# Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps

Authors: Guodong Liu; Xiaohong Wang; Xinzhu Meng; Shujing Gao
Journal: Complexity
 (2017)

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## Abstract

In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.


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## Body

## 1. Introduction

With the development of the economy, environmental pollution is caused by various industries and other activities of human, which has been one of the most important social problems in the world today. Many species have gone extinct due to the toxicant in the environment. Therefore, controlling the environmental pollution has been the important topics around the world. There are many researchers which have investigated the pollution models in recent years [1–3]. In addition, a lot of animal populations suffer from infectious disease, so some scholars investigated the predator-prey systems with diseases [4–8]. For example, a deterministic predator-prey model with infected predator in an impulsive polluted environment is described by the following equation:(1)dXt=Xtr-δ1c1t-a11Xt-a12Stdt,dSt=St-μ1-δ2c2t+a21Xt-βIt-a22Stdt,dIt=It-μ2-δ3c3t+βSt-a33Itdt,c˙1t=k1cet-g1c1t-m1c1t,c˙2t=k2cet-g2c2t-m2c2t,c˙3t=k3cet-g3c3t-m3c3t,c˙et=-hcet,t≠nT,ΔXt=0,ΔSt=0,ΔIt=0,Δcit=0,Δcet=u,i=1,2,3,t=nT,n∈Z+,where ΔX(t)=X(t+)-X(t), ΔS(t)=S(t+)-S(t), ΔI(t)=I(t+)-I(t), Δci(t)=ci(t+)-ci(t), Δce(t)=ce(t+)-ce(t), X(t) is the density of the prey at time t, and S(t) and I(t) represent the density of susceptible predator and infected predator at time t, respectively. ci(t)(i=1,2,3) and ce(t) stand for the concentrations of the toxicant in the organism and the environment at time t, respectively. δi(i=1,2,3) are dose-response parameter to the toxicant. r stands for the intrinsic growth rate of the prey. μi(i=1,2) denote the death rates of S(t) and I(t), respectively. β is the infection rate. aii(i=1,2,3) are density-dependent coefficients, and a12 and a21 represent predation rate and ingestion rate, respectively. ki(i=1,2,3) are environmental toxicant uptake rates, mi(i=1,2,3) denote organismal net depuration rates, gi(i=1,2,3) represent organismal net ingestion rates, h is the loss rate of toxicant from environment, and u denotes the amount of pulsed input concentration of the toxicant at every time. The above parameters are all positive constants. Next, we propose a new mathematical model by taking more factors into account based on model (1).In the natural world, time delay often occurs in almost every situation. Thus it is significant to take time delay into consideration [9–14]. As we know, deterministic model is not enough to describe the species activities. Sometimes, the species activities may be disturbed by environmental noises. May [15] revealed that the birth rates, death rates, carrying capacities, competition coefficients, and other parameters involved in the system should exhibit random fluctuation to a greater or lesser extent. Hence some parameters should be stochastic [16–26]. First, we assume that the intrinsic growth rate and the death rates of species are disturbed by white noise, then r and μi can be replaced by(2)r⟶r+σ1B˙1t,-μ1⟶-μ1+σ2B˙2t,-μ2⟶-μ2+σ3B˙3t,where Bi(t)(i=1,2,3) are standard Brownian motions and σi are the intensities of Bi(t). Bi(t) are mutually independent defined on a complete probability space (Ω,F,{F}t≥0,P).Furthermore, populations may suffer from sudden environmental fluctuations, such as floods and earthquakes, which cannot be described by Brownian motions. To explain these phenomena, introducing a jump process into the underlying population dynamics is one of the important methods. Thus, there are many scholars introduce Lévy jumps into the population system [27–31]. Taking all above influences into consideration, we focus on the infected stochastic predator-prey system with Lévy jumps and delays in a polluted environment(3)dXt=Xt-r-δ1c1t-a11Xt--a12e-d1τ1St--τ1dt+σ1XtdB1t+∫YXt-γ1uN~dt,du,dSt=St--μ1-δ2c2t+a21e-d2τ2Xt--τ2-βe-d3τ3It--τ3-a22St-dt+σ2StdB2t+∫YSt-γ2uN~dt,du,dIt=It--μ2-δ3c3t+βe-d4τ4St--τ4-a33It-dt+σ3ItdB3t+∫YIt-γ3uN~dt,du,c˙1t=k1cet-g1c1t-m1c1t,c˙2t=k2cet-g2c2t-m2c2t,c˙3t=k3cet-g3c3t-m3c3t,c˙et=-hcet,t≠nT,ΔXt=0,ΔSt=0,ΔIt=0,Δcit=0,Δcet=u,i=1,2,3,t=nT,n∈Z+,where X(t-), S(t-), and I(t-) stand for the left limits of X(t), S(t), and I(t), respectively. N(dt,du) denotes a Poisson counting measure with characteristic measure ν which defines a measurable bounded subset Y of (0,∞) with ν(Y)<∞ and N~(dt,du)=N(dt,du)-ν(du)dt, and γi is bounded and continuous with respect to ν and is B(Y)×Ft-measurable, and 1+γi>0(i=1,2,3) (see [27–30]). Moreover, Bi(t)(i=1,2,3) are independent of N, di are death rates of species, and τi≥0(i=1,2,3,4) represent the time delay. Other parameters are defined as system (1).The rest of this paper is arranged as follows. Section2 introduces some lemmas which will be used in our main results. In Section 3, we show the main results. We examine the extinction of system (3) in Section 3.1; in Section 3.2 we also prove the permanence in mean of this system. Finally, we present some simulations and conclusions in Section 4.


