# Analysis of a Stochastic Competitive Model with Distributed Time Delays and Jumps in a Polluted Environment

Authors: Taolin Zhang; Yuanfu Shao; Xiaowan She
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, a stochastic competitive model with distributed time delays and Lévy jumps is formulated. With or without a polluted environment, the model is denoted by (M) or (M0), respectively. The existence of positive solution, persistence in mean, and extinction of species for (M) and (M0) are both studied. The sufficient criteria of stability in distribution for model (M) is obtained. Finally, some numerical simulations are given to illustrate our theoretical results.


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## Body

## 1. Introduction

The dynamics of the biological system has attracted many researchers and has no interruption in the past few decades. This includes the study of the persistence and extinction, stability in distribution of biological systems, optimal harvesting effects of renewable resources (for example, fish and plants), and so on. These studies have implications for the management of biological resources. The dynamics behaviors from the initial deterministic model to stochastic model have been extensively studied and a lot of nice results have been reported [1–4]. It has been verified that the growth rates of species are inevitably subject to white noise. And whether to consider the white noise is the difference between the stochastic model and the deterministic model. Following the method adopted in [4], we will model a stochastic system with white noise. For the biological system, usually there are three kinds of population relationship, i.e., predator-prey, mutualistic, and competitive scenarios, where the competitive scenario between populations is relatively popular [5, 6]. The general competitive model between two populations with white noise is as follows:(1)dy1t=y1tr10−a11y1t−a12y2tdt+σ1y1tdB1t,dy2t=y2tr20−a21y1t−a22y2tdt+σ2y2tdB2t,where yit is the size of the i-th population at time t; ri0 represents the growth rate of i-th population; aii>0 denotes the intraspecific competitive coefficients of yi; a12 and a21 are positive and represent the competitive rates between y1 and y2, respectively; Bit stands for the standard Brownian motion defined on a complete probability space Ω,ℱ,ℱt≥0,P with a filtration ℱt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous and ℱ0 contains all P null sets); σi2 is the intensity of the white noise, i=1,2.However, the world economy is developing more and more rapidly and the economic development will inevitably destroy the ecological environment. With the increasing toxins and pollution into the ecological environment, the quality of human living environment is becoming worse and worse. Therefore, the study on the impact of toxin importation and environmental pollution on biological populations has become one of the most popular topics in the world [7–12], which is of vital significance to the development of sustainable economy and the protection of human’s only living environment. Based on model (1) and considering the environmental pollution factors, then we derive the following model:(2)dy1t=y1tr10−r11C1t−a11y1t−a12y2tdt+σ1y1tdB1t,dy2t=y2tr20−r21C2t−a21y1t−a22y2tdt+σ2y2tdB2t,dC1t=k1CEt−g1+m1C1tdt,dC2t=k2CEt−g2+m2C2tdt,dCEt=−hCEt+utdt,where C1t,C2t, and CEt are the concentrations of the toxicant in the organism of species y1 and y2 and environment at time t, respectively; r11andr21 denote the dose-response of species y1 and y2 to the organismal toxicant, respectively; ki and gi represent the absorbing and excretion rates of the toxicant from the environment respectively, −mi is depuration rate of the toxicant, i=1,2. −hCEt denotes the loss rate of the toxicant because of volatilization; ut represents the exogenous rate of toxicant inputting into the environment and is always assumed to be bounded.On the other hand, the behavior between predator and prey is often not always continuous. For example, in some cases, young predators cannot engage in predation; that is, young prey cannot be preyed on. These phenomena are called time delays. Similar phenomena include hibernation, pregnancy, and migration. In fact, time delays exist not only in biological systems, but in other domains as well. For example, R. Manivannan has studied a control system with probabilistic time-delay signals [13]. Therefore, time delays are very important to reveal the real world and should be taken into account in our system. Some scholars pointed out that discrete delays and continuous delays do not include each other, but the S-type distributed delays can be done [14, 15]. Therefore, taking S-type delays into account in above model is interesting. In addition, in nature there are some environmental perturbations such as earthquakes, epidemics, and hurricanes, which differ from white noise because of its sudden and destructive nature, so Lévy jumps are introduced to simulate them in mathematical modeling [16–21]. For example, Liu and Wang [18] studied the persistence and extinction of the two-species model with Lévy jumps. Liu and Bai [21] investigated the stability in distribution of a stochastic model with Lévy noises by Lyapunov functional approach.Motivated by these, taking theS-type distributed time delays and Lévy noises into the above model, we get the following stochastic predator-prey model (M):(3)Mdy1t=y1t−r10−r11C1t−a11y1t−a12∫−τ20y2t−+θdμ2θdt+σ1y1tdB1t+y1t−∫ℤγ1uΓ˜dt,du,dy2t=y2t−r20−r21C2t−a21∫−τ10y1t−+θdμ1θ−a22y2tdt+σ2y2tdB2t+y2t−∫ℤγ2uΓ˜dt,du,dC1t=k1CEt−g1+m1C1tdt,dC2t=k2CEt−g2+m2C2tdt,dCEt=−hCEt+utdt,with initial data(4)yiθ=ξiθ,θ∈−τ,0,τ=maxτi,i=1,2,where yit− denotes the left limit of yit; Γ˜dt,du=Γdt,du−λdudt represents a compensated Poisson process, where Γ is a Poisson counting measure, λ is the characteristic measure of Γ on a measurable subset ℤ in 0,+∞ with λℤ<+∞,λdu is the measure of ℤ; γiu is a bounded function with γiu>−1,u∈ℤ; the term ∫−τi0yit−+θdμiθ denotes the Lebesgue–Stieltjes integral, where μiθ denotes a nonnegative variation function defined on −τ,0 with τ=maxτi satisfying ∫−τi0dμiθ=1,i=1,2. The biological meanings of other parameters are the same as before. If r11=r21=0, the corresponding model is denoted by (M0), which means that the population is not contaminated.We aim to study the dynamical behaviors of (M) and (M0) such as the extinction and persistence in mean for all species and explore the impacts on the dynamics of time delays and Lévy noise.The article is structured as follows. For preliminaries, we give some notations and important lemmas in Section2. In Section 3, we establish the sufficient criteria for the persistence in mean and nonpersistence of M and M0 and investigate the stable in distribution of (M). In Section 4, some numerical simulations are presented to verify our main results. Finally, conclusion and discussion are given to end this article in Section 5.


