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Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps

  • This paper studies a stochastic delay logistic model with Lévy jumps and impulsive perturbations. We show that the model has a unique global positive solution. Sufficient conditions for extinction, non-persistence in the mean, weak persistence, stochastic permanence and global asymptotic stability are established. The threshold between weak persistence and extinction is obtained. The results demonstrate that impulsive perturbations which may represent human factor play an important role in protecting the population even if it suffers sudden environmental shocks that can be discribed by Lévy jumps.

    Citation: Chun Lu, Bing Li, Limei Zhou, Liwei Zhang. Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3251-3271. doi: 10.3934/mbe.2019162

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  • This paper studies a stochastic delay logistic model with Lévy jumps and impulsive perturbations. We show that the model has a unique global positive solution. Sufficient conditions for extinction, non-persistence in the mean, weak persistence, stochastic permanence and global asymptotic stability are established. The threshold between weak persistence and extinction is obtained. The results demonstrate that impulsive perturbations which may represent human factor play an important role in protecting the population even if it suffers sudden environmental shocks that can be discribed by Lévy jumps.



    In the real world, "all species should exhibit time delay" [1]. As is known to all, logistic model with delay is one of the most important and classical model in mathematical biology. The deterministic logistic model with infinite delay is described by the differential equation:

    dy(t)dt=y(t)[ba1y(t)+a2y(tτ)+a30y(t+θ)dζ(θ)], (1.1)

    where y(t) is the population size at time t, b denotes the intrinsic growth rate, τ represents the time delay and ζ(θ) is a probability measure on (,0]. Model (1.1) and its various forms have been investigated extensively (see [1,2,3,4,5,6]). Particularly, Gopalsamy [2] and Kuang [6] have obtained the classical result: if b>0 and a1>a2+a3, then the positive equilibrium of model (1.1) is globally asymptotically stable.

    Biological populations are inevitably subject to environmental noises which are the important component in an ecosystem(see [7,8,9,10,11,12,13,14,15,16]). And time delay and system uncertainty are commonly encountered and are often the sources of instability[4]. Considering the effect of stochastic factors for infinite delay systems have attracted great attentions in recent years(see [9,10,11,12,14,15,16]). Wu et al.[14] studied the effects of environmental noise on the asymptotic properties of population model with infinite delay for the first time.

    For the intrinsic growth rate b of model (1.1), there are many methods of introducing random perturbations into the models from biological and mathematical perspectives [17,18,19,20,21,22,23,24,25]. The corresponding non-autonomous system of (1.1) can be described as follows:

    dy(t)=y(t)[b(t)a1(t)y(t)+a2(t)y(tτ)+a3(t)0y(t+θ)dζ(θ)]dt+σ1(t)y1+ϱ(t)dB(t)+y(t)Yβ(u)˜N(dt,du), (1.2)

    where B(t) is standard Brownian motion defined on a complete probability space (Ω,F,P), σ1(t) is positive continuous bounded function on ¯R+=[0,+), ϱ is a constant, y(t)=limsty(s), N(dt,du) is a real-valued Poisson counting measure with characteristic measure λ on a measurable subset Y of ¯R+ with λ(Y)<+,˜N(dt,du)=N(dt,du)λ(du)dt, β(u) is bounded function and β(u)>1,uY. Also, B(t) is independent of N(dt,du). Recently, Liu et al.[22] get the results of the extinction, non-persistence in the mean and weak persistence for the following model:

    dy(t)=y(t)[b(t)a1(t)y(t)+a2(t)y(tτ(t))+a3(t)0y(t+θ)dζ(θ)]dt+σ1(t)y(t)dω1(t)+σ2(t)y1+α(t)dω2(t)+σ3(t)yβ(tτ(t))dω3(t)+σ4(t)y(t)0y(t+θ)dζ(θ)dω4(t)+y(t)Yγ(t,y)˜N(dt,dy), (1.3)

    where α,β are positive constants, ωi(t) are standard Brownian motions, i=1,2,3,4, and N(dt,du) is a real-valued Poisson counting measure. Obviously, model (1.2) is a special case of model (1.3).

    Moreover, impulsive stochastic differential equations have been considered extensively because impulsive perturbations can describe the activities of human exploitation, such as planting and harvesting (see [26,27,28,29,30,31,32,33,34]). Liu et.al [26] investigated the impact of impulsive perturbation in the stochastic model driven by Brownian motion for the first time. However, there are little attention on the effects of impulsive perturbation in the stochastic delay model driven by Lévy jumps. This is because the impulsive perturbations can not classified as Lévy jumps(see [35,36]). Motivated by the above, we will study the following stochastic delay logistic model with Lévy jumps and impulsive perturbations:

    {dy(t)=y(t)[b(t)a1(t)y(t)+a2(t)y(tτ)+a3(t)0y(t+θ)dζ(θ)]dt+σ1(t)y1+ϱ(t)dB(t)+y(t)Yβ(u)˜N(dt,du),ttk,KN,y(t+k)y(tk)=Jky(tk),kN, (1.4)

    where y(tk)=limttky(t),y(t+k)=limtt+ky(t) (see [37]), N denotes the set of positive integers, 0<t1<t2,limk+tk=+. The other parameters are as defined above.