## 2. Preliminary Results

Throughout the paper, we assume thatX(t), S(t), I(t), and ci(t) are continuous at t=nT and ce(t) is left continuous at t=nT and ce(nT+)=limt→nT+ce(t). Moreover, let (Ω,F,{F}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the common conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets).For the sake of convenience, we introduce some notions and some lemmas which will be used for the main results. We define(4)τ=max⁡τ1,τ2,τ3,τ4,b1=r-12σ12-∫Yγ1u-ln1+γ1uνdu,b2=μ1+12σ22+∫Yγ2u-ln1+γ2uνdu,b3=μ2+12σ32+∫Yγ3u-ln1+γ3uνdu,Δ1=b1-δ1c¯1a21e-d2τ2-b2+δ2c¯2a11,Δ′=a11a22+a12a21e-d1τ1-d2τ2,Δ2=b1-δ1c¯1a22a33+β2e-d3τ3-d4τ4+b2+δ2c¯2a33a12e-d1τ1-b3+δ3c¯3a12βe-d1τ1-d3τ3,Δ3=a33Δ1+b3+δ3c¯3a11βe-d3τ3,Δ4=βe-d4τ4Δ1-b3+δ3c¯3Δ′,Δ=a33Δ′+a11β2e-d3τ3+d4τ4,R=βe-d4τ4Δ1b3+δ3c¯3a11a22,ft=1t∫0tftdt,f∗=limsupt→+∞ft,f∗=liminft→+∞ft,where f(t) is a bounded continuous function on [0,+∞).Then we show some basic properties of the subsystem of system (3)(5)c˙1t=k1cet-g1c1t-m1c1t,c˙2t=k2cet-g2c2t-m2c2t,c˙3t=k3cet-g3c3t-m3c3t,c˙et=-hcet,t≠nT,n∈Z+,Δcit=0,Δcet=u,i=1,2,3,t=nT,n∈Z+.Lemma 1 (see [3]).
System (5) has a unique positive T-periodic solution (c~1(t),c~2(t),c~3(t),c~e(t))T which is globally asymptotical stable. Furthermore, if ci(0)>c~i(0) and ce(0)>c~e(0), then ci(t)>c~i(t) and ce(t)>c~e(t) for all t≥0, where (6)c~it=c~i0e-gi+mit-nT+kiue-gi+mit-nT-e-ht-nTh-gi-mi1-e-hT,c~et=ue-ht-nT1-e-hT,c~i0=kiue-gi+miT-e-hTh-gi-mi1-e-gi+miT1-e-hT,c~e0=u1-e-hT,for t∈(nT,(n+1)T], i=1,2,3, and n∈Z+.It can be obtained from a simple calculation that(7)∫nTn+1Tc~itdt=kiuhgi+mi.Since c~i(t) is a periodic function, then we get(8)limt→∞cit∗≤limn→∞1nT∫0n+1Tc~itdt=limn→∞n+1nT∫nTn+1Tc~itdt=kiuhgi+miT,limt→∞cit∗≥limn→∞1n+1T∫0nTc~itdt=limn→∞nn+1T∫n-1TnTc~itdt=kiuhgi+miT.Thus we have (9)limt→+∞cit=kiuhgi+miT≔c¯i.From Lemma1, we know that the long time dynamical behaviors of system (3) can be replaced by the dynamical behaviors of the following limiting system:(10)dXt=Xt-r-δ1c~1t-a11Xt--a12e-d1τ1St--τ1dt+σ1XtdB1t+∫YXt-γ1uN~dt,du,dSt=St--μ1-δ2c~2t+a21e-d2τ2Xt--τ2-βe-d3τ3It--τ3-a22St-dt+σ2StdB2t+∫YSt-γ2uN~dt,du,dIt=It--μ2-δ3c~3t+βe-d4τ4St--τ4-a33It-dt+σ3ItdB3t+∫YIt-γ3uN~dt,du.Now we give an assumption which will be used in the following proof.Assumption 2.
There exist constantsCi such that (11)∫Yγi-ln1+γiνdu≤Ci,i=1,2,3.Lemma 3.
For any given initial value((ϕ1(t),ϕ2(t),ϕ3(t)):-τ≤t≤0,τ=max{τ1,τ2,τ3,τ4})∈C([-τ,0];R+3), there is a unique solution (X(t),S(t),I(t)) of (10) on t≥τ and the solution will remain in R+2 with probability 1.Proof.
This proof is the same as Theorem3.1 in [11] by defining (12)VX,S,I=V1X,S,I+V2X,S,I,where (13)V1X,S,I=Xt-1-ln⁡Xt+St-1-ln⁡St+It-1-ln⁡It,V2X,S,I=a12e-d1τ1∫tt+τ1Ss-τ1ds+12∫tt+τ2X2s-τ2ds+βe-d3τ3∫tt+τ3Is-τ3ds+12∫tt+τ4S2s-τ4ds.Thus, we omit it here.The stochastic comparison theorem and limit superior and limit inferior theory are given as follows.Lemma 4 (see [27]).
Suppose thatY(t)∈C(Ω×[0,∞),R+).
(i) If there exist three positive constantsT, λ, and λ0 such that(14)ln⁡Yt≤λt-λ0∫0tYsds+αBt+∑i=1nηi∫0t∫Yln⁡1+γiuN~ds,dua.s.for all t≥T, where α and ηi are constants, then(15)limsupt→∞Yt=Yt∗≤λλ0a.s.,  ifλ≥0,limt→∞Yt=0a.s.,  ifλ<0.
(ii) If there exist three positive constantsT, λ, and λ0 such that(16)ln⁡Yt≥λt-λ0∫0tYsds+αBt+∑i=1nηi∫0t∫Yln1+γiuN~ds,dua.s.for all t≥T, then liminft→∞Y(t)=Y(t)∗≥λ/λ0a.s.First, we explore the following auxiliary system:(17)dX1t=Xt-r-δ1c~1t-a11X1t-dt+σ1X1tdB1t+∫YX1t-γ1uN~dt,du,dS1t=S1t--μ1-δ2c~2t+a21e-d2τ2X1t--τ2-a22S1t-dt+σ2S1tdB2t+∫YS1t-γ2uN~dt,du,dI1t=I1t--μ2-δ3c~3t+βe-d4τ4S1t--τ4-a33I1t-dt+σ3I1tdB3t+∫YI1t-γ3uN~dt,du.Lemma 5.
For system (17), let (X1(t),S1(t),I1(t)) be the solution of this system with initial value ((ϕ1(t),ϕ2(t),ϕ3(t)):-τ≤t≤0)∈C([-τ,0];R+3).
