# Stability of a Three-Species Cooperative System with Time Delays and Stochastic Perturbations

Authors: Jinxing Zhao; Yuanfu Shao
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Considering the impacts of time delays and different kinds of stochastic perturbations, we propose two three-species delayed cooperative systems with stochastic perturbations in this paper. We establish the sufficient criteria of the asymptotical stability and stability in probability by constructing a neutral stochastic differential equation and some suitable functionals. The impacts of time delays and stochastic perturbations to the system dynamics are revealed by some numerical simulations at the end.


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## Body

## 1. Introduction

In biological world, mutualism is an important interaction and many cooperative models have been proposed to describe such biological phenomena (see [1–3] and references cited therein).In nature, time delays usually appear and bring some important influence to the dynamics of ecosystem models. Kuang [4] says that ignoring time delays means ignoring the reality. It is essential to take the influence of time delays into account in mathematical modelling [5–9]. Furthermore, in view of the complexity of natural world, single-species or two-species ecological models often cannot describe some natural phenomena accurately and many vital behaviours can only be exhibited by systems with three or more species [10–13]. Considering the effect of time delays to three-species cooperative system, we propose the following deterministic model:(1)dN1t=N1tr1−a11N1t+a12N2t−τ12+a13N3t−τ13dt,dN2t=N2tr2+a21N1t−τ21−a22N2t+a23N3t−τ23dt,dN3t=N3tr3+a31N1t−τ31+a32N2t−τ32−a33N3tdt.For (1), we define(2)α1=a11,−a12,−a13T,α2=−a21,a22,−a23T,α3=−a31,−a32,a33T,R=r1,r2,r3T,A=detα1,α2,α3,A1=detR,α2,α3,A2=detα1,R,α3,A3=detα1,α2,R,and assume that there exists a unique positive equilibrium point as follows:(3)N1∗=A1A,N2∗=A2A,N3∗=A3A.On the contrary, the growth of populations is often subject to environmental fluctuation, so it is necessary to consider stochastic perturbation in the process of mathematical modelling [14–18]. Usually, there are many kinds of stochastic perturbations. The authors in [19] proposed that the stochastic perturbation of state variable around the steady-state N1∗, N2∗, and N3∗ was Brownian white noise, which was proportional to the distance from the equilibrium state. Consequently, we obtain the following three-species stochastic cooperative system with time delays:(4)dN1t=N1tr1−a11N1t+a12N2t−τ12+a13N3t−τ13dt+σ1N1t−N1∗dω1t,dN2t=N2tr2+a21N1t−τ21−a22N2t+a23N3t−τ23dt+σ2N2t−N2∗dω2t,dN3t=N3tr3+a31N1t−τ31+a32N2t−τ32−a33N3tdt+σ3N3t−N3∗dω3t,with initial data(5)Niθ=ϕiθ>0,θ∈−τ,0,i=1,2,3,where Nit stands for the population size of the ith species at time t, ri>0 is the growth rate of Nit, aii>0 represents the intraspecific competitive coefficient of Nit, aij is the interspecific cooperative rate, τij>0 is time delay, τ=maxτ12,τ13,τ21,τ23,τ31,τ32, ϕiθ is positive and continuous function defined on −τ,0, σi2 denotes the intensity of white noise, and ωitt>0 is the standard independent Brownian motion defined on a complete probability space Ω,ℱ,ℱt≥0,P,i,j=1,2,3,j≠i.Furthermore, Liu [20] proposed that the stochastic perturbation of state variable was proportional to NiNi−Ni∗i=1,2,3, and then, we get the following model:(6)dN1t=N1tr1−a11N1t+a12N2t−τ12+a13N3t−τ13dt+σ1N1tN1t−N1∗dω1t,dN2t=N2tr2+a21N1t−τ21−a22N2t+a23N3t−τ23dt+σ2N2tN2t−N2∗dω2t,dN3t=N3tr3+a31N1t−τ31+a32N2t−τ32−a33N3tdt+σ3N3tN3t−N3∗dω3t.For the deterministic system, whether there exists a positive equilibrium state is an important topic, which attracts many attentions of researchers, while stochastic model cannot tend to a positive fixed point; i.e., there exists no traditional positive equilibrium state, so it is popular to study the dynamics of stochastic system around the equilibrium state of its corresponding deterministic system. Furthermore, some kinds of delays are considered in biological systems, but many obtained results have no relation with delays and the effects of time delays are not revealed clearly [9, 11]. Time delays are the sources of instability in population dynamics, and they can cause population fluctuations, so it is interesting to study the effects of delays to the system dynamics. Motivated by these, we propose two three-species delayed cooperative systems with stochastic perturbation and aim to investigate how time delays affect the stability in probability around the equilibrium state and then by numerical examples to validate our theoretical results. To the best of our knowledge, this paper is the first attempt to investigate the influence of time delays on the stability in probability of a stochastic three-species cooperative system.The rest work of this paper is structured as follows. Section2 begins with definitions and some important lemmas and notations. Section 3 focuses on the asymptotical stability and stability in probability of (4) and (6), respectively. Some numerical simulations are given in Section 4 to validate our theoretical results. Finally, a brief conclusion and future direction are given in Section 5 to conclude the paper.