    The main aims of the work are to discuss how impulsive perturbations impact on the persistence and extinction of (1.4). We need to obtain the sufficient conditions of stochastic permanence and global asymptotic stability for model (1.4), which is not studied in [22]. For model (1.4) without Lévy jumps and impulsive perturbations, in Ref.[16], Lu et.al have investigated the stochastic permanence when the parameter a3(t)>0. In this paper we will discuss the global asymptotic stability of (1.4) without Lévy jumps and impulsive perturbations when the parameter a3(t)<0, which are good supplements for the deficiency of [16].

    To proceed, we introduce the notations and definitions into here.

    gu=supt¯R+g(t),gl=inft¯R+g(t),g(t)=1tt0g(s)ds,
    g=lim inft+g(t),g=lim supt+g(t),R+=(0,+).

    The following definitions are commonly used and we list them here.

    Definition 1. The population y(t) is said to be extinction if limt+y(t)=0.

    Definition 2. The population y(t) is said to be non-persistence in the mean(see e.g., [38]) if lim supt+y(t)=0.

    Definition 3. The population y(t) is said to be weak persistence(see e.g., Hallam and Ma [39]) if lim supt+y(t)>0.

    Definition 4. The population y(t) is said to be stochastic permanence [13] if for an arbitrary ε>0, there are constants α1>0,α2>0 such that lim inft+P{y(t)α1}1ε and lim inft+P{y(t)α2}1ε.

    Definition 5. Let y(t) and y(t) be two arbitrary solutions with initial values ξ and ξ, respectively. If limt+|y(t)y(t)|=0 a.s., then we say that solution of model (1.4) without Lévy jumps and impulsive perturbations is globally asymptotically stable.

    The rest of the paper is arranged as follows. In Section 2, we obtain the sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence, respectively. In Section 3, we get the sufficient condition of global asymptotic stability of solution for model (1.4) without Lévy jumps and impulsive perturbation. The last section, we illustrate the main results with figure testification.

    In this section, we shall present the extinction and permanence of model (1.4). For simply research, we make the following assumptions:

    (A1): Based on biological meanings, we assume that 1+Jk>0.

    (A2): b(t), a1(t), a2(t), a3(t) and σ1(t) are continuous and bounded functions on ¯R+, where ¯R+=[0,+). al>0, σl>0 and ϱ>0.75.

    (A3): There exists a positive constant c such that |ln(1+β(u))|c for β(u)>1.

    (A4): μr=0e2rθdζ(θ)<+. In this article, K denotes a positive constant and its value may be different in various conditions. Let the initial value ξ be positive and ξCg (see [2,22,40,41]) which is defined by

    Cg={φC((,0];R):∥φcg=sup<s0ers|φ(s)|<+},

    where g(s)=ers,r>0.

    Lemma 1. Suppose (A1)-(A4) hold. For model (1.4), with any given initial value ξCg, there is a unique solution y(t) on tR with probability 1.

    Proof. Now consider the stochastic functional differential equation with infinite delay and Lévy jumps:

    dz(t)=z(t)[b(t)0<tk<t(1+Jk)a1(t)z(t)+0<tk<tτ(1+Jk)a2(t)z(tτ)+a3(t)00<tk<t+θ(1+Jk)z(t+θ)dζ(θ)]dt+σ1(t)(0<tk<t(1+Jk))ϱz1+ϱ(t)dB(t)+z(t)Yβ(u)˜N(dt,du) (2.1)

    with the same initial value as model (1.4). Obviously, tt1, 0<tk<t(1+Jk)=1 holds and y(t)=0<tk<t(1+Jk)z(t),t>0. By the work of Mao et al. [10], (2.1) has a unique maximal local solution, on (,τe), where τe is the explosion time. The rest of proof is standard and hence is omitted(see [26,32]). This completes the proof.

    Theorem 1. Suppose (A1)-(A4) hold, if G<0 and inft¯R+{a1(t)a2(t+τ)au3}0, where G=lim supt+t1[0<tk<tln(1+Jk)+t0b(s)ds]Y[β(u)ln(1+β(u))]λ(du), then population y(t) modeled by (1.4) will go to extinction almost surely (a.s.).

    Proof. For (2.1), using the Itô's formula, we get

    dlnz=dzz(dz)22z2=[b(t)0<tk<t(1+Jk)a1(t)z+0<tk<tτ(1+Jk)a2(t)z(tτ)+a3(t)00<tk<t+θ(1+Jk)z(t+θ)dζ(θ)(0<tk<t(1+Jk))2ϱσ2(t)z2ϱ(t)2Y(β(u)ln(1+β(u))λ(du)]dt+σ1(t)(0<tk<t(1+Jk))ϱzϱ(t)dB(t)+Yln(1+β(u))˜N(dt,du).

    Integrating both sides from 0 to t, where t[0,t1] or t(tk,tk+1],k=1,2,, we get

    lnz(t)lnz(0)=t0[b(s)0<tk<s(1+Jk)a1(s)z(s)+0<tk<sτ(1+Jk)a2(s)z(sτ)+a3(s)00<tk<s+θ(1+Jk)z(s+θ)dζ(θ)(0<tk<s(1+Jk))2σ2(s)z2ϱ(s)2]ds+t0(0<tk<s(1+Jk))ϱσ1(s)zϱ(s)dB(s)tY[β(u)ln(1+β(u))]λ(du)+t0Yln(1+β(u))˜N(ds,du).