(i) Ifb1<δ1c¯1, then (18)limt→∞X1t=0,limt→∞S1t=0,limt→∞I1t=0,a.s.
(ii) Ifb1>δ1c¯1, when Δ1<0, then(19)limt→∞X1t=b1-δ1c¯1a11,limt→∞S1t=0,limt→∞I1t=0,limt→∞ln⁡X1tt=0,a.s.
(iii) Ifb1>δ1c¯1, Δ1>0, when R<1, then(20)limt→∞X1t=b1-δ1c¯1a11,limt→∞S1t=Δ1a11a22,limt→∞I1t=0,limt→∞ln⁡S1tt=0,a.s.
(iv) Ifb1>δ1c¯1, when R>1, then(21)limt→∞X1t=b1-δ1c¯1a11,limt→∞S1t=Δ1a11a22,a.s,limt→∞I1t=βe-d4τ4Δ1-b3+δ3c¯3a11a22a11a22a33,limt→∞ln⁡I1tt=0,a.s.Proof.
Applying Itô’s formula to system (17) leads to(22)dln⁡X1=b1-δ1c~1t-a11X1tdt+σ1dB1t+∫Yln1+γ1uN~dt,du,dln⁡S1=-b2-δ2c~2t+a21e-d2τ2X1t-τ2-a22S1tdt+σ2dB2t+∫Yln1+γ2uN~dt,du,dln⁡I1=-b3-δ3c~3t+βe-d4τ4S1t-τ4-a33I1tdt+σ3dB3t+∫Yln1+γ3uN~dt,du.
Integrating both sides of (22), we have(23)ln⁡X1t-ln⁡X10=b1t-δ1∫0tc~1sds-a11∫0tX1sds+σ1B1t+∫0t∫Yln1+γ1uN~ds,du,ln⁡S1t-ln⁡S10=-b2t-δ2∫0tc~2sds+a21e-d2τ2∫0tX1t-τ2-a22∫0tS1sds+σ2B2t+∫0t∫Yln1+γ2uN~ds,du=-b2t-δ2∫0tc~2sds+a21e-d2τ2∫0tX1sds-a22∫0tS1sds-a21e-d2τ2∫tt+τ2X1s-τ2ds-∫0τ2X1s-τ2ds+σ2B2t+∫0t∫Yln1+γ2uN~ds,du,ln⁡I1t-ln⁡I10=-b3t-δ3∫0tc~3sds+βe-d4τ4∫0tS1s-τ4ds-a33∫0tI1sds+σ3B3t+∫0t∫Yln1+γ3uN~ds,du=-b3t-δ3∫0tc~3sds+βe-d4τ4∫0tS1sds-a33∫0tI1sds-βe-d4τ4∫tt+τ4S1s-τ4ds-∫0τ4S1s-τ4ds+σ3B3t+∫0t∫Yln1+γ3uN~ds,du.Then we can obtain(24)ln⁡X1t-ln⁡X10t=b1-δ1c~1t-a11X1t+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du,(25)ln⁡S1t-ln⁡S10t+a21e-d2τ2∫tt+τ2X1s-τ2ds-∫0τ2X1s-τ2dst=-b2-δ2c~2t+a21e-d2τ2X1t-a22S1t+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du,(26)ln⁡I1t-ln⁡I10t+βe-d4τ4∫tt+τ4S1s-τ4ds-∫0τ4S1s-τ4dst=-b3-δ3c~3t+βe-d4τ4S1t-a33I1t+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du.
Case (i). From Lemma 4 and (24), if b1<δ1c¯1, then we have (27)limt→∞X1t=0a.s.Obviously, we find(28)limt→∞S1t=0,limt→∞I1t=0,a.s.
Case (ii). By Lemma 4, it is derived from (24) and conditions that(29)X1t∗≤b1-δ1c~1t∗a11,X1t∗≥b1-δ1c~1t∗a11.Since (30)limt→∞c~it∗=limt→∞c~it∗=c¯i,i=1,2,3,then we obtain(31)limt→∞X1t=b1-δ1c¯1a11.Using Assumption 2 and the strong law of large numbers for local martingales, one has(32)limt→+∞1t∫0t∫Yln1+γiuN~ds,du=0,limt→+∞σiBitt=0,a.s,i=1,2,3.Then, substituting (31) into (24) yields(33)limt→∞ln⁡X1tt=0,a.s.Since(34)limt→∞∫tt+τ2X1s-τ2dst=limt→∞1t∫t-τ2tX1sds=0a.s.,(35)limt→∞∫0τ2X1s-τ2dst=limt→∞∫-τ20X1sdst=0a.s.,then we have that, for any 0<ε1<Δ1, there exists T1>0 such that(36)-ε1<a21e-d2τ2∫tt+τ2X1s-τ2ds-∫0τ2X1s-τ2dst<ε1,t≥T1.Combining (25) with (36) yields(37)ln⁡S1t-ln⁡S10t≤-b2-δ2c~2t∗+a21e-d2τ2X1t∗+ε1-a22S1t+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du.From Lemma 4 and conditions, we have (38)limt→∞S1t=0a.s.Obviously, we have (39)limt→∞I1t=0a.s.
Case (iii). By (25), we obtain(40)ln⁡S1t-ln⁡S10t≥-b2-δ2c~2t∗+a21e-d2τ2X1t∗-ε1-a22S1t+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du.It can infer from (37), (40), and Lemma 4 that (41)S1t∗≤Δ1+ε1a11a22,S1t∗≥Δ1-ε1a11a22.Since ε1 is an arbitrary number, then we get (42)limt→∞S1t=Δ1a11a22a.s.Combining this equality with (25) leads to(43)limt→∞ln⁡S1tt=0,a.s.Similar to (34) and (35), we have that, for any 0<ε2<βe-d4τ4Δ1-(b3+δ3c¯3)a11a22, there exists T2>0 such that(44)ln⁡I1t-ln⁡I10t≤-b3-δ3c~3t∗+βe-d4τ4S1t∗+ε2-a33I1t+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du.Note that R<1, so we have (45)limt→∞I1t=0a.s.
Case (iv). By (26), we get(46)ln⁡I1t-ln⁡I10t≥-b3-δ3c~3t∗+βe-d4τ4S1t∗-ε2-a33I1t+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du.When R>1, from (44) and (46), using Lemma 4 results in(47)I1t∗≤βe-d4τ4Δ1-b3+δ3c~3t∗a11a22+ε2a11a22a33,I1t∗≥βe-d4τ4Δ1-b3+δ3c~3t∗a11a22-ε2a11a22a33.Since ε2 is an arbitrary number, then we can obtain (48)limt→∞I1t=βe-d4τ4Δ1-b3+δ3c¯3a11a22a11a22a33a.s.Combining this equality with (26) leads to(49)limt→∞ln⁡I1tt=0,a.s.This completes the proof of Lemma 1.