## 2. Preliminaries

For the simplicity, we first make the following notations:(5)bi=ri0−0.5σi2−∫ℤγiu−ln1+γiuλdu,i=1,2,Rit=∫0t∫ℤln1+γiuΓ˜ds,du,i=1,2,ft¯=t−1∫0tfsds,f∗=limt⟶∞supft,f∗=limt⟶∞infft,ft¯∗=limt⟶∞supt−1∫0tfsds,ft¯∗=limt⟶∞inft−1∫0tfsds,Δ=a11a22−a12a21,Δ1=a22b1−a12b2,Δ2=a11b2−a21b1,Δ31=a22b1−r11C1t¯∗−a12b2−r21C2t¯∗,Δ32=a22b1−r11C1t¯∗−a12b2−r21C2t¯∗,Δ41=−a21b1−r11C1t¯∗+a11b2−r21C2t¯∗,Δ42=−a21b1−r11C1t¯∗+a11b2−r21C2t¯∗.Our later discussion are based on the following technological hypotheses.Assumption 1.
There exists a positive constantL such that(6)∫ℤln1+γiu2λdu<L,i=1,2.Assumption 2.
Suppose thatΔ>0, which means the internal competition is greater than the external competition (see [22]).
The following lemmas are necessary for our later proof.Lemma 1.
Let Assumption1 hold, then for any given initial data ξ1t,ξ2t∈C−τ,0,R+2, there exists a unique solution remaining in R+2 with probability 1.Proof.
It is obvious that the coefficients of model (M) are locally Lipschitz. By [5], model (M) has a unique local solution yt=y1t,y2t∈R+2a.s. for any initial data ξ1t,ξ2t∈C−τ,0,R+2 and t∈0,τe, where τe>0 is the explosion time. It needs only to verify that τe=∞a.s. Let m0>0 be sufficiently large such that ξ10,ξ20∈1/m0,m0 for each integer m≥m0. Define the stopping time(7)τm=inft∈0,τe:y1t∉1m,m,y2t∉1m,m.
Clearly,τm is strictly increasing with m. Let τ∞=limm⟶∞τma.s., then τ∞≤τma.s. Then, we only need to prove τ∞=∞a.s. If the statement is not true, then there exist T>0 and 0<ε<1 such that Pτ∞<T>ε and an integer m1>m0 such that(8)Pτm≤T>ε,for anym>m1.
Define(9)Vy=βV1y1+V2y2+V3t,where(10)y=y1,y2,V1y1=y1−1−lny1,V2y2=y2−1−lny2,V3t=β2n2a12∫−τ20∫t+θty22s−dsdμ2θ+a212n2∫−τ10∫t+θty12s−dsdμ1θ.
Choose a constantβ>0 and integer n>0 such that(11)a212n2−βa11<0,βa122n2−a21<0.
For model (M), by Ito’s formula, we get(12)dVy=βLV1y1+LV2y2+ddtV3tdt+βσ1y1−1dB1t+β∫ℤy1γ1u−ln1+γ1uΓ˜ds,du+σ2y2−1dB2t+∫ℤy2γ2u−ln1+γ2uΓ˜ds,du,where(13)LV1y1=y1−1r10−r11C1t−a11y1t−−a12∫−τ20y2t−+θdμ2θ+0.5σ12+∫ℤγ1u−ln1+γ1uλdu,LV2y2=y2−1r20−r21C2t−a21∫−τ10y1t−+θdμ1θ−a22y2t−+0.5σ22+∫ℤγ2u−ln1+γ2uλdu.
By basic inequalitya2+b2≥2ab, we have(14)a12∫−τ20y2t−+θdμ2θ≤12a12n2+1n2∫−τ20y22t−+θdμ2θ,(15)a21∫−τ10y1t−+θdμ1θ≤12a21n2+1n2∫−τ10y12t−+θdμ1θ.
Substituting (14) and (15) into LV1y1,LV2y2, then(16)LV1y1≤y1r10−r11C1t−a11y12−r10+r11C1t+a11y1+0.5σ12+∫ℤγ1u−ln1+γ1uλdu+a12∫−τ20y2t−+θdμ2θ≤y1r10−r11C1t−a11y12−r10+r11C1t+a11y1+0.5σ12+∫ℤγ1u−ln1+γ1uλdu+n22a12+a122n2∫−τ20y22t−+θdμ2θ,LV2y2≤y2r20−r21C2t−a22y22−r20+r21C2t+a22y2+0.5σ22+∫ℤγ2u−ln1+γ2uλdu+n22a21+a212n2∫−τ10y12t−+θdμ1θ.
Therefore,(17)βLV1y1+LV2y2+ddtV3t≤βr10y1−r11C1ty1−a11y12−r10+r11C1t+a11y1+0.5σ12+n22a12+∫ℤγ1u−ln1+γ1uλdu+a122n2y22+r20y2−r21C2ty2−a22y22−r20+r21C2t+a22y2+0.5σ22+∫ℤγ2u−ln1+γ2uλdu+n22a21+a212n2y12.
Substituting (17) into dVy and together with (11), there exists a constant K>0 such that(18)dVy=Kdt+βσ1y1−1dB1t+β∫ℤy1γ1u−ln1+γ1uΓ˜ds,du+σ2y2−1dB2t+∫ℤy2γ2u−ln1+γ2uΓ˜ds,du.
By this result and according to the argument in [23], we have(19)∞≤KT+V0y10,y20≤∞,which leads to a contradiction, and hence, τ∞=∞a.s. Therefore, τe=∞a.s. The proof is completed.Lemma 2.
Lety1t,y2t be a positive solution of (14) initial data ξ1θ,ξ2θ∈C−τ,0,R+2. Then, for any p>0, there exists a constant Kip>0 such that(20)limt⟶+∞supEyipt≤Kip,i=1,2.Proof.
We only provelimt⟶+∞supEy1pt≤K1p. The rest is similar and omitted. Defining Q1t=ety1pt, by Ito’s formula, we have(21)dQ1t=LQ1tdt+pety1pσ1dB1t+∫ℤety1p1+γ1upΓ˜ds,du,where(22)LQ1t=ety1p1+pp−1σ122+∫ℤ1+γ1up−pγ1uλdu+pr10−r11C1t−a11y1t−−a12∫−τ20y2t−+θdμ2θ.
LetK1p=maxy1≥0y1p1+pp−1σ12/2+∫ℤ1+γ1up−pγ1uλdu+pr10−pa11y1p+1, then(23)LQ1t≤K1pet.
Integrating both sides of (21) from 0 to t and taking expectation leads to(24)Eety1p−ξ1p0≤K1pet−1.
Then,limt⟶+∞supEy1pt≤K1p. The proof is completed.Lemma 3.
(see [18]). Let Y∈CR+×Ω,R+,Z∈CR+×Ω,R, and ρ∈CR+,R satisfying limt⟶∞Zt/t=0a.s.(1)
If there exist two constantsT>0 and ρ0>0 such thatlnYt=∫0tρsds−ρ0∫0tYsds+Zta.s. for all t≥T,then (i)limt⟶∞Yt=0 and limt⟶∞suplnYt/t≤0a.s. if ρ¯<0;limt⟶∞Yt¯=ρ/ρ0 and limt⟶∞lnYt/t=0a.s. if ρ¯≥0.(2)
If there exist two constantsT>0 and ρ0>0 such thatlnYt≤∫0tρsds−ρ0∫0tYsds+Zta.s. for all t≥Tthen (i)limt⟶∞Yt¯∗≤ρ/ρ0a.s. if ρ¯≥0;limt⟶∞Yt=0,a.s. if ρ¯<0.(3)
If there exist two constantsT>0 and ρ0>0 such that for all(25)lnYt≥∫0tρsds−ρ0∫0tYsds+Zt,a.s.t≥T.Then,limt⟶∞Yt¯∗≥ρ/ρ0a.s.Lemma 4.
Lety1t,y2t be any positive solution of model (M), then(1)
limt⟶∞suplnyit/t≤0a.s.,i=1,2;(2)
For any positive constantτ, limt⟶∞t−1∫t−τtyisds=0a.s.,i=1,2.The proof of Lemma4 is standard and is omitted (see, e.g., [24]).