    Then

    lnz(t)lnz(0)=t0[b(s)a1(s)y(s)+a2(s)y(sτ)+a3(s)0y(s+θ)dζ(θ)σ2(s)y2ϱ(s)2]ds+t0σ1(s)yϱ(s)dB(s)tY[β(u)ln(1+β(u))]λ(du)+t0Yln(1+β(u))˜N(ds,du). (2.2)

    On the other hand,

    t0a2(s)y(sτ)ds=tττa2(s+τ)y(s)ds=0τa2(s+τ)y(s)ds+tτ0a2(s+τ)y(s)ds0τa2(s+τ)y(s)ds+t0a2(s+τ)y(s)ds.

    In other word, t¯R+, we have

    lnz(t)lnz(0)=t0[b(s)(a1(s)a2(s+τ))y(s)+a3(s)0y(s+θ)dζ(θ)σ2(s)y2ϱ(s)2]dstY[β(u)ln(1+β(u))]λ(du)+0τa2(s+τ)y(s)ds+M1(t)+M2(t)

    where M1(t)=t0σ1(s)xϱ(s)dB(s) and M2(t)=t0Yln(1+β(u))˜N(ds,du). By (A1), we compute that

    t0a3(s)0y(s+θ)dζ(θ)ds=t0a3(s)[sy(s+θ)dζ(θ)ds+0sy(s+θ)dζ(θ)]ds=t0a3(s)dsser(s+θ)y(s+θ)er(s+θ)dζ(θ)+0tdζ(θ)tθa3(s)y(s+θ)dsau3||ξ||cgt0ersds0erθdζ(θ)+au30dζ(θ)t0y(s)dsau3||ξ||cgt0ersds0e2rθdζ(θ)+au30dζ(θ)t0y(s)ds1rau3||ξ||cgμr(1ert)+au3t0y(s)ds.

    Consequently,

    ttτa2(s+τ)y(s)ds0τa2(s+τ)y(s)ds+lnz(t)lnz(0)t0[b(s)(a1(s)a2(s+τ)au3)y(s)σ2(s)y2ϱ(s)2]ds+1rau3||ξ||cgμr(1ert)tY[β(u)ln(1+β(u))]λ(du)+M1(t)+M2(t), (2.3)

    where M1(t) is a real-valued continues local martingale vanishing at t=0 and its quadratic form is given by M1(t),M1(t)=t0σ2(s)y2ϱ(s)ds. By virtue of the exponential martingale inequality, for any positive constants T0,α and β, we have

    P{sup0tT0[M1(t)α2M1(t),M1(t)]>β}eαβ.

    Choose T0=k,α=1,β=2lnn. Then it follows that

    P{sup0tn[M1(t)12M1(t),M1(t)]>2lnn}1k2.

    By using the Borel-Cantelli lemma, for almost all ωΩ, there is a random integer n0=n0(ω) such that for nn0,

    sup0tn[M1(t)12M1(t),M1(t)]2lnn.

    This is to say

    M1(t)2lnn+12M1(t),M1(t)=2lnn+12t0σ2(s)y2ϱ(s)ds

    for all 0tn,nn0 a.s. Substituting this inequality into (2.3), we can obtain that

    ttτa2(s+τ)y(s)ds+lnz(t)lnz(0)0τa2(s+τ)y(s)ds+t0[b(s)(a1(s)a2(s+τ)au3)y(s)]dsY[β(u)ln(1+β(u))]λ(du)+2lnn+1rau3||ξ||cgμr(1ert)+M2(t) (2.4)

    for all 0tn,nn0 a.s. In addition, it follows from (2.4) that

    0<tk<tln(1+Jk)+lnz(t)lnz(0)0<tk<tln(1+Jk)+0τa2(s+τ)y(s)ds+t0[b(s)(a1(s)a2(s+τ)au3)y(s)]ds+2lnn+1rau3||ξ||cgμr(1ert)tY[β(u)ln(1+β(u))]λ(du)+M2(t)

    for all 0tn,nn0 a.s. In other words, we have shown that

    lny(t)lny(0)0<tk<tln(1+Jk)+0τa2(s+τ)y(s)ds+t0[b(s)(a1(s)a2(s+τ)a3(s))y(s)]ds+2lnn+1rau3||ξ||cgμr(1ert)tY[β(u)ln(1+β(u))]λ(du)+M2(t) (2.5)

    for all n1tn,nn0 a.s. Therefore, t¯R+, we get

    lny(t)lny(0)0<tk<tln(1+Jk)+0τa2(s+τ)y(s)ds+t0b(s)ds+2lnn+1rau3||ξ||cgμr(1ert)tY[β(u)ln(1+β(u))]λ(du)+M2(t)

    for all 0tn,nn0 a.s. By (A3), one can get M2(t)=t0Y(ln(1+β(u))2λ(du)dsc2tλ(Y), by the strong law of large numbers for local martingales(see, e.g., [43]), then we obtain

    limt+M2(t)t=0a.s., (2.6)

    which is the required assertion by (A2).

    Theorem 2. Suppose (A1)-(A4) hold, if G=0 and inft¯R+{a1(t)a2(t+τ)au3}>0, then y(t) modeled by (1.4) is non-persistent in the mean a.s.

    Proof. If G=0, satisfying (2.5) and (A2), for arbitrarily ε>0, there exists a constant T such that t1[0<tk<tln(1+Jk)+t0b(s)ds]+t10τa2(s+τ)y(s)ds+t11rau3||ξ||cgμr(1ert)+2lnktY[β(u)ln(1+β(u))]λ(du)+M2(t)t<ε for all Tk1tk,kk0 a.s.