## 3. Main Results

### 3.1. Extinction

Now we are going to show our main results. By Lemma5, we can get the extinction of system (10).Theorem 6.
For system (10), let (X(t),S(t),I(t)) be the solution of this system with initial value ((ϕ1(t),ϕ2(t),ϕ3(t)):-τ≤t≤0)∈C([-τ,0];R+3).
(i) Ifb1<δ1c¯1, then (50)limt→∞Xt=0,limt→∞St=0,limt→∞It=0,a.s.
(ii) Ifb1>δ1c¯1, when Δ1<0, then(51)limt→∞Xt=b1-δ1c¯1a11,limt→∞St=0,limt→∞It=0,a.s.
(iii) Ifb1>δ1c¯1, Δ1>0, when R<1, then (52)limt→∞Xt=b1-δ1c¯1a22+b2+δ2c¯2a12e-d1τ1Δ′a.s,limt→∞St=Δ1Δ′a.s,limt→∞It=0a.s.Proof.
By stochastic comparison theorem, we have(53)Xt≤X1t,St≤S1t,It≤I1t.By (34), we derive(54)limt→∞∫tt+τ2Xs-τ2dst=0.In the same way, we can verify that(55)limt→∞∫tt+τ2Ss-τ2dst=0,limt→∞∫tt+τ2Is-τ2dst=0.It follows from (33), (43), and (49) that(56)ln⁡Xtt∗≤limt→∞ln⁡X1tt=0,(57)ln⁡Stt∗≤limt→∞ln⁡S1tt=0,(58)ln⁡Itt∗≤limt→∞ln⁡I1tt=0.Applying Itô’s formula to system (10) leads to(59)dln⁡X=b1-δ1c~1t-a11Xt-a12e-d1τ1St-τ1dt+σ1dB1t+∫Yln1+γ1uN~dt,du,dln⁡S=-b2-δ2c~2t+a21e-d2τ2Xt-τ2-βe-d3τ3It-τ3-a22Stdt+σ2dB2t+∫Yln1+γ2uN~dt,du,dln⁡I=-b3-δ3c~3t+βe-d4τ4St-τ4-a33Itdt+σ3dB3t+∫Yln1+γ3uN~dt,du.Then we can obtain(60)ln⁡Xt-ln⁡X0t=b1-δ1c~1t-a12e-d1τ1St-a11Xt-a12e-d1τ1∫0τ1Xs-τ1ds-∫tt+τ1Xs-τ1dst+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du,(61)ln⁡St-ln⁡S0t=-b2-δ2c~2t+a21e-d2τ2Xt-βe-d3τ3It-a22St-a21e-d2τ2∫tt+τ2Xs-τ2ds-∫0τ2Xs-τ2dst+βe-d3τ3∫tt+τ2Is-τ2ds-∫0τ2Is-τ2dst+σ2B2tt+1t∫0t∫Yln⁡1+γ2uN~ds,du,(62)ln⁡It-ln⁡I0t=-b3-δ3c~3t+βe-d4τ4St-a33It-βe-d4τ4∫tt+τ4Ss-τ4ds-∫0τ4Ss-τ4dst+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du.Computing (60) ×  a21e-d2τ2 + (61) ×  a11 leads to(63)a21e-d2τ2ln⁡Xt-ln⁡X0t+a11ln⁡St-ln⁡S0t=b1-δ1c~1ta21e-d2τ2-b2+δ2c~2ta11-Δ′St-a11βe-d3τ3It+Γ1,where Γ1 is given in Appendix.
Case (i). By Lemma 5, we know that limt→∞X1(t)=0, if b1<δ1c¯1. From (53), we have (64)limt→∞Xt=0a.s.Obviously, we get (65)limt→∞St=0,limt→∞It=0,a.s.
Case (ii). By Lemma 5, we know that limt→∞S1(t)=0, if b1>δ1c¯1 and Δ1<0. From (53), we have (66)limt→∞St=0a.s.Then we obtain (67)limt→∞It=0a.s.By using (60) and Lemma 4, we can prove (68)limt→∞Xt=b1-δ1c¯1a11a.s.