## 3. Main Results

### 3.1. Persistence in Mean

Definition 1.
(see [25]). The system is said to be persistence in mean if there are positive constants vi and sii=1,2 such that(26)vi≤yit¯∗≤yit¯∗≤si,i=1,2,holds for any solution y1t,y2t of model (M) with initial data(27)yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2.Theorem 1.
AssumeΔ1>0,Δ2>0, then for any initial data yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2, the solution y1t,y2t of model (M0) has the following properties:(28)limt⟶∞y1t¯=Δ1Δ,limt⟶∞y2t¯=Δ2Δ.Proof.
Fori=1,2, we compute(29)∫0t∫−τi0yis−+θdμiθds−∫−τi0dμiθ∫0tyis−ds=∫−τi0∫θ0yis−dsdμiθ−∫−τi0∫t+θtyis−dsdμiθ.
For model (M0), noticing ∫−τi0dμiθ=1i=1,2 and using Ito’s formula, we get(30)lny1t=b1t−a11∫0ty1s−ds+σ1B1t+∫0t∫ℤln1+γ1uΓ˜ds,du+lny10−a12∫0t∫−τ20y2s−+θμ2θds=b1t−a11∫0ty1s−ds−a12∫0ty2s−ds+Φ1t,(31)lny2t=b2t−a22∫0ty2s−ds−a21∫0ty1s−ds+Φ2t,where(32)Φ1t=a12∫−τ20∫t+θty2s−dsdμ2θ−a12∫−τ20∫θ0y2s−dsdμ2θ+lny10+σ1B1t+R1t,(33)Φ2t=a21∫−τ10∫t+θty1s−dsdμ1θ−a21∫−τ10∫θ0y1s−dsdμ1θ+lny20+σ2B2t+R2t.
Since(34)∫−τ20∫t+θty2s−dsdμ2θ≤∫−τ20dμ2θ∫t−τ2ty2s−ds,∫−τ20∫θ0y2s−dsdμ2θ≤∫−τ20dμ2θ∫−τ20y2s−ds.
By Lemma4, we get(35)limt⟶∞∫−τ20∫t+θty2s−dsdμ2θ=0,limt⟶∞∫−τ20∫θ0y2s−dsdμ2θ=0.
By the same way, we can derive that(36)limt⟶∞∫−τ10∫t+θty1s−dsdμ1θ=0,limt⟶∞∫−τ10∫θ0y1s−dsdμ1θ=0.
Substituting (35) and (36) into (32) and (33), respectively, then(37)limt⟶∞t−1Φ1t=0,limt⟶∞t−1Φ2t=0a.s.
Computing(38)a22lny1t−a12lny2t=a22b1−a12b2t−a11a22−a12a21∫0ty1s−ds+Φ3t=Δ1t−Δ∫0ty1s−ds+Φ3t,where Φ3t=a22Φ1t−a12Φ2t. From (37), we can easily get limt⟶∞t−1Φ3t=0a.s.
By Lemma4 again, for any ε>0, there is a T0>0 such that(39)a22lny1t≤Δ1+a12εt−Δ∫0ty1s−ds+Φ3t,for anyt>T0.
According to Lemma3, then(40)limt⟶∞y1t¯∗≤Δ1Δa.s.
Thus, for anyε>0 and sufficiently large t, there is(41)a21y1t¯≤a21y1t¯∗+ε≤a21Δ1Δ+ε.
Using (41) in (31), we get(42)lny2t≥b2−a21Δ1Δ−εt−a22∫0ty2s−ds+Φ2t=a22Δ2Δ−εt−a22∫0ty2s−ds+Φ2t,
Lemma3 implies that(43)limt⟶∞y2t¯∗≥Δ2Δa.s.
Similarly, we have(44)a11lny2t−a21lny1t=a11b2−a21b1t−a11a22−a12a21∫0ty2s−ds+Φ4t=Δ2t−Δ∫0ty2s−ds+Φ4t,where Φ4t=a11Φ2t−a21Φ1t. Obviously, limt⟶∞Φ4t=0.
From Lemma4, for any ε>0, there is a T1>0 such that(45)a11lny2t≤Δ2+a21εt−Δ∫0ty2s−ds+Φ4t,fort>T1.
It follows from Lemma3 that(46)limt⟶∞y2t¯∗≤Δ2Δa.s.
Combining (43) and (46) leads to(47)limt⟶∞y2t¯=Δ2Δa.s.
Substituting (47) into (30) and using Lemma 2, we get(48)limt⟶∞y1t¯=Δ1Δa.s.
The proof is completed.Next, let us consider model (M).Theorem 2.
IfΔ31>0,Δ41>0 hold, then for any initial yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0;R+2, the solution y1t,y2t of model (M) has the properties that(49)Δ31Δ≤y1t¯∗≤y1t¯∗≤Δ32Δ,Δ41Δ≤y2t¯∗≤y2t¯∗≤Δ42Δ.That is to say, model (M) will be persistence in mean.Proof.
Using Ito’s formula to computelnyiti=1,2, we have(50)t−1lny1t=b1−r11C1t¯−a11y1t¯−a12y2t¯+t−1Φ1t,(51)t−1lny2t=b2−r21C2t¯−a21y1t¯−a22y2t¯+t−1Φ2t.
Then,(52)a11t−1lny2t−a21t−1lny1t=a11b2−r21C2t¯−a21b1−r11C1t¯−a11a22−a12a21y2t¯+t−1Φ4t≥a11b2−r21C2t¯∗−a21b1−r11C1t¯∗−a11a22−a12a21y2t¯+t−1Φ4t=Δ41−Δy2t¯+t−1Φ4t.
We can get from Lemma3 that(53)y2t¯∗≥Δ41Δa.s.
For anyε>0, there is sufficiently T2>0 such that(54)a12y2t¯≥a12y2t¯∗−ε≥a12Δ41Δ−ε,t≥T2.
Substituting (54) into (50), we get(55)t−1lny1t≤b1−r11C1t¯∗−a12Δ41Δ+ε−a11y1t¯+t−1Φ1t=a11Δ32Δ+ε−a11y1t¯+t−1Φ1t.
Therefore,(56)y1t¯∗≤Δ32Δa.s.
Similar computation leads to(57)a22t−1lny1t−a12t−1lny2t=a22b1−r11C1t¯−a12b2−r21C2t¯−a11a22−a12a21y1t¯+t−1Φ3t≥Δ31−Δy1t¯+t−1Φ3t.
In view of Lemmas3 and 4, we have(58)y1t¯∗≥Δ31Δa.s.
For anyε>0, there is a T3>0 such that(59)a21y1t¯≥a21y1t¯∗−ε≥a21Δ31Δ−ε,t≥T3.
Substituting (59) into (51), we get(60)t−1lny2t≤b2−r21C2t¯∗−a21Δ31Δ+ε−a22y2t¯+t−1Φ2t=a22Δ42Δ+ε−a22y2t¯+t−1Φ2t,and then we have(61)y2t¯∗≤Δ42Δ.
Combing (53), (56), (58), and (61) leads to the result. The proof is completed.Remark 1.
If the limit ofC1t¯ and C2t¯ exist, that is, C1t¯∗=C1t¯∗ and C2t¯∗=C2t¯∗, then Theorem 2 will be simplified as the following case.
IfΔ31>0,Δ41>0 hold, then model (M) will have the properties that(62)limt⟶∞y1t¯=Δ31Δ,limt⟶∞y2t¯=Δ41Δ.


