# Analysis of Stochastic Delay Predator-Prey System with Impulsive Toxicant Input in Polluted Environments

Authors: Meng Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A stochastic delay predator-prey model in a polluted environment with impulsive
toxicant input is proposed and studied. The thresholds between stability
in time average and extinction of each population are obtained. Some
recent results are extended and improved greatly. Several simulation figures
are introduced to support the conclusions.


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## Body

## 1. Introduction

Environmental pollution by industries, agriculture, and other human activities is one of the most important socio-ecological problems in the world today. Due to toxins in the environment, lots of species have gone extinct, and many are on the verge of extinction. Thus, controlling the environmental pollution and the conservation of biodiversity are the major focus areas of all the countries around the world. This motivates scholars to study the effects of toxins on populations and to find out a theoretical persistence-extinction threshold.Recently, a lot of population models in a polluted environment have been proposed and investigated; here, we may mention, among many others, [1–23]. Particularly, Yang et al. [15] pointed out that in many cases toxicants should be emitted in regular pulses, for example, the use of pesticides and the pollution by heavy metals (see, e.g., [24]). Thus, they proposed the following two-species Lotka-Volterra predator-prey system in a polluted environment with impulsive toxicant input:
(1)dx1(t)dt=x1(t)[r10-r11C10(t)-a11x1(t)-a12x2(t)],dx2(t)dt=x2(t)[-r20-r21C20(t)+a21x1(t)-a22x2(t)],dC10(t)dt=k1Ce(t)-(g1+m1)C10(t),dC20(t)dt=k2Ce(t)-(g2+m2)C20(t),dCe(t)dt=-hCe(t),t≠nγ,n∈Z+,Δxi(t)=0,ΔCi0(t)=0,ΔCe(t)=b,t=nγ,n∈Z+,i=1,2,
where all the parameters are positive constants andΔf(t)=f(t+)-f(t), Z+={1,2,…};x1(t) and x2(t): the size of prey population and the predator population, respectively; ri0: the intrinsic growth rate of the ith population without toxicant; ri1: the ith population response to the pollutant present in the organism; Ci0(t): the concentration of toxicant in the ith organism; Ce(t): the concentration of toxicant in the environment; kCe(t): the organism’s net uptake of toxicant from the environment; gCi0(t)+mCi0(t): the egestion and depuration rates of the toxicant in the ith organism; hCe(t): the toxicant loss from the environment itself by volatilization and so on; γ: the period of the impulsive effect about the exogenous input of toxicant; b: the toxicant input amount at every time.Yang et al. [15] showed that in the following Lemma holds.Lemma 1.
For system (1), define
(2)Δ2=r10a21-r20a11,Δ¯2:=a21r11K1γ+a11r21K2γ,Ki=kibh(gi+mi).(a)
Ifr10<r11K1/γ, then limt→+∞xi(t)=0, i=1,2.(b)
Ifr10>r11K1/γ and Δ2<Δ¯2, then limsupt→+∞t-1∫0t‍x1(s)ds>0 and x2(t) goes to extinction.(c)
IfΔ2>Δ¯2, then limsupt→+∞t-1∫0t‍xi(s)ds>0,i=1,2.Some interesting and important problems arise naturally.(Q1) In the real world, the growth of species depends on various environmental factors, such as temperature, humidity and parasites and so forth. Therefore population models should be stochastic rather than deterministic (May [25]). Thus, what happens if model (1) is subject to stochastic noises?(Q2) In addition, time delays occur in almost every situation. Kuang [26] has pointed out that ignoring time delays means ignoring reality. Therefore, what happens if model (1) takes time delays into account?(Q3) Can we improve the results given in Lemma 1?The aim of this paper is to study the above problems. Suppose that stochastic noises mainly affect the growth rates, withri0→ri0+αiB˙i(t) (see, e.g., [27–39]), where B˙i(t) is a white noise and αi2 is the intensity of the noise. Moreover, taking time delays into account, we obtain the following model:
(3)dx1(t)=x1(t)[r10-r11C10(t)-a11x1(t)-a12x2(t-τ1)]dt+α1x1(t)dB1(t),dx2(t)=x2(t)[-r20-r21C20(t)-a22x2(t)+a21x1(t-τ2)]dt+α2x2(t)dB2(t),dC10(t)dt=k1Ce(t)-(g1+m1)C10(t),dC20(t)dt=k2Ce(t)-(g2+m2)C20(t),dCe(t)dt=-hCe(t),t≠nγ,n∈Z+,Δxi(t)=0,ΔCi0(t)=0,ΔCe(t)=b,t=nγ,n∈Z+,i=1,2,
with initial condition
(4)xi(t)=ϕi(t)>0,t∈[-τ,0];ϕi(0)>0,i=1,2,
where τi≥0, τ=max{τ1,τ2}, ϕi(t) is continuous on [-τ,0]. Our main result is the following theorem.Theorem 2.