Case (iii). Similarly, by Lemma 5, we get that limt→∞I1(t)=0, if b1>δ1c¯1, Δ1>0, and R<1. By (53), we obtain (69)limt→∞It=0a.s.Combining Lemma 4 with (56) and (63), we get (70)St∗≥Δ1Δ′.From (60), we have(71)ln⁡Xt-ln⁡X0t≤b1-δ1c~1t∗-a12e-d1τ1St∗-a11Xt+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du.Then we infer from Lemma 4 that (72)Xt∗≤b1-δ1c~1t∗a22+b2+δ2c~2t∗a12e-d1τ1Δ′.By (61), we get (73)ln⁡St-ln⁡S0t≤-b2-δ2c~2t∗+a21e-d2τ2Xt∗-a22St+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du.From Lemma 4, we obtain (74)St∗≤Δ1Δ′.By (60), we have (75)ln⁡Xt-ln⁡X0t≥b1-δ1c~1t∗-a12e-d1τ1St∗-a11Xt+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du.In view of Lemma 4, we can see that (76)Xt∗≥b1-δ1c~1t∗a22+b2+δ2c~2t∗a12e-d1τ1Δ′.Consequently, (77)limt→∞Xt=b1-δ1c¯1a22+b2+δ2c¯2a12e-d1τ1Δ′a.s,limt→∞St=Δ1Δ′a.s.This completes the proof of Theorem 6.


### 3.2. Permanence in Mean

In this section, we prove the permanence in mean of system (10).Theorem 7.
Let(X(t),S(t),I(t)) be the solution of system (10) with initial value ((ϕ1(t),ϕ2(t),ϕ3(t)):-τ≤t≤0)∈C([-τ,0];R+3). If Δ2>0 and Δ3>0, when R^=(a11a22/Δ′)R>1, then (78)limt→∞Xt=Δ2Δ,limt→∞St=Δ3Δ,limt→∞It=Δ4Δ,a.s.Proof.
Computing (62) ×  Δ′ + (63) ×  βe-d4τ4, we obtain (79)Δ′ln⁡It-ln⁡I0t+βa21e-d2τ2-d4τ4ln⁡Xt-ln⁡X0t+a11βe-d4τ4ln⁡St-ln⁡S0t=-b3+δ3c~3tΔ′+b1-δ1c~1ta21βe-d2τ2-d4τ4-b2+δ2c~2ta11βe-d4τ4-a33Δ′+a11β2e-d3τ3-d4τ4It+Γ2,where Γ2 is given in Appendix.
WhenR^=(a11a22/Δ′)R>1, by Lemma 4, we have (80)It∗≥Δ4Δ.So (81)limt→∞ln⁡Itt=0a.s.Computing (63) ×  a33-  (62) ×  a11βe-d3τ3, we have (82)a33a21e-d2τ2ln⁡Xt-ln⁡X0t+a11a33ln⁡St-ln⁡S0t-a11βe-d3τ3ln⁡It-ln⁡I0t=b3+δ3c~3ta11βe-d3τ3+b1-δ1c~1ta33a21e-d2τ2-b2+δ2c~2ta11a33-a33Δ′+a11β2e-d3τ3-d4τ4St+Γ3,where Γ3 is given in Appendix.
WhenΔ3>0, by Lemma 4, we have (83)St∗≥Δ3Δ.From (71), when Δ2>0, we obtain (84)Xt∗≤Δ2Δ.By (61), we get (85)ln⁡St-ln⁡S0t≤-b2-δ2c~2t∗+a21e-d2τ2Xt∗-βe-d3τ3It∗-a22St+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du,then from Lemma 4 we prove that (86)St∗≤Δ3Δ.By (60), we have (87)ln⁡Xt-ln⁡X0t≥b1-δ1c~1t∗-a12e-d1τ1St∗-a11Xt+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du,and then from Lemma 4 we get (88)Xt∗≥Δ2Δ.It can be inferred from (62) that(89)ln⁡It-ln⁡I0t≤-b3-δ3c~3t∗+βe-d4τ4St∗-a33It-βe-d4τ4∫tt+τ4Ss-τ4ds-∫0τ4Ss-τ4dst+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du,and then from Lemma 4 we have (90)It∗≤Δ4Δ.Consequently, (91)limt→∞Xt=Δ2Δ,limt→∞St=Δ3Δ,limt→∞It=Δ4Δ,a.s.This completes the proof of Theorem 7.