### 3.2. Nonpersistence

Theorem 3.
If0<a11/a21b2−r21C2t¯∗<b1−r11C1t¯∗≤b1−r11C1t¯∗ holds, then for any initial data yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2, the solution y1t,y2t of model (M) has the properties that(63)b1−r11C1t¯∗a11≤y1t¯∗≤y1t¯∗≤b1−r11C1t¯∗a11,limt⟶∞y2t=0a.s.That is to say, model (M) is nonpersistent.Proof.
From (50), we get(64)t−1lny1t≤b1−r11C1t¯∗−a11y1t¯+t−1Φ1t.
According to Lemma3, we get(65)y1t¯∗≤b1−r11C1t¯∗a11a.s.
We can easily compute(66)a11t−1lny2t−a21t−1lny1t=a11b2−r21C2t¯−a21b1−r11C1t¯−Δy2t¯+t−1Φ4t.
By Lemma4, for enough large t and any ε>0, there is(67)lnyitt<ε,i=1,2.
Then,(68)a11t−1lny2t≤a11b2−r21C2t¯∗−a21b1−r11C1t¯∗−Δy2t¯+t−1Φ4t.
By the assumption(69)a11a21b2−r21C2t¯∗<b1−r11C1t¯∗,and Lemma 3, we obtain from (68) that(70)limt⟶∞y2t=0a.s.
Substituting (70) into (50), then(71)t−1lny1t=b1−r11C1t¯−a11y1t¯+t−1Φ1t≥b1−r11C1t¯∗−a11y1t¯+t−1Φ1t.
From Lemma3, we get(72)y1t¯∗≥b1−r11C1t¯∗a11,and thus,(73)b1−r11C1t¯∗a11≤y1t¯∗≤y1t¯∗≤b1−r11C1t¯∗a11.
The proof is finished.Theorem 4.
Ifa11/a21b2−r21C2t¯∗<b1−r11C1t¯∗≤b1−r11C1t¯∗<0 holds, then for any initial data yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2, the solution y1t,y2t of model (M) has the properties that(74)limt⟶∞y1t=0,limt⟶∞y2t=0a.s.That is to say, model (M) is nonpersistent.Proof.
Under the assumption thatb1−r11C1t¯∗<0, we can get from Lemma 3 and (64) that(75)limt⟶∞y1t=0a.s.
Substituting (75) into (51) reads(76)t−1lny2t=b2−r21C2t¯−a22y2t¯+t−1Φ2t≤b2−r21C2t¯∗−a22y2t¯+t−1Φ2t.
The assumptiona11/a21b2−r21C2t¯∗<b1−r11C1t¯∗<0 implies that b2−r21C2t¯∗<0. Then, from (76) and Lemma 3, it is easy to verify that(77)limt⟶∞y2t=0a.s.
This completes the proof.


