# Dynamical Behavior of the Stochastic Delay Mutualism System

Authors: Peiyan Xia; Daqing Jiang; Xiaoyue Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We discuss the dynamical behavior of the stochastic delay three-specie mutualism system. We develop the technique for stochastic differential equations to deal with the asymptotic property. Using it we obtain the existence of the unique positive solution, the asymptotic properties, and the nonpersistence. Finally, we give the numerical examinations to illustrate our results.


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## Body

## 1. Introduction

The classical Lotka-Volterra model for two mutualistic species is described by the ordinary differential equation (ODE)(1)x˙1(t)=x1(t)[r1-a11x1(t)+a12x2(t)],x˙2(t)=x2(t)[r2+a21x1(t)-a22x2(t)].
There are many extensive literatures concerned with the dynamics of this model and we do not mention them here except [1]. Goh [1] showed that if a11a22>a12a21 holds, system (1) has a stable and globally attractive equilibrium point x*=(x1*,x2*) with the following property:
(2)x1(t)⟶x1*,x2(t)⟶x2*,ast⟶∞,
where
(3)x1*=r1a22+r2a12a11a22-a12a21>0,x2*=r2a11+r1a21a11a22-a12a21>0.In fact, in many physical as well as biological systems, many studies indicate that time delay widely exists in nature, for examples, in [2–7]. When the growth rate of each specie is affected by the time delay, as a result, (1) becomes a delay differential equation (DDE)
(4)x˙1(t)=x1(t)[r1-a11x1(t-τ)+a12x2(t-τ)],x˙2(t)=x2(t)[r2+a21x1(t-τ)-a22x2(t-τ)].
In [4], He and Gopalsamy obtained a supercritical Hopf-bifurcation of (4) at τ=τ* (a constant) and proved that the equilibrium (x1*,x2*) is no longer asymptotically stable as the delay increases to τ*. If only the interspecific positive feedback terms are affected by the delay, (1) becomes
(5)x˙1(t)=x1(t)[r1-a11x1(t)+a12x2(t-τ)],x˙2(t)=x2(t)[r2+a21x1(t-τ)-a22x2(t)].
The positive equilibrium (x1*,x2*) of (5) is globally attractive if a11a22>a12a21 holds, which implies the delay is harmless.Population systems are often subject to environmental noise and many authors have investigated the dynamical behaviors of stochastic population systems, for examples, in [8–26]. May [27] revealed that the parameters of the stochastic systems always fluctuate around their average values and the solution also fluctuates around its average value. If we still use ri to denote the average growth rate, then the intrinsic growth rate becomes
(6)ri⟶ri+σiB˙i(t),
where B˙i(t) is white noise and σi is a positive constant representing the intensity of the noise. As a result, the mutualism system (1) becomes a stochastic differential equation (SDE)
(7)x˙1(t)=x1(t)[(r1-a11x1(t)+a12x2(t))dt+σ1dB1(t)],x˙2(t)=x2(t)[(r2+a21x1(t)-a22x2(t))dt+σ2dB2(t)].
Ji et al. in [14] analyzed the long-time asymptotic behavior of the system (7) and obtained the ergodic property and its stationary distribution.Let us take a further step by considering a 3-dimensional mutualism system(8)x˙1(t)=x1(t)[r1-a11x1(t)+a12x2(t-τ)+a13x3(t-τ)],x˙2(t)=x2(t)[r2+a21x1(t-τ)-a22x2(t)+a23x3(t-τ)],x˙3(t)=x3(t)[r3+a31x1(t-τ)+a32x2(t-τ)-a33x3(t)],
subject to the white noise. As a result, it becomes a stochastic delay differential equation (SDDE)
(9)dx1(t)=x1(t)[(r1-a11x1(t)+a12x2(t-τ)hhhhhhhh+a13x3(t-τ))dt+σ1dB1(t)],dx2(t)=x2(t)[(r2+a21x1(t-τ)-a22x2(t)hhhhhhhh+a23x3(t-τ))dt+σ2dB2(t)],dx3(t)=x3(t)[(r3+a31x1(t-τ)+a32x2(t-τ)hhhhhhhh-a33x3(t))dt+σ2dB2(t)].Our aim is to investigate the long-time asymptotic behavior of SDDE (9). This paper is organized as follows. In order to obtain better dynamic properties of SDDE (9), we show that there exists a unique global positive solution with any initial positive value under some assumptions in Section 2. Then, we estimate the expectation in time average of the distance between the solution of (9) and the positive equilibrium point of the deterministic model (8); namely,
(10)limsup⁡t→∞1t∫0t‍E∥x(t)-x*∥≤K(σ),
where x* is the unique positive equilibrium point of system (8). In Section 3, we prove that system (9) is persistent in time average as the intensity of the white noise is small and yields the limit of the solution in time average. In Section 4, we obtain the nonpersistence of system (9) as the intensity of noise is big. Finally, in Section 5, we illustrate our results by some numerical examinations.Throughout this paper, unless otherwise specified, let(Ω,{Ft}t≥0,P) denote a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). Denote by R+3 the positive cone in R3; namely, R+3={x=(x1,x2,x3)T∈R3:xi>0,i=1,2,3}. Let τ>0 and denote by C([-τ,0];R+3) the family of the continuous functions from [-τ,0] to R+3. For x∈R3, its norm is denoted by |x|=|x1|+|x2|+|x3|.To discuss the dynamical behavior of (9), we impose the following assumption.Assumption 1.
Consideraij>0,i,j=1,2,3 and a11>a12+a13, a22>a21+a23, a33>a31+a32.Assumption 2.
Considerri>σi2/2,i=1,2,3.