    Substituting this inequality into (2.5) yields that

    lny(t)lny(0)0<tk<tln(1+Jk)+0τa2(s+τ)y(s)ds+t0[b(s)(a1(s)a2(s+τ)a3(s))y(s)]ds+2lnk+1rau3||ξ||cgμr(1ert)Y[β(u)ln(1+β(u))]λ(du)+M2(t)<εtt0(a1(s)a2(s+τ)au3)y(s)ds

    for all Tk1tk,kk0 a.s.

    Define h(t)=t0y(s)ds and I=inft¯R+[a1(t)a2(t+τ)au3]. The rest of the proof is similar to Theorem 3 in [13], so is omitted.

    Theorem 3. Suppose (A1)-(A4) hold. If G>0, then y(t) of model (1.4) is weak persistence a.s.

    Proof. If the assertion is not true, let F be the set F={lim supt+y(t)=0}, then P(F)>0, on the basis of (2.2), we derive

    0<tk<tln(1+Jk)+lnz(t)lnz(0)=0<tk<tln(1+Jk)+t0[b(s)a1(s)y(s)+a2(s)y(sτ)+a3(s)0y(s+θ)dζ(θ)σ2(s)y2ϱ(s)2]dstY[β(u)ln(1+β(u))]λ(du)+M1(t)+M2(t),

    then divide by t, we get

    t1lny(t)t1lny(0)=t10<tk<tln(1+Jk)+t1t0[b(s)a1(s)y(s)+a2(s)y(sτ)+a3(s)0y(s+θ)dζ(θ)σ2(s)y2ϱ(s)2]dsY[β(u)ln(1+β(u))]λ(du)+M1(t)t+M2(t)t. (2.7)

    On the other hand, for ωF, we have limt+x(t,ω)=0. Consequently, by the law of large numbers for local martingales, we obtain that limt+Mi(t)/t=0,i=1,2. Substituting this equality into (2.7), one can get a contradiction

    0lim supt+[t1lny(t,ω)]=G+lim supt+t1t0a2(s)y(sτ)ds+lim supt+t1t0a3(s)0y(s+θ)dζ(θ)dsG>0.

    Remark 1. Based on Theorems 1-3, we can point out that G is the threshold between weak persistence and extinction for the population y(t) by (A1)-(A4) if inft¯R+{a1(t)a2(t+τ)au3}0 holds.

    Lastly we show that y(t) modeled by (1.4) is stochastic permanence in some cases. We also need the assumption as follows:

    (A5): There exist two positive constants m and M such that m0<tk<t(1+Jk)M for all t>0.

    Theorem 4. Suppose (A1)-(A5) hold, if bYβ2(u)1+β(u)λ(du)>0, ϱ<1, and a2(t),a3(t)0, then y(t) represented by (1.4) will be stochastic permanence.

    Proof. First, we prove that for arbitrary ε>0, there exists a constant ϑ1>0 such that lim inft+P{y(t)ϑ1}1ε.

    Let 0.5<p=2ϱ1<1 and choose ρ1(0,2r), we obtain

    dzp(t)=pzp1(t)dz(t)+12p(p1)zp2(t)(dz(t))2=pzp1(t)[(z(t)(b(t)0<tk<t(1+Jk)a1(t)z(t)+0<tk<tτ(1+Jk)a2(t)z(tτ)+a3(t)00<tk<t+θ(1+Jk)z(t+θ)dζ(θ)))dt+(0<tk<t(1+Jk))ϱσ1(t)z1+ϱ(t)dB(t)]+12p(p1)(0<tk<t(1+Jk))2ϱσ2(t)zp+2ϱ(t)dt+Y[(1+β(u))p1pβ(u)]λ(du)zp(t)+Y[(1+β(u))p1]˜N(dt,du)zp(t)
    pzp1(t)[(z(t)(b(t)ma1(t)z(t)+Ma2(t)z(tτ)+Ma3(t)0z(t+θ)dζ(θ)))dt+(0<tk<t(1+Jk))ϱσ1(t)z1+ϱ(t)dB(t)]+12p(p1)(0<tk<t(1+Jk))2ϱσ2(t)zp+2ϱ(t)dt+Y[(1+β(u))p1pβ(u)]λ(du)zp(t)+Y[(1+β(u))p1]˜N(dt,du)zp(t)[b(t)pzp(t)+p2M2a22(t)z2p(t)4+z2(tτ)+p2M2a23(t)z2p(t)4+0z2(t+θ)dζ(θ)]dt+(0<tk<t(1+Jk))ϱpσ1(t)zp+ϱ(t)dB(t)12p(1p)m2ϱσ2(t)zp+2ϱ(t)dt+Y[(1+β(u))p1pβ(u)]λ(du)zp(t)+Y[(1+β(u))p1]˜N(dt,du)zp(t)=F(z(t))dt[ρ1yp(t)+eρ1τz2(t)z2(tτ)0z2(t+θ)dζ(θ)+μrz2(t)]dt+(0<tk<t(1+Jk))ϱpσ1(t)zp+ϱ(t)dB(t)+Y[(1+β(u))p1pβ(u)]λ(du)zp(t)+Y[(1+β(u))p1]˜N(dt,du)zp(t)

    where

    F(z)=eρ1τz2+μrz2+(ρ1+b(t)p)zp+p2a22(t)z2p+p2a23(t)z2p12p(1p)m2ϱσ2(t)zp+2ϱ+Y[(1+β(u))p1pβ(u)]λ(du)zp.