## 3.1. Extinction

Now we are going to show our main results. By Lemma5, we can get the extinction of system (10).Theorem 6.
For system (10), let (X(t),S(t),I(t)) be the solution of this system with initial value ((ϕ1(t),ϕ2(t),ϕ3(t)):-τ≤t≤0)∈C([-τ,0];R+3).
(i) Ifb1<δ1c¯1, then (50)limt→∞Xt=0,limt→∞St=0,limt→∞It=0,a.s.
(ii) Ifb1>δ1c¯1, when Δ1<0, then(51)limt→∞Xt=b1-δ1c¯1a11,limt→∞St=0,limt→∞It=0,a.s.
(iii) Ifb1>δ1c¯1, Δ1>0, when R<1, then (52)limt→∞Xt=b1-δ1c¯1a22+b2+δ2c¯2a12e-d1τ1Δ′a.s,limt→∞St=Δ1Δ′a.s,limt→∞It=0a.s.Proof.
By stochastic comparison theorem, we have(53)Xt≤X1t,St≤S1t,It≤I1t.By (34), we derive(54)limt→∞∫tt+τ2Xs-τ2dst=0.In the same way, we can verify that(55)limt→∞∫tt+τ2Ss-τ2dst=0,limt→∞∫tt+τ2Is-τ2dst=0.It follows from (33), (43), and (49) that(56)ln⁡Xtt∗≤limt→∞ln⁡X1tt=0,(57)ln⁡Stt∗≤limt→∞ln⁡S1tt=0,(58)ln⁡Itt∗≤limt→∞ln⁡I1tt=0.Applying Itô’s formula to system (10) leads to(59)dln⁡X=b1-δ1c~1t-a11Xt-a12e-d1τ1St-τ1dt+σ1dB1t+∫Yln1+γ1uN~dt,du,dln⁡S=-b2-δ2c~2t+a21e-d2τ2Xt-τ2-βe-d3τ3It-τ3-a22Stdt+σ2dB2t+∫Yln1+γ2uN~dt,du,dln⁡I=-b3-δ3c~3t+βe-d4τ4St-τ4-a33Itdt+σ3dB3t+∫Yln1+γ3uN~dt,du.Then we can obtain(60)ln⁡Xt-ln⁡X0t=b1-δ1c~1t-a12e-d1τ1St-a11Xt-a12e-d1τ1∫0τ1Xs-τ1ds-∫tt+τ1Xs-τ1dst+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du,(61)ln⁡St-ln⁡S0t=-b2-δ2c~2t+a21e-d2τ2Xt-βe-d3τ3It-a22St-a21e-d2τ2∫tt+τ2Xs-τ2ds-∫0τ2Xs-τ2dst+βe-d3τ3∫tt+τ2Is-τ2ds-∫0τ2Is-τ2dst+σ2B2tt+1t∫0t∫Yln⁡1+γ2uN~ds,du,(62)ln⁡It-ln⁡I0t=-b3-δ3c~3t+βe-d4τ4St-a33It-βe-d4τ4∫tt+τ4Ss-τ4ds-∫0τ4Ss-τ4dst+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du.Computing (60) ×  a21e-d2τ2 + (61) ×  a11 leads to(63)a21e-d2τ2ln⁡Xt-ln⁡X0t+a11ln⁡St-ln⁡S0t=b1-δ1c~1ta21e-d2τ2-b2+δ2c~2ta11-Δ′St-a11βe-d3τ3It+Γ1,where Γ1 is given in Appendix.
Case (i). By Lemma 5, we know that limt→∞X1(t)=0, if b1<δ1c¯1. From (53), we have (64)limt→∞Xt=0a.s.Obviously, we get (65)limt→∞St=0,limt→∞It=0,a.s.
Case (ii). By Lemma 5, we know that limt→∞S1(t)=0, if b1>δ1c¯1 and Δ1<0. From (53), we have (66)limt→∞St=0a.s.Then we obtain (67)limt→∞It=0a.s.By using (60) and Lemma 4, we can prove (68)limt→∞Xt=b1-δ1c¯1a11a.s.
Case (iii). Similarly, by Lemma 5, we get that limt→∞I1(t)=0, if b1>δ1c¯1, Δ1>0, and R<1. By (53), we obtain (69)limt→∞It=0a.s.Combining Lemma 4 with (56) and (63), we get (70)St∗≥Δ1Δ′.From (60), we have(71)ln⁡Xt-ln⁡X0t≤b1-δ1c~1t∗-a12e-d1τ1St∗-a11Xt+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du.Then we infer from Lemma 4 that (72)Xt∗≤b1-δ1c~1t∗a22+b2+δ2c~2t∗a12e-d1τ1Δ′.By (61), we get (73)ln⁡St-ln⁡S0t≤-b2-δ2c~2t∗+a21e-d2τ2Xt∗-a22St+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du.From Lemma 4, we obtain (74)St∗≤Δ1Δ′.By (60), we have (75)ln⁡Xt-ln⁡X0t≥b1-δ1c~1t∗-a12e-d1τ1St∗-a11Xt+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du.In view of Lemma 4, we can see that (76)Xt∗≥b1-δ1c~1t∗a22+b2+δ2c~2t∗a12e-d1τ1Δ′.Consequently, (77)limt→∞Xt=b1-δ1c¯1a22+b2+δ2c¯2a12e-d1τ1Δ′a.s,limt→∞St=Δ1Δ′a.s.This completes the proof of Theorem 6.