## 2. Preliminaries

LetΩ,σ,P be a probability space and ft,t≥0 be a family of σ-algebras. We consider the following neutral stochastic differential equation:(7)dxt−Ft,xt=at,xtdt+bt,xtdωt,x0=ϕ∈H,where H represents the space with all f0-adapted functions ϕs∈Rn and xts=xt+s,s≤0,ωt denotes the m-dimensional ft-adapted Brownian process, at,ϕ and bt,ϕ are n-dimensional vector and n×m-dimensional matrix, respectively. Define(8)ϕ0=sups≤0ϕs,ϕ1=sups≤0Eϕs2,where E denotes the mathematical expectation.Definition 1.
(see [19]). The zero solution of (7) is said to be stable in probability if for any ε,ε>0, there exists a number δ>0 such that the solution xt=xt,ϕ satisfies Pxt,ϕ>ε<ε for any initial function ϕ∈H such that Pϕ0≤δ=1,in whichP is the probability of an event.Lemma 1.
Systems (4) and (6) have a unique globally positive solution on t>−τ for any initial data given above, respectively.Remark 1.
The proof is standard. For more details, refer to [20, 21].Lemma 2.
(see [19, 22]). If there exists a functional Vt,x such that(9)λ1xt2≤Vt,xt≤λ2xt02,λ1,λ2>0,LVt,xt≤0,for any function ϕ∈H such that Pϕ0≤δ=1, δ>0, is sufficiently small positive constant, and then, the zero solution of (7) is stable in probability.
For convenience in the later, if there is no special mention, we always denoteft,t∈R, by f and give the following notations to end this section:(10)ξ1=a21a21+a23,η1=a31a31+a32,ξ2=a12a12+a13,η2=a32a31+a32,ξ3=a13a12+a13,η3=a23a21+a23,Δ1=2a11−a12−a13−σ12N1∗,Δ2=2a22−a21−a23−σ22N2∗,Δ3=2a33−a31−a32−σ12N3∗,Δ˜1=2a11−a12−a13−σ12N1∗,Δ˜2=2a22−a21−a23−σ22N2∗,Δ˜3=2a33−a31−a32−σ12N3∗.