### 3.3. Stable in Distribution

Theorem 5.
If all the assumptions hold, then model (M) is stable in distribution.Proof.
The proof of this result is divided into three steps.
Step 1. We first prove model (M) is globally attractive.
Letyt=y1t;φ,y2t;φ,y∧t=y1t;φ∧,y2t;φ∧ be any two solutions of (M) with initial data φ,φ∧∈C−τ,0,R+2, respectively. Denote y˜it=yit;φ−yit;φ∧,i=1,2. We only need to verify that(78)limt⟶+∞Ey˜it=limt⟶+∞Eyit;φ−yit;φ∧=0,i=1,2.
DefineVit;φ,φ∧=lnyit;φ−lnyit;φ∧,i=1,2. By Ito’s formula, we get(79)LV1t;φ,φ∧=signy˜1t−a11y1t;φ−y1t;φ∧−a12∫−τ20y2t+θ;φ−y2t+θ;φ∧dμ2θ≤−a11y˜1t+a12∫−τ20y˜2t+θdμ2θ,LV2t;φ,φ∧=signy˜2t−a22y2t;φ−y2t;φ∧−a21∫−τ10y1t+θ;φ−y1t+θ;φ∧dμ1θ≤−a22y˜2t+a21∫−τ10y˜1t+θdμ1θ.
Let(80)Vt;φ,φ∧=∑i=12Vit;φ,φ∧+V3t;φ,φ∧,where V3t;φ,φ∧=a12∫−τ20∫t+θty˜2sdμ2θds+a21∫−τ10∫t+θty˜1sdμ1θds.
Using Ito’s formula to (80), we can easily compute that(81)LVt;φ,φ∧=LV1t;φ,φ∧+LV2t;φ,φ∧+dV3t;φ,φ∧dt≤−a11−a21∫−τ10dμ1θy˜1t−a22−a12∫−τ20dμ2θy˜2t.
According to (81), we get(82)EVt;φ,φ∧≤V0;φ,φ∧−a11−a21∫−τ10dμ1θ∫0tEy˜1sds−a22−a12∫−τ20dμ2θ∫0tEy˜2sds,which means(83)EVt;φ,φ∧+a11−a21∫−τ10dμ1θ∫0tEy˜1sds+a22−a12∫−τ20dμ2θ∫0tEy˜2sds≤V0;φ,φ∧<+∞.
Consequently,(84)Ey˜it∈L10,+∞,i=1,2.
Furthermore, considering the continuity ofEyit,i=1,2, and combining (M), we have(85)EBit=0,ERit=0,i=1,2.
Therefore,(86)Ey1t=y10+∫0tEr10y1s−r11C1ty1s−a11y12s−a12y1s∫−τ20y2s+θdμ2θds.
It is not difficult to see thatEy1t is differential. From Lemma 2, we get(87)dEy1tdt=Er10y1t−r11C1ty1t−a11y12t−a12y1t∫−τ20y2t+θdμ2θ≤Ey1tr10≤r10K1,where K1 is a positive constant. Therefore, Ey1t is uniformly continuous. Similarly, we can also obtain that Ey2t is uniformly continuous. By virtue of (84) and Barbalat’s conclusion in [26], we get(88)limt⟶+∞Ey˜it=0,i=1,2.
Step 2. For anyφ∈C−τ,0,R+2, it is denoted by pt,φ,dz the transition probability of the process zt, Pt,φ,R+2 the probability of y1t;φ,y2t;φT∈R+2, and P−τ,0,R+2 the space of all probability measures on C−τ,0,R+2. For any P1,P2∈P−τ,0,R+2, define(89)dBLP1,P2=supg∈BL∫R+2gzP1dz−∫R+2gzP2dz,where BL=g:C−τ,0,R+2⟶R:gz1−gz2≤z1−z2,g⋅≤1. Thanks to Lemma 2 and Chebyshev’s inequality, for any φ∈C−τ,0,R+2, the family pt,φ,dz is tight. That is, for any ε∈0,1, there exists a compact subset D⊆R+2 such that Pt,φ,D≥1−ε for any t≥0..
We compute(90)dBLpt+s,φ,⋅,pt,φ,⋅=supg∈BL∫R+2gzt+s;φpt+s,φ,dz−∫R+2gzt;φpt,φ,dz=supg∈BLEgzt+s;φ−Egzt;φ=supg∈BLEEgzt+s;φ|ℱs−Egzt;φ=supg∈BL∫R+2Egzt;ψps,φ,dψ−Egzt;φ=supg∈BL∫R+2Egzt;ψ−gzt;φps,φ,dψ≤supg∈BL∫R+2Egzt;ψ−gzt;φps,φ,dψ≤supg∈BL∫UBEgzt;ψ−gzt;φps,φ,dψ+supg∈BL∫R+2\UBEgzt;ψ−gzt;φps,φ,dψ,where UB=x,yT∈R+2:x2+y2≤B.(91)supg∈BL∫R+2\UBEgzt;ψ−gzt;φps,φ,dψ≤2Ps,φ,R+2\UB≤2ε.Therefore, for sufficiently large t and any ε>0, we can derive that(92)dBLpt+s,φ,⋅,pt,φ,⋅≤3ε.
That is to say,pt,φ,⋅:t≥0 is Cauchy in P−τ,0,R+2 with initial data φ∈C−τ,0,R+2.
Step 3. We provelimt⟶+∞dBLpt,φ,⋅,v⋅=0.
According to (92), for φ0∈C−τ,0,R+2,pt,φ0,⋅:t≥0 is Cauchy in P−τ,0,R+2, then there exists a unique v⋅ such that(93)limt⟶+∞dBLpt,φ0,⋅,v⋅=0.
By virtue of (78), we derive
γ2u=0.1837. That is, specie y2 will die out at some point.(94)limt⟶+∞dBLpt,φ,⋅,pt,φ0,⋅=supg∈BLEgzt;φ−Egzt;φ0≤supg∈BLEgzt;φ−gzt;φ0≤limt⟶+∞Ezt;φ−zt;φ0=0.
By the triangle inequality and together with (93) and (94), we have(95)limt⟶+∞dBLpt,φ,⋅,v⋅≤limt⟶+∞dBLpt,φ,⋅,pt,φ0,⋅+limt⟶+∞dBLpt,φ0,⋅,v⋅=0.
This completes the proof.