    According to p>0, (A2) and (A4), we have F(z) is bounded in R+, namely

    K=supzR+F(z)<+.

    Thus we show

    dzp(t)=[Kρ1zp(t)eρ1τz2(t)+z2(tτ)]dt+0z2(t+θ)dζ(θ)dtμrz2(t)dt+(0<tk<t(1+Jk))ϱpσ1(t)zp+ϱ(t)dB(t)+Y[(1+β(u))p1]˜N(dt,du)zp(t).

    Applying Itô's formula leads to

    d[eρ1tzp(t)]=eρ1t[ρ1zp(t)dt+dzp(t)]eρ1t[Keρ1τz2(t)+z2(tτ)+0z2(t+θ)dζ(θ)μrz2(t)]dt+eρ1t(0<tk<t(1+Jk))ϱpσ1(t)zp+ϱ(t)dB(t).

    Hence we derive that

    eρ1tE[zp(t)]ξp(0)+eρ1tKρ1Kρ1Et0eρ1s+ρ1τz2(s)ds+Et0eρ1sz2(sτ)ds+Et0eρ1s0z2(s+θ)dζ(θ)dsEt0μreρ1sz2(s)ds=ξp(0)+eρ1tKρ1Kρ1Et0eρ1s+ρ1τz2(s)ds+Etττeρ1s+ρ1τz2(s)ds+Et0eρ1s0z2(s+θ)dζ(θ)dsEt0μreρ1sz2(s)dsξp(0)+eρ1tKρ1Kρ1+0τeρ1s+ρ1τz2(s)ds+Et0eρ1s0z2(s+θ)dζ(θ)dsEμrt0eρ1sz2(s)ds.

    By (A4), we have

    t0eρ1s0z2(s+θ)dζ(θ)ds=t0eρ1s[sz2(s+θ)dζ(θ)+0sz2(s+θ)dζ(θ)]ds=t0eρ1sdsse2r(s+θ)z2(s+θ)e2r(s+θ)dζ(θ)+0tdζ(θ)tθeρ1(s)z2(s+θ)ds=t0eρ1sdsse2r(s+θ)z2(s+θ)e2r(s+θ)dζ(θ)+0tdζ(θ)t+θ0eρ1(sθ)z2(s)ds||ξ||2cgt0e(ρ12r)sds0e2rθdζ(θ)+0eρ1θdζ(θ)t0eρ1sz2(s)ds||ξ||2cgμrt+μrt0eρ1sz2(s)ds.

    This immediately implies that

    \begin{equation} \begin{aligned} &\limsup\limits_{t\rightarrow+\infty}E[z^{p}(t)]\leq\frac{K}{\rho_{1}} = H. \end{aligned} \end{equation} (2.8)

    Consequently,

    \limsup\limits_{t\rightarrow+\infty}E(y^{p}(t)) = \limsup\limits_{t\rightarrow+\infty}\Big[\prod\limits_{0 \lt t_{k} \lt t}(1+J_{k})\Big]^{p}E(z^{p}(t))\leq\Big[M^{p}\frac{K}{\rho_{1}}\Big] = M_{1}.

    Thus for any given \varepsilon > 0 , let \vartheta_{1} = M_{1}^{\frac{1}{p}}/\varepsilon^{\frac{1}{p}} , by virtue of Chebyshev's inequality, we can derive that

    \mathcal {P}\{y(t) \gt \vartheta_{1}\} = \mathcal {P}\{y^{p}(t) \gt \vartheta_{1}^{p}\}\leq E[y^{p}(t)]/\vartheta_{1}^{p}.

    That is to say \limsup\limits_{t\rightarrow+\infty}\mathcal {P}\{y(t) > \vartheta_{1}\}\leq \varepsilon . Consequently, \liminf\limits_{t\rightarrow+\infty}\mathcal {P}\{y(t)\leq\vartheta_{1}\}\geq 1-\varepsilon .

    Next, we claim that for arbitrary \varepsilon > 0 , there exists a constant \vartheta_{2} > 0 such that \liminf\limits_{t\rightarrow +\infty}\mathcal {P}\{y(t)\geq\vartheta_{2}\}\geq 1-\varepsilon . One can obtain that

    \begin{equation} \begin{aligned} d\Big( \frac{1}{z(t)}\Big) = &\Big[-\frac{b(t)-\int_{\mathbb{Y}}\frac{\beta^{2}(u)}{1+\beta(u)}\lambda(du)}{z(t)}-\frac{a_{2}(t)z(t-\tau)}{z(t)}\\ &-\frac{a_{3}(t)\int\limits_{-\infty}^{0}\prod\limits_{0 \lt t_{k} \lt t+\theta}(1+J_{k})z(t+\theta)d\zeta(\theta)}{z(t)}+a_{1}(t)+\sigma^{2}(t)\Big(\prod\limits_{0 \lt t_{k} \lt t}(1+J_{k})\Big)^{2\varrho}\\ &\times z^{2\varrho-1}(t)\Big]dt-\sigma_{1}(t)\Big(\prod\limits_{0 \lt t_{k} \lt s}(1+J_{k})\Big)^{\varrho}z^{\varrho-1}(t)dB(t)+ \frac{1}{z(t)}\int_{\mathbb{Y}}\Big[\frac{1}{(1+\beta(u))}\\ &-1\Big]\tilde{N}(dt, du). \end{aligned} \end{equation} (2.9)