For system (3), define
(5)θ1=r10-0.5α12,θ2=r20+0.5α22,Δ=a11a22+a12a21,Δ~1:=θ1a22+θ2a12,Δ~2:=θ1a21-θ2a11,Δ¯1:=a22r11K1γ-a12r21K2γ.(i)
Ifθ1<r11K1/γ, then both x1 and x2 go to extinction almost surely (a.s.); that is, limt→+∞xi(t)=0a.s.,i=1,2.(ii)
Ifθ1>r11K1/γ and Δ~2<Δ¯2, then x2(t) goes to extinction and x1 is stable in time average a.s.; that is,
(6)limt→+∞t-1∫0t‍x1(s)ds=θ1-r11K1/γa11>0,a.s.(iii)
IfΔ~2>Δ¯2, then both x1 and x2 are stable in time average a.s.
(7)limt→+∞t-1∫0t‍x1(s)ds=Δ~1-Δ¯1Δ>0,a.s.limt→+∞t-1∫0t‍x2(s)ds=Δ~2-Δ¯2Δ>0,a.s.Remark 3.
By comparing Lemma1 with our Theorem 2, we can see that on the one hand, if α1=α2=0 and τ1=τ2=0, then θi=ri0, Δ~i=Δi, i=1,2, and our stochastic delay system (3) becomes model (1); on the other hand, our results in Theorem 2 improve that in Lemma 1. Lemma 1 shows that the superior limit is positive, while Theorem 2 reveals that the limit exists and gives the explicit form of the limit. The contribution of this paper is therefore clear.


## 2. Proof

For the sake of simplicity, we introduce some notations:(8)R+2={a=(a1,a2)∈R2∣ai>0,i=1,2},〈f(t)〉=t-1∫0t‍f(s)ds;〈f(t)〉*=limsupt→+∞t-1∫0t‍f(s)ds,〈f(t)〉*=liminft→+∞t-1∫0t‍f(s)ds.Lemma 4.
For any given initial valueϕ(t)=(ϕ1(t),ϕ2(t))∈C([-τ,0],R+2), there is a unique global positive solution x(t)=(x1(t),x2(t))T to the first two equations of system (3) a.s.Proof.
The proof is similar to Hung [29] by defining
(9)V(x)=[x1-1-lnx1]+a11a22a212[x2-1-lnx2]+0.5a11∫t-τ2t‍x12(s)ds
and hence is omitted.To begin with, let us consider the following subsystem of (3):
(10)dC10(t)dt=k1Ce(t)-(g1+m1)C10(t),dC20(t)dt=k2Ce(t)-(g2+m2)C20(t),dCe(t)dt=-hCe(t),t≠nγ,n∈Z+,ΔC10(t)=0,ΔC20(t)=0,ΔCe(t)=b,t=nγ,n∈Z+,0≤C0(0)≤1,0≤Ce(0)≤1.Lemma 5 (see [13, 15]).