## 3.1. Persistence in Mean

Definition 1.
(see [25]). The system is said to be persistence in mean if there are positive constants vi and sii=1,2 such that(26)vi≤yit¯∗≤yit¯∗≤si,i=1,2,holds for any solution y1t,y2t of model (M) with initial data(27)yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2.Theorem 1.
AssumeΔ1>0,Δ2>0, then for any initial data yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2, the solution y1t,y2t of model (M0) has the following properties:(28)limt⟶∞y1t¯=Δ1Δ,limt⟶∞y2t¯=Δ2Δ.Proof.
Fori=1,2, we compute(29)∫0t∫−τi0yis−+θdμiθds−∫−τi0dμiθ∫0tyis−ds=∫−τi0∫θ0yis−dsdμiθ−∫−τi0∫t+θtyis−dsdμiθ.
For model (M0), noticing ∫−τi0dμiθ=1i=1,2 and using Ito’s formula, we get(30)lny1t=b1t−a11∫0ty1s−ds+σ1B1t+∫0t∫ℤln1+γ1uΓ˜ds,du+lny10−a12∫0t∫−τ20y2s−+θμ2θds=b1t−a11∫0ty1s−ds−a12∫0ty2s−ds+Φ1t,(31)lny2t=b2t−a22∫0ty2s−ds−a21∫0ty1s−ds+Φ2t,where(32)Φ1t=a12∫−τ20∫t+θty2s−dsdμ2θ−a12∫−τ20∫θ0y2s−dsdμ2θ+lny10+σ1B1t+R1t,(33)Φ2t=a21∫−τ10∫t+θty1s−dsdμ1θ−a21∫−τ10∫θ0y1s−dsdμ1θ+lny20+σ2B2t+R2t.
Since(34)∫−τ20∫t+θty2s−dsdμ2θ≤∫−τ20dμ2θ∫t−τ2ty2s−ds,∫−τ20∫θ0y2s−dsdμ2θ≤∫−τ20dμ2θ∫−τ20y2s−ds.
By Lemma4, we get(35)limt⟶∞∫−τ20∫t+θty2s−dsdμ2θ=0,limt⟶∞∫−τ20∫θ0y2s−dsdμ2θ=0.
By the same way, we can derive that(36)limt⟶∞∫−τ10∫t+θty1s−dsdμ1θ=0,limt⟶∞∫−τ10∫θ0y1s−dsdμ1θ=0.
Substituting (35) and (36) into (32) and (33), respectively, then(37)limt⟶∞t−1Φ1t=0,limt⟶∞t−1Φ2t=0a.s.
Computing(38)a22lny1t−a12lny2t=a22b1−a12b2t−a11a22−a12a21∫0ty1s−ds+Φ3t=Δ1t−Δ∫0ty1s−ds+Φ3t,where Φ3t=a22Φ1t−a12Φ2t. From (37), we can easily get limt⟶∞t−1Φ3t=0a.s.
By Lemma4 again, for any ε>0, there is a T0>0 such that(39)a22lny1t≤Δ1+a12εt−Δ∫0ty1s−ds+Φ3t,for anyt>T0.
According to Lemma3, then(40)limt⟶∞y1t¯∗≤Δ1Δa.s.
Thus, for anyε>0 and sufficiently large t, there is(41)a21y1t¯≤a21y1t¯∗+ε≤a21Δ1Δ+ε.
Using (41) in (31), we get(42)lny2t≥b2−a21Δ1Δ−εt−a22∫0ty2s−ds+Φ2t=a22Δ2Δ−εt−a22∫0ty2s−ds+Φ2t,
Lemma3 implies that(43)limt⟶∞y2t¯∗≥Δ2Δa.s.
Similarly, we have(44)a11lny2t−a21lny1t=a11b2−a21b1t−a11a22−a12a21∫0ty2s−ds+Φ4t=Δ2t−Δ∫0ty2s−ds+Φ4t,where Φ4t=a11Φ2t−a21Φ1t. Obviously, limt⟶∞Φ4t=0.
From Lemma4, for any ε>0, there is a T1>0 such that(45)a11lny2t≤Δ2+a21εt−Δ∫0ty2s−ds+Φ4t,fort>T1.
It follows from Lemma3 that(46)limt⟶∞y2t¯∗≤Δ2Δa.s.
Combining (43) and (46) leads to(47)limt⟶∞y2t¯=Δ2Δa.s.
Substituting (47) into (30) and using Lemma 2, we get(48)limt⟶∞y1t¯=Δ1Δa.s.
The proof is completed.Next, let us consider model (M).Theorem 2.
IfΔ31>0,Δ41>0 hold, then for any initial yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0;R+2, the solution y1t,y2t of model (M) has the properties that(49)Δ31Δ≤y1t¯∗≤y1t¯∗≤Δ32Δ,Δ41Δ≤y2t¯∗≤y2t¯∗≤Δ42Δ.That is to say, model (M) will be persistence in mean.Proof.
Using Ito’s formula to computelnyiti=1,2, we have(50)t−1lny1t=b1−r11C1t¯−a11y1t¯−a12y2t¯+t−1Φ1t,(51)t−1lny2t=b2−r21C2t¯−a21y1t¯−a22y2t¯+t−1Φ2t.
Then,(52)a11t−1lny2t−a21t−1lny1t=a11b2−r21C2t¯−a21b1−r11C1t¯−a11a22−a12a21y2t¯+t−1Φ4t≥a11b2−r21C2t¯∗−a21b1−r11C1t¯∗−a11a22−a12a21y2t¯+t−1Φ4t=Δ41−Δy2t¯+t−1Φ4t.
We can get from Lemma3 that(53)y2t¯∗≥Δ41Δa.s.
For anyε>0, there is sufficiently T2>0 such that(54)a12y2t¯≥a12y2t¯∗−ε≥a12Δ41Δ−ε,t≥T2.
Substituting (54) into (50), we get(55)t−1lny1t≤b1−r11C1t¯∗−a12Δ41Δ+ε−a11y1t¯+t−1Φ1t=a11Δ32Δ+ε−a11y1t¯+t−1Φ1t.
Therefore,(56)y1t¯∗≤Δ32Δa.s.
Similar computation leads to(57)a22t−1lny1t−a12t−1lny2t=a22b1−r11C1t¯−a12b2−r21C2t¯−a11a22−a12a21y1t¯+t−1Φ3t≥Δ31−Δy1t¯+t−1Φ3t.
In view of Lemmas3 and 4, we have(58)y1t¯∗≥Δ31Δa.s.
For anyε>0, there is a T3>0 such that(59)a21y1t¯≥a21y1t¯∗−ε≥a21Δ31Δ−ε,t≥T3.
Substituting (59) into (51), we get(60)t−1lny2t≤b2−r21C2t¯∗−a21Δ31Δ+ε−a22y2t¯+t−1Φ2t=a22Δ42Δ+ε−a22y2t¯+t−1Φ2t,and then we have(61)y2t¯∗≤Δ42Δ.
Combing (53), (56), (58), and (61) leads to the result. The proof is completed.Remark 1.
If the limit ofC1t¯ and C2t¯ exist, that is, C1t¯∗=C1t¯∗ and C2t¯∗=C2t¯∗, then Theorem 2 will be simplified as the following case.
IfΔ31>0,Δ41>0 hold, then model (M) will have the properties that(62)limt⟶∞y1t¯=Δ31Δ,limt⟶∞y2t¯=Δ41Δ.


