Processing math: 100%
Research article Special Issues

Dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment


  • In this paper, a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment is investigated. We obtain the sufficient and necessary conditions for persistence in mean and extinction of each species. The results show that the stochastic dynamics of the system are closely correlated with both time delays and environmental noises. Some numerical examples are introduced to illustrate the main results.

    Citation: Zeyan Yue, Sheng Wang. Dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 186-213. doi: 10.3934/mbe.2024009

    Related Papers:

    [1] Chun Lu, Bing Li, Limei Zhou, Liwei Zhang . Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps. Mathematical Biosciences and Engineering, 2019, 16(5): 3251-3271. doi: 10.3934/mbe.2019162
    [2] Sheng Wang, Lijuan Dong, Zeyan Yue . Optimal harvesting strategy for stochastic hybrid delay Lotka-Volterra systems with Lévy noise in a polluted environment. Mathematical Biosciences and Engineering, 2023, 20(4): 6084-6109. doi: 10.3934/mbe.2023263
    [3] Linni Li, Jin-E Zhang . Input-to-state stability of stochastic nonlinear system with delayed impulses. Mathematical Biosciences and Engineering, 2024, 21(2): 2233-2253. doi: 10.3934/mbe.2024098
    [4] Yongxin Gao, Shuyuan Yao . Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ predator-prey stochastic model in polluted environments with impulsive toxicant input. Mathematical Biosciences and Engineering, 2021, 18(4): 4894-4918. doi: 10.3934/mbe.2021249
    [5] Jian Meng, Bin Zhang, Tengda Wei, Xinyi He, Xiaodi Li . Robust finite-time stability of nonlinear systems involving hybrid impulses with application to sliding-mode control. Mathematical Biosciences and Engineering, 2023, 20(2): 4198-4218. doi: 10.3934/mbe.2023196
    [6] Meng Gao, Xiaohui Ai . A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps. Mathematical Biosciences and Engineering, 2024, 21(3): 4117-4141. doi: 10.3934/mbe.2024182
    [7] Yuanpei Xia, Weisong Zhou, Zhichun Yang . Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays. Mathematical Biosciences and Engineering, 2020, 17(5): 6149-6180. doi: 10.3934/mbe.2020326
    [8] Dongmei Li, Tana Guo, Yajing Xu . The effects of impulsive toxicant input on a single-species population in a small polluted environment. Mathematical Biosciences and Engineering, 2019, 16(6): 8179-8194. doi: 10.3934/mbe.2019413
    [9] Ting Kang, Yanyan Du, Ming Ye, Qimin Zhang . Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment. Mathematical Biosciences and Engineering, 2020, 17(6): 6702-6719. doi: 10.3934/mbe.2020349
    [10] Tainian Zhang, Zhixue Luo, Hao Zhang . Optimal harvesting for a periodic n-dimensional food chain model with size structure in a polluted environment. Mathematical Biosciences and Engineering, 2022, 19(8): 7481-7503. doi: 10.3934/mbe.2022352
  • In this paper, a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment is investigated. We obtain the sufficient and necessary conditions for persistence in mean and extinction of each species. The results show that the stochastic dynamics of the system are closely correlated with both time delays and environmental noises. Some numerical examples are introduced to illustrate the main results.



    The predator-prey model is one of the hotspots in biomathematics. For example, Yavuz and Sene [1] considered a fractional predator-prey model with harvesting rate, Chatterjee and Pal [2] studied a predator-prey model for the optimal control of fish harvesting through the imposition of a tax and Ghosh et al. [3] presented a three-component model consisting of one prey and two predator species using imprecise biological parameters as interval numbers and applied a functional parametric form in the proposed prey-predator system. Because of its important role in the ecosystem, the food chain model has been extensively studied [4,5,6,7,8]. Specifically, the classical four-species food chain model can be expressed as follows:

    {dx1(t)=x1(t)[r1a11x1(t)a12x2(t)]dt,dx2(t)=x2(t)[r2+a21x1(t)a22x2(t)a23x3(t)]dt,dx3(t)=x3(t)[r3+a32x2(t)a33x3(t)a34x4(t)]dt,dx4(t)=x4(t)[r4+a43x3(t)a44x4(t)]dt, (1.1)

    where x1(t), x2(t), x3(t) and x4(t) represent the densities of prey, primary predator, intermediate predator and top predator at time t, respectively. r1 is the growth rate of prey, r2, r3 and r4 are the death rates of primary predator, intermediate predator and top predator, respectively. aij and aji (i<j) are the capture rates and food conversion rates, respectively. aii are the intraspecific competition rates of species i. All parameters in system (1.1) are positive constants.

    In ecology, biology, physics, engineering and other areas of applied sciences, continuous-time models, fractional-order models as well as discrete-time models have been widely adopted [9,10]. However, "time delays occur so often that to ignore them is to ignore reality" [11,12], and in the models of population dynamics, the delay differential equations are much more realistic [13,14,15]. We know that systems with discrete time delays and those with continuously distributed time delays do not contain each other but systems with S-type distributed time delays contain both. Introducing S-type distributed time delays into system (1.1) yields

    {dx1(t)=x1(t)[r1D11(x1)(t)D12(x2)(t)]dt,dx2(t)=x2(t)[r2+D21(x1)(t)D22(x2)(t)D23(x3)(t)]dt,dx3(t)=x3(t)[r3+D32(x2)(t)D33(x3)(t)D34(x4)(t)]dt,dx4(t)=x4(t)[r4+D43(x3)(t)D44(x4)(t)]dt, (1.2)

    where Dji(xi)(t)=ajixi(t)+0τjixi(t+θ)dμji(θ), 0τjixi(t+θ)dμji(θ) are Lebesgue-Stieltjes integrals, τji>0 are time delays, μji(θ) are nondecreasing bounded variation functions defined on [τ,0], τ=max{τji}.

    On the other hand, the deterministic system has its limitation in mathematical modeling of ecosystems since the parameters involved in the system are unable to capture the influence of environmental noises [16,17]. Introducing Gaussian white noises into the corresponding deterministic model is one common way to characterize environmental noises [18,19,20,21,22,23,24,25]. As we all know, Gaussian white noise ξ(t) is a stationary and ergodic stochastic process with ξ(t)=0 and ξ(t)ξ(s)=σ2δ(ts), where σ2 is the noise intensity [26]. The readers can refer to [27,28,29,30,31,32,33,34] for more related works. In this paper, we assume that ri are affected by Gaussian white noises, i.e., r1r1+σ1˙W1(t), r2r2+σ2˙W2(t), r3r3+σ3˙W3(t) and r4r4+σ4˙W4(t). Then, system (1.2) becomes

    {dx1(t)=x1(t)[r1D11(x1)(t)D12(x2)(t)]dt+σ1x1(t)dW1(t),dx2(t)=x2(t)[r2+D21(x1)(t)D22(x2)(t)D23(x3)(t)]dt+σ2x2(t)dW2(t),dx3(t)=x3(t)[r3+D32(x2)(t)D33(x3)(t)D34(x4)(t)]dt+σ3x3(t)dW3(t),dx4(t)=x4(t)[r4+D43(x3)(t)D44(x4)(t)]dt+σ4x4(t)dW4(t), (1.3)

    where Wi(t) are mutually independent standard Wiener processes defined on a complete probability space (Ω,F,P) satisfying the usual statistical properties, namely dWi(t)=0 and dWi(t)dWj(s)=δijδ(ts)dt [35].

    Besides, population system may be affected by telephone noises which can cause the system to switch from one environmental regime to another [36,37,38]. So, telephone noises should be taken into consideration in system (1.3), resulting the following model:

    {dx1(t)=x1(t)[r1(ρ(t))D11(x1)(t)D12(x2)(t)]dt+σ1(ρ(t))x1(t)dW1(t),dx2(t)=x2(t)[r2(ρ(t))+D21(x1)(t)D22(x2)(t)D23(x3)(t)]dt+σ2(ρ(t))x2(t)dW2(t),dx3(t)=x3(t)[r3(ρ(t))+D32(x2)(t)D33(x3)(t)D34(x4)(t)]dt+σ3(ρ(t))x3(t)dW3(t),dx4(t)=x4(t)[r4(ρ(t))+D43(x3)(t)D44(x4)(t)]dt+σ4(ρ(t))x4(t)dW4(t), (1.4)

    where ρ(t) is a continuous time Markov chain with finite state space S={1,2,...,S}, which describes the telephone noises.