System (10) has a unique positive γ-periodic solution (C~10(t),C~20(t),C~e(t))T, and for each solution (C10(t),C20(t),Ce(t))T of (10), C10(t)→C~10(t), C20(t)→C~20(t), and Ce(t)→C~e(t) as t→∞. Moreover, Ci0(t)>C~i0(t) and Ce(t)>C~e(t) for all t≥0 if Ci0(0)>C~i0(0) and Ce(0)>C~e(0), i=1,2, where
(11)C~10(t)=C~10(0)e-(g1+m1)(t-nγ)+k1b(e-(g1+m1)(t-nγ)-e-h(t-nγ))(h-g1-m1)(1-e-hγ),C~20(t)=C~20(0)e-(g2+m2)(t-nγ)+k2b(e-(g2+m2)(t-nγ)-e-h(t-nγ))(h-g2-m2)(1-e-hγ),C~e(t)=be-h(t-nγ)1-e-hγ,C~10(0)=k1b(e-(g1+m1)γ-e-hγ)(h-g1-m1)(1-e-(g1+m1)γ)(1-e-hγ),C~20(0)=k2b(e-(g2+m2)γ-e-hγ)(h-g2-m2)(1-e-(g2+m2)γ)(1-e-hγ),C~e(0)=b1-e-hγ
for t∈(nγ,(n+1)γ] and n∈Z+. In addition,
(12)limt→+∞t-1∫0t‍C~i0(s)ds=kibh(gi+mi)γ=Kiγ,i=1,2.Lemma 6 (see [34]).
Suppose thatx(t)∈C[Ω×[0,+∞),R+]. (I) If there existσ and positive constants σ0,T such that
(13)lnx(t)≤σt-σ0∫0t‍x(s)ds+∑i=1n‍βiBi(t)fort≥T, where Bi(t) are independent standard Brownian motions and βi are constants, 1≤i≤n, then one has the following: if σ≥0, then 〈x〉*≤σ/σ0 a.s.; if σ<0, then limt→+∞x(t)=0a.s.(II) If there exist positive constantsσ0,T and σ such that
(14)lnx(t)≥σt-σ0∫0t‍x(s)ds+∑i=1n‍βiBi(t)fort≥T, then 〈x〉*≥σ/σ0  a.s.Now, let us consider the following auxiliary system:(15)dy1(t)=y1(t)[r10-r11C10(t)-a11y1(t)]dt+α1y1(t)dB1(t),dy2(t)=y2(t)[-r20-r21C20(t)+a21y1(t-τ2)-a22y2(t)]dt+α2y2(t)dB2(t),
with initial value ϕ(t)∈C([-τ,0],R+2).Lemma 7.
Ifθ1=r10-0.5α12>r11K1/γ, then the solution y(t) of system (15) obeys
(16)limt→+∞〈y1(t)〉=θ1-r11K1/γa11;limt→+∞y2(t)=0a.s.,ifΔ~2<Δ¯2;limt→+∞〈y2(t)〉=Δ~2-Δ¯2a11a22a.s.,ifΔ~2>Δ¯2.Proof.
By Lemma5,
(17)limt→+∞t-1∫0t‍Ci0(s)ds=limt→+∞t-1∫0t‍C~i0(s)ds=Kiγ,i=1,2.
Then, for allε>0, there exists T>0 such that
(18)Kiγ-ε≤〈Ci0(t)〉≤Kiγ+ε,t>T,i=1,2.
An application of Itô’s formula to (15) yields
(19)lny1(t)-lny1(0)=θ1t-r11∫0t‍C10(s)ds-a11∫0t‍y1(s)ds+α1B1(t),lny2(t)-lny2(0)=-θ2t-r21∫0t‍C20(s)ds+a21∫0t‍y1(s-τ2)ds-a22∫0t‍y2(s)ds+α2B2(t)=-θ2t-r21∫0t‍C20(s)ds+a21∫0t‍y1(s)ds-a21[∫t-τ2t‍y1(s)ds-∫-τ20‍y1(s)ds]-a22∫0t‍y2(s)ds+α2B2(t).