## 3. Stochastic Stability

In this section, we investigate the asymptotical stability and stable in probability of (4) and (6) around the equilibrium state of (1), respectively. Let x1=N1−N1∗, x2=N2−N2∗, and x3=N3−N3∗, then system (4) is transformed to the following equivalent system:(11)dx1t=−a11N1∗x1t+a12N1∗x2t−τ12+a13N1∗x3t−τ13−a11x12+a12x1x2t−τ12+a13x1x3t−τ13dt+σ1x1dω1t,dx2t=a21N2∗x1t−τ21−a22N2∗x2t+a23N2∗x3t−τ23−a22x22+a21x2x1t−τ21+a23x2x3t−τ23dt+σ2x2dω2t,dx3t=−a33N3∗x3t+a31N3∗x3t−τ31+a31N3∗x3t−τ32−a33x32+a31x3x1t−τ31+a32x3x2t−τ32dt+σ3x3dω3t.By the equivalent property of above transformation, the stability of (4) around the equilibrate state of (1) is equivalent to the stability of zero solution of corresponding equivalent system (11). Consequently, we only need to study the stability of zero solution of system (11).By use of the definition of derivative, system (11) is equivalent to the following neutral system:(12)dx1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds=−a11N1∗x1+a12N1∗x2+a13N1∗x3−a11x12+a12x1x2t−τ12+a13x1x3t−τ13dt+σ1x1dω1t,dx2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds=a21N2∗x1−a22N2∗x2+a23N2∗x3−a22x22+a21x2x1t−τ21+a23x2x3t−τ23dt+σ2x2dω2t,dx3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds=a31N3∗x1+a32N3∗x2−a33N3∗x3−a33x32+a31x3x1t−τ31+a32x3x2t−τ32dt+σ3x3dω3t.The linear case of (12) is as follows:(13)dx1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds=−a11N1∗x1+a12N1∗x2+a13N1∗x3dt+σ1x1dω1t,dx2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds=a21N2∗x1−a22N2∗x2+a23N2∗x3dt+σ2x2dω2t,dx3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds=a31N3∗x1+a32N3∗x2−a33N3∗x3dt+σ3x3dω3t.For the linear and neutral system (13), the following statement is true.Theorem 1.
If the following condition holds,(14)H1Δ1−a21−a31−a12Δ2−a32−a13−a23Δ3N1∗N2∗N3∗>2τ0ξ1η1ξ20η2ξ3η30N1∗2N2∗2N3∗2,then the zero solution of (13) is globally asymptotically stable, almost surely, i.e., limt⟶∞xit=0,i=1,2,3.Proof.
First, by the characteristics of neutral functional differential system (13), we define(15)V1=x1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds2,V2=x2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds2,V3=x3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds2.
Applying Itô formula toV1, we have(16)LV1=2x1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds×−a11N1∗x1+a12N1∗x2+a13N1∗x3+σ12x12=−2a11N1∗x12+2a12N1∗x1x2+2a13N1∗x1x3+σ12x12−2a11a22N1∗2∫t−τ12tx1tx2sds+2a122N1∗2∫t−τ12tx2tx2sds+2a12a13N1∗2∫t−τ12tx3tx2sds−2a11a13N1∗2×∫t−τ13tx1tx3sds+2a12a13N1∗2∫t−τ13tx2tx3sds+2a132N1∗2∫t−τ13tx3tx3sds≤−2a11N1∗x12+a12N1∗x12+x22+a13N1∗x12+x32+σ12x12+a12a12+a13N1∗2x22τ+a13a13+a12N1∗2x32τ+a12a12+a13N1∗2∫t−τ12tx22sds+a13a12+a13N1∗2∫t−τ13tx32sds.