## 2. Existence and Uniqueness of the Positive Solution

In population dynamics, the existence of the global positive solution is necessary. In order for a SDE to have a unique global (i.e., no explosion at any finite time) solution for any given initial value, its coefficients are generally required to satisfy the linear growth condition and local Lipschitz condition (Arnold et al. [28], Mao [23]). However, the coefficients of system (9) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (9) may explode at a finite time. So we prepare the following useful lemma and then yield the existence of the positive solution by using it.Lemma 3.
Under Assumption1, then we have
(11)(1)a22a33-a23a32>0,a11a33-a13a31>0,a11a22-a12a21>0,(2)(a22a33-a23a32)+(a33a21+a31a23)+(a32a21+a31a22)>0,(3)(a11a33-a13a31)+(a33a12+a32a13)+(a11a32+a31a12)>0,(4)(a11a22-a12a21)+(a12a23+a22a13)+(a11a23+a21a13)>0,(5)|-a11a12a13a21-a22a23a31a32-a33|=-a11a22a33+a11a23a32+a22a13a31hhhhhhhhhhhhhhhhhhh+a33a12a21+a12a23a31+a21a32a13hhhhhhhhhhhhhhhhh<0.Proof.
The assertions(1)–(5) are obviously obtained from a11>a12+a13, a22>a21+a23, a33>a31+a32, and aij>0,i,j=1,2,3. Here we omit the proof.Theorem 4.
Under Assumption1, for any given initial value x(·)∈C([-τ,0];R+3), there is a unique positive solution x(t) to system (9) on t≥-τ and the solution will remain in R+3 with probability  1; namely, x(t)∈R+3  for all t≥-τ a. s.Proof.
Firstly, define aC2 function V1:R+3→R+ by
(12)V1(x1,x2,x3)=c1[x1(t)-1-log⁡x1(t)]+c2[x2(t)-1-log⁡x2(t)]+c3[x3(t)-1-log⁡x3(t)],
where
(13)c1=((a22a33-a23a32)+(a33a21+a31a23)+(a32a21+a31a22))×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1,c2=((a11a33-a13a31)+(a33a12+a32a13)+(a11a32+a31a12))×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1,c3=((a11a22-a12a21)+(a12a23+a22a13)+(a11a23+a21a13))×(a11a22a33-a11a23a32-a22a13a31hhhhh-a33a12a21-a12a23a31-a21a32a13)-1.
From Lemma 3, ci>0,i=1,2,3. So, V1(x1,x2,x3) is nonnegative. By Itô’s formula, we obtain
(14)dV1(x1,x2,x3)=LV1(x1,x2,x3)dt+∑i=13‍ci[xi(t)-1]σidBi(t),
where
(15)LV1(x1,x2,x3)=c1[x1(t)-1][r1-a11x1(t)+a12x2(t-τ)+a13x3(t-τ)]+c2[x2(t)-1][r2+a21x1(t-τ)-a22x2(t)+a23x3(t-τ)]+c3[x3(t)-1][r3+a31x1(t-τ)+a32x2(t-τ)-a33x3(t)]+c1σ12+c2σ22+c3σ322.
By Young’s inequality, we have