    Moreover, the behaviour of real biological species, in different ecosystems, is affected by Lévy noises [39]. Lévy processes are characterized by stationary independent increments [40]. Assume that L(t) (t0) is a Lévy process, using the decomposition [41]

    L(t)=L(tn)+[L(2tn)L(tn)]++[L(ntn)L((n1)tn)],

    one can observe that the probability distribution of L(t) is infinitely divisible. The most general expression for the characteristic function of L(t) is

    φ(k)=exp{ikμ|σk|α[1iβsgn(k)Φ]},

    where sgn(k) is the sign function with

    Φ={tan(πα/2),forallα1,(2/π)log|k|,forallα=1,

    where α(0,2] is the stability parameter, σ is the scale parameter, σα is the noise intensity, μR is the location parameter and β[1,1] is the skewness parameter [39]. In addition, Lévy noises are statistically independent with zero mean. Now, let us further improve system (1.4) by considering Lévy noises. Some scholars pointed out that Lévy noises can be used to describe some sudden environmental perturbations, for instance, earthquakes and hurricanes [42,43,44,45,46,47]. In the context of an epidemic situation, random jumps could refer to sudden and significant increases in the number of cases or spread of the disease that occur unpredictably [48]. System (1.4) with Lévy noises can be expressed as follows:

    {dx1(t)=x1(t)[(r1(ρ(t))D11(x1)(t)D12(x2)(t))dt+S1(t,ρ(t))],dx2(t)=x2(t)[(r2(ρ(t))+D21(x1)(t)D22(x2)(t)D23(x3)(t))dt+S2(t,ρ(t))],dx3(t)=x3(t)[(r3(ρ(t))+D32(x2)(t)D33(x3)(t)D34(x4)(t))dt+S3(t,ρ(t))],dx4(t)=x4(t)[(r4(ρ(t))+D43(x3)(t)D44(x4)(t))dt+S4(t,ρ(t))], (1.5)

    where Si(t,ρ(t))=σi(ρ(t))dWi(t)+Zγi(μ,ρ(t))˜N(dt,dμ), N is a Poisson counting measure with characteristic measure λ on a measurable subset Z of [0,+), where λ(Z)<+ and ˜N(dt,dμ)=N(dt,dμ)λ(dμ)dt, γj(μ,ρ(t))>1 (μZ) are bounded functions (j=1,2,3,4).

    Finally, environmental pollution caused by agriculture, industries and other human activities has become a big challenge that is commonly concerned by international society. For example, with the rapid development of industrial and agricultural production, some chemical plants and other industries often periodically discharge sewage or other pollutants into rivers, soil and air [49]. These pollutants can cause direct damage to ecosystems, such as species extinction, desertification and the greenhouse effect. Hence, we extend system (1.5) into the following form:

    {dx1(t)=x1(t)[(r1(ρ(t))r11C10(t)D11(x1)(t)D12(x2)(t))dt+S1(t,ρ(t))],dx2(t)=x2(t)[(r2(ρ(t))r22C20(t)+D21(x1)(t)D22(x2)(t)D23(x3)(t))dt+S2(t,ρ(t))],dx3(t)=x3(t)[(r3(ρ(t))r33C30(t)+D32(x2)(t)D33(x3)(t)D34(x4)(t))dt+S3(t,ρ(t))],dx4(t)=x4(t)[(r4(ρ(t))r44C40(t)+D43(x3)(t)D44(x4)(t))dt+S4(t,ρ(t))],dCi0(t)=[kiCe(t)(gi+mi)Ci0(t)]dt,dCe(t)=hCe(t)dt,}tnγ,Δxi(t)=0,ΔCi0(t)=0,ΔCe(t)=b,t=nγ,nZ+(i=1,2,3,4), (1.6)

    where Δxi(t)=xi(t+)xi(t), ΔCi0(t)=Ci0(t+)Ci0(t) and ΔCe(t)=Ce(t+)Ce(t). For other parameters in system (1.6), see Table 1.

    Table 1.  Definition of some parameters in system (1.6).
    Parameter Definition
    Ci0(t) the toxicant concentration in the organism of species i at time t
    Ce(t) the toxicant concentration in the environment at time t
    rii the dose-response rate of species i to the organismal toxicant
    ki the toxin uptake rate per unit biomass
    gi the organismal net ingestion rate of toxin
    mi the organismal deportation rate of toxin
    h the rate of toxin loss in the environment
    γ the period of the impulsive toxicant input
    b the toxicant input amount at every time

     | Show Table
    DownLoad: CSV

    To the best of our knowledge to date, results about a stochastic hybrid delay four-species food chain model with jumps have not been reported. So, in this paper we investigate the dynamics of a stochastic hybrid delay four-species food chain model with jumps in an impulsive polluted environment. The organization of this paper is as follows: In Section 2, some basic preliminaries are presented. In Section 3, the sufficient and necessary conditions for stochastic persistence in mean and extinction of each species are obtained. In Section 4, some numerical examples are provided to illustrate our main results. Finally, we conclude the paper with a brief conclusion and discussion in Section 5.

    We have four fundamental assumptions for system (1.6).

    Assumption 1. W1(t), W2(t), W3(t), W4(t), ρ(t) and N are mutually independent. ρ(t), taking values in S={1,2,...,S}, is irreducible with one unique stationary distribution π=(π1,π2,...,πS)T.

    Assumption 2. rj(i)>0, ajk>0 and there exist γj(i)γj(i)>1 such that γj(i)γj(μ,i)γj(i) (μZ), iS, j,k=1,2,3,4.

    Remark 1. Assumption 2 implies that the intensities of Lévy jumps are not too big to ensure that the solution will not explode in finite time.

    Assumption 3. 0<kigi+mi (i=1,2,3,4), 0<b1ehγ.

    Remark 2. Assumption 3 means 0Ci0(t)<1 and 0Ce(t)<1, which must be satisfied to be realistic because Ci0(t) and Ce(t) are concentrations of the toxicant (i=1,2,3,4).

    Assumption 4. A22A33A44|A||Ξ|>A12A21A23A32A44|Ξ|+A23A32A34A43|A|2.

    Lemma 1. [50,51] Ci0(t) involved in system (1.6) satisfies

    limt+t1t0Ci0(s)ds=kibh(gi+mi)γ=Ki(i=1,2,3,4).

    Denote

    {Aij=aij+0τijdμij(θ),Ki=kibh(gi+mi)γ,b1()=r1()σ21()2Z[γ1(μ,)ln(1+γ1(μ,))]λ(dμ),bj()=rj()+σ2j()2+Z[γj(μ,)ln(1+γj(μ,))]λ(dμ)(j=2,3,4),Σ1=Si=1πib1(i)r11K1,Σj=Si=1πibj(i)rjjKj(j=2,3,4),B1=Σ1,B2=Σ2+A21A11B1,B3=Σ3+A32A22B2,B4=Σ4+A43A33B3,A=(A11A12A21A22),Ξ=(A11A120A21A22A230A32A33),Δ=(A11A1200A21A22A2300A32A33A3400A43A44).

    Denote Σ(2)=(Σ1,Σ2)T, Σ(3)=(Σ1,Σ2,Σ3)T, Σ=(Σ1,Σ2,Σ3,Σ4)T. Denote Aj is A with column j replaced by Σ(2) (j=1,2); Ξj is Ξ with column j replaced by Σ(3) (j=1,2,3); Δj is Δ with column j replaced by Σ (j=1,2,3,4).

    Theorem 1. For any initial condition ϕC([τ,0],R4+), system (1.6) has a unique global solution (x1(t),x2(t),x3(t),x4(t))TR4+ on tR+ a.s. Moreover, for any constant p>0, there exists Ki(p)>0 such that suptR+E[xpi(t)]Ki(p) (i=1,2,3,4).

    Proof. The proof is rather standard and hence is omitted (see e.g., [52]).

    Lemma 2. [53] Suppose Z(t)C(Ω×[0,+),R+) and limt+o(t)t=0.

    (i) If there exists constant δ0>0 such that for t1,

    lnZ(t)δtδ0t0Z(s)ds+o(t),

    then

    {lim supt+t1t0Z(s)dsδδ0a.s.(δ0);limt+Z(t)=0a.s.(δ<0).

    (ii) If there exist constants δ>0 and δ0>0 such that for t1,

    lnZ(t)δtδ0t0Z(s)ds+o(t),

    then

    lim inft+t1t0Z(s)dsδδ0a.s.

    Lemma 3. If |Δ4|>0, then |Δj|>0 (j=1,2,3).

    Proof. Compute

    A44|Δ4|A43|Δ3|=[(A33A44+A34A43)|A|+A11A23A32A44]Σ4.
    A32|Δ2|=A33|Δ3|+A34|Δ4|[(A33A44+A34A43)|A|+A11A23A32A44]Σ3.
    A21|Δ1|=A22|Δ2|+A23|Δ3|[A11A44(A22A33+A23A32)+A34A43|A|+A12A21A33A44]Σ2.

    Noting that Σj<0 (j=2,3,4), we obtain the desired assertion.