By the same way, we have(17)LV2=2x2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds×a21N2∗x1−a22N2∗x2+a23N2∗x3+σ22x22=−2a22N2∗x22+2a21N2∗x1x2+2a23N2∗x2x3+σ22x22+2a212N2∗2∫t−τ21tx1tx1sds−2a21a22N2∗2∫t−τ21tx2tx1sds+2a21a23N2∗2∫t−τ21tx3tx1sds+2a21a23N2∗2×∫t−τ23tx1tx3sds−2a22a23N2∗2∫t−τ23tx2tx3sds+2a232N2∗2∫t−τ13tx3tx3sds≤−2a22N2∗x22+a21N2∗x12+x22+a23N2∗x22+x32+σ22x22+a21a21+a232N2∗2x12τ+a23a21+a23N2∗2x32τ+a21a21+a23N2∗2∫t−τ21tx12sds+a23a21+a23N2∗2∫t−τ23tx32sds,LV3=2x3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds×a31N3∗x1+a32N3∗x2−a33N3∗x3+σ32x32=−2a33N3∗x32+2a31N3∗x1x3+2a32N3∗x2x3+σ32x32+2a312N3∗2∫t−τ31tx1tx1sds+2a31a32N3∗2∫t−τ31tx2tx1sds−2a31a33N3∗2∫t−τ31tx3tx1sds+2a32a31N3∗2×∫t−τ32tx1tx2sds+2a322N3∗2∫t−τ32tx2tx2sds−2a32a33N3∗2∫t−τ32tx3tx2sds≤−2a33N3∗x32+a31N3∗x12+x32+a32N3∗x22+x32+σ32x32+a32a31+a32N3∗2x22τ+a31a31+a32N3∗2x12τ+a31a31+a32N3∗2∫t−τ31tx12sds+a32a31+a32N3∗2∫t−τ32tx22sds.
Adding both sides ofLV1, LV2, and LV3 yields(18)LV1+V2+V3≤−2a11−a12−a13N1∗+σ12+a21N2∗+a31N3∗x12+τa21a21+a23N2∗2+a31a31+a32N3∗2x12+a31a31+a32N3∗2∫t−τ31tx12sds+a21a21+a23N2∗2∫t−τ21tx12sds−2a22−a21−a23N2∗+σ22+a12N1∗+a32N3∗x22+τa12a12+a13N1∗2+a32a31+a32N3∗2x22+a12a12+a13N1∗2∫t−τ12tx22sds+a32a31+a32N3∗2∫t−τ32tx22sds−2a33−a31−a32N3∗+σ12+a13N1∗+a23N2∗x32+τa23a21+a23N2∗2+a13a13+a12N1∗2x32+a23a21+a23N2∗2∫t−τ23tx32sds+a13a12+a13N1∗2∫t−τ13tx32sds.
Define(19)V4=a21a21+a23N2∗2∫t−τ21tθ−t+τ21x12θdθ+a31a31+a32N3∗2∫t−τ31tθ−t+τ31x12θdθ,V5=a12a12+a13N1∗2∫t−τ12tθ−t+τ12x22θdθ+a32a31+a32N3∗2∫t−τ32tθ−t+τ32x22θdθ,V6=a13a12+a13N1∗2∫t−τ13tθ−t+τ13x32θdθ+a23a21+a23N2∗2∫t−τ23tθ−t+τ23x32θdθ.
LetV=∑i=16Vi, then(20)LV≤−2a11−a12−a13−σ12N1∗N1∗+a21N2∗+a31N3∗+2τa21a21+a23N2∗2+2τa31a31+a32N3∗2x12t+−2a22−a21−a23−σ22N2∗N2∗+a12N1∗+a32N3∗+2τa12a12+a13N1∗2+2τa32a31+a32N3∗2x22+−2a33−a31−a32−σ12N3∗N3∗+a13N1∗+a23N2∗+2τa23a21+a23N2∗2+2τa13a12+a13N1∗2x32.
By the hypothesisH1, it is easy to get LV<0 along all trajectories in R+3 except Ni∗. Hence, by the stability theory of stochastic functional differential equations [23], the zero solution of (13) is globally asymptotically stable. This completes the proof.Remark 2.
The globally asymptotical stability of zero solution of (13) means the globally asymptotical stability of the solution of (4) around the equilibrate state of (1), that is, limt⟶∞Nit=Ni∗,i=1,2,3.Remark 3.
The proof method is motivated in [19]. We apply the theory of neutral functional differential equation and define some suitable functionals V1,V2, and V3 to obtain the sufficient conditions assuring the globally asymptotical stability of (13) around the equilibrium state Ni∗, which are much different from those of [20].Theorem 2.
IfH1 holds, then the zero solution of system (12) is stable in probability; that is, system (4) around the equilibrium state of (1) is stable in probability.Proof.
For system (12), define V1, V2, and V3 as before and assume that there exists a number δ>0 such that supt≥τxis<δ,i=1,2,3. By the Itô formula, we calculate LV1,LV2, and LV3 along system (12), respectively, then(21)LV1=2x1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds−a11N1∗x1+a12N1∗x2+a13N1∗x3−a11x12+a12x1x2t−τ12+a13x1x3t−τ13+σ12x12≤−2a11N1∗x12+a12N1∗x12+x22+a13N1∗x12+x32+σ12x12+a12a12+a13N1∗2x22τ+a13a13+a12N1∗2x32τ+a12a12+a13N1∗2∫t−τ12tx22sds+a13a12+a13N1∗2∫t−τ13tx32sds+2δa12+a13x12+a12+a132N1∗τδx12+δa122N1∗∫t−τ12tx22sds+δa132N1∗∫t−τ13tx32sds,LV2=2x2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds−a22N2∗x2+a21N2∗x1+a23N2∗x3+a21x2x1t−τ21−a22x22+a23x2x3t−τ23+σ22x22≤−2a22N2∗x22+a21N2∗x12+x22+a23N2∗x22+x32+σ22x22+a21a21+a23N2∗2x12τ+a23a21+a23N2∗2x32τ+a21a21+a23N2∗2∫t−τ21tx12sds+a23a21+a23N2∗2∫t−τ23tx32sds+2δa21+a23x22+a21+a232N2∗τδx22+δa212N2∗∫t−τ23tx12sds+δa232N2∗∫t−τ23tx32sds,LV3=2x3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sdsa31N3∗x1+a32N3∗x2−a33N3∗x3+a31x3x1t−τ31+a32x3x2t−τ32−a33x32+σ32x32≤−2a33N3∗x32+a31N3∗x12+x32+a32N3∗x22+x32+σ32x32+a32a31+a32N3∗2x22τ+a31a31+a32N3∗2x12τ+a31a31+a32N3∗2∫t−τ31tx12sds+a32a31+a32N3∗2∫t−τ32tx22sds+2δa31+a32x32+a31+a322N3∗τδx32+δa312N3∗∫t−τ31tx12sds+δa322N3∗∫t−τ32tx22sds.
DefineV˜4,V˜5, and V˜6 as follows:(22)V˜4=a21a21+a23N2∗2+δa212N2∗∫t−τ21tθ−t+τ21x12θdθ+a31a31+a32N3∗2+δa312N3∗∫t−τ31tθ−t+τ31x12θdθ,V˜5=a12a12+a13N1∗2+δa122N1∗∫t−τ12tθ−t+τ12x22θdθ+a32a31+a32N3∗2+δa322N3∗∫t−τ32tθ−t+τ32x22θdθ,V˜6=a13a12+a13N1∗2+δa132N1∗∫t−τ13tθ−t+τ13x32θdθ+a23a21+a23N2∗2+δa232N2∗∫t−τ23tθ−t+τ23x32θdθ.
LetV=∑i=13Vi+∑j=46V˜j. Computing the derivatives of V˜jj=4,5,6 and adding both sides of LVii=1,2,3 and LVjj=4,5,6 reads(23)LV≤−2a11−a12−a13−σ12N1∗N1∗+a21N2∗+a31N3∗+2τa21a21+a23N2∗2+2τa31a31+a32N3∗2+τδa122N1∗+a322N3∗x12t+−2a22−a21−a23−σ22N2∗N2∗+a12N1∗+a32N3∗+2τa12a12+a13N1∗2+2τa32a31+a32N3∗2+τδa132N1∗+a232N2∗x22+−2a33−a31−a32−σ12N3∗N3∗+a13N1∗+a23N2∗+2τa23a21+a23N2∗2+2τa13a12+a13N1∗2+τδa212N2∗+a312N3∗x32.
By choosing sufficiently smallδ>0 such that H1 holds, then we have LV<0. Thus, it follows from Lemma 2 that the zero solution of (12) is stable in probability. The proof is completed.Remark 4.
In the process of our proof, the same method applied in linear case (Theorem1) is generalized to nonlinear case. We define V˜jj=4,5,6 and compute their derivatives so as to eliminate the integral items appeared in LVii=1,2,3. By Lemma 2, we obtain the sufficient conditions assuring the stability in probability of (4) around the equilibrium state Ni∗, which is relatively new in some sense.
Next, we consider system (6). By setting x1=N1−N1∗, x2=N2−N2∗, and x3=N3−N3∗, and then, (6) is transformed to the following equivalent system:(24)dx1t=−a11N1∗x1t+a12N1∗x2t−τ12+a13N1∗x3t−τ13−a11x12+a12x1x2t−τ12+a13x1x3t−τ13dt+σ1x1x1+N1∗dω1t,dx2t=−a22N2∗x2t+a21N2∗x1t−τ21+a23N2∗x3t−τ23−a22x22+a21x2x1t−τ21+a23x2x3t−τ23dt+σ2x2x2+N2∗dω2t,dx3t=−a33N3∗x3t+a31N3∗x3t−τ31+a31N3∗x3t−τ32−a33x32+a31x3x1t−τ31+a32x3x2t−τ32dt+σ3x3x3+N3∗dω3t.
Similarly, the stability of (6) around the equilibrate state of (1) is equivalent to the stability of zero solution of (24). Consequently, we only need to study the stability of zero solution of (24). By use of transformation, system (24) is equivalent to the following system:(25)dx1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds=−a11N1∗x1+a12N1∗x2+a13N1∗x3−a11x12+a12x1x2t−τ12+a13x1x3t−τ13dt+σ1x1x1+N1∗dω1t,dx2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds=a21N2∗x1−a22N2∗x2+a23N2∗x3−a22x22+a21x2x1t−τ21+a23x2x3t−τ23dt+σ2x2x2+N2∗dω2t,dx3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds=a31N3∗x1+a32N3∗x2−a33N3∗x3−a33x32+a31x3x1t−τ31+a32x3x2t−τ32dt+σ3x3x3+N3∗dω3t.