(16)LV1(x1,x2,x3)=c1[σ122(r1+a11)x1(t)-a12x2(t-τ)hhhhhhh-a13x3(t-τ)-a11x12(t)+a12x1(t)x2(t-τ)hhhhhhh+a13x1(t)x3(t-τ)-r1+σ122]+c2[σ222(r2+a22)x2(t)-a21x1(t-τ)hhhhhhh-a23x3(t-τ)-a22x22(t)+a21x2(t)x1(t-τ)hhhhhhh+a23x2(t)x3(t-τ)-r2+σ222]+c3[σ322(r3+a33)x3(t)-a31x1(t-τ)hhhhhhh-a32x2(t-τ)-a33x32(t)+a31x3(t)x1(t-τ)hhhhhhh+a32x3(t)x2(t-τ)-r3+σ322]≤c1[σ122(r1+a11)x1(t)-a12x2(t-τ)hhhhhhh-a13x3(t-τ)+(-a11+a122+a132)x12(t)hhhhhhh+a122x22(t-τ)+a132x32(t-τ)-r1+σ122]+c2[σ222(r2+a22)x2(t)-a21x1(t-τ)-a23x3(t-τ)hhhhhhh+(-a22+a212+a232)x22(t)hhhhhhh+a212x12(t-τ)+a232x32(t-τ)-r2+σ222]+c3[σ322(r3+a33)x3(t)-a31x1(t-τ)hhhhhhh-a32x2(t-τ)+(-a33+a312+a322)x32(t)hhhhhhh+a312x12(t-τ)+a322x22(t-τ)-r3+σ322].
Secondly, define
(17)V2(x1,x2,x3)=∫tt+τ‍[(a12c12+a32c32)x22(s-τ)hhhhh+(a13c12+a23c22)x32(s-τ)hhhhh+(a21c22+a31c32)x12(s-τ)]ds.
By Itô’s formula, we have
(18)dV2(x1,x2,x3)=[(a12c12+a32c32)x22(t)+(a13c12+a23c22)x32(t)+(a21c22+a31c32)x12(t)-(a12c12+a32c32)x22(t-τ)-(a13c12+a23c22)x32(t-τ)-(a21c22+a31c32)x12(t-τ)]dt.
Therefore, let
(19)V(x1,x2,x3)=V1(x1,x2,x3)+V2(x1,x2,x3).
Equalities (16) and (18) imply that
(20)dV(x1,x2,x3)=LV(x1,x2,x3)+∑i=13‍ci[xi(t)-1]σidBi(t),
where
(21)LV(x1,x2,x3)≤c1[σ122-x12(t)(a11-a12-a13)+(r1+a11)x1(t)-a12x2(t-τ)-a13x3(t-τ)-r1+σ122]+c2[σ222-x22(t)(a22-a21-a23)+(r2+a22)x2(t)-a21x1(t-τ)-a23x3(t-τ)-r2+σ222]+c3[σ322-x32(t)(a33-a31-a32)+(r3+a33)x3(t)-a31x1(t-τ)-a32x2(t-τ)-r3+σ322]≤K,
where K is a positive constant. By the method similar to that in [24], the proof is therefore completed.Under Assumption1, DDE (8) has a positive equilibrium x*=(x1*,x2*,x3*) which is globally attractive while the system with the stochastic perturbation has a unique global positive solution. It is natural to ask how to estimate the distance between the solution of the deterministic system and the solution of the stochastic system. The following theorem gives us an answer.Theorem 5.
Under Assumption1, system (9) has the following property:
(22)limsup⁡t→∞1tE∫0t‍{(a11-a12-a13)c1+12[x1(s)-x1*]2hhhhh.hhhhhh+(a22-a21-a23)c2+12[x2(s)-x2*]2hhhhh.hhhhhh+(a33-a31-a32)c3+12[x3(s)-x3*]2}ds≤∑i=13cixi*σi22,
where x(t) is a solution on t≥τ to (9) with an initial value x(·)∈C([-τ,0];R+3), ci(i=1,2,3) is defined in the proof of Theorem 4, and x*=(x1*,x2*,x3*) is the unique positive equilibrium of system (8); namely,
(23)x1*=((a22a33-a23a32)r1+(a12a33+a13a32)r2+(a12a23+a13a22)r3)×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1,x2*=((a21a33+a23a31)r1+(a11a33-a13a31)r2+(a11a23+a13a21)r3)×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1,x3*=((a21a32+a22a31)r1+(a11a32+a12a31)r2+(a11a22-a12a21)r3)×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1.For brevity, we will give the proof in the appendix.