    Integrating from 0 to t and taking expectations on the both sides of (2.9) we get

    \begin{align*} E\Big[\frac{1}{z(t)}\Big] = &E\Big[\frac{1}{z(0)}\Big]+\int\limits_{0}^{t}\bigg(E\Big[\frac{-\Big(b(s)-\int_{\mathbb{Y}}\frac{\beta^{2}(u)}{1+\beta(u)}\lambda(du)\Big)}{z(s)}\Big]+a_{1}(s)\\ &+\sigma^{2}(s)\Big(\prod\limits_{0 \lt t_{k} \lt t}(1+J_{k})\Big)^{2\varrho}E[z^{2\varrho-1}(s)]-a_{2}(s)E\Big[ \frac{z(s-\tau)}{z(s)}\Big]\\ &-a_{3}(s)E\Big[\frac{\int\limits_{-\infty}^{0}\prod\limits_{0 \lt t_{k} \lt t+\theta}(1+J_{k})z(s+\theta)d\zeta(\theta)}{z(s)}\Big]\bigg)ds. \end{align*}

    Hence

    \begin{equation} \begin{aligned} \frac{d E[1/z(t)]}{dt} = & -\Big(b(t)-\int_{\mathbb{Y}}\frac{\beta^{2}(u)}{1+\beta(u)}\lambda(du)\Big)E\Big[1/z(t)\Big]+a_{1}(t)\\ &+\sigma^{2}(t)\Big(\prod\limits_{0 \lt t_{k} \lt t}(1+J_{k})\Big)^{2\varrho}E[z^{2\varrho-1}(t)]-a_{2}(t)E\Big[\frac{z(t-\tau(t))}{z(t)}\Big]\\ &-a_{3}(t)E\Big[\frac{\int\limits_{-\infty}^{0}\prod\limits_{0 \lt t_{k} \lt t+\theta}z(t+\theta)d\zeta(\theta)}{z(t)}\Big]. \end{aligned} \end{equation} (2.10)

    Considering the equation

    \begin{equation} \begin{aligned} \frac{d E[1/z_{1}(t)]}{dt} = -\Big(b(t)-\int_{\mathbb{Y}}\frac{\beta^{2}(u)}{1+\beta(u)}\lambda(du)\Big)E\Big[1/z_{1}(t)\Big]+a_{1}(t)+\sigma^{2}(t)M^{2\varrho}H. \end{aligned} \end{equation} (2.11)

    For \forall \varepsilon > 0 , there exists T_{1} > 0 , such that b(t) > b_{*}-\varepsilon for all t > T_{1} . Using the same method (Theorem 4.5 in [44]), we have

    \lim\limits_{t\rightarrow +\infty}E\Big[\frac{1}{z_{1}(t)}\Big]\leq d,

    where

    d = \frac{2(\sigma^{2})^{u}M^{2\varrho}H}{b_{*}-\int_{\mathbb{Y}}\frac{\beta^{2}(u)}{1+\beta(u)}\lambda(du)}+\frac{\exp\{-\int_{0}^{T}\Big(b(s)-\int_{\mathbb{Y}}\frac{\beta^{2}(u)}{1+\beta(u)}\lambda(du)\Big)ds\}}{z_{1}(0)}.

    Taking into consideration (2.8), there exists T > T_{1} > 0 such that E[z^{2\varrho-1}(t)]\leq H for all t > T . Thus from (2.10) and (2.11), using the comparison theorem for ODEs yields that

    E\Big[\frac{1}{z(t)}\Big]\leq E\Big[\frac{1}{z_{1}(t)}\Big]

    for t > T , which implies that

    \limsup\limits_{t\rightarrow +\infty}E\Big[\frac{1}{z(t)}\Big]\leq\limsup\limits_{t\rightarrow +\infty}E\Big[\frac{1}{z_{1}(t)}\Big] = \lim\limits_{t\rightarrow +\infty}E\Big[\frac{1}{z_{1}(t)}\Big]\leq d.

    Thus

    \limsup\limits_{t\rightarrow +\infty}E\Big[\frac{1}{y(t)}\Big] = \lim\limits_{t\rightarrow +\infty}E\Big[\frac{1}{\prod\limits_{0 \lt t_{k} \lt t}(1+J_{k})z(t)}\Big] = m^{-1}d.

    So for any given \varepsilon > 0 , let \vartheta_{2} = \frac{m\varepsilon}{d} , then the desired assertion follows from the Chebyshev inequality.

    Remark 2. It is easy to see that the Theorem 2-4 coincide with the Theorem 2-4 in Liu et al. [13] when a_{2}(t) = a_{3}(t)\equiv 0, J_{k} = 0, \beta(u) = 0 and \varrho = 1 in model (1.4), respectively.

    Remark 3. Taking notice of G^{*} in Theorem 1-3, we can find that the impulsive perturbation does not affect extinction, non-persistent in the mean and weak persistence satisfying the conditions that impulsive perturbations exist upper bound.

    Remark 4. Unlike earlier studies, we do not employ Lyapunov methods in establishing the sufficient conditions of stochastic permanence. As we know, in general, it is not easy to construct suitable Lyapunov function to deal with stochastic permanence for stochastic differential equation with infinite delay. Therefore, we use the comparison theorem of ordinary differential equation to derive the sufficient condition of stochastic permanence.