That is to say, we have shown that
(20)t-1lny1(t)y1(0)=θ1-r11〈C10(t)〉-a11〈y1(t)〉+t-1α1B1(t),(21)t-1lny2(t)y2(0)+t-1a21[∫t-τ2t‍y1(s)ds-∫-τ20‍y1(s)ds]=-θ2-r21〈C20(t)〉+a21〈y1(t)〉-a22〈y2(t)〉+t-1α2B2(t).
When (18) is used in (20), we can see that for t>T,
(22)t-1lny1(t)y1(0)≤θ1-r11K1γ+r11ε-a11〈y1(t)〉+t-1α1B1(t),(23)t-1lny1(t)y1(0)≥θ1-r11K1γ-r11ε-a11〈y1(t)〉+t-1α1B1(t).
Let ε be sufficiently small such that θ1-r11K1/γ-r11ε>0. Making use of (I) and (II) in Lemma 6 to (22) and (23), respectively, we have
(24)〈y1(t)〉*≤θ1-r11K1/γ+r11εa11,〈y1(t)〉*≥θ1-r11K1/γ-r11εa11.
It then follows from the arbitrariness of ε that
(25)limt→+∞〈y1(t)〉=θ1-r11K1/γa11.
Substituting (17) and (25) into (20) and noting that limt→+∞t-1B1(t)=0, one can derive that
(26)limt→+∞t-1lny1(t)=0,a.s.
Employing (20) and (21) in the expression a21ln(y1(t)/y1(0))+a11ln(y2(t)/y2(0)) yields
(27)a11t-1lny2(t)y2(0)+a21t-1lny1(t)y1(0)=Δ~2-r11a21〈C10(t)〉-r21a11〈C20(t)〉-a11a22〈y2(t)〉-t-1a11a21[∫t-τ2t‍y1(s)ds-∫-τ20‍y1(s)ds]+t-1[a21α1B1(t)+a11α2B2(t)].
In view of (25), we get
(28)limt→+∞t-1∫t-τ2t‍y1(s)ds=limt→+∞t-1(∫0t‍y1(s)ds-∫0t-τ2‍y1(s)ds)=0,a.s.
By (17), (26), (27), and (28), for allε>0, there exists T>0 such that, for t≥T,
(29)a11t-1lny2(t)y2(0)≤Δ~2-Δ¯2+ε-a11a22〈y2(t)〉+t-1[a21σ1B1(t)+a11σ2B2(t)],(30)a11t-1lny2(t)y2(0)≥Δ~2-Δ¯2-ε-a11a22〈y2(t)〉+t-1[a21σ1B1(t)+a11σ2B2(t)].
If Δ~2<Δ¯2, then we can choose ε sufficiently small such that Δ~2-Δ¯2+ε<0. Then, by (29) and (I) in Lemma 6, we obtain limt→+∞y2(t)=0a.s. If Δ~2>Δ¯2, then we can choose ε sufficiently small such that Δ~2-Δ¯2-ε>0. An application of (I) and (II) in Lemma 6 to (29) and (30), respectively, makes one observe that
(31)Δ~2-Δ¯2-εa11a22≤〈y2(t)〉*≤〈y2(t)〉*≤Δ~2-Δ¯2+εa11a22,a.s.
Therefore, using the arbitrariness of ε results in
(32)limt→+∞〈y2(t)〉=Δ~2-Δ¯2a11a22a.s.
This completes the proof.We are now in the position to prove our main results.Proof of Theorem2.