## 3. Persistence in Time Average

For convenience, we denote the unique global solution of system (9) by x(t,ξ) with an initial data ξ∈C([-τ,0];R3+). Theorem 4 shows that the solution of system (9) will remain positive under Assumption 1. This property gives us an opportunity to investigate how the solution varies in R3+. In population dynamics, one of the most attractive properties is persistence which means all species will coexist. Now we give the definition of persistence in time average.Definition 6.
System (9) is persistent in time average, if, for any initial data ξ∈C([-τ,0];R+3), the solution x(t,ξ) has the property that
(24)0<liminf⁡t→∞1t∫0t‍xi(s)ds≤limsup⁡t→∞1t∫0t‍xi(s)ds<+∞a.s.i=1,2,3.To prove that system (9) is persistent in time average, we will cite a lemma. Jiang and Shi in [17] discussed a stochastic nonautonomous logistic equation
(25)dN(t)=N(t)[(a(t)-b(t)N(t))dt+α(t)dB(t)],
where B(t) is 1-dimensional standard Brownian motion, N(0)=N0, and N0 is independent of B(t). They obtained the following result.Lemma 7 (see [17]).
Assume thata(t),b(t), and α(t) are bounded continuous functions defined on [0,∞), a(t)>0, and b(t)>0. Then there exists a unique continuous positive solution of (25) for any initial value N(0)=N0>0, which is global and represented by
(26)N(t)=(exp⁡{∫0t‍[a(s)-α2(s)2]ds+α(s)dB(s)})×(1N0+∫0t‍b(s)exp⁡{∫0s‍[a(τ)-α2(τ)2]dτhhhhh.hhhhhhhhhhhh+α(τ)dB(τ)∫0s‍}ds)-1,t≥0.
Moreover, the solution N(t) has the property that
(27)lim⁡t→∞log⁡N(t)t=0,a.s.Remark 8.
Letϕi(t), i=1,2,3, be the solution of the following equation:
(28)dϕi(t)=ϕi(t)[(ri-aiiϕi(t))dt+σidBi(t)],ffffffffffffffffffffft≥0,i=1,2,3,
with initial value ϕi(0)=xi(0). From Lemma 7, we have
(29)ϕi(t)=e(ri-(σi2/2))t+σiBi(t)(1/xi(0))+aii∫0t‍e(ri-(σi2/2))s+σiBi(s)ds,t≥0.
From the result in [13], we know
(30)lim⁡t→∞1t∫0t‍ϕi(s)ds=ri-σi2/2aii,lim⁡t→∞log⁡ϕi(t)t=0,a.s.i=1,2,3
provided Assumption 2.Theorem 9.
Under Assumptions1 and 2, system (9) is persistent in time average.Proof.
From Lemma7, we know
(31)xi(t)=e∫0t‍(ri+∑j=1,j≠i3aijxj(s-τ)-(σi2/2))ds+σiBi(t)(1/xi(0))+aii∫0t‍e∫0s‍(ri+∑j=1,j≠i3aijxj(u-τ)-(σi2/2))du+σiBi(s)ds=(1)(∫0t‍e-∫st‍(ri+∑j=1,j≠i3aijxj(u-τ)-(σi2/2))du+σiBi(s)ds]e-σiBi(t)hhhhhh×[1xi(0)e-∫0t‍(ri+∑j=1,j≠i3aijxj(s-τ)-(σi2/2))ds+aiihhhhhhhhh×∫0t‍e-∫st‍(ri+∑j=1,j≠i3aijxj(u-τ)-(σi2/2))du+σiBi(s)ds])-1.
From Remark 8, yields
(32)xi(t)≥ϕi(t),i=1,2,3.
Together with Lemma 7, it is easy to obtain
(33)ri-σi2/2aii≤liminf⁡t→∞1t∫0t‍xi(s)ds,liminf⁡t→∞log⁡xi(t)t≥0a.s.i=1,2,3.
The inequality limsup⁡t→∞(1/t)∫0t‍xi(s)ds<+∞ a.s. i=1,2,3 will be shown in Theorem 11.Theorem9 shows that system (9) is persistent in time average if the intensity of noise is small. Next we want to obtain the limit of the solution in time average of system (9). We begin from the lemma in [29].Lemma 10.
Letf∈C[[0,∞)×Ω,(0,∞)], F(t)∈((0,∞)×Ω,R). If there exist positive constants λ0 and λ such that
(34)log⁡f(t)≥λt-λ0∫0t‍f(s)ds+F(t),t≥0,a.s.,
and lim⁡t→∞(F(t)/t)=0 a.s., then
(35)liminf⁡t→∞1t∫0t‍f(s)ds≥λλ0,a.s.Theorem 11.
Under Assumptions1 and 2, for any initial data ξ∈C([-τ,0];R+3), the solution x(t,ξ) has the property that
(36)lim⁡t→∞1t∫0t‍xi(s)ds=x~i*a.s.i=1,2,3,
where
(37)x~1*=((a22a33-a23a32)(r1-σ122)+(a12a33+a13a32)×(r2-σ222)+(a12a23+a13a22)(r3-σ322))×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1,x~2*=((a21a33+a23a31)(r1-σ122)+(a11a33-a13a31)×(r2-σ222)+(a11a23+a13a21)(r3-σ322))×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1,x~3*=((a21a32+a22a31)(r1-σ122)+(a11a32+a12a31)×(r2-σ222)+(a11a22-a12a21)(r3-σ322))×(a11a22a33-a11a23a32-a22a13a31-a33a12a21-a12a23a31-a21a32a13)-1.The mathematical derivations are lengthy; we will give the proof in the appendix.