    Remark 5. Paying attention to G^{*} in Theorem 1-3, where \int_{\mathbb{Y}}[\beta(u)-\ln(1+\beta(u))]\lambda(du) > 0 (see Lemma 1.2 in Ref.[17]), we obtain the result that the jump process may exert a considerable negative impact on the population and can lead to the extinction (see case 1 in section 4), which accords with biological significance.

    In this section, we will gain sufficient criteria of the global asymptotic stability for model (1.4) without Lévy jumps and impulsive perturbation:

    \begin{equation} \left\{ \begin{split} dy(t) = &y(t)\Big[b-a_{1}y(t)+a_{2}y(t-\tau)+a_3\int\limits_{-\infty}^{0}y(t+\theta)d\zeta(\theta)\Big]dt\\ &+\sigma_{1}(t)y^{1+\varrho}(t)dB(t)\\ \end{split} \right. \end{equation} (3.1)

    with the same initial value of model (1.4).

    Theorem 5. Suppose (A2)-(A4) hold. If a_{2}\leq 0, a_{3}\leq 0, a_{1}+a_{2}+a_{3} > 0 , then the positive solution of model (3.1) is globally asymptotically stable.

    Proof. Let y(t) and y^{*}(t) be two arbitrary solutions of model (1.4). Define

    \begin{align*} V(t) = &|\ln y(t)-\ln y^{*}(t)|-a_{2}\int\limits_{t-\tau}^{t}|y(s)-y^{*}(s)|ds-a_{3}\int\limits_{-\infty}^{0}\int\limits_{t+\theta}^{t}|y(s)-y^{*}(s)|dsd\zeta(\theta). \end{align*}

    A calculation of the right differential D^{+}V(t) , and then making use of the generalized Itô's formula, we can observe

    \begin{align*} D^{+}V(t) = &\mathrm{sgn}(y(t)-y^{*}(t))\Big(-a_{1}(y(t)-y^{*}(t))+a_{2}(y(t-\tau)-y^{*}(t-\tau))\\ &+a_{3}\Big( \int\limits_{-\infty}^{0}\Big(y(t+\theta)-y^{*}(t+\theta)\Big)d\zeta(\theta)-\frac{\sigma^{2}}{2}(u^{2\nu}(t)-(u^{*}(t))^{2\nu})\Big)dt\\ &-a_{2}|y(t)-y^{*}(t)|dt+a_{2}|y(t-\tau)-y^{*}(t-\tau)|dt\\ &-a_{3}\int\limits_{-\infty}^{0}|y(t)-y^{*}(t)|d\nu(\theta)dt+a_{3}\int\limits_{-\infty}^{0}|y(t+\theta)-y^{*}(t+\theta)|d\nu(\theta)dt\\ = &-a_{1}|y(t)-y^{*}(t)|dt-a_{2}|y(t-\tau)-y^{*}(t-\tau)|dt\\ &-a_{3}\Big( \int\limits_{-\infty}^{0}|y(t+\theta)-y(t+\theta)|d\nu(\theta)\Big)dt-\frac{\sigma^{2}(t)}{2}|y^{2\nu}(t)-(y^{*}(t))^{2\nu}|dt\\ &-a_{2}|y(t)-y^{*}(t)|dt+a_{2}|y(t-\tau)-y^{*}(t-\tau)|dt-a_{3}|y(t)-y^{*}(t)|dt\\ &+a_{3}\int\limits_{-\infty}^{0}|y(t+\theta)-y^{*}(t+\theta)|d\nu(\theta)dt\\ = &-(a_{1}+a_{2}+a_{3})|y(t)-y^{*}(t)|dt-\frac{\sigma^{2}(t)}{2}|y^{2\nu}(t)-(y^{*}(t))^{2\nu}|dt. \end{align*}

    Integrating both sides and then taking the expectation yields

    V(t)\leq V(0)-\int\limits_{0}^{t}(a_{1}+a_{2}+a_{3})|y(s)-y^{*}(s)|ds.

    In other words, we have already shown that

    V(t)+\int\limits_{0}^{t}(a_{1}+a_{2}+a_{3})|y(s)-y^{*}(s)|ds\leq V(0) \lt \infty.

    Because of a_{1}+a_{2}+a_{3} > 0 , we derive

    |y(t)-y^{*}(t)|\in L^{1}[0, +\infty).

    Similar to Theorem 11 in [26] and Theorem 3.7 in [45], we have

    \lim\limits_{t\rightarrow+\infty}|y(t)-y^{*}(t)| = 0.

    This completes the proof of Theorem 5.

    In this section, we present an example to illustrate the results. Here let the probability measure \zeta(\theta) be e^{\theta} on (-\infty, 0] . Hence the model (1.4) will be rewritten as

    \begin{equation} \left\{ \begin{split} dy(t) = &y(t)\Big[b(t)-a_{1}(t)y(t)+a_{2}(t)y(t-\tau)+a_{3}(t)e^{-t}\int\limits_{-\infty}^{0}e^{\theta}\xi(\theta)d\theta\\ &+a_{3}(t)e^{-t}\int\limits_{0}^{t}e^{\theta}y(\theta)d\theta\Big]dt+\sigma_{1}(t) y^{1.8}(t)dB(t)\\ &+y(t^{-})\int\limits_{\mathbb{Y}}\beta(u)\tilde{N}(dt, du), \\ &t\neq t_{k}, \; \; K\in N, \\ y(t_{k}^{+})-&y(t_{k}) = J_{k}y(t_{k}), \; \; \; \; k\in N. \end{split} \right. \end{equation} (4.1)