    First, let us consider the following auxiliary system:

    { dX1(t)=X1(t)[(r1(ρ(t))r11C10(t)D11(X1)(t))dt+S1(t,ρ(t))],dX2(t)=X2(t)[(r2(ρ(t))r22C20(t)+D21(X1)(t)D22(X2)(t))dt+S2(t,ρ(t))],dX3(t)=X3(t)[(r3(ρ(t))r33C30(t)+D32(X2)(t)D33(X3)(t))dt+S3(t,ρ(t))],dX4(t)=X4(t)[(r4(ρ(t))r44C40(t)+D43(X3)(t)D44(X4)(t))dt+S4(t,ρ(t))],dCi0(t)=[kiCe(t)(gi+mi)Ci0(t)]dt,dCe(t)=hCe(t)dt,}tnγ,ΔXi(t)=0,ΔCi0(t)=0,ΔCe(t)=b,t=nγ,nZ+(i=1,2,3,4). (3.1)

    Lemma 4. System (3.1) satisfies Table 2, where

    ¯XT()=limt+t1(t0X1(s)ds,t0X2(s)ds,t0X3(s)ds,t0X4(s)ds).
    Table 2.  Stochastic persistence in mean and extinction of system (3.1).
    B4 B3 B2 B1 ¯XT()
    0 0 0 0 (B1A11,B2A22,B3A33,B4A44)
    <0 0 0 0 (B1A11,B2A22,B3A33,0)
    <0 0 0 (B1A11,B2A22,0,0)
    <0 0 (B1A11,0,0,0)
    <0 (0,0,0,0)

     | Show Table
    DownLoad: CSV

    Proof. Consider the following stochastic hybrid delay logistic model with Lévy jump in an impulsive polluted environment:

    {dX1(t)=X1(t)[(r1(ρ(t))h1r11C10(t)D11(X1)(t))dt+S1(t,ρ(t))],dC10(t)=[k1Ce(t)(g1+m1)C10(t)]dt,dCe(t)=hCe(t)dt,}tnγ,ΔX1(t)=0,ΔC10(t)=0,ΔCe(t)=b,t=nγ,nN+. (3.2)

    Thanks to Lemma 1 and Lemma 2.3 in [54], system (3.2) satisfies

    {limt+X1(t)=0a.s.(B1<0);limt+t1t0X1(s)ds=B1A11a.s.(B10). (3.3)

    By Itô's formula, we compute

    lnX(t)=ΣtA0t0X(s)ds+(T11(X1)(t)T21(X1)(t)T22(X2)(t)T32(X2)(t)T33(X3)(t)T43(X3)(t)T44(X4)(t))+o(t), (3.4)

    where

    lnX(t)=(lnX1(t)lnX2(t)lnX3(t)lnX4(t)),X(s)ds=(X1(s)dsX2(s)dsX3(s)dsX4(s)ds),A0=(A11000A21A22000A32A33000A43A44),o(t)=o(t)(1111),Tji(Xi)(t)=0τji0θXi(s)dsdμji(θ)0τjitt+θXi(s)dsdμji(θ).

    Case(1): B1<0. Based on Eq (3.3), limt+X1(t)=0 a.s. Therefore, for ϵ(0,1) and t1,

    lnX2(t)(Σ2+ϵ)a22t0X2(s)ds.

    Since Σ2<0, then limt+X2(t)=0 a.s. Similarly, limt+Xj(t)=0 a.s. (j=3,4).

    Case(2): B10. Then,

    limt+t1t0X1(s)ds=B1A11a.s. (3.5)

    Consider the following auxiliary function:

    d~X2(t)=~X2(t)[(r2(ρ(t))r22C20(t)+D21(X1)(t)a22~X2(t))dt+S2(t,ρ(t))].

    Then X2(t)~X2(t) a.s. By Itô's formula, we get

    ln~X2(t)=B2ta22t0~X2(s)ds+o(t).

    In view of Lemma 2, we obtain:

    If B10, B2<0, then limt+~X2(t)=0 a.s.

    If B10, B20, then

    limt+t1t0~X2(s)ds=B2a22a.s.

    Therefore, for arbitrary ζ>0, we have

    limt+t1ttζXi(s)ds=0a.s.(i=1,2). (3.6)

    According to system (3.4) and Eq (3.6), we obtain

    lnX2(t)=B2tA22t0X2(s)ds+o(t).

    Thanks to Lemma 2, we deduce:

    If B10, B2<0, then limt+Xj(t)=0 a.s. (j=2,3,4).

    If B10, B20, then

    limt+t1t0X2(s)ds=B2A22a.s.

    Case(3): B10, B20. Consider the following SDE:

    d~X3(t)=~X3(t)[(r3(ρ(t))r33C30(t)+D32(X2)(t)a33~X3(t))dt+S3(t,ρ(t))].

    Then X3(t)~X3(t) a.s. By Itô's formula, we get

    ln~X3(t)=B3ta33t0~X3(s)ds+o(t).

    In the light of Lemma 2, we obtain:

    If B10, B20, B3<0, then limt+~X3(t)=0 a.s.

    If B10, B20, B30, then

    limt+t1t0~X3(s)ds=B3a33a.s.

    Hence, for arbitrary ζ>0,

    limt+t1ttζXi(s)ds=0a.s.(i=1,2,3). (3.7)

    Thanks to system (3.4) and Eq (3.7), we obtan

    lnX3(t)=B3tA33t0X3(s)ds+o(t).

    Based on Lemma 2, we obtain:

    If B10, B20, B3<0, then limt+Xj(t)=0 a.s. (j=3,4).

    If B10, B20, B30, then

    limt+t1t0X3(s)ds=B3A33a.s.

    Case(4): B10, B20, B30. Consider the following SDE:

    d~X4(t)=~X4(t)[(r4(ρ(t))r44C40(t)+D43(X3)(t)a44~X4(t))dt+S4(t,ρ(t))].

    Then X4(t)~X4(t) a.s. By Itô's formula, we get

    ln~X4(t)=B4ta44t0~X4(s)ds+o(t).

    In view of Lemma 2, we obtain:

    If B10, B20, B30, B4<0, then limt+~X4(t)=0 a.s.

    If B10, B20, B30, B40, then

    limt+t1t0~X4(s)ds=B4a44a.s.

    Hence, for arbitrary ζ>0,

    limt+t1ttζXi(s)ds=0a.s.(i=1,2,3,4). (3.8)

    Thanks to systems (3.4) and (3.8), we deduce

    lnX4(t)=B4tA44t0X4(s)ds+o(t).

    Based on Lemma 2 and the arbitrariness of ϵ, we obtain:

    If B10,B20,B3<0,B4<0, then limt+X4(t)=0a.s.

    If B10,B20,B30,B40, then

    limt+t1t0X4(s)ds=B4A44a.s.

    The proof is complete.

    Lemma 5. For system (1.6):

    (i) lim supt+t1lnxi(t)0 a.s. (i=1,2,3,4).

    (ii) limt+xi(t)=0limt+xj(t)=0 a.s. (1i<j4).

    Proof. From Lemma 4, system (3.1) satisfies limt+t1lnXi(t)=0 a.s. (i=1,2,3,4). By the stochastic comparison theorem, we obtain the desired assertion (i). The proof of (ii) is similar to that of Lemma 4 and here is omitted.

    Theorem 2. Under Assumption 4 system (1.6) satisfies Table 3, where

    ¯xT()=limt+t1(t0x1(s)ds,t0x2(s)ds,t0x3(s)ds,t0x4(s)ds).
    Table 3.  Stochastic persistence in mean and extinction of system (1.6).
    |Δ4| |Ξ3| |A2| B1 ¯xT()
    + (|Δ1||Δ|,|Δ2||Δ|,|Δ3||Δ|,|Δ4||Δ|)
    + (|Ξ1||Ξ|,|Ξ2||Ξ|,|Ξ3||Ξ|,0)
    + (|A1||A|,|A2||A|,0,0)
    + (B1A11,0,0,0)
    (0,0,0,0)

     | Show Table
    DownLoad: CSV

    Proof. Compute |Δ4|<A43|Ξ3|<A32A43|A2|<A21A32A43B1. By Eq (3.8), for any ζ>0,

    limt+t1ttζxi(s)ds=0a.s.(i=1,2,3,4).

    By Itô's formula, we compute

    lnx(t)=ΣtΔt0x(s)ds+o(t). (3.9)

    Case(i): |Δ4|>0. According to system (3.9), we compute

    limt+t1(A21A32A43lnx1(t)+A11A32A43lnx2(t)+A43|A|lnx3(t)+|Ξ|lnx4(t)+|Δ|t0x4(s)ds)=|Δ4|. (3.10)

    In view of Lemma 5 (i) and Lemma 2, we get

    lim inft+t1t0x4(s)ds|Δ4||Δ|a.s. (3.11)

    Based on system (3.9), we compute

    limt+t1(A22A43lnx1(t)A12A43lnx2(t)A12A23lnx4(t)+A43|A|t0x1(s)dsA12A23A44t0x4(s)ds)=A43|A1|A12A23Σ4. (3.12)

    By Lemma 5 (i) and Eq (3.12), for ϵ(0,1) and t1,

    A22A43lnx1(t)(A43|A1|A12A23Σ4+A12A23A44lim supt+t1t0x4(s)ds+ϵ)tA43|A|t0x1(s)ds.