Consider the following linear case of (25):(26)dx1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds=−a11N1∗x1+a12N1∗x2+a13N1∗x3dt+σ1x1N1∗dω1t,dx2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds=a21N2∗x1−a22N2∗x2+a23N2∗x3dt+σ2x2N2∗dω2t,dx3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds=a31N3∗x1+a32N3∗x2−a33N3∗x3dt+σ3x3N3∗dω3t.
For system (25), the following result holds.Theorem 3.
If the followingH2 holds,(27)H2Δ˜1−a21−a31−a12Δ˜2−a32−a13−a23Δ˜3N1∗N2∗N3∗>2τ0ξ1η1ξ20η2ξ3η30N1∗2N2∗2N3∗2,then system (25) around the equilibrium state Ni∗ is stable in probability; i.e., system (6) around the equilibrium state Ni∗ is stable in probability.Proof.
Applying the same manner as before, consider the linear system (26) and define V1,V2, and V3 as before. By using the Itô formula and computing the derivatives of V1, V2, and V3 along solutions of (26), then(28)LV1=2x1+a12N1∗∫t−τ12tx2sds+a13N1∗∫t−τ13tx3sds×−a11N1∗x1+a12N1∗x2+a13N1∗x3+σ12N1∗2x12≤−2a11−a12−a13−σ12N1∗N1∗x12+a12N1∗x22+a13N1∗x32+a12a12+a13N1∗2x22τ+a13a13+a12N1∗2x32τ+a12a12+a13N1∗2∫t−τ12tx22sds+a13a12+a13N1∗2∫t−τ13tx32sds,LV2=2x2+a21N2∗∫t−τ21tx1sds+a23N2∗∫t−τ23tx3sds×a21N2∗x1−a22N2∗x2+a23N2∗x3+σ2N2∗2x2≤−2a22−a21−a23−σ22N2∗N2∗x22+a21N2∗x22+a23N2∗x32+a21a21+a23N2∗2x12τ+a23a21+a23N2∗2x32τ+a21a21+a23N2∗2∫t−τ21tx12sds+a23a21+a23N2∗2∫t−τ23tx32sds,LV3=2x3+a31N3∗∫t−τ31tx1sds+a32N3∗∫t−τ32tx2sds×a31N3∗x1+a32N3∗x2−a33N3∗x3+σ32N3∗2x32≤−2a33−a31−a32−σ32N3∗N3∗x32+a31N3∗x32+a32N3∗x32+a32a31+a32N3∗2x22τ+a31a31+a32N3∗2x12τ+a31a31+a32N3∗2∫t−τ31tx12sds+a32a31+a32N3∗2∫t−τ32tx22sds.
The rest proof is similar to proofs of Theorems1 and 2, so we omit it. The proof is completed.Remark 5.
Theorems1–3 show that both time delays (τij,i,j=1,2,3) and stochastic disturbances (σi,i=1,2,3) have important impacts on the globally asymptotical stability and stability in probability around the positive equilibrium state. Conditions H1 and H2 imply that too large time delays or white noises may destroy the stability of above systems, which are also verified by numerical simulations (see Figures 1 and 2 in the next section).Figure 1
Unstable cases with large time delay, whereσ1=0.2,σ2=0.1,σ3=0.3,andτ=0.8. Simulations reveal that too large time delay destroys the stability of (4) and (6), respectively. (a) The state graph of system (4); (b) the state graph of system (6).
(a)(b)Figure 2
Unstable cases for (4) and (6) with large stochastic perturbations, respectively, which show that too large stochastic disturbances destroy the stability of above systems. (a) For system (4), with σ1=2,σ2=2,σ3=2,andτ=0.5; (b) for system (6), with σ1=0.9,σ2=0.9,σ3=0.9,andτ=0.5.
(a)(b)Remark 6.
Theorems1–3 highlight the effects of time delays and stochastic disturbances on the stability around the positive equilibrium state, where conditions H1 and H2 seem complexity, but they are not difficult to be verified by computation software like Matlab.