## 4. Nonpersistence

In this section, we will show that the system (9) is nonpersistent if the intensity of the noise is big enough; however, it does not occur to the deterministic system. First of all, we give the definition of nonpersistence.Definition 12.
System (9) is nonpersistent, if there are positive constants b1,b2,b3 such that
(38)lim⁡t→∞∏i=13‍xibi(t)=0a.s.Theorem 13.
Under Assumption1, if  C<0 holds, system (9) is nonpersistent, where C=b11(r1-(σ12/2))+b12(r2-(σ22/2))+b13(r3-(σ32/2)),b11>0; b12,b13 is defined by (B.24).Proof.
It follows from(B.25) that
(39)b11log⁡x1(t)+b12log⁡x2(t)+b13log⁡x3(t)≤(b11r-1+b12r-2+b13r-3)t-m1∫0t‍x1(s)ds+∑i=13‍b1iσiBi(t)+∑i=13‍b1iUi≤(b11r-1+b12r-2+b13r-3)t+∑i=13‍b1iσiBi(t)+∑i=13‍b1iUi.
Together with lim⁡t→∞(Bi(t)/t)=0andlim⁡t→∞(Ui/t)=0,i=1,2,3, we have
(40)limsup⁡t→∞1t∑i=13‍b1ilog⁡xi(t)≤b11r-1+b12r-2+b13r-3.
If b11r-1+b12r-2+b13r-3<0 holds, it follows
(41)lim⁡t→∞∏i=13‍xib1i(t)=0a.s.
Hence, system (9) is nonpersistent. The proof is completed.By a similar method, we can yield the following theorems.Theorem 14.
Under Assumption1, if  C<0 holds, system (9) is nonpersistent, where C=b21(r1-(σ12/2))+b22(r2-(σ22/2))+b23(r3-(σ32/2)), b22>0, b21=((-a21a33-a23a31)/(a13a31-a11a33))b22,b23=((-a13a21-a11a23)/(a13a31-a11a33))b22.Theorem 15.
Under Assumption1, if  C<0 holds, system (9) is nonpersistent, where C=b31(r1-(σ12/2))+b32(r2-(σ22/2))+b23(r3-(σ32/2)),b33>0,b31=((-a31a22-a32a21)/(a12a21-a11a22))b33,b32=((-a31a12-a32a11)/(a12a21-a11a22))b33.