    In view of Eq (3.11), we deduce

    A43|A1|A12A23Σ4+A12A23A44lim supt+t1t0x4(s)dsA43|A1|A12A23Σ4+A12A23A44lim inft+t1t0x4(s)dsA43|A1|A12A23Σ4+A12A23A44|Δ4||Δ|=A43|A||Δ1||Δ|, (3.13)

    where |A|>0 and |Δ|>0. From Lemma 3, we have |Δ1|>0. Based on Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x1(s)dsA143|A|1(A43|A1|A12A23Σ4+A12A23A44lim supt+t1t0x4(s)ds)=Γsupx1a.s. (3.14)

    According to system (3.9), we compute

    limt+t1(A21A32lnx1(t)+A11A32lnx2(t)+|A|lnx3(t)+A34|A|t0x4(s)ds+|Ξ|t0x3(s)ds)=|Ξ3|. (3.15)

    Thanks to Lemma 5 (i) and Eq (3.15), for ϵ(0,1) and t1,

    |A|lnx3(t)(|Ξ3|A34|A|lim supt+t1t0x4(s)dsϵ)t|Ξ|t0x3(s)ds.

    If

    |Ξ3|A34|A|lim supt+t1t0x4(s)ds>0,

    then by Lemma 2 and the arbitrariness of ϵ, we obtain

    lim inft+t1t0x3(s)ds|Ξ|1(|Ξ3|A34|A|lim supt+t1t0x4(s)ds)=Γinfx3a.s. (3.16)

    If

    |Ξ3|A34|A|lim supt+t1t0x4(s)ds0,

    since lim inft+t1t0x3(s)ds0, we also obtain Eq (3.16).

    According to system (3.9), Eq (3.14) and Eq (3.16), for ϵ(0,1) and t1,

    lnx2(t)(Σ2+A21Γsupx1A23Γinfx3+ϵ)tA22t0x2(s)ds.

    On the basis of Eq (3.13), Eq (3.14) and Eq (3.16), we have

    Σ2+A21Γsupx1A23Γinfx3Σ2+A21|Δ1||Δ|A23|Ξ|1(|Ξ3|A34|A|lim inft+t1t0x4(s)ds)Σ2+A21|Δ1||Δ|A23|Ξ|1(|Ξ3|A34|A||Δ4||Δ|)=A22|Δ2||Δ|. (3.17)

    From Lemma 3, we have |Δ2|>0. By Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x2(s)dsA122(Σ2+A21Γsupx1A23Γinfx3)=Γsupx2a.s. (3.18)

    By system (3.9), Eq (3.11) and Eq (3.18), for ϵ(0,1) and t1,

    lnx3(t)(Σ3+A32Γsupx2A34|Δ4||Δ|+ϵ)tA33t0x3(s)ds.

    In view of Eq (3.17) and Eq (3.18), we obtain

    Σ3+A32Γsupx2A34|Δ4||Δ|Σ3+A32|Δ2||Δ|A34|Δ4||Δ|=A33|Δ3||Δ|. (3.19)

    In the light of Lemma 3, we have |Δ3|>0. Thanks to Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x3(s)dsA133(Σ3+A32Γsupx2A34|Δ4||Δ|)=Γsupx3a.s. (3.20)

    By system (3.9) and Eq (3.20), for ϵ(0,1) and t1,

    lnx4(t)(Σ4+A43Γsupx3+ϵ)tA44t0x4(s)ds.

    Thanks to Eq (3.19), we obtain

    Σ4+A43Γsupx3Σ4+A43|Δ3||Δ|=A44|Δ4||Δ|.

    In the light of Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x4(s)dsA144(Σ4+A43Γsupx3)a.s.

    In other words, we have

    A22A33A44|A||Ξ|A12A21A23A32A44|Ξ|A23A32A34A43|A|2A22A33|A||Ξ|lim supt+t1t0x4(s)dsΣ4+A43A133[Σ3+A32A122(Σ2+A21|A1||A|+A12A21A23Σ4A43|A|A23|Ξ3||Ξ|)A34|Δ4||Δ|]. (3.21)

    According to Eq (3.21) and Assumption 4, we obtain

    lim supt+t1t0x4(s)ds|Δ4||Δ|a.s. (3.22)

    Combining Eq (3.11) and Eq (3.22) yields

    limt+t1t0x4(s)ds=|Δ4||Δ|a.s. (3.23)

    Substituting Eq (3.22) into Eq (3.14) yields

    lim supt+t1t0x1(s)dsA143|A|1(A43|A1|A12A23Σ4+A12A23A44|Δ4||Δ|)=|Δ1||Δ|. (3.24)

    Substituting Eq (3.22) into Eq (3.16) yields

    lim inft+t1t0x3(s)ds|Ξ|1(|Ξ3|A34|A||Δ4||Δ|)=|Δ3||Δ|. (3.25)

    Substituting Eq (3.24) and Eq (3.25) into Eq (3.18) yields

    lim supt+t1t0x2(s)dsA122(Σ2+A21|Δ1||Δ|A23|Δ3||Δ|)=|Δ2||Δ|. (3.26)

    Substituting Eq (3.26) into Eq (3.20) yields

    lim supt+t1t0x3(s)dsA133(Σ3+A32|Δ2||Δ|A34|Δ4||Δ|)=|Δ3||Δ|. (3.27)

    Combining Eq (3.25) and Eq (3.27) yields

    limt+t1t0x3(s)ds=|Δ3||Δ|a.s. (3.28)

    In view of system (3.9), we have

    limt+t1(lnx1(t)+A11t0x1(s)ds+A12t0x2(s)ds)=B1. (3.29)

    By Eq (3.26) and Eq (3.29), for ϵ(0,1) and t1,

    lnx1(t)(B1A12|Δ2||Δ|ϵ)tA11t0x1(s)ds,

    where B1A12|Δ2||Δ|=A11|Δ1||Δ|. From Lemma 3, we have |Δ1|>0. According to Lemma 2 and the arbitrariness of ϵ, we have

    lim inft+t1t0x1(s)ds|Δ1||Δ|a.s. (3.30)

    Combining Eq (3.24) with Eq (3.30) yields

    limt+t1t0x1(s)ds=|Δ1||Δ|a.s. (3.31)

    Substituting Eq (3.31) into system (3.9) yields

    limt+t1(lnx2(t)+A22t0x2(s)ds)=A22|Δ2||Δ|a.s.

    From Lemma 3, we have |Δ2|>0. By Lemma 2, we obtain

    limt+t1t0x2(s)ds=|Δ2||Δ|a.s. (3.32)

    Case(ii): |Ξ3|>0>|Δ4|. Thanks to Eq (3.10), we deduce

    lim supt+t1ln(xA21A32A431(t)xA11A32A432(t)xA43|A|3(t)x|Ξ|4(t))|Δ4|<0a.s.

    which implies

    limt+xA21A32A431(t)xA11A32A432(t)xA43|A|3(t)x|Ξ|4(t)=0a.s. (3.33)

    From Lemma 5 (ii) and Eq (3.33), we obtain

    limt+x4(t)=0a.s. (3.34)

    In other words,

    limt+t1t0x4(s)ds=0a.s.

    According to system (3.9), we compute

    limt+t1(A21A32lnx1(t)+A11A32lnx2(t)+|A|lnx3(t)+|Ξ|t0x3(s)ds)=|Ξ3|. (3.35)

    Combining Lemma 5 (i) with Lemma 2 yields

    lim inft+t1t0x3(s)ds|Ξ3||Ξ|a.s. (3.36)

    Based on system (3.9), we compute

    limt+t1(A22lnx1(t)A12lnx2(t)+|A|t0x1(s)dsA12A23t0x3(s)ds)=|A1|. (3.37)

    By Lemma 5 (i) and Eq (3.37), for ϵ(0,1) and t1,

    A22lnx1(t)(|A1|+A12A23lim supt+t1t0x3(s)ds+ϵ)t|A|t0x1(s)ds.

    On the basis of Eq (3.36), we deduce

    |A1|+A12A23lim supt+t1t0x3(s)ds|A1|+A12A23|Ξ3||Ξ|=|A||Ξ1||Ξ|>0.