## 4. Numerical Simulations

In this section, we present some numerical simulations to verify the theoretical result. After giving values of all parameters, by applying the method [24] and writing some suitable Matlab code, we can obtain the numerical results as follows. Let a11=1,a12=0.2,a13=0.3,a21=0.3,a22=1,a23=0.2,a31=0.2,a32=0.3,a33=1,r1=1,r2=1.2,andr3=1.5. Then, an easy computation yields that A=0.785,A1=1.783,A2=1.928,A3=2.098 and N1∗=2.2713,N2∗=2.4561,N3∗=2.6726.Ifσ1=0.2,σ2=0.1,σ3=0.3,andτ=0.5, with the help of Matlab, by computation, we have(29)Δ1=1.4824,Δ2=1.4959,Δ3=1.4663,Δ˜1=1.4091,Δ˜2=1.4754,Δ˜3=1.2592,and hence,(30)Δ1−a21−a31−a12Δ2−a32−a13−a23Δ3N1∗N2∗N3∗=2.09562.41802.7462,Δ˜1−a21−a31−a12Δ˜2−a32−a13−a23Δ˜3N1∗N2∗N3∗=1.92912.36772.1927,2τ0ξ1η1ξ20η2ξ3η30N1∗2N2∗2N3∗2=1.61911.58731.37719.By verification, bothH1 and H2 hold, and then, Theorems 2 and 3 hold. Theorem 2 implies system (4) is stable around the equilibrium of (1). The state graphs of each population are illustrated in Figure 3. Furthermore, Theorem 3 indicates system (6) around the equilibrium Ni∗ is also stable. The state graphs of each population are illustrated in Figure 4.Figure 3
Stable case of system (4) with σ1=0.2,σ2=0.1,σ3=0.3,andτ=0.5: (a) equilibrium of (1) with N1∗=2.2713,N2∗=2.4561,andN3∗=2.6726; (b) the state graph of three species of (4); (c), (d), and (e) are state graphs of populations N1,N2, and N3 of (4) around their corresponding equilibrium points N1∗,N2∗, and N3∗, respectively.
(a)(b)(c)(d)(e)Figure 4
Stable case of system (6) with σ1=0.2,σ2=0.1,σ3=0.3,andτ=0.5: (a) the state graph of three species; (b), (c), and (d) are the state graphs of the populations N1,N2, and N3 around their corresponding equilibrium points N1∗,N2∗, and N3∗, respectively.
(a)(b)(c)(d)Ifτ=0.8, other parameters are same as before, then(31)2τ0ξ1η1ξ20η2ξ3η30N1∗2N2∗2N3∗2=2.59062.53972.2033.An easy computation yields that neitherH1 nor H2 holds; hence, (4) and (6) around the equilibrium point may be unstable (see Figures 1(a) and 1(b), respectively). The state graphs of each population in Figure 1 show that too large time delays destroy the stability of (4) and (6).Finally, ifσ1=2,σ2=2,σ3=2,τ=0.5, then Δ1=−0.2611,Δ2=−0.1286,Δ3=0.0434. Hence, H1 does not hold and system (4) may be unstable, which is shown in Figure 2(a). If σ1=0.9,σ2=0.9,σ3=0.9,τ=0.5, then Δ˜1=−0.3398,Δ˜2=−0.4894,Δ˜3=−0.6648. Hence, H2 does not hold and system (6) may be unstable too (see Figure 2(b)). The state graphs of each population in Figure 2 show that too large stochastic disturbances destroy the stability of above systems.


## 5. Conclusion and Future Direction

In this paper, we investigate the stability of two three-species cooperative systems with time delays and stochastic disturbances. Theorem1 gives the sufficient conditions assuring the globally asymptotical stability around the equilibrium state of the corresponding deterministic system. Theorems 2 and 3 imply that systems (4) and (6) are stable in probability around the equilibrium states under some conditions, respectively. By giving Remarks 3–5, we clarify the difference of our proof from some existing methods and, particularly, show that time delays and stochastic perturbations bring some significant impacts on the globally asymptotical stability and stability in probability.In this paper, we assume time delays are constants, whereas in practice, time delays may be time-varying (see [25–27]), so it is necessary to study the dynamics with time-varying delays. Furthermore, for multispecies predator-prey system or multispecies competitive system, whether some similar results can be obtained? These are interesting and left for our future work.



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