Applying Itô’s formula to (3) leads to
(33)lnx1(t)-lnx1(0)=θ1t-r11∫0t‍C10(s)ds-a11∫0t‍x1(s)ds-a12∫0t‍x2(s-τ1)ds+α1B1(t)=θ1t-r11∫0t‍C10(s)ds-a12∫0t‍x2(s)ds+a12[∫t-τ1t‍x2(s)ds-∫-τ10‍x2(s)ds]-a11∫0t‍x1(s)ds+α1B1(t).(34)lnx2(t)-lnx2(0)=-θ2t-r21∫0t‍C20(s)ds+a21∫0t‍x1(s)ds-a21[∫t-τ2t‍x1(s)ds-∫-τ20‍x1(s)ds]-a22∫0t‍x2(s)ds+α2B2(t).
(
i
) It follows from (17) and (33) that
(35)t-1lnx1(t)-t-1lnx1(0)≤θ1-r11〈C10(t)〉-a11〈x1(t)〉+α1B1(t)t≤θ1-r11K1λ+ε-a11〈x1(t)〉+α1B1(t)t
for sufficiently large t. Since θ1-r11K1/λ<0, then we can choose ε sufficiently small such that θ1-r11K1/λ+ε<0. Then, by (I) in Lemma 6,
(36)limt→+∞x1(t)=0,a.s.
When (36) is used in (34), one can see that
(37)t-1lnx2(t)-lnx2(0)≤-θ2+ε-a22〈x2(t)〉+α2B2(t)t
for sufficiently large t, where ε>0 obeys -θ2+ε<0. In view of Lemma 6 again, limt→+∞x2(t)=0,a.s.
(
ii
) By the stochastic comparison theorem [40], one can observe that
(38)x1(t)≤y1(t),x2(t)≤y2(t).
Note that θ1>r11K1/γ and Δ~2<Δ¯2; it then follows from Lemma 7 that limt→+∞y2(t)=0,a.s. Making use of (38) gives limt→+∞x2(t)=0,a.s. Thus, for allε>0, there exists T>0 such that, for t≥T,
(39)ε2≤a12x2(t)≤ε2.
Substituting the above inequalities into (33) and then using (18), we obtain
(40)t-1lnx1(t)≤t-1lnx1(0)+θ1-r11〈C10(t)〉-a11〈x1(t)〉+ε2+α1B1(t)t≤θ1-r11K1γ+2ε-a11〈x1(t)〉+α1B1(t)t,(41)t-1lnx1(t)≥t-1lnx1(0)+θ1-r11〈C10(t)〉-a11〈x1(t)〉-ε2+α1B1(t)t≥θ1-r11K1γ-2ε-a11〈x1(t)〉+α1B1(t)t.
Let ε be sufficiently small such that θ1-r11K1/γ-ε>0, and then, applying (I) and (II) in Lemma 6 to (40) and (41), respectively, one can see that
(42)θ1-r11K1/γ-2εa11≤〈x1(t)〉*≤〈x1(t)〉*≤θ1-r11K1/γ+2εa11a.s.
An application of the arbitrariness of ε gives
(43)limt→+∞〈x1(t)〉=θ1-r11K1/γa11,a.s.
(
iii
) Clearly, Δ~2>Δ¯2 implies θ1>r11K1/γ, and then, by Lemma 7,
(44)limt→+∞〈y2(t)〉=Δ~2-Δ¯2a11a22.
Thus, similar to the proof of (28), we get
(45)limt→+∞t-1∫t-τ1t‍y2(s)ds=0a.s.
Therefore, by (26), (28), and (38), we can observe that
(46)limsupt→+∞t-1lnx1(t)≤limt→+∞t-1lny1(t)=0,(47)limt→+∞t-1∫t-τ2t‍x1(s)ds=0,limt→+∞t-1∫t-τ1t‍x2(s)ds=0,a.s.