    In view of Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x1(s)ds|A|1(|A1|+A12A23lim supt+t1t0x3(s)ds)=Υsupx1a.s. (3.38)

    According to system (3.9), Eq (3.36) and Eq (3.38), for ϵ(0,1) and t1,

    lnx2(t)(Σ2+A21Υsupx1A23|Ξ3||Ξ|+ϵ)tA22t0x2(s)ds. (3.39)

    Combining Eq (3.38) with system (3.39) yields

    Σ2+A21Υsupx1A23|Ξ3||Ξ|Σ2+A21|A|1(|A1|+A12A23|Ξ3||Ξ|)A23|Ξ3||Ξ|=A22|Ξ2||Ξ|>0. (3.40)

    In the light of Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x2(s)dsA122(Σ2+A21Υsupx1A23|Ξ3||Ξ|)a.s. (3.41)

    From system (3.9) and Eq (3.41), for ϵ(0,1) and t1,

    lnx3(t)(Σ3+A32A22(Σ2+A21Υsupx1A23|Ξ3||Ξ|)+ϵ)tA33t0x3(s)ds.

    Thanks to Eq (3.40), we obtain

    Σ3+A32A22(Σ2+A21Υsupx1A23|Ξ3||Ξ|)Σ3+A32|Ξ2||Ξ|=A33|Ξ3||Ξ|>0.

    In the light of Lemma 2 and the arbitrariness of ϵ, we obtain

    lim supt+t1t0x3(s)dsA133(B3A21A32A11A22B1+A32A22(A21Υsupx1A23|Ξ3||Ξ|))a.s.

    In other words, we have

    A22A33|A|A12A21A23A32A22|A|lim supt+t1t0x3(s)dsB3A21A32A11A22B1+A32A22(A21|A1||A|A23|Ξ3||Ξ|). (3.42)

    In view of Eq (3.42) and Assumption 4, we obtain

    lim supt+t1t0x3(s)ds|Ξ3||Ξ|a.s. (3.43)

    Combining Eq (3.36) and Eq (3.43) yields

    limt+t1t0x3(s)ds=|Ξ3||Ξ|a.s. (3.44)

    Substituting Eq (3.44) into Eq (3.38) yields

    lim supt+t1t0x1(s)ds|A|1(|A1|+A12A23|Ξ3||Ξ|)=|Ξ1||Ξ|. (3.45)

    Substituting Eq (3.45) into Eq (3.41) yields

    lim supt+t1t0x2(s)dsA122(Σ2+A21|Ξ1||Ξ|A23|Ξ3||Ξ|)=|Ξ2||Ξ|. (3.46)

    In view of system (3.9), we compute

    limt+t1(A21lnx1(t)+A11lnx2(t)+|A|t0x2(s)ds+A11A23t0x3(s)ds)=|A2|. (3.47)

    By Lemma 5 (i) and Eq (3.47), for ϵ(0,1) and t1,

    A11lnx2(t)(|A2|A11A23|Ξ3||Ξ|ϵ)t|A|t0x2(s)ds. (3.48)

    From Eq (3.40), we have |Ξ2|>0 and |Ξ|>0. Based on system (3.48) and Lemma 2,

    lim inft+t1t0x2(s)ds|A|1(|A2|A11A23|Ξ3||Ξ|)=|Ξ2||Ξ|a.s. (3.49)

    Combining Eq (3.46) with Eq (3.49) yields

    limt+t1t0x2(s)ds=|Ξ2||Ξ|a.s. (3.50)

    Substituting Eq (3.50) into system (3.9) yields

    limt+t1(lnx1(t)+A11t0x1(s)ds)=A11|Ξ1||Ξ|a.s.

    From Eq (3.38), we have |Ξ1|>0 and |Ξ|>0. Therefore, by Lemma 2, we obtain

    limt+t1t0x1(s)ds=|Ξ1||Ξ|a.s. (3.51)

    Case(iii): |A2|>0>|Ξ3|. Then limt+x4(t)=0 a.s. Thanks to Eq (3.35), we deduce

    lim supt+t1ln(xA21A321(t)xA11A322(t)x|A|3(t))|Ξ3|<0a.s.

    which implies

    lim supt+xA21A321(t)xA11A322(t)x|A|3(t)=0a.s. (3.52)

    According to Lemma 5 (ii) and Eq (3.52), we obtain

    limt+x3(t)=0a.s. (3.53)

    In other words, we derive

    limt+t1t0x3(s)ds=0a.s. (3.54)

    In view of Eq (3.47) and Eq (3.54), we get

    limt+t1(A21lnx1(t)+A11lnx2(t)+|A|t0x2(s)ds)=|A2|. (3.55)

    Based on Lemma 5 (i) and Lemma 2, we have

    lim inft+t1t0x2(s)ds|A2||A|a.s. (3.56)

    In the light of Eq (3.37) and Eq (3.54), we obtain

    limt+t1(A22lnx1(t)A12lnx2(t)+|A|t0x1(s)ds)=|A1|.

    By Lemma 5 (i) and Lemma 2, we obtain

    lim supt+t1t0x1(s)ds|A1||A|a.s. (3.57)

    Substituting Eq (3.54) and Eq (3.57) into system (3.9) yields

    lnx2(t)(Σ2+A21|A1||A|+ϵ)tA22t0x2(s)ds.

    On the basis of Lemma 2 and the arbitrariness of ϵ, we have

    lim supt+t1t0x2(s)dsA122(Σ2+A21|A1||A|)=|A2||A|a.s. (3.58)

    Combining Eq (3.56) with Eq (3.58) yields

    limt+t1t0x2(s)ds=|A2||A|a.s. (3.59)

    By system (3.9) and Eq (3.59), we compute

    limt+t1(lnx1(t)+A11t0x1(s)ds)=B1A12|A2||A|=A11|A1||A|a.s.

    In the light of Lemma 2, we obtain

    limt+t1t0x1(s)ds=|A1||A|a.s. (3.60)

    Case(iv): B1>0>|A2|. Then

    limt+xi(t)=0a.s.(i=3,4). (3.61)

    According to Eq (3.55), we gain

    lim supt+t1ln(xA211(t)xA112(t))|A2|<0a.s. (3.62)

    Hence, lim supt+xA211(t)xA112(t)=0. By Lemma 5 (ii) and Eq (3.62),

    limt+x2(t)=0a.s. (3.63)

    In other words, we derive

    limt+t1t0x2(s)ds=0a.s. (3.64)

    Substituting Eq (3.64) into system (3.9) yields

    lnx1(t)=B1tA11t0x1(s)ds+o(t).

    On the basis of Lemma 2 and the arbitrariness of ϵ, we obtain

    limt+t1t0x1(s)ds=B1A11a.s. (3.65)

    Case(v): B1<0. Compute

    limt+t1(lnx1(t)+A11t0x1(s)ds+A12t0x2(s)ds)=B1. (3.66)

    By Eq (3.66), we have

    lim supt+t1(lnx1(t)+A11t0x1(s)ds)B1.

    In view of Lemma 2, we obtain limt+x1(t)=0 a.s. According to Lemma 5 (ii), we get

    limt+xj(t)=0a.s.(j=2,3,4). (3.67)

    The proof is complete.

    In this section we introduce some numerical examples to illustrate our main results. For simplicity, we suppose that S={1,2}. Then system (1.6) is a hybrid system of the following two subsystems:

    {dx1(t)=x1(t)[(r1(1)r11C10(t)D11(x1)(t)D12(x2)(t))dt+S1(t,1)],dx2(t)=x2(t)[(r2(1)r22C20(t)+D21(x1)(t)D22(x2)(t)D23(x3)(t))dt+S2(t,1)],dx3(t)=x3(t)[(r3(1)r33C30(t)+D32(x2)(t)D33(x3)(t)D34(x4)(t))dt+S3(t,1)],dx4(t)=x4(t)[(r4(1)r44C40(t)+D43(x3)(t)D44(x4)(t))dt+S4(t,1)],dCi0(t)=[0.1Ce(t)(0.1+0.1)Ci0(t)]dt,dCe(t)=0.5Ce(t)dt,}t12n,Δxi(t)=0,ΔCi0(t)=0,ΔCe(t)=0.6,t=12n,nN+(i=1,2,3,4), (4.1)

    and

    {dx1(t)=x1(t)[(r1(2)r11C10(t)D11(x1)(t)D12(x2)(t))dt+S1(t,2)],dx2(t)=x2(t)[(r2(2)r22C20(t)+D21(x1)(t)D22(x2)(t)D23(x3)(t))dt+S2(t,2)],dx3(t)=x3(t)[(r3(2)r33C30(t)+D32(x2)(t)D33(x3)(t)D34(x4)(t))dt+S3(t,2)],dx4(t)=x4(t)[(r4(2)r44C40(t)+D43(x3)(t)D44(x4)(t))dt+S4(t,2)],dCi0(t)=[0.1Ce(t)(0.1+0.1)Ci0(t)]dt,dCe(t)=0.5Ce(t)dt,}t12n,Δxi(t)=0,ΔCi0(t)=0,ΔCe(t)=0.6,t=12n,nN+(i=1,2,3,4), (4.2)

    with initial conditions x1(θ)=2eθ, x2(θ)=1.5eθ, x3(θ)=0.8eθ x4(θ)=0.5eθ and θ[ln2,0].