## 5. Numerical Examinations

In this section, we give the numerical examinations to illustrate above results. By the method mentioned in [30], consider the discrete equation:
(42)x1,k+1=x1,k+x1,k[12(r1-a11x1,k+a12x2,k-m+a13x3,k-m)Δthhhhh.hhh+σ1ε1,kΔt+12σ12(ε1,k2Δt-Δt)],x2,k+1=x2,k+x2,k[12(r2+a21x1,k-m-a22x2,k+a23x3,k-m)Δthhhhhhhh+σ2ε2,kΔt+12σ22(ε2,k2Δt-Δt)],x3,k+1=x3,k+x3,k[12(r3+a31x1,k-m+a32x2,k-m-a33x3,k)Δthhhhhhhh+σ3ε3,kΔt+12σ32(ε3,k2Δt-Δt)],
where m represents the integer part of τ/Δt-1. Choosing suitable parameters in the system, by Matlab we get the simulation figures with initial value (x1(t),x2(t),x3(t))≡(0.7,0.3,0.9),t∈[-τ,0]. (For convenience we let the initial value be a constant function; otherwise we have to give m+1 values.) The time step Δt=0.005; we always choose
(43)(a11a12a13a21a22a23a31a32a33)=(0.70.20.40.10.50.30.20.40.8),(r1r2r3)=(0.40.20.3).
Then a11-a12-a13=0.1,a22-a21-a23=0.2,a33-a31-a32=0.2,(x1*,x2*,x3*)=(2.2679,2.0268,1.9554). By choosing different intensities of the noise and time delays, we obtain the following cases to illustrate our results.Case 1 (persistence).
Choosingτ=20.2σ1=0.04,σ2=0.08,σ3=0.07, then we have ri>σi2/2,i=1,2,3. Hence all assumptions of Theorem 11 are satisfied. Figure 1 shows that the solution fluctuates in a small zone.The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=0.04,σ2=0.08,σ3=0.07,Δt=0.005.
(a)
(b)
(c)
(d)
(e)
(f)Case 2 (nonpersistence (we only illustrate the first situation)).
(1)
x
1
(
t
)
Is Disturbed by a Big Noise, Which Leads to Nonpersistence. Choosing τ=20.2,σ1=1.53,σ2=0.08,σ3=0.07,b11=0.10,b12=0.1143,b13=0.0929, then we have ri>σi2,i=2,3 but b11(r1-(σ12/2))+b12(r2-(σ22/2))+b13(r3-(σ32/2))<0. For convenience we shorten b1i to bi. Hence the assumptions of Theorem 13 are satisfied, and system (9) is nonpersistent (see Figure 2(d)). In addition, since ri>σi2,i=2,3, x2(t),x3(t) do not tend to zero in time average by Theorem 9. So lim⁡t→∞x1(t)=0 a.s.  (see Figures 2(a)–2(c)).
(2) The Second Specie
x
2
(
t
)
Is Disturbed by the Big White Noise, Which Leads to the Nonpersistence. Choosing τ=20.2,σ1=0.04,σ2=1.17,σ3=0.07,b1=0.10,b2=0.1143,b3=0.0929, then we have ri>σi2,i=1,3, but b1(r1-(σ12/2))+b2(r2-(σ22/2))+b3(r3-(σ32/2))<0. Hence the assumptions of Theorem 13 are satisfied, and system (9) is nonpersistent (see Figure 3(d)). In addition, since ri>σi2,i=1,3,x1(t),x3(t) do not tend to zero in time average by Theorem 9. So lim⁡t→∞x2(t)=0 a.s. (see Figures 3(a)–3(c)).
(3) The Third Specie
x
3
(
t
)
Is Disturbed by the Big White Noise, Which Leads to the Nonpersistence. Choosing τ=20.2,σ1=0.04,σ2=0.08,σ3=1.36,b1=0.10,b2=0.1143,b3=0.0929, then we have ri>σi2,i=1,2, but b1(r1-(σ12/2))+b2(r2-(σ22/2))+b3(r3-(σ32/2))<0. Hence the assumptions of Theorem 13 are satisfied, and system (9) is nonpersistent (see Figure 4(d)). In addition, since ri>σi2,i=1,2, x1(t),x2(t) do not tend to zero in time average by Theorem 9. So we have lim⁡t→∞x3(t)=0 a.s. (see Figures 4(a)–4(c)).