Employing (33) and (34) in the expression a21ln(x1(t)/x1(0))+a11ln(x2(t)/x2(0)) yields
(48)t-1a21lnx1(t)x1(0)+t-1a11lnx2(t)x2(0)=a12a21t-1[∫t-τ1t‍x2(s)ds-∫-τ10‍x2(s)ds]-a11a21t-1[∫t-τ2t‍x1(s)ds-∫-τ20‍x1(s)ds]+Δ~2-a21r11〈C10(t)〉-a11r21〈C20(t)〉-Δ〈x2(t)〉+t-1a21α1B1(t)+t-1a11α2B2(t).
When (18), (46) and (47), are used in (48), one can obtain
(49)t-1a11lnx2(t)x2(0)≥Δ~2-Δ¯2-ε-Δ〈x2(t)〉+t-1a21α1B1(t)+t-1a11α2B2(t)
for sufficiently large t, where ε>0 obeys Δ~2-Δ¯2-ε>0. It then follows from (II) in Lemma 6 that
(50)〈x2(t)〉*≥Δ~2-Δ¯2-εΔ.
By virtue of the arbitrariness of ε, we can see that
(51)〈x2(t)〉*≥Δ~2-Δ¯2Δ.
Consequently, for every 0<ε<a12(Δ~2-Δ¯2)/Δ, there is T>0 such that
(52)a12〈x2(t)〉≥a12〈x2〉*-ε≥a12(Δ~2-Δ¯2)Δ-ε,t>T.
Substituting the above inequality into (33) and then using (18) and (47), one can see that
(53)t-1lnx1(t)x1(0)≤θ1-a12(Δ~2-Δ¯2)Δ+3ε-a11t-1∫0t‍x1(s)ds+t-1α1B1(t)=a11(Δ~1-Δ¯1)Δ+3ε-a11t-1∫0t‍x1(s)ds+t-1α1B1(t)
for sufficiently large t. Since Δ~1-Δ¯1>0, and then, by Lemma 6 and the arbitrariness of ε, one can observe that
(54)〈x1(t)〉*≤Δ~1-Δ¯1Δ.
When this inequality, (18) and (47), are used in (34), we can see that
(55)t-1lnx2(t)x2(0)≤-θ2+a21Δ~1-Δ¯1Δ+3ε-a22t-1∫0t‍x2(s)ds+t-1α2B2(t)=a22(Δ~2-Δ¯2)Δ+3ε-a22t-1∫0t‍x2(s)ds+t-1α2B2(t)
for sufficiently large t. Then, it follows from Lemma 6 and the arbitrariness of ε that
(56)〈x2(t)〉*≤Δ~2-Δ¯2Δ.
Substituting the above inequality and (18) into (33), we get
(57)t-1lnx1(t)x1(0)≥θ1-a12Δ~2-Δ¯2Δ-3ε-a11t-1∫0t‍x1(s)ds+t-1α1B1(t)=a11(Δ~1-Δ¯1)Δ-3ε-a11t-1∫0t‍x1(s)ds+t-1α1B1(t)
for sufficiently large t. By (II) in Lemma 6 and the arbitrariness of ε again, we obtain
(58)〈x1(t)〉*≥Δ~1-Δ¯1Δ.
Then, the required assertion follows from (51), (54), (56), and (58).


## 3. Numerical Simulations

Let us use the famous Milstein method (see, e.g., [41]) to illustrate the analytical results.To begin with, we chooser10=0.85, r20=0.05, r11=r21=1,  a11=0.4,   a12=0.4, a21=0.3, a22=0.3, τ1=3,τ2=8, α22=0.1, ki=gi=mi=0.1, i=1,2, h=0.5, b=0.6, and γ=12. Then,
(59)Ki=kibh(gi+mi)=0.6,Δ2=r10a21-r20a11=0.235>Δ¯2=a21r11K1γ+a11r21K2γ=0.035.
By (c) in Lemma 1, the solution of model (1) obeys
(60)limsupt→+∞t-1∫0t‍x1(s)ds>0,limsupt→+∞t-1∫0t‍x2(s)ds>0.
