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Research article

Modelling and analysis of a stochastic nonautonomous predator-prey model with impulsive effects and nonlinear functional response

  • Received: 23 November 2020 Accepted: 18 January 2021 Published: 28 January 2021
  • In this paper, a new stochastic predator-prey model with impulsive perturbation and Crowley-Martin functional response is proposed. The dynamical properties of the model are systematically investigated. The existence and stochastically ultimate boundedness of a global positive solution are derived using the theory of impulsive stochastic differential equations. Some sufficient criteria are obtained to guarantee the extinction and a series of persistence in the mean of the system. Moreover, we provide conditions for the stochastic permanence and global attractivity of the model. Numerical simulations are performed to support our qualitative results.

    Citation: Yan Zhang, Shujing Gao, Shihua Chen. Modelling and analysis of a stochastic nonautonomous predator-prey model with impulsive effects and nonlinear functional response[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1485-1512. doi: 10.3934/mbe.2021077

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  • In this paper, a new stochastic predator-prey model with impulsive perturbation and Crowley-Martin functional response is proposed. The dynamical properties of the model are systematically investigated. The existence and stochastically ultimate boundedness of a global positive solution are derived using the theory of impulsive stochastic differential equations. Some sufficient criteria are obtained to guarantee the extinction and a series of persistence in the mean of the system. Moreover, we provide conditions for the stochastic permanence and global attractivity of the model. Numerical simulations are performed to support our qualitative results.



    Predator-prey models are highly important in general and mathematical ecology [1]. In the past decades, many factors have been considered to describe the ecological predator-prey system more correctly and reasonably [2,3]. Notably, population models in the real world is inevitably influenced by numerous unpredictable environmental noise, and deterministic systems are fairly challenged in describing the fluctuation accurately [4,5]. Hence, an increasing number of researchers have paid attention to stochastic models and proposed various population models with stochastic perturbations, such as in [6,7,8,9,10]. Liu and Wang [10] introduced a stochastic non-autonomous predator-prey model for one species with white noise as follows:

    dx(t)=r(t)[x(t)a(t)x(t)]dt+σ(t)x(t)dB(t). (1.1)

    The group analyzed the conditions for extinction and species persistence in Eq (1.1). On the basis of the theoretical and practical significance of this stochastic model, many results have been presented, particularly, in [11,12,13]. However, the influence of the functional response to systems has been rarely considered in previous stochastic population models.

    Generally, two types of functional response exist, that is, prey- and predator-dependent responses. The first functional response considers only the prey density, whereas the other accounts for both prey and predator densities [2]. When investigating biological phenomena, one must not ignore the predator's functional response to prey because of such response's effect on dynamical system properties [14,15,16,17,18]. Among many different forms of predator-dependent functional responses, the three classical ones include the Beddington-DeAngelis, Hassell-Varley and Crowley-Martin types. We let x1(t) and x2(t) denote the prey and predator population densities, respectively, at time t. Then, ω(t)x1(t)1+a(t)x1(t)+b(t)x2(t)+a(t)b(t)x1(t)x2(t) becomes the Crowley-Martin functional response, where a(t), b(t) and ω(t) represent the effects of handling time, the magnitude of interference among predators, and capture rate, respectively. Interestingly, if a(t)=0 and b(t)=0, then the Crowley-Martin functional response becomes a linear mass-action functional response. If a(t)=0 and b(t)>0, the response represents a saturation response; if a(t)>0 and b(t)=0, then the response becomes a Michaelis-Menten functional response (or Holling type-II functional response) [1,19,20]. Given its importance and appeal, some scholars have studied stochastic predator-prey models incorporating Crowley-Martin functional response [21,22,23] and in this paper, we consider the Crowley-Martin functional response to embody interference among predators and provide insight into the dynamics of the predator-prey population model.

    Meanwhile, the theory of impulsive differential equation was well developed recently, and impulsive differential equations were found as a more effective method for describing species and the ecological systems more realistically. Many important and peculiar results have been obtained regarding the dynamical behavior of these systems, including the permanence, extinction of positive solution and dynamical complexity. However, few studies have addressed the population dynamics of two species both with stochastic and impulsive perturbations, except in [24,25,26]. In [25], Zhang and Tan considered a stochastic autonomous predator-prey model in a polluted environment with impulsive perturbations and analyzed the extinction and persistence of the system. By contrast, the model proposed is autonomous, that is, the parameters are assumed as constants and independent of time. In [26], Wu, Zou, and Wang proposed a stochastic Lotka-Volterra model with impulsive perturbations. The asymptotic properties of the model were examined. However, their model was based on the prey-dependent functional response and did not consider the predator's functional response to prey. In addition, there are four approaches to introduce stochastic perturbations to the model as usual, through time Markov chain model, parameter perturbation, being proportional to the variables, and robusting the positive equilibria of deterministic models [27]. In this paper, we adopt the third approach to include stochastic effects to Eq (1.2).

    Inspired by the above discussion and [11,25,26], we consider the possible effects of impulsive and stochastic perturbations on the system and propose the following non-autonomous stochastic differential equation:

    {dx1(t)=x1(r(t)k(t)x1ω(t)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2)dt+x1σ1(t)dB1(t),dx2(t)=x2(g(t)h(t)x2+f(t)x11+a(t)x1+b(t)x2+a(t)b(t)x1x2)dt+x2σ2(t)dB2(t),tτk,x1(τ+k)=(1+ρ1k)x1(τk),x2(τ+k)=(1+ρ2k)x2(τk),t=τk,k=1,2,3, (1.2)

    The parameters are defined as follows: r(t) and g(t) denote the intrinsic growth rate of the prey and predator population at time t; k(t) and h(t) are the density-dependent coefficients of prey and predator populations, respectively; f(t) represents the conversion rate of nutrients into the reproductive predator population; σ2i(t) (i=1,2) refers to the intensities of the white noises at time t; ˙B1(t) and ˙B2(t) are standard white noises, in particular, B1(t),B2(t) are Brownian motions defined on a complete probability space (Ω,F,P) [6].

    Throughout this paper, all the coefficients are assumed to be positive and continuously bounded on R+=[0,+). The impulsive points satisfy 0<τ1<τ2<<τk< and limk+τk=+. According to biological meanings, ρ1k>1,ρ2k>1. Moreover, we assume that there are some positive constants m, M, ˜m and ˜M that satisfy 0<m0<τk<t(1+ρ1k)M and 0<˜m0<τk<t(1+ρ2k)˜M, for all t>0.

    The remaining portion of this paper is arranged as follows. In the next section, some preliminaries are introduced. We analyze the impulsive stochastic differential model and obtain the existence, uniqueness and stochastically ultimate boundness of the positive solution in Section 3. In Section 4, sufficient conditions for extinction and a set of persistence in the mean, including non-persistence, weak persistence, and strong persistence in the mean, are presented. Additionally, we provide conditions to guarantee the stochastic permanence of the system. In Section 5, the global attractiveness of Eq (1.2) is studied. Finally, some numerical simulations, which verify our theoretical results, are given in Section 6. We compare the results of stochastic models under positive or negative impulsive perturbations with those without such disturbances, as well as, the figures with different stochastic perturbations and same impulse. By doing so, we clearly show that the impulsive and stochastic perturbations are of great importance to species permanence and extinction.

    To proceed, we list some appropriate definitions, notations, and lemmas as follows. For convenience, we denote

    fu=supt0f(t), fl=inft0f(t),f(t)=1tt0f(s)ds,f=lim supt+f(t),f=lim inft+f(t),

    where f(t) is a continuous and bound function defined on [0,+). X(t) represents (x1(t),x2(t)) and |X(t)|=(x21(t)+x22(t))12. N is the set of positive integers and Rn+={xRn:xi>0,i=1,2,3...n}.

    Definition 2.1.

    (a) If limt+x(t)=0 a.s., then the species x(t) is said to go to extinction.

    (b) If x=0 a.s., then the population x(t) is said to be non-persistent in the mean.

    (c) If x>0 a.s., then the population x(t) is said to be weakly persistent in the mean.

    (d) If x>0 a.s., then the population x(t) is said to be strongly persistent in the mean.

    (e) If x>0 a.s., then the population x(t) is said to be weakly persistent.

    Definition 2.2 ([28]) Solution X(t)=(x1(t),x2(t)) of Eq (1.2) is said to be stochastically ultimately bound, if for arbitrary ε(0,1), a positive constant δ=δ(ε) exists, such that for any given initial value X0=(x1(0),x2(0))R2+, the solution X(t) to Eq (1.2) satisfies lim suptP{|X(t)|>δ}<ε.

    Definition 2.3 ([28]) Solution X(t)=(x1(t),x2(t)) of Eq (1.2) is said to be stochastically permanent, if for any ε(0,1), there is a pair of positive constants δ=δ(ε) and χ=χ(ε) such that for any initial value X0=(x1(0),x2(0))R2+, the solution X(t) to Eq (1.2) satisfies lim inftP{|X(t)|δ}1ε, lim inftP{|X(t)|χ}1ε.

    Definition 2.4. Eq (1.2) is said to be globally attractive if

    limt+|x1(t)¯x1(t)|=0,limt+|x2(t)¯x2(t)|=0,

    for any two solutions (x1(t),x2(t)), (¯x1(t),¯x2(t)) of Eq (1.2).

    Definition 2.5. ([29]) Consider the impulsive stochastic equation

    dx(t)=F(t,x(t))dt+G(t,x(t))dB(t),ttk,t>0,x(t+k)x(tk)=αkx(tk),t=tk,k=1,2, (2.1)

    with the initial value x(0)=x0Rn. A stochastic process x(t)=(x1(t),x2(t),,xn(t))T, t[0,+) is the solution of Eq (2.1) if

    (a) x(t) is Ft adapted and continuous on (0,t1) and each interval (tk,tk+1), kN and F(t,x(t))L1(R+,Rn), G(t,x(t))L2(R+,Rn).

    (b) For each tk, x(t+k)=limtt+kx(t) and x(tk)=limttkx(t) and x(tk)=x(tk) a.s..

    (c) x(t) obeys the equivalent integral Eq (2.1) for almost every tR+tk and satisfies the impulsive conditions at t=tk a.s..

    Lemma 2.1. ([5]) Suppose that x(t)C[Ω×R+,R0+], where R0+=(0,+) and Bi(t) (i=1,2,3,,n) are independent Brownian motions defined on a complete probability space (Ω,F,P), then

    (a) If there are positive constants λ0,T and λ0 satisfying

    lnx(t)λtλ0t0x(s)ds+ni=1βiBi(t),

    for all tT, where βi is a constant, 1in, then, xλ/λ0 a.s.

    (b) If there are positive constants λ0,T and λ0 satisfying

    lnx(t)λtλ0t0x(s)ds+ni=1βiBi(t),

    for all tT, where βi is a constant, 1in, then, xλ/λ0 a.s.

    Lemma 2.2. ([30]) Let f be a non-negative function defined on R+ such that f is integrate and is uniformly continuous. Then limt+f(t)=0.

    In this section, the existence, uniqueness, and stochastically ultimate boundedness of the global positive solution are obtained.

    Firstly, we denote

    x1(t)=0<τk<t(1+ρ1k)y1(t),x2(t)=0<τk<t(1+ρ2k)y2(t),
    λ(t)=1+a(t)0<τk<t(1+ρ1k)y1+b(t)0<τk<t(1+ρ2k)y2+a(t)b(t)0<τk<t(1+ρ1k)0<τk<t(1+ρ2k)y1y2,

    then by virtue of Lemma 2.1 in [30], the following lemma can be obtained.

    Lemma 3.1. For the stochastic equations without impulses

    dy1(t)=y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)dt+y1σ1(t)dB1(t),dy2(t)=y2(g(t)h(t)0<ρ2k<t(1+ρ1k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1)dt+y2σ2(t)dB2(t), (3.1)

    (y1(t),y2(t)) is a solution of Eq (3.1) if and only if (x1(t),x2(t)) is a solution of Eq (1.2) with initial value (x1(0),x2(0))=(y1(0),y2(0)).

    The proof can be given easily as in [31], but such approach is not applied herein.

    Theorem 3.1 For any given value (x1(0),x2(0))=X0R2+, a unique solution (x1(t),x2(t)) exists for Equation (1.2) on t0 and the solution will remain in R2+ with probability one.

    The proof of Theorem 3.1 is standard and is presented in Appendix A.

    Theorem 3.2 The solutions of Eq (1.2) are stochastically ultimately bounded for any initial value X0=(x1(0),x2(0))R2+.

    The proof of Theorem 3.2 is presented in Appendix A.

    In this section, sufficient conditions for extinction and a series of persistence in the mean, such as non-persistence, weak persistence and strong persistence in the mean, are established. Furthermore, we obtain conditions to guarantee the stochastic permanence of the system. Before giving the main theorems, we introduce a lemma essential to our proofs.

    Lemma 4.1. If lim supt0<τk<t(1+ρ1k)lnt<, lim supt0<τk<t(1+ρ2k)lnt< hold, then for any initial value (x1(0),x2(0))R2+, the solution X1(t)=(x1(t),x2(t)) of Eq (1.2) satisfies

    lim suptlnx1(t)t0,lim suptlnx2(t)t0,a.s.

    The proof of Lemma 4.1 is given in Appendix A.

    The results about persistence in the mean and extinction of the prey and predator populations are presented in Theorems 4.1.1 and 4.1.2.

    Theorem 4.1.1 For the prey population x1 of Eq (1.2), we have

    (a) If ˆr1<0, then the prey population x1 is extinct with probability 1, where r1(t)=r(t)0.5σ21(t), ˆr1=lim supt+1t[0<τk<tln(1+ρ1k)+t0r1(s)ds].

    (b) If ˆr1=0, then the prey population x1 is non-persistent in the mean with probability 1.

    (c) If ˆr1>0 and ˆr2<0, then the prey population x1 is weakly persistent in the mean with probability 1, where ˆr2=lim supt+1t[0<τk<tln(1+ρ2k)+t0r2(s)ds].

    (d) If ˇr1ωb>0, then the prey population x1 is strongly persistent in the mean with probability 1, where ˇr1=lim inft+1t[0<τk<tln(1+ρ1k)+t0r1(s)ds].

    (e) If ˆr1>0, the prey population x1(t) holds a superior bound in time average, that is, x1(t)^r1klMx.

    Proof. (a) According to Itˆo's formula and Eq (3.1), the function can be expressed as

    dlny1=[r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2]dt0.5σ21(t)dt+σ1(t)dB1(t)=[r1(t)k(t)x1ω(t)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+σ1(t)dB1(t),dlny2=[g(t)h(t)0<τk<t(1+ρ2k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1]dt0.5σ22(t)dt+σ2(t)dB2(t)=[r2(t)h(t)x2+f(t)x11+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+σ2(t)dB2(t). (4.1)

    Taking integral on both sides of Eq (4.1) results in

    lny1(t)lny1(0)=t0r1(s)dst0k(s)x1(s)dst0ω(s)x2(s)1+a(s)x1(s)+b(s)x2(s)+a(s)b(s)x1(s)x2(s)ds+N1(t),lny2(t)lny2(0)=t0r2(s)dst0h(s)x2(s)ds+t0f(s)x1(s)1+a(s)x1(s)+b(s)x2(s)+a(s)b(s)x1(s)x2(s)ds+N2(t).

    We let N1(t)=t0σ1(s)dB1(s), N2(t)=t0σ2(s)dB2(s), where Ni(t)(i=1,2) is a local martingale with a quadratic variation satisfying N1,N1t=t0σ21(s)ds(σu1)2t, N2,N2t=t0σ22(s)ds(σu2)2t. Using the strong law of large numbers for martingales, we show that

    lim suptNi(t)t=0,a.s.i=1,2. (4.2)

    Thus

    lnx1(t)lnx1(0)t=0<τk<tln(1+ρ1k)t+lny1(t)lny1(0)t=0<τk<tln(1+ρ1k)+t0r1(s)dst1tt0k(s)x1(s)ds+N1(t)t1tt0ω(s)x2(s)1+a(s)x1(s)+b(s)x2(s)+a(s)b(s)x1(s)x2(s)ds,
    lnx2(t)lnx2(0)t=0<τk<tln(1+ρ2k)t+lny2(t)lny2(0)t=0<τk<tln(1+ρ2k)+t0r2(s)dst1tt0h(s)x2(s)ds+N2(t)t+1tt0f(s)x1(s)1+a(s)x1(s)+b(s)x2(s)+a(s)b(s)x1(s)x2(s)ds. (4.3)

    Therefore,

    lnx1(t)lnx1(0)t0<τk<tln(1+ρ1k)+t0r1(s)dst+N1(t)t. (4.4)

    Making use of Eq (4.2) and the superior limit as t in Eq (4.4), we get [lnx1(t)lnx1(0)t]ˆr1<0 which results in limtx1(t)=0.

    (b). According to the definition of superior limit and Eq (4.2), for an arbitrary ε>0, there is a T>0 satisfying 1t[0<τk<tln(1+ρ1k)+t0r1(s)ds]ˆr1+ε2, N1(t)tε2 for all t>T. Substituting the above inequalities into the first equation of (4.4), we easily show that

    lnx1(t)lnx1(0)tˆr1kx1+εεkx1. (4.5)

    By virtue of Lemma 2.1, we obtain x1(t)εk. In accordance with the arbitrariness of ε, we achieve the result.

    (c). By virtue of Eq (4.2), superior limit and Lemma 4.1, we show that

    kux1+ωux2[lnx1(t)lnx1(0)t]+k(t)x1+ω(t)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2ˆr1>0. (4.6)

    Thus, x1(t)>0 a.s. By reduction to absurdity, we can assume that for any υ{x1(t,υ)=0}, by Equation (4.6), we obtain x2(t,υ)>0. Meanwhile, using the superior limit for the second equation of (4.3) and x1(t,v)=0 leads to

    [lnx2(t,υ)lnx2(0)t]ˆr2+fux1(t,υ)+hlx2(t,υ)<0.

    Therefore, limtx2(t,υ)=0. This expression is a contradiction. Then, x1(t)>0 a.s.

    (d). Under the condition ˇr1ωb>0, an arbitrary ε>0 satisfying ˇr1ωbε>0 exists. According to the definition of superior limit, interior limit and Eq (4.2), for the above-mentioned positive constant ε, there is a T>0 satisfying 1t(0<τk<tln(1+ρ1k)+t0r1(s)ds)>ˇr1ε3,ωb<ωb+ε3,N1(t)t>ε3, for all t>T. Then, from Eq (4.3),

    lnx1(t)lnx1(0)tˇr1ωbεkux1.

    Using Lemma 2.1 and the arbitrariness of ε, we have that

    x1(t)ˇr1ωbkumx>0. (4.7)

    (e). Passing to the first equation of (4.3), we yield

    lnx1(t)lnx1(0)t1t(0<τk<tln(1+ρ1k)+t0r1(s)ds)klx1(t)+N1(t)t. (4.8)

    Thus, x1(t)ˆr1klMx, which is obtained by a similar process in the proof of conclusion (2) and is omitted.

    Let (ˉx1(t),ˉx2(t)) be the solution of the following comparison equation

    dˉx1=ˉx1(r(t)k(t)ˉx1)dt+ˉx1σ1(t)dB1(t),dˉx2=ˉx2(h(t)ˉx2+f(t)a(t))dt+ˉx2σ2(t)dB2(t). (4.9)

    with initial value (x1(0),x2(0))R2+, then we hold the following theorem.

    Theorem 4.1.2. For the predator population x2 of Eq (1.2),

    (a) if kˆr2+fˆr1<0, then the predator population x2 is extinct with probability 1;

    (b) if kˆr2+fˆr1=0, then the predator population x2 is non-persistent in the mean with probability 1;

    (c) if ˆr2+fˉx11+aˉx1+bˉx2+abˉx1ˉx2>0, then the predator population x2 is weakly persistent in the mean with probability 1;

    (d) if ˆr2+fa>0, then the predator population x2(t) has a superior bound in time average, that is, x2(t)ˆr2+fahlMy;

    (e) if ˆr2>0,ˆr10, then the predator population x2 is weakly persistent.

    Proof. (a). Case I. If ˆr10, then by virtue of Theorem 4.1.1, we obtain x1(t)=0. According to the definition of superior limit, for an arbitrary ε>0, there is a T>0 satisfying 1t[0<τk<tln(1+ρ2k)+t0r2(s)ds]<ˆr2+ε2, N2(t)t<ε2 for all t>T. By the second equation of (4.3), we noted

    [lnx2(t)lnx2(0)t]ˆr2+fx1(t)+ε=ˆr2+ε<0,

    then limtx2(t)=0.

    Case II. If ˆr1>0, by Eq (4.3), for the above constant ε>0, there is a T1>0 such that lnx1(t)lnx1(0)tˆr1kx1(t)+ε, for all t>T1. Applying Lemma 2.1, we show that

    x1(t)ˆr1+εk. (4.10)

    Substituting the above inequality into the second equation of (4.3) and using the arbitrariness of ε, we yield

    [lnx2(t)lnx2(0)t]ˆr2+fx1(t)kˆr2+f(ˆr1+ε)k<0. (4.11)

    Thus, limtx2(t)=0.

    (b). In (1), we prove that if ˆr10, then limtx2(t)=0, consequently, x2(t)=0. At this point, we only need to show that if ˆr1>0, then x2(t)=0 is also valid. Otherwise, x2(t)>0, and by Lemma 4.1, we obtain that [lnx2(t)t]=0. From Eq (4.11), we note that

    0=[lnx2(t)lnx2(0)t]ˆr2+fx1(t).

    Meanwhile, for any constant ε>0, there is a T>0 satisfying 1t[0<τk<tln(1+ρ2k)+t0r2(s)ds]<ˆr2+ε3, f(t)x1(t)fx1(t)+ε3, N2(t)tε3 for all t>T. By the second equation of (4.3),

    lnx2(t)lnx2(0)t1t[0<τk<tln(1+ρ2k)+t0r2(s)ds]+f(t)x1(t)h(t)x2(t)+N2(t)tˆr2+fx1(t)+εhx2(t).

    Then, making use of Lemma 2.1, we achieve x2(t)ˆr2+fx1(t)+εh, which indicates that x2(t)ˆr2+fx1(t)h. By virtue of Eq (4.10) and the arbitrariness of ε, we obtain x2(t)kˆr2+fˆr1hk=0. This is a contradiction. Therefore, x2(t)=0a.s.

    (c). In the following, we show that x2(t)>0 a.s.. By reduction to absurdity, for arbitrary ε1>0 and initial value (x1(0),x2(0))R2+, there is a solution (ˇx1(t),ˇx2(t)) of Eq (1.2) satisfying P{ˇx2(t)<ε1}>0. Let ε1 be sufficiently small that

    ^r2+fˉx11+aˉx1+bˉx2+abˉx1ˉx2>2(fuωukl+hu+1)ε1.

    From the second equation of (4.3), it can be shown that

    lnˇx2(t)lnx2(0)t=1t[0<τk<tln(1+ρ2k)+t0r2(s)ds]+f(t)ˉx11+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx2+N2(t)t+f(t)ˇx11+a(t)ˇx1+b(t)ˇx2+a(t)b(t)ˇx1ˇx2f(t)ˉx11+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx2h(t)ˇx2(t).

    Herein, ˇx1(t)ˉx1(t), ˇx2(t)ˉx2(t), a.s. for t[0,+). Note that

    f(t)ˇx11+a(t)ˇx1+b(t)ˇx2+a(