## 3.2. Nonpersistence

Theorem 3.
If0<a11/a21b2−r21C2t¯∗<b1−r11C1t¯∗≤b1−r11C1t¯∗ holds, then for any initial data yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2, the solution y1t,y2t of model (M) has the properties that(63)b1−r11C1t¯∗a11≤y1t¯∗≤y1t¯∗≤b1−r11C1t¯∗a11,limt⟶∞y2t=0a.s.That is to say, model (M) is nonpersistent.Proof.
From (50), we get(64)t−1lny1t≤b1−r11C1t¯∗−a11y1t¯+t−1Φ1t.
According to Lemma3, we get(65)y1t¯∗≤b1−r11C1t¯∗a11a.s.
We can easily compute(66)a11t−1lny2t−a21t−1lny1t=a11b2−r21C2t¯−a21b1−r11C1t¯−Δy2t¯+t−1Φ4t.
By Lemma4, for enough large t and any ε>0, there is(67)lnyitt<ε,i=1,2.
Then,(68)a11t−1lny2t≤a11b2−r21C2t¯∗−a21b1−r11C1t¯∗−Δy2t¯+t−1Φ4t.
By the assumption(69)a11a21b2−r21C2t¯∗<b1−r11C1t¯∗,and Lemma 3, we obtain from (68) that(70)limt⟶∞y2t=0a.s.
Substituting (70) into (50), then(71)t−1lny1t=b1−r11C1t¯−a11y1t¯+t−1Φ1t≥b1−r11C1t¯∗−a11y1t¯+t−1Φ1t.
From Lemma3, we get(72)y1t¯∗≥b1−r11C1t¯∗a11,and thus,(73)b1−r11C1t¯∗a11≤y1t¯∗≤y1t¯∗≤b1−r11C1t¯∗a11.
The proof is finished.Theorem 4.
Ifa11/a21b2−r21C2t¯∗<b1−r11C1t¯∗≤b1−r11C1t¯∗<0 holds, then for any initial data yt=ξ1t,ξ2t:−τ≤t≤0∈C−τ,0:R+2, the solution y1t,y2t of model (M) has the properties that(74)limt⟶∞y1t=0,limt⟶∞y2t=0a.s.That is to say, model (M) is nonpersistent.Proof.
Under the assumption thatb1−r11C1t¯∗<0, we can get from Lemma 3 and (64) that(75)limt⟶∞y1t=0a.s.
Substituting (75) into (51) reads(76)t−1lny2t=b2−r21C2t¯−a22y2t¯+t−1Φ2t≤b2−r21C2t¯∗−a22y2t¯+t−1Φ2t.
The assumptiona11/a21b2−r21C2t¯∗<b1−r11C1t¯∗<0 implies that b2−r21C2t¯∗<0. Then, from (76) and Lemma 3, it is easy to verify that(77)limt⟶∞y2t=0a.s.
This completes the proof.


