^1dw2t,where D=max⁡c+e^y^1+η+gy^2,c+e^x^+2fy^2,η+gx^+2fy^1.
Define(108)V2x,y1,y2=x+y1+y2.Note that there exist constants l1>0, l2>0, m1>0, and m2>0 such that for any z>0(109)az-bz2≤-l1z+m1,dz-fz2≤-l2z+m2.Thus, by Ito^’s formula, we have(110)dV2=ax-bx2-cxy1-ηxy2+dy1-fy12-e^xy1-2fy1y2-gxy2-fy22dt+σ1xdw1t+σ2y1dw2t≤ax-bx2+dy1-fy12+dy2-fy22dt+σ1xdw1t+σ2y1dw2t≤-l1x+m1-l2y1+m2-l2y2+m2dt+σ1xdw1t+σ2y1dw2t≤-l1∧l2x+y1+y2+m1+2m2dt+σ1xdw1t+σ2y1dw2t.Then define(111)V3x,y1,y2=V1x,y1,y2+Dl1∧l2V2x,y1,y2.Using Ito^’s formula, and noting that (107) and (110) and b∧f>0, one can get(112)dV3=dV1+Dl1∧l2dV2≤-bx-x^2-fy1-y^12-fy2-y^22+12σ12x^+12σ22y^1+Dm1+2m2l1∧l2dt+σ1x-x^dw1t+σ2y1-y^1dw2t+Dσ1l1∧l2xdw1t+Dσ2l1∧l2y1dw2t≤-b∧fx-x^2+y1-y^12+y2-y^22+12σ12x^+12σ22y^1+Dm1+2m2l1∧l2dt+σ1x-x^dw1t+σ2y1-y^1dw2t+Dσ1l1∧l2xdw1t+Dσ2l1∧l2y1dw2t.Integrating both sides of (112) from 0 to t and taking the expectation, we have(113)0≤EV3≤-b∧fE∫0tx-x^2+y1-y^12+y2-y^22ds+12σ12x^+12σ22y^1+Dm1+2m2l1∧l2t.Noting that b∧f>0, so dividing both sides by (b∧f)t and letting t→∞, we get(114)lim⁡supt→∞1tE∫0tx-x^2+y1-y^12+y2-y^22ds≤1/2σ12x^+1/2σ22y^1+Dm1+2m2/l1∧l2b∧f.The proof is therefore complete.