## 3.2. Permanence in Mean

In this section, we prove the permanence in mean of system (10).Theorem 7.
Let(X(t),S(t),I(t)) be the solution of system (10) with initial value ((ϕ1(t),ϕ2(t),ϕ3(t)):-τ≤t≤0)∈C([-τ,0];R+3). If Δ2>0 and Δ3>0, when R^=(a11a22/Δ′)R>1, then (78)limt→∞Xt=Δ2Δ,limt→∞St=Δ3Δ,limt→∞It=Δ4Δ,a.s.Proof.
Computing (62) ×  Δ′ + (63) ×  βe-d4τ4, we obtain (79)Δ′ln⁡It-ln⁡I0t+βa21e-d2τ2-d4τ4ln⁡Xt-ln⁡X0t+a11βe-d4τ4ln⁡St-ln⁡S0t=-b3+δ3c~3tΔ′+b1-δ1c~1ta21βe-d2τ2-d4τ4-b2+δ2c~2ta11βe-d4τ4-a33Δ′+a11β2e-d3τ3-d4τ4It+Γ2,where Γ2 is given in Appendix.
WhenR^=(a11a22/Δ′)R>1, by Lemma 4, we have (80)It∗≥Δ4Δ.So (81)limt→∞ln⁡Itt=0a.s.Computing (63) ×  a33-  (62) ×  a11βe-d3τ3, we have (82)a33a21e-d2τ2ln⁡Xt-ln⁡X0t+a11a33ln⁡St-ln⁡S0t-a11βe-d3τ3ln⁡It-ln⁡I0t=b3+δ3c~3ta11βe-d3τ3+b1-δ1c~1ta33a21e-d2τ2-b2+δ2c~2ta11a33-a33Δ′+a11β2e-d3τ3-d4τ4St+Γ3,where Γ3 is given in Appendix.
WhenΔ3>0, by Lemma 4, we have (83)St∗≥Δ3Δ.From (71), when Δ2>0, we obtain (84)Xt∗≤Δ2Δ.By (61), we get (85)ln⁡St-ln⁡S0t≤-b2-δ2c~2t∗+a21e-d2τ2Xt∗-βe-d3τ3It∗-a22St+σ2B2tt+1t∫0t∫Yln1+γ2uN~ds,du,then from Lemma 4 we prove that (86)St∗≤Δ3Δ.By (60), we have (87)ln⁡Xt-ln⁡X0t≥b1-δ1c~1t∗-a12e-d1τ1St∗-a11Xt+σ1B1tt+1t∫0t∫Yln1+γ1uN~ds,du,and then from Lemma 4 we get (88)Xt∗≥Δ2Δ.It can be inferred from (62) that(89)ln⁡It-ln⁡I0t≤-b3-δ3c~3t∗+βe-d4τ4St∗-a33It-βe-d4τ4∫tt+τ4Ss-τ4ds-∫0τ4Ss-τ4dst+σ3B3tt+1t∫0t∫Yln1+γ3uN~ds,du,and then from Lemma 4 we have (90)It∗≤Δ4Δ.Consequently, (91)limt→∞Xt=Δ2Δ,limt→∞St=Δ3Δ,limt→∞It=Δ4Δ,a.s.This completes the proof of Theorem 7.