    Let rii=0.3, τji=ln2, μji(θ)=μjieθ, γj(μ,i)=γj(i) and λ(Z)=1, see Table 4. Denote

    Param(i)=(a11a1200μ11μ1200σ1(i)γ1(i)a21a22a230μ21μ22μ230σ2(i)γ2(i)0a32a33a340μ32μ33μ34σ3(i)γ3(i)00a43a4400μ43μ44σ4(i)γ4(i)).
    Table 4.  Source of some parameter values in system (1.6).
    Parameter Value Source
    ki 0.1 [55]
    gi 0.1 [55]
    mi 0.1 [55]
    h 0.5 [55]
    γ 12 [55]
    b 0.6 [55]
    τji ln2 [56]
    λ(Z) 1 [56]

     | Show Table
    DownLoad: CSV

    Then system (1.6) may be regarded as the result of regime switching between subsystems (4.1) and (4.2) with the following estimated parameters, respectively,

    Param(1)=(0.20.1000.20.1000.10.10.50.30.100.20.10.100.10.100.40.30.200.20.20.20.10.1000.40.3000.10.20.10.1),
    Param(2)=(0.20.1000.20.1001.20.20.50.30.100.20.10.100.20.200.40.30.200.20.20.20.20.2000.40.3000.10.20.20.2).

    Compute |Δ|=0.066525, |Ξ|=0.1005 and |A|=0.195. Denote

    γ(1)=(γ1(1),γ2(1),γ3(1),γ4(1)),r(j)=(r1(j),r2(j),r3(j),r4(j))(j=1,2).

    Let r(1)=(0.9,0.5,0.3,0.2). Compute

    |Δ1|=0.1384,|Δ2|=0.1113,|Δ3|=0.0614,|Δ4|=0.0317>0.

    Based on Theorem 2, all species in subsystem (4.1) are persistent in mean and

    {limt+t1t0x1(s)ds=|Δ1||Δ|=2.0811a.s.limt+t1t0x2(s)ds=|Δ2||Δ|=1.6731a.s.limt+t1t0x3(s)ds=|Δ3||Δ|=0.9226a.s.limt+t1t0x4(s)ds=|Δ4||Δ|=0.4762a.s. (4.3)

    Let r(2)=(0.6,0.3,0.2,0.1). Then B1=0.1527<0. From Theorem 2, all species in subsystem (4.2) are extinctive.

    Case1: (π1,π2)=(0.8,0.2). Compute

    |Δ1|=0.1096,|Δ2|=0.0779,|Δ3|=0.0390,|Δ4|=0.0090>0.

    By Theorem 2, all species in system (1.6) are persistent in mean and

    {limt+t1t0x1(s)ds=|Δ1||Δ|=1.6469a.s.limt+t1t0x2(s)ds=|Δ2||Δ|=1.1709a.s.limt+t1t0x3(s)ds=|Δ3||Δ|=0.5870a.s.limt+t1t0x4(s)ds=|Δ4||Δ|=0.1346a.s. (4.4)

    Case2: (π1,π2)=(0.6,0.4). Compute

    |Δ4|=0.0138<0,|Ξ1|=0.1205,|Ξ2|=0.0700,|Ξ3|=0.0132>0.

    From Theorem 2, x1(t), x2(t) and x3(t) are persistent in mean, while x4(t) is extinctive and

    {limt+t1t0x1(s)ds=|Ξ1||Ξ|=1.1988a.s.limt+t1t0x2(s)ds=|Ξ2||Ξ|=0.6965a.s.limt+t1t0x3(s)ds=|Ξ3||Ξ|=0.1309a.s. (4.5)

    Case3: (π1,π2)=(0.5,0.5). Compute

    |Ξ3|=0.0137<0,|A1|=0.1923,|A2|=0.0852>0.

    Thanks to Theorem 2, x1(t) and x2(t) are persistent in mean, while x3(t) and x4(t) are extinctive and

    {limt+t1t0x1(s)ds=|A1||A|=0.9860a.s.limt+t1t0x2(s)ds=|A2||A|=0.4368a.s. (4.6)

    Case4: (π1,π2)=(0.3,0.7). Compute

    |A2|=0.0279<0,B1=0.1557>0.

    Based on Theorem 2, x1(t) is persistent in mean, while x2(t), x3(t) and x4(t) are extinctive and

    limt+t1t0x1(s)ds=B1A11=0.5191a.s. (4.7)

    Case5: (π1,π2)=(0.1,0.9). Compute B1=0.0499<0. On the basis of Theorem 2, all species in system (1.6) are extinctive.

    Let r(1)=(0.7,0.5,0.3,0.2). We study the effects of Lévy jumps on the persistence in mean and extinction of system (4.1) by changing the values of γj(1) and setting the remaining parameters of the examples to be the same as those in system (4.1). Denote I4={0.3,0.4}, α4I4; I3={0.6,1.1}, α3I3; I2={0.9,1.9}, α2I2; I1={0.8,1.7}, α1I1.

    Case1: Let γ(1)=(0.1,0.1,0.1,α4). Then |Δ4|<0, |Ξ3|=0.0606>0. According to Theorem 2, x1(t), x2(t) and x3(t) are persistent in mean, while x4(t) is extinctive.

    Let γ(1)=(0.1,0.1,0.1,0.1). Then |Δ4|=0.0047>0. By Theorem 2, all species in system (4.1) are persistent in mean. See Table 5.

    Table 5.  Changes of γ4(1) when γ1(1)=γ2(1)=γ3(1)=0.1.
    γ1(1) γ2(1) γ3(1) γ4(1) ¯xT()
    0.1 0.1 0.1 α4 (1.6852,1.1316,0.6027,0)
    0.1 0.1 0.1 0.1 (1.6805,1.1410,0.5618,0.0703)

     | Show Table
    DownLoad: CSV

    Case2: Let γ(1)=(0.1,0.1,α3,α4). Then |Ξ3|<0, |A2|=0.2478>0. Based on Theorem 2, x1(t) and x2(t) are persistent in mean, while x3(t) and x4(t) are extinctive. See Table 6.

    Table 6.  Changes of γ3(1) when γ1(1)=γ2(1)=0.1 and γ4(1)I4.
    γ1(1) γ2(1) γ3(1) γ4(1) ¯xT()
    0.1 0.1 α3 I4 (1.6157,1.2707,0,0)
    0.1 0.1 0.1 I4 (1.6852,1.1316,0.6027,0)

     | Show Table
    DownLoad: CSV

    Case3: Let γ(1)=(0.1,α2,α3,α4). Then |A2|<0, B1=0.6753>0. From Theorem 2, x1(t) is persistent in mean, while x2(t), x3(t) and x4(t) are extinctive. See Table 7.

    Table 7.  Changes of γ2(1) when γ1(1)=0.1, γ3(1)I3 and γ4(1)I4.
    γ1(1) γ2(1) γ3(1) γ4(1) ¯xT()
    0.1 α2 I3 I4 (2.2510,0,0,0)
    0.1 0.1 I3 I4 (1.6157,1.2707,0,0)

     | Show Table
    DownLoad: CSV

    Case4: Let γ(1)=(α1,α2,α3,α4). Then B1<0. Thanks to Theorem 2, all species are extinctive. See Table 8.

    Table 8.  Changes of γ1(1) when γ2(1)I2, γ3(1)I3 and γ4(1)I4.
    γ1(1) γ2(1) γ3(1) γ4(1) ¯xT()
    α1 I2 I3 I4 (0,0,0,0)
    0.1 I2 I3 I4 (2.2510,0,0,0)

     | Show Table
    DownLoad: CSV

    Case1: Let γ1(1)=0.8. Then B1=0.1294<0. According to Theorem 2, all species in system (4.1) are extinctive.