## 3.3. Stable in Distribution

Theorem 5.
If all the assumptions hold, then model (M) is stable in distribution.Proof.
The proof of this result is divided into three steps.
Step 1. We first prove model (M) is globally attractive.
Letyt=y1t;φ,y2t;φ,y∧t=y1t;φ∧,y2t;φ∧ be any two solutions of (M) with initial data φ,φ∧∈C−τ,0,R+2, respectively. Denote y˜it=yit;φ−yit;φ∧,i=1,2. We only need to verify that(78)limt⟶+∞Ey˜it=limt⟶+∞Eyit;φ−yit;φ∧=0,i=1,2.
DefineVit;φ,φ∧=lnyit;φ−lnyit;φ∧,i=1,2. By Ito’s formula, we get(79)LV1t;φ,φ∧=signy˜1t−a11y1t;φ−y1t;φ∧−a12∫−τ20y2t+θ;φ−y2t+θ;φ∧dμ2θ≤−a11y˜1t+a12∫−τ20y˜2t+θdμ2θ,LV2t;φ,φ∧=signy˜2t−a22y2t;φ−y2t;φ∧−a21∫−τ10y1t+θ;φ−y1t+θ;φ∧dμ1θ≤−a22y˜2t+a21∫−τ10y˜1t+θdμ1θ.
Let(80)Vt;φ,φ∧=∑i=12Vit;φ,φ∧+V3t;φ,φ∧,where V3t;φ,φ∧=a12∫−τ20∫t+θty˜2sdμ2θds+a21∫−τ10∫t+θty˜1sdμ1θds.
Using Ito’s formula to (80), we can easily compute that(81)LVt;φ,φ∧=LV1t;φ,φ∧+LV2t;φ,φ∧+dV3t;φ,φ∧dt≤−a11−a21∫−τ10dμ1θy˜1t−a22−a12∫−τ20dμ2θy˜2t.
According to (81), we get(82)EVt;φ,φ∧≤V0;φ,φ∧−a11−a21∫−τ10dμ1θ∫0tEy˜1sds−a22−a12∫−τ20dμ2θ∫0tEy˜2sds,which means(83)EVt;φ,φ∧+a11−a21∫−τ10dμ1θ∫0tEy˜1sds+a22−a12∫−τ20dμ2θ∫0tEy˜2sds≤V0;φ,φ∧<+∞.
Consequently,(84)Ey˜it∈L10,+∞,i=1,2.
Furthermore, considering the continuity ofEyit,i=1,2, and combining (M), we have(85)EBit=0,ERit=0,i=1,2.
Therefore,(86)Ey1t=y10+∫0tEr10y1s−r11C1ty1s−a11y12s−a12y1s∫−τ20y2s+θdμ2θds.
It is not difficult to see thatEy1t is differential. From Lemma 2, we get(87)dEy1tdt=Er10y1t−r11C1ty1t−a11y12t−a12y1t∫−τ20y2t+θdμ2θ≤Ey1tr10≤r10K1,where K1 is a positive constant. Therefore, Ey1t is uniformly continuous. Similarly, we can also obtain that Ey2t is uniformly continuous. By virtue of (84) and Barbalat’s conclusion in [26], we get(88)limt⟶+∞Ey˜it=0,i=1,2.
Step 2. For anyφ∈C−τ,0,R+2, it is denoted by pt,φ,dz the transition probability of the process zt, Pt,φ,R+2 the probability of y1t;φ,y2t;φT∈R+2, and P−τ,0,R+2 the space of all probability measures on C−τ,0,R+2. For any P1,P2∈P−τ,0,R+2, define(89)dBLP1,P2=supg∈BL∫R+2gzP1dz−∫R+2gzP2dz,where BL=g:C−τ,0,R+2⟶R:gz1−gz2≤z1−z2,g⋅≤1. Thanks to Lemma 2 and Chebyshev’s inequality, for any φ∈C−τ,0,R+2, the family pt,φ,dz is tight. That is, for any ε∈0,1, there exists a compact subset D⊆R+2 such that Pt,φ,D≥1−ε for any t≥0..
We compute(90)dBLpt+s,φ,⋅,pt,φ,⋅=supg∈BL∫R+2gzt+s;φpt+s,φ,dz−∫R+2gzt;φpt,φ,dz=supg∈BLEgzt+s;φ−Egzt;φ=supg∈BLEEgzt+s;φ|ℱs−Egzt;φ=supg∈BL∫R+2Egzt;ψps,φ,dψ−Egzt;φ=supg∈BL∫R+2Egzt;ψ−gzt;φps,φ,dψ≤supg∈BL∫R+2Egzt;ψ−gzt;φps,φ,dψ≤supg∈BL∫UBEgzt;ψ−gzt;φps,φ,dψ+supg∈BL∫R+2\UBEgzt;ψ−gzt;φps,φ,dψ,where UB=x,yT∈R+2:x2+y2≤B.(91)supg∈BL∫R+2\UBEgzt;ψ−gzt;φps,φ,dψ≤2Ps,φ,R+2\UB≤2ε.Therefore, for sufficiently large t and any ε>0, we can derive that(92)dBLpt+s,φ,⋅,pt,φ,⋅≤3ε.
That is to say,pt,φ,⋅:t≥0 is Cauchy in P−τ,0,R+2 with initial data φ∈C−τ,0,R+2.
Step 3. We provelimt⟶+∞dBLpt,φ,⋅,v⋅=0.
According to (92), for φ0∈C−τ,0,R+2,pt,φ0,⋅:t≥0 is Cauchy in P−τ,0,R+2, then there exists a unique v⋅ such that(93)limt⟶+∞dBLpt,φ0,⋅,v⋅=0.
By virtue of (78), we derive
γ2u=0.1837. That is, specie y2 will die out at some point.(94)limt⟶+∞dBLpt,φ,⋅,pt,φ0,⋅=supg∈BLEgzt;φ−Egzt;φ0≤supg∈BLEgzt;φ−gzt;φ0≤limt⟶+∞Ezt;φ−zt;φ0=0.
By the triangle inequality and together with (93) and (94), we have(95)limt⟶+∞dBLpt,φ,⋅,v⋅≤limt⟶+∞dBLpt,φ,⋅,pt,φ0,⋅+limt⟶+∞dBLpt,φ0,⋅,v⋅=0.
This completes the proof.


## 4. Numerical Simulations

In this section, some numerical simulations are given to verify our theoretical results. Takea11=0.52,a12=0.02,a21=0.03,a22=0.8,r10=0.71,r20=1.35,σ12=0.22,σ22=0.59,τ1=0.1,τ2=0.1. It is easy to check that Δ=0.4154, which means Assumption 2 holds.(1)
Setγ1u=1.0187,γ2u=0.1332, and then it is easy to count that Δ1=0.2061,Δ2=0.5358. Theorem 1 implies limt⟶∞y1t¯=0.4961,limt⟶∞y2t¯=1.2900, that is, the species are both persistent in mean. Simulation also validates the result, see Figure 1.(2)
Setγ1u=0.1303,γ2u=0.1837,r11=0.48,r21=0.86,C1t=0.1+0.1sint,C2t=0.4+0.1sint. By computation, we have Δ31=0.3813,Δ41=0.2994. By Theorem 2 and Remark 1, model (M) is persistent in mean, shown in Figure 2.(3)
Setγ1u=0.5528,γ2u=1.0391,r11=0.48,r21=0.86,C1t=0.1+0.1sint,C2t=0.4+0.1sint, and then these parameters satisfy the conditions of Theorem 3. By Theorem 3, we know that y1 is permanent, but y2 will die out, see Figure 3.(4)
Setγ1u=1.0429,γ2u=1.6876,r11=0.1,r21=0.56,C1t=0.4+0.1sint,C2t=0.4+0.1sint,σ12=0.87,σ22=1.35. Obviously, Theorem 4 shows that both y1 and y2 will die out, see Figure 4.Figure 1
The persistent in mean of model (M0) with γ1u=1.0187,γ2u=0.1332. (a) The time series of y1,y2, respectively; (b) the probability density function of γ1u=1.0429,γ2u=1.6876, respectively.
(a)(b)Figure 2
The persistent in mean of model (M) with γ1u=0.1303,γ2u=0.1837. (a) Time series of y1andy2; (b) the probability density function of y1andy2.
(a)(b)Figure 3
The nonpersistent in mean for model (M) with γ1u=0.1303,Figure 4
The extinction of model (M) with γ1u=1.0429,γ2u=1.6876. Both species y1 and y2 will die out at some point.


## 5. Conclusions

The study of biological dynamics has been a popular topic in the field of biomathematics in recent years. With the development of economy, the environmental pollution is becoming more and more serious, which has become an important factor affecting the population relationship. Time delays are also important factors affecting the relationship. In this paper, we formulate a delayed predator-prey model with Lévy noise. Theorem1 and 2 give the sufficient criteria of persistent in mean for cases (M) and (M0), respectively. Theorem 3 and 4 obtain the sufficient conditions of nonpersistence. Theorem 5 investigates the stable in distribution. Finally, numerical simulations are given to validate our conclusion.In view of the complexity of the environments, other factors such as the telephone noise and impulsive input may bring important influence to the dynamics, which needs further research in the future.



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