## 4. Conclusion and Simulations

This paper explores the dynamics of a stochastic predator-prey model with time delays in the polluted environment. We show some properties of the subsystem of the predator-prey system. Then by using comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. From Theorem6, we know that if environmental noises are large enough, the species will be extinct (see Figure 1) and if the amount of pulsed input concentration of the toxicant is large enough or the pulse period is small enough, the species will also be extinct (see Figures 2 and 3). In addition, Theorem 7 indicates that the permanence of populations has high correlation with the intensity of environmental noises (see Figure 2(a)). The main results show that time delays can lead to extinction of this system. Therefore, we realize that the environmental noises, the toxicant input, and delays are harmful to the permanence of the populations. Now we show some simulations to verify our main results.Figure 1
In (a),γ1=0, γ2=0, γ3=0, and the species are permanent; in (b), γ1=2.2, γ2=1.5, γ3=1.2, and then the species are extinct; in (c), γ1=0.5, γ2=1.5, γ3=1.2, X(t) is permanent, S(t) and I(t) are extinct, and limt→∞X(t)=2.953; in (d), γ1=0.5, γ2=0.5, γ3=1.2, X(t) and S(t) are permanent, I(t) is extinct, limt→∞X(t)=2.144, and limt→∞S(t)=0.892.
(a)
(b)
(c)
(d)Figure 2
Keepγ1=0.5, γ2=0.5, γ3=0.1, and T=1. In (a), u=0.4, X(t), S(t), and I(t) are permanent, limt→∞X(t)=2.072, limt→∞S(t)=0.680, and limt→∞I(t)=0.345; in (b), u=0.6, X(t) and S(t) are permanent, I(t) is extinct, limt→∞X(t)=1.719, and limt→∞S(t)=0.313; in (c), u=0.8, X(t) is permanent, S(t) and I(t) are extinct, and limt→∞X(t)=0.952; in (d), u=1 and then the species are extinct.
(a)
(b)
(c)
(d)Figure 3
Keepγ1=0.5, γ2=0.5, γ3=0.1, and u=0.4. In (a), T=1, X(t), S(t) and I(t) are permanent, limt→∞X(t)=2.072, limt→∞S(t)=0.680, and limt→∞I(t)=0.345; in (b), T=0.7, X(t) and S(t) are permanent, I(t) is extinct, limt→∞X(t)=1.743, and limt→∞S(t)=0.386; in (c), T=0.5, X(t) is permanent, S(t) and I(t) are extinct, and limt→∞X(t)=0.952; in (d), u=0.4 and then the species are extinct.
(a)
(b)
(c)
(d)Choose some parameters as followsr=1.6, μ1=0.1, μ2=0.1, δ1=0.3, δ2=0.1, δ3=0.1, a11=0.3, a12=0.3, a21=0.3, a22=0.3, a33=0.1, d1=0.1, d2=0.1, d3=0.1, d4=0.1, τ1=1, τ2=1, τ3=1, τ4=1, h=0.3, σ1=0.2, σ2=0.2, σ3=0.1, k1=0.3, g1=0.1, m1=0.1, k2=0.3, g2=0.2, m2=0.2, k3=0.3, g3=0.2, and m3=0.2. Based on the above parameters, we give simulations to explain the biological implications.In Figure1, we keep u=0.4 and T=1. In Figure 1(a), choose γ1=0, γ2=0, and γ3=0, there is no Lévy noises, and we can see that the species are permanent. In Figure 1(b), if γ1=2.2, γ2=1.5, and γ3=1.2, the intensities of Lévy noises are large and then we get b1=0.543<δ1c¯1=0.600, which means that the productiveness of the prey is less than its death loss rate; from Theorem 6, the species go extinction. In Figure 1(c), if γ1=0.5, γ2=1.5, and γ3=1.2, then b1=0.886>δ1c¯1=0.600 and Δ1=-0.009<0, from Theorem 6 we have that X(t) is permanent and S(t) and I(t) are extinct; furthermore, we have limt→∞X(t)=2.953. In Figure 1(d), if γ1=0.5, γ2=0.5, and γ3=1.2, then b1=0.886>δ1c¯1=0.600, Δ1=0.146>0, and R=0.946<1; from Theorem 6 we obtain that X(t) and S(t) are permanent and I(t) is extinct and we get limt→∞X(t)=2.144 and limt→∞St=0.892.In Figure2, we keep γ1=0.5, γ2=0.5, γ3=0.1, and T=1. In Figure 2(a), we choose u=0.4, and then Δ2=0.114>0, Δ3=0.037>0, and R=2.809>(a11a22/Δ′)R=1.559>1. From Theorem 7, we have that X(t), S(t), and I(t) are permanent and we obtain that limt→∞X(t)=2.072, limt→∞S(t)=0.680 and limt→∞I(t)=0.345. In Figure 2(b), if u=0.6, then Δ1=0.050>0 and R=0.768<1, and from Theorem 6, we get that X(t) and S(t) are permanent and I(t) is extinct and limt→∞X(t)=1.719 and limt→∞S(t)=0.313. In Figure 2(c), if u=0.8, then Δ1=-0.047<0, X(t) is permanent, S(t) and I(t) are extinct by Theorem 6, and limt→∞X(t)=0.952. In Figure 2(d), we choose u=1, and then b1=1.4855<δ1c¯1=1.5; from Theorem 6 the species are extinct.In Figure3, we keep γ1=0.5, γ2=0.5, γ3=0.1, and u=0.4. In Figure 3(a), we choose T=1 and then Δ2=0.114>0, Δ3=0.037>0, and R=2.809>(a11a22/Δ′)R=1.559>1. From Theorem 7, we have that X(t), S(t), and I(t) are permanent and we obtain that limt→∞X(t)=2.072, limt→∞S(t)=0.680, and limt→∞I(t)=0.345. In Figure 3(b), if T=0.7, then Δ1=0.063>0 and R=0.710<1, and from Theorem 6, we get that X(t) and S(t) are permanent and I(t) is extinct and limt→∞X(t)=1.743 and limt→∞S(t)=0.386. In Figure 3(c), if T=0.5, then Δ1=-0.047<0, X(t) is permanent, S(t) and I(t) are extinct by Theorem 6, and limt→∞X(t)=0.952. In Figure 3(d), we choose T=0.4 and then b1=1.4855<δ1c¯1=1.5; from Theorem 6 the species are extinct.From Figures1–3, we can obtain the following conclusions:(1)
Large stochastic disturbance can cause the populations to go to extinction; that is, the persistent population of a deterministic system can become extinct due to the white noise stochastic disturbance.(2)
Large impulsive input concentration of the toxicant or small impulsive period of the exogenous input of toxicant can cause the populations to go to extinction.Therefore, the above numerical simulations illustrate the performance of the theoretical results, and the biological results show that the white noise stochastic disturbance and impulsive toxicant input are disadvantage for the permanence of system.



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*Source: 1950970-2017-06-20.xml*