(4) The First Two Species
x
1
(
t
)
,
x
2
(
t
)
Are Disturbed by the Big White Noises, Which Leads to the Nonpersistence. Choosing τ=20.2,σ1=1.25,σ2=1.01,σ3=0.07,b1=0.10,b2=0.1143,b3=0.0929, then we have r3>σ32, but b1(r1-(σ12/2))+b2(r2-(σ22/2))+b3(r3-(σ32/2))<0. Hence the assumptions of Theorem 13 are satisfied; then system (9) is nonpersistent (see Figure 5(d)). In addition, since r3>σ32, x3(t) does not tend to zero in time average by Theorem 9. So we have lim⁡t→∞xi(t)=0,i=1,2 a.s. (see Figures 5(a)–5(c)).
(5) The First and the Third Species
x
1
(
t
)
,
x
3
(
t
)
Are Disturbed by the Big White Noises, Which Leads to the Nonpersistence. Choosing τ=20.2, σ1=1.25, σ2=0.08,σ3=1.12,b1=0.10, b2=0.1143, b3=0.0929, then we have r2>σ22, but b1(r1-(σ12/2))+b2(r2-(σ22/2))+b3(r3-(σ32/2))<0. Hence the assumptions of Theorem 13 are satisfied, and system (9) is nonpersistent (see Figure 6(d)). In addition, since r2>σ22, x2(t) does not tend to zero in time average by Theorem 9. So we have lim⁡t→∞xi(t)=0,i=1,3 a.s. (see Figures 6(a)–6(c)).
(6) The Last Two Species
x
2
(
t
)
,
x
3
(
t
)
Are Disturbed by the Big White Noises, Which Leads to Nonpersistence. Choosing τ=20.2,σ1=0.04,σ2=1.17,σ3=1.36,b1=0.10,b2=0.1143,b3=0.0929, then we have r1>σ12, but b1(r1-(σ12/2))+b2(r2-(σ22/2))+b3(r3-(σ32/2))<0. Hence the assumptions of Theorem 13 are satisfied, and system (9) is nonpersistent (see Figure 7(d)). In addition, since r1>σ12, x1(t) does not tend to zero in time average by Theorem 9. So we have lim⁡t→∞xi(t)=0,i=2,3 a.s. (see Figures 7(a)–7(c)).
(7) All the Three Species
x
1
(
t
)
,
x
2
(
t
)
,
x
3
(
t
)
Are Disturbed by the Big White Noise, Which Leads to the Nonpersistence. Choosing τ=20.2,σ1=1.25,σ1=1.01,σ1=1.36,b1=0.10,b2=0.1143,b3=0.0929, then we have b1(r1-(σ12/2))+b2(r2-(σ22/2))+b3(r3-(σ32/2))<0. Hence the assumptions of Theorem 13 are satisfied, and system (9) is nonpersistent (see Figure 8(d)). So we have lim⁡t→∞xi(t)=0,i=1,2,3 a.s. (see Figures 8(a)–8(c)).The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=1.53,σ2=0.08,σ3=0.07,Δt=0.005.
(a)
(b)
(c)
(d)The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=0.04,σ2=1.17,σ3=0.07,Δt=0.005.
(a)
(b)
(c)
(d)The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=0.04,σ2=0.08,σ3=1.36,Δt=0.005.
(a)
(b)
(c)
(d)The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=1.25,σ2=1.01,σ3=0.07,Δt=0.005.
(a)
(b)
(c)
(d)The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=1.25,σ2=0.08,σ3=1.12,Δt=0.005.
(a)
(b)
(c)
(d)The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=0.04,σ2=1.17,σ3=1.36,Δt=0.005.
(a)
(b)
(c)
(d)The solution (imaginary line) of system (8) and the solution (real line) of system (9) with σ1=1.25,σ2=1.01,σ3=1.36,Δt=0.005.
(a)
(b)
(c)
(d)



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*Source: 781394-2014-05-29.xml*