    Let γ1(1)=0.7. Then |A2|=0.0518<0, B1=0.1760>0. By Theorem 2, x1(t) is persistent in mean, while x2(t), x3(t) and x4(t) are extinctive and

    limt+t1t0x1(s)ds=B1A11=0.5868a.s. (4.8)

    Let γ1(1)=0.6. Then |Ξ3|=0.0329<0, |A2|=0.0608>0. Based on Theorem 2, x1(t) and x2(t) are persistent in mean, while x3(t) and x4(t) are extinctive and

    {limt+t1t0x1(s)ds=|A1||A|=1.0564a.s.limt+t1t0x2(s)ds=|A2||A|=0.3119a.s. (4.9)

    Let γ1(1)=0.3. Then |Δ4|=0.0023<0, |Ξ3|=0.0450>0. In view of Theorem 2, x1(t), x2(t) and x3(t) are persistent in mean, while x4(t) is extinctive and

    {limt+t1t0x1(s)ds=|Ξ1||Ξ|=1.5740a.s.limt+t1t0x2(s)ds=|Ξ2||Ξ|=1.0074a.s.limt+t1t0x3(s)ds=|Ξ3||Ξ|=0.4476a.s. (4.10)

    Let γ1(1)=0.1. Then |Δ4|=0.0046>0. From Theorem 2, all species are persistent in mean and

    {limt+t1t0x1(s)ds=|Δ1||Δ|=1.6792a.s.limt+t1t0x2(s)ds=|Δ2||Δ|=1.1392a.s.limt+t1t0x3(s)ds=|Δ3||Δ|=0.5606a.s.limt+t1t0x4(s)ds=|Δ4||Δ|=0.0690a.s. (4.11)

    Case2: Let γ1(1)=0.2. Then |Δ4|=0.0029>0. Thanks to Theorem 2, all species are persistent in mean and

    {limt+t1t0x1(s)ds=|Δ1||Δ|=1.6545a.s.limt+t1t0x2(s)ds=|Δ2||Δ|=1.1065a.s.limt+t1t0x3(s)ds=|Δ3||Δ|=0.5384a.s.limt+t1t0x4(s)ds=|Δ4||Δ|=0.0440a.s. (4.12)

    Let γ1(1)=0.6. Then |Δ4|=0.0122<0, |Ξ3|=0.0230>0. On the basis of Theorem 2, x1(t), x2(t) and x3(t) are persistent in mean, while x4(t) is extinctive and

    {limt+t1t0x1(s)ds=|Ξ1||Ξ|=1.4172a.s.limt+t1t0x2(s)ds=|Ξ2||Ξ|=0.8323a.s.limt+t1t0x3(s)ds=|Ξ3||Ξ|=0.2287a.s. (4.13)

    Let γ1(1)=0.9. Then |Ξ3|=0.0155<0, |A2|=0.0957>0. By Theorem 2, x1(t) and x2(t) are persistent in mean, while x3(t) and x4(t) are extinctive and

    {limt+t1t0x1(s)ds=|A1||A|=1.1608a.s.limt+t1t0x2(s)ds=|A2||A|=0.4908a.s. (4.14)

    Let γ1(1)=1.3. Then |A2|=0.0297<0, B1=0.2129>0. From Theorem 2, x1(t) is persistent in mean, while x2(t), x3(t) and x4(t) are extinctive and

    limt+t1t0x1(s)ds=B1A11=0.7097a.s. (4.15)

    Let γ1(1)=1.7. Then B1=0.0267<0. In view of Theorem 2, all species are extinctive.

    This paper concerns the dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment. Theorem 2 establishes sufficient and necessary conditions for persistence in mean and extinction of each species. Our results reveal that the stochastic dynamics of the system is closely correlated with both time delays and environmental noises.

    Some interesting topics deserve further investigation, for instance, it is meaningful to consider the optimal harvesting problem of the stochastic hybrid delay food chain model with Lévy noises in an impulsive polluted environment. One may also propose some more realistic systems, such as considering the generalized functional response and the influences of impulsive perturbations. We will leave investigation of these problems to the future.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by National Natural Science Foundation of China (No. 11901166).

    The authors declare that there are no conflicts of interest.