However, when the white noises are taken into account, the properties of the system may be changed greatly. In Figure 1, we let the coefficients be same with the above. The only difference between conditions of Figures 1(a), 1(b), and 1(c) is that the value of α12 is different. In Figure 1(a), we choose α12/2=0.82. Therefore,
(61)θ1=r10-α122=0.03<r11K1γ=0.05.
Then, by (i) in Theorem 2, both  x1 and x2 are extinctive. Figure 1(a) confirms these. In Figure 1(b), we choose α12/2=0.65. That is to say θ1=0.2>r11K1/γ=0.05 and Δ~2=0.02<Δ¯2=0.035. It then follows from (ii) in Theorem 2 that x2 is extinctive and x1 is stable in time average:
(62)limt→+∞〈x1(t)〉=θ1-r11K1/γa11=0.375.
See Figure 1(b). In Figure 1(c), we choose α12/2=0.2. Then, Δ~2=0.155>Δ¯2=0.035. In view of (iii) in Theorem 2, we can obtain that both x1 and x2 are stable in time average:
(63)limt→+∞〈x1(t)〉=Δ~1-Δ¯1Δ=0.240.24=1,limt→+∞〈x2(t)〉=Δ~2-Δ¯2Δ=0.120.24=0.5.
Figure 1(c) confirms these.Solutions of system (3) for r10=0.85, r20=0.05, r11=r21=1, a11=0.4, a12=0.4, a21=0.3, a22=0.3, τ1=3, τ2=8, α22=0.1, ki=gi=mi=0.1, i=1,2, h=0.5, b=0.6, γ=12, x1(0)=0.9, x2(0)=0.5, C0(0)=Ce(0)=0.1, and step size Δt=0.001. (a) is with α12/2=0.82;  (b) is with α12/2=0.65;  (c) is with α12/2=0.2.
(a)
(b)
(c)In Figure2, we choose r10=0.85, r20=0.05, r11=r21=1, a11=0.4, a12=0.4, a21=0.3, a22=0.3, τ1=3, τ2=8, α12=0.4, α22=0.1, ki=gi=mi=0.1, i=1,2, h=0.5, and b=0.6. The only difference between conditions of Figures 1(c) and 2 is that the value of γ is different. In Figure 2, we choose γ=0.8. Then, θ1=0.65<r11K1/γ=0.75. It follows from (i) in Theorem 2 that both x1 and x2 are extinctive. Figure 2 confirms these. By comparing Figure 1(c) with Figure 2, one can see that the impulsive period γ plays a key role in determining the stability in time average and the extinction of the species.Figure 2
Solutions of system (3) for r10=0.85, r20=0.05, r11=r21=1, a11=0.4, a12=0.4, a21=0.3, a22=0.3, τ1=3, τ2=8, α12=0.4, α22=0.1, ki=gi=mi=0.1, i=1,2, h=0.5, b=0.6, γ=0.8, x1(0)=0.9, x2(0)=0.5, C0(0)=Ce(0)=0.1, and step size Δt=0.001.


## 4. Conclusions and Future Directions

This paper is concerned with stochastic delay predator-prey model in a polluted environment with impulsive toxicant input. For each species, the threshold between stability in time average and extinction is established. Some recent results are improved and extended. Our Theorem2 reveals some interesting and important results.  (A)
Firstly, time delay is harmless for stability in time average and extinction of the stochastic system (3).(B)
The white noiseα1dB1(t) and α2dB2(t) can change the properties of the system greatly.(C)
The impulsive periodγ plays an important role in determining the stability in time average and the extinction of the species.Some interesting questions deserve further investigations. One may consider some more realistic but more complex systems, for example, stochastic delay model with Markov switching (see, e.g., [30, 32, 39]). It is also interesting to investigate what happens if aij is stochastic.



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*Source: 139216-2013-10-10.xml*