    Using the Euler scheme to discretize this equation [46], choosing \xi(\theta) = e^{-0.5\theta} and \tau\equiv 0.3 , we can obtain the discrete approximate solution of (3.1):

    \begin{equation} \left\{ \begin{split} y_{k+1} = &y_{k}+y_{k}\Big[r(k\Delta t)-a(k\Delta t)y_{k}+b(k\Delta t)y_{k-300}+c(k\Delta t)e^{-k\Delta t}\int\limits_{-\infty}^{0}e^{0.5\theta}d\theta\\ &+c(k\Delta t)e^{-k\Delta t}\sum\limits_{j = 0}^{k}\omega_{j}^{(k)}e^{j\Delta t}y_{j}\Big]\Delta t+\sigma(k\Delta t)y^{1.8}_{k}\Delta B_{k}+y_{k}\beta(u)\Delta\tilde{N_{k}}, \\ &t\neq t_{k}, \; \; K\in N, \\ y_{k+1}-&y_{k} = J_{k}y_{k}, \; \; t = t_{k}, \; \; k\in N, \end{split} \right. \end{equation} (4.2)

    where \Delta B_k = B((k+1)\Delta t)-B(k\Delta t), \Delta\tilde{N}_k = \tilde{N}((k+1)\Delta t)-\tilde{N}(k\Delta t), k = 0, 1, 2, \cdots. The general composite \theta -rule has weights

    \{\omega_{0}^{(k)}, \omega_{1}^{(k)}, \cdots, \omega_{k}^{(k)}\} = \{ \theta, 1, \cdots, 1-\theta\}, \; \theta \in [0, 1]

    and \sum\limits_{j = 0}^{k}\omega_{j}^{(k)} = k, \; k\geq 0 .

    Here, we choose b(t) = 0.25+0.02\cos t, a_{1}(t) = 0.24+0.01\cos t, a_{2}(t) = 0.04, a_{3}(t) = 0.08, \sigma_{1}(t) = 0.02, \beta(u) = 0.98, \varrho = -0.2, t_{k} = 10k and step size \Delta t = 0.001 . The only difference between conditions of Figure 1(A)(C) is that the representation of J_{k} is different.

    Figure 1.  The horizontal axis in this and following figures represent the time t . (A) is with J_{k} = 0 ; (B) is with J_{k} = e^{0.5}-1 ; (C) is with J_{k} = e^{0.7}-1 ; (D) is with J_{k} = e^{\frac{(-1)^{k+1}}{k}}-1 .

    Case 1. If J_{k} = 0 and the conditions of Theorem 1 have been satisfied, then the population y(t) will be extinct(see Figure 1(A)).

    Case 2. Considering J_{k} = e^{0.5}-1 , and the conditions of Theorem 2 hold, then the population y(t) will be non-persistent in the mean(see Figure 1(B)).

    Case 3. If J_{k} = e^{0.7}-1 , and the conditions of Theorem 3 are fulfilled, then the population y(t) will be weak persistence(see Figure 1(C)).

    Case 4. Considering J_{k} = e^{\frac{(-1)^{k+1}}{k}}-1 , and the conditions of Theorem 4 hold, then the population y(t) will be stochastic permanence(see Figure 1(D)). By comparing with Figure 1(A)(C), we can find that the impulsive perturbations have ability to transform the properties of the model.

    Generally speaking, the above results show that the impulse perturbations which are bounded never influence the persistence and extinction. However, if the impulsive perturbations are unbounded, the persistence and extinction are severely affected. Usually, we think that making great efforts to continuous exploitation and harvesting may be described by unbounded impulsive perturbations.

    In this paper, we study the basic features of a stochastic logistic model with infinite delay in presence of Lévy jumps and impulsive perturbations to understand the dynamics in a complicated environment. We have established sufficient conditions for the existence of global positive solution and obtained sufficient conditions for extinction, non-persistence in the mean, weak persistence, stochastic permanence and global asymptotic stability of model. Moreover, our investigation also reveals that impulsive perturbations which may represent natural and human factors play an important role in protecting the population, even if population suffers sudden environmental changes described by Lévy jumps, such as earthquakes, hurricanes, epidemics, etc.

    Finally, we would like to talk about some interesting topics deserving further discussion. On the one hand, one may propose the realistic but complex model, such as considering multi-dimensional system. The motivation of investigating this is that biological systems are composed of multiple populations, such as prey, predator, etc. On the other hand, it is also significant to incorporate the telegraph noise, which can be modeled by a continuous-time Markov chain [23], into model (1.4). We will look into these topics in following work.

    The authors are very grateful to the editor and the three anonymous reviewers for their valuable comments and suggestions, which greatly improved the presentation of this work. CL is supported by grants from the Natural Science Foundation of Shandong Province of China (Nos. ZR2018MA023, ZR2017MA008, ZR2017BA007, ZR2016AM02), a Project of Shandong Province Higher Educational Science and Technology Program of China (Nos. J16LI09, J18KA218), a Scientific Research Fund of Heilongjiang Provincial Education Department in China (No. 12541893), National Natural Science Foundation of China (No. 61803220). BL is supported by the National Natural Science Foundation of China(No.11471089), and Doctor Start-up Fund Program of Harbin Normal University(XKB201806).

    All authors declare no conflicts of interest in this paper.



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