## 7. Numerical Simulations

In this section, we make numerical simulations to illustrate our results by using the Milstein method (see, e.g., [27]). The numerical simulations of population dynamics are carried out for the academic tests with the arbitrary values of the vital rates and other parameters which do not correspond to some specific biological populations and exhibit only the theoretical properties of numerical solutions of the considered model. In the figures, the red lines, blue lines, and green lines represent the trajectories of populations x(t), y1(t), and y2(t), respectively. Here we give numerical simulations of model (6) with the same initial values x0=1, y10=0.9, and y20=0.1.In Figure1, we choose a=0.3, b=0.15, c=0.2, η=0.2, d=0.3, e^=0.15, f=0.5, δ=0.1, g=0.5, σ12=1, and σ22=1. It is easy to see that a<0.5σ12, d<0.5σ22, and δ<f. By Theorem 9, populations x, y1, and y2 will go to extinction.Figure 1
Numerical simulation of the solution of model (6). The parameters of model (6) are a=0.3, b=0.15, c=0.2, η=0.2, d=0.3, e^=0.15, f=0.5, δ=0.1, g=0.5, σ12=1, and σ22=1.
(a) (b) (c)In Figure2, we choose a=0.3, b=0.15, c=0.2, η=0.2, d=0.3, e^=0.15, f=0.15, δ=0.2, g=0.5, σ12=1, and σ22=1. It is easy to see that a<0.5σ12, d<0.5σ22, and δ>f. By Theorem 9, populations x, y1, and y2 will go to extinction.Figure 2
Numerical simulation of the solution of model (6). The parameters of model (6) are a=0.3, b=0.15, c=0.2, η=0.2, d=0.3, e^=0.15, f=0.15, δ=0.2, g=0.5, σ12=1, and σ22=1.
(a) (b) (c)In Figure3, we choose a=0.55, b=0.3, c=0.38, η=0.6, d=0.5, e^=0.23, f=0.3, δ=0.1, g=0.5, and σ12=σ22=0.1. It is easy to see that Δ=0.0026>0, Δ1=-0.021<0, Δ2=-0.028<0, and Δ3=0.006>0. By (i) of Theorem 10, populations x and y2 will go to extinction and population y1 will be weakly persistent in mean almost surely.Figure 3
Numerical simulation of the solution of model (6). The parameters of model (6) are a=0.55, b=0.3, c=0.38, η=0.6, d=0.5, e^=0.23, f=0.3, δ=0.1, g=0.5, and σ12=σ22=0.1.
(a) (b) (c)In Figure4, we choose a=0.55, b=0.3, c=0.38, η=0.5, d=0.4, e^=0.23, f=0.3, δ=0.1, g=0.1, and σ12=σ22=0.1. It is easy to see that Δ=0.0026>0, Δ4=-0.01<0, Δ5=-0.005<0, and Δ6=0.029>0. By (ii) of Theorem 10, populations y1 and y2 will go to extinction and population x will be weakly persistent in mean almost surely.Figure 4
Numerical simulation of the solution of model (6). The parameters of model (6) are a=0.55, b=0.3, c=0.38, η=0.5, d=0.4, e^=0.23, f=0.3, δ=0.1, g=0.1, and σ12=σ22=0.1.
(a) (b) (c)As done in [4], in Figure 5, we choose a=1, b=1, c=0.1, η=0.1, d=2, e^=0.1, f=1, δ=2.5, g=0.2, and σ1=σ2=0.5. For this example, we get the following interior equilibrium: E^=(0.9163,0.4080,0.4287). Moreover, we have a-σ12/2>0 and d-σ22/2>0. Thus, the conditions of Theorem 14 hold. Then, by Theorem 14, the solution of model (6) oscillates around the equilibrium E^ of the corresponding deterministic model (2) (see Figure 5).Figure 5
Numerical simulation of the solution of model (6). The parameters of model (6) are a=1, b=1, c=0.1, η=0.1, d=2, e^=0.1, f=1, δ=2.5, g=0.2, σ1=0.5, and σ2=0.5.
(a) (b) (c)As can be seen from Figures1 and 2, regardless of the size of the interspecific competition rate f and the rate of infection δ, as long as a<0.5σ12 and d<0.5σ22, we know that the healthy individuals x(t), the healthy individuals y1(t), and the infected individuals y2(t) will go to extinction.From Figures3 and 4, we can see that although the environmental noise is relatively small, population extinction can also occur when the competition coefficients satisfy certain conditions. It can be seen from Figure 2 that the population y is dominant in competition, while the population x is at a disadvantage, and the individuals with infectious diseases in population y will go to extinction. Moreover, populations x and y2 will go to extinction and population y1 will be weakly persistent in mean almost surely. However, from Figure 3, we can see that the population x is dominant in competition, while the population y is at a disadvantage, and the individuals with infectious diseases in population y will go to extinction. That is, populations y1 and y2 will go to extinction and population x will be weakly persistent in mean almost surely.


## 8. Conclusion and Discussion

In this paper, we consider a stochastic two-species competition model under the effect of disease. By the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model. Then, an important asymptotic property of model is given via the exponential martingale inequality and Borel-Cantelli lemma. Next, we prove the boundedness of thepth moment of the global positive solution. Then, sufficient conditions for extinction and persistence in mean are obtained. Furthermore, by constructing suitable Lyapunov function, we investigate the asymptotic behavior of the stochastic system around the interior equilibrium of the deterministic model.From Remark13, we know that our work can be seen as the extension of [15]. Some interesting problems deserve further consideration. One may incorporate the Markovian switching into the stochastic population model (6). Since model (6) may be perturbed by the telegraph noise which can make the model switch from one environmental regime to another, one may also introduce the jumps into the stochastic model (6). Since population systems may suffer severe environmental perturbations, such as tsunami, volcanoes, avian influenza, SARS, floods, hurricanes, earthquakes, and toxic pollutants, we leave this for future consideration.



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