    [1] M. Yavuz, N. Sene, Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate, Fractal Fract., 4 (2020), 35. https://doi.org/10.3390/fractalfract4030035 doi: 10.3390/fractalfract4030035
    [2] A. Chatterjee, S. Pal, A predator-prey model for the optimal control of fish harvesting through the imposition of a tax, Int. J. Optim. Control Theor. Appl., 13 (2023), 68–80. https://doi.org/10.11121/ijocta.2023.1218 doi: 10.11121/ijocta.2023.1218
    [3] D. Ghosh, P. K. Santra, G. S. Mahapatra, A three-component prey-predator system with interval number, Math. Model. Numer. Simul. Appl., 3 (2023), 1–16. https://doi.org/10.53391/mmnsa.1273908 doi: 10.53391/mmnsa.1273908
    [4] M. Liu, C. Bai, Optimal harvesting policy of a stochastic food chain population model, Appl. Math. Comput., 245 (2014), 265–270. https://doi.org/10.1016/j.amc.2014.07.103 doi: 10.1016/j.amc.2014.07.103
    [5] J. Yu, M. Liu, Stationary distribution and ergodicity of a stochastic food-chain model with Lévy jumps, Phys. A, 482 (2017), 14–28. https://doi.org/10.1016/j.physa.2017.04.067 doi: 10.1016/j.physa.2017.04.067
    [6] T. Zeng, Z. Teng, Z. Li, J. Hu, Stability in the mean of a stochastic three species food chain model with general Lévy jumps, Chaos Solitons Fractals, 106 (2018), 258–265. https://doi.org/10.1016/j.chaos.2017.10.025 doi: 10.1016/j.chaos.2017.10.025
    [7] H. Li, H. Li, F. Cong, Asymptotic behavior of a food chain model with stochastic perturbation, Phys. A, 531 (2019), 121749. https://doi.org/10.1016/j.physa.2019.121749 doi: 10.1016/j.physa.2019.121749
    [8] Q. Yang, X. Zhang, D. Jiang, Dynamical behaviors of a stochastic food chain system with ornstein-uhlenbeck process, J. Nonlinear Sci., 32 (2022), 34. https://doi.org/10.1007/s00332-022-09796-8 doi: 10.1007/s00332-022-09796-8
    [9] P. A. Naik, Z. Eskandari, M. Yavuz, J. Zu, Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect, J. Comput. Appl. Math., 413 (2022), 114401. https://doi.org/10.1016/j.cam.2022.114401 doi: 10.1016/j.cam.2022.114401
    [10] P. A. Naik, Z. Eskandari, H. E. Shahraki, Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model, Math. Model. Numer. Simul. Appl., 1 (2021), 95–101. https://doi.org/10.53391/mmnsa.2021.01.009 doi: 10.53391/mmnsa.2021.01.009
    [11] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993.
    [12] F. Shakeri, M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model., 48 (2008), 486–498. https://doi.org/10.1016/j.mcm.2007.09.016 doi: 10.1016/j.mcm.2007.09.016
    [13] K. Golpalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992.
    [14] F. A. Rihan, H. J. Alsakaji, Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species, Discrete. Contin. Dyn. Syst. Ser. S, 15 (2020), 245–263. https://doi.org/10.3934/dcdss.2020468 doi: 10.3934/dcdss.2020468
    [15] H. J. Alsakaji, S. Kundu, F. A. Rihan, Delay differential model of one-predator two-prey system with Monod-Haldane and holling type Ⅱ functional responses, Appl. Math. Comput., 397 (2021), 125919. https://doi.org/10.1016/j.amc.2020.125919 doi: 10.1016/j.amc.2020.125919
    [16] J. Roy, D. Barman, S. Alam, Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment, Biosystems, 197 (2020), 104176. https://doi.org/10.1016/j.biosystems.2020.104176 doi: 10.1016/j.biosystems.2020.104176
    [17] Q. Liu, D. Jiang, Influence of the fear factor on the dynamics of a stochastic predator-prey model, Appl. Math. Lett., 112 (2021), 106756. https://doi.org/10.1016/j.aml.2020.106756 doi: 10.1016/j.aml.2020.106756
    [18] N. Tuerxun, Z. Teng, Global dynamics in stochastic n-species food chain systems with white noise and Lévy jumps, Math. Methods Appl. Sci., 45 (2022), 5184–5214. https://doi.org/10.1002/mma.8101 doi: 10.1002/mma.8101
    [19] Q. Zhang, D. Jiang, Z. Liu, D. O'Regan, Asymptotic behavior of a three species eco-epidemiological model perturbed by white noise, J. Math. Anal. Appl., 433 (2016), 121–148. https://doi.org/10.1016/j.jmaa.2015.07.025 doi: 10.1016/j.jmaa.2015.07.025
    [20] Y. Zhao, L. You, D. Burkow, S. Yuan, Optimal harvesting strategy of a stochastic inshore-offshore hairtail fishery model driven by Lévy jumps in a polluted environment, Nonlinear Dyn., 95 (2019), 1529–1548. https://doi.org/10.1007/s11071-018-4642-y doi: 10.1007/s11071-018-4642-y
    [21] L. Liu, X. Meng, T. Zhang, Optimal control strategy for an impulsive stochastic competition system with time delays and jumps, Physica A, 477 (2017), 99–113. https://doi.org/10.1016/j.physa.2017.02.046 doi: 10.1016/j.physa.2017.02.046
    [22] J. R. Beddington, R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463–465. https://doi.org/10.1126/science.197.4302.463 doi: 10.1126/science.197.4302.463
    [23] X. Zou, W. Li, K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170–174. https://doi.org/10.1016/j.aml.2012.08.006 doi: 10.1016/j.aml.2012.08.006
    [24] X. Zou, K. Wang, Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal. Hybrid Syst., 13 (2014), 32–44. https://doi.org/10.1016/j.nahs.2014.01.001 doi: 10.1016/j.nahs.2014.01.001
    [25] H. Qiu, W. Deng, Optimal harvesting of a stochastic delay logistic model with Lévy jumps, J. Phys. A: Math. Theor., 49 (2016), 405601. https://doi.org/10.1088/1751-8113/49/40/405601 doi: 10.1088/1751-8113/49/40/405601
    [26] G. Denaro, D. Valenti, A. L. Cognata, B. Spagnolo, A. Bonanno, G. Basilone, et al., Spatio-temporal behaviour of the deep chlorophyll maximum in Mediterranean Sea: Development of a stochastic model for picophytoplankton dynamics, Ecol. Complex., 13 (2013), 21–34. https://doi.org/10.4414/pc-d.2013.00242 doi: 10.4414/pc-d.2013.00242
    [27] D. Valenti, A. Fiasconaro, B. Spagnolo, Stochastic resonance and noise delayed extinction in a model of two competing species, Phys. A, 331 (2004), 477–486. https://doi.org/10.1016/j.physa.2003.09.036 doi: 10.1016/j.physa.2003.09.036
    [28] R. Grimaudo, P. Lazzari, C. Solidoro, D. Valenti, Effects of solar irradiance noise on a complex marine trophic web, Sci. Rep., 12 (2022), 12163. https://doi.org/10.1038/s41598-022-12384-1 doi: 10.1038/s41598-022-12384-1
    [29] P. Lazzari, R. Grimaudo, C. Solidoro, D. Valenti, Stochastic 0-dimensional biogeochemical flux model: Effect of temperature fluctuations on the dynamics of the biogeochemical properties in a marine ecosystem, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 105994. https://doi.org/10.1016/j.cnsns.2021.105994 doi: 10.1016/j.cnsns.2021.105994
    [30] D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio, S. Mazzola, et al., Noise induced phenomena in the dynamics of two competing species, Math. Model. Nat. Phenom., 11 (2016), 158–174. https://doi.org/10.1051/mmnp/201611510 doi: 10.1051/mmnp/201611510
    [31] D. Valenti, G. Denaro, B. Spagnolo, S. Mazzola, G. Basilone, F. Conversano, et al., Stochastic models for phytoplankton dynamics in Mediterranean Sea, Ecol. Complex., 27 (2016), 84–103. https://doi.org/10.1016/j.ecocom.2015.06.001 doi: 10.1016/j.ecocom.2015.06.001
    [32] D. Valenti, G. Denaro, F. Conversano, C. Brunet, A. Bonanno, G. Basilone, et al., The role of noise on the steady state distributions of phytoplankton populations, J. Stat. Mech., 2016 (2016), 054044. https://doi.org/10.1088/1742-5468/2016/05/054044 doi: 10.1088/1742-5468/2016/05/054044
    [33] G. Denaro, D. Valenti, B. Spagnolo, A. Bonanno, G. Basilone, S. Mazzola, et al., Dynamics of two picophytoplankton groups in mediterranean sea: Analysis and prediction of the deep chlorophyll maximum by a stochastic reaction-diffusion-taxis model, PLoS One, 8 (2013), e66765. https://doi.org/10.1371/journal.pone.0066765 doi: 10.1371/journal.pone.0066765
    [34] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, 1985.
    [35] B. Spagnolo, D. Valenti, A. Fiasconaro, Noise in ecosystems: A short review, Math. Biosci. Eng., 1 (2004), 185–211. https://doi.org/10.3934/mbe.2004.1.185 doi: 10.3934/mbe.2004.1.185
    [36] Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84. https://doi.org/10.1016/j.jmaa.2006.12.032 doi: 10.1016/j.jmaa.2006.12.032
    [37] X. Li, A. Gray, D. Jiang, X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11–28. https://doi.org/10.1016/j.jmaa.2010.10.053 doi: 10.1016/j.jmaa.2010.10.053
    [38] Y. Cai, S. Cai, X. Mao, Stochastic delay foraging arena predator-prey system with Markov switching, Stoch. Anal. Appl., 38 (2020), 191–212. https://doi.org/10.1080/07362994.2019.1679645 doi: 10.1080/07362994.2019.1679645
    [39] A. La Cognata, D. Valenti, B. Spagnolo, A. A. Dubkov, Two competing species in super-diffusive dynamical regimes, Eur. Phys. J. B, 77 (2010), 273–279. https://doi.org/10.1140/epjb/e2010-00239-6 doi: 10.1140/epjb/e2010-00239-6
    [40] J. Bertoin, Lévy Processes, Cambridge University Press, 1996.
    [41] A. L. Cognata, D. Valenti, A. A. Dubkov, B. Spagnolo, Dynamics of two competing species in the presence of Lévy noise sources, Phys. Rev. E, 82 (2010), 011121. https://doi.org/10.1103/PhysRevE.82.011121 doi: 10.1103/PhysRevE.82.011121
    [42] J. Bao, X. Mao, G. Yin, C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601–6616. https://doi.org/10.1016/j.na.2011.06.043 doi: 10.1016/j.na.2011.06.043
    [43] J. Bao, C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363–375. https://doi.org/10.1016/j.jmaa.2012.02.043 doi: 10.1016/j.jmaa.2012.02.043
    [44] M. Liu, K. Wang, Dynamics of a Leslie-Gower Holling-type Ⅱ predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204–213. https://doi.org/10.1016/j.na.2013.02.018 doi: 10.1016/j.na.2013.02.018
    [45] M. Liu, K. Wang, Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410 (2014), 750–763. https://doi.org/10.1016/j.jmaa.2013.07.078 doi: 10.1016/j.jmaa.2013.07.078
    [46] M. Liu, M. Deng, B. Du, Analysis of a stochastic logistic model with diffusion, Appl. Math. Comput., 266 (2015), 169–182. https://doi.org/10.1016/j.amc.2015.05.050 doi: 10.1016/j.amc.2015.05.050
    [47] X. Zhang, W. Li, M. Liu, K. Wang, Dynamics of a stochastic Holling Ⅱ one-predator two-prey system with jumps, Phys. A, 421 (2015), 571–582. https://doi.org/10.1016/j.physa.2014.11.060 doi: 10.1016/j.physa.2014.11.060
    [48] Y. Sabbar, Asymptotic extinction and persistence of a perturbed epidemic model with different intervention measures and standard Lévy jumps, Bull. Biomath., 1 (2023), 58–77. https://doi.org/10.59292/bulletinbiomath.2023004 doi: 10.59292/bulletinbiomath.2023004
    [49] Y. Zhao, S. Yuan, Q. Zhang, The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment, Appl. Math. Model., 40 (2016), 7583–7600. https://doi.org/10.1016/j.apm.2016.01.056 doi: 10.1016/j.apm.2016.01.056
    [50] B. Liu, L. Chen, Y. Zhang, The effects of impulsive toxicant input on a population in a polluted environment, J. Biol. Syst., 11 (2003), 265–274. https://doi.org/10.1142/S0218339003000907 doi: 10.1142/S0218339003000907
    [51] X. Yang, Z. Jin, Y. Xue, Week average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input, Chaos Solitons Fractals, 31 (2007), 726–735. https://doi.org/10.1016/j.chaos.2005.10.042 doi: 10.1016/j.chaos.2005.10.042
    [52] S. Wang, L. Wang, T. Wei, Optimal harvesting for a stochastic predator-prey model with S-type distributed time delays, Methodol. Comput. Appl. Probab., 20 (2018), 37–68. https://doi.org/10.1007/s11009-016-9519-2 doi: 10.1007/s11009-016-9519-2
    [53] M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969–2012. https://doi.org/10.1007/s11538-010-9569-5 doi: 10.1007/s11538-010-9569-5
    [54] S. Wang, L. Wang, T. Wei, Optimal harvesting for a stochastic logistic model with S-type distributed time delay, J. Differ. Equations Appl., 23 (2017), 618–632. https://doi.org/10.1080/10236198.2016.1269761 doi: 10.1080/10236198.2016.1269761
    [55] Q. Liu, Q. Chen, Dynamics of stochastic delay Lotka-Volterra systems with impulsive toxicant input and Lévy noise in polluted environments, Appl. Math. Comput., 256 (2015), 52–67. https://doi.org/10.1016/j.amc.2015.01.009 doi: 10.1016/j.amc.2015.01.009
    [56] S. Wang, G. Hu, T. Wei, On a three-species stochastic hybrid Lotka-Volterra system with distributed delay and Lévy noise, Filomat, 36 (2022), 4737–4750. https://doi.org/10.2298/FIL2214737W doi: 10.2298/FIL2214737W
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1588) PDF downloads(67) Cited by(0)

Figures and Tables

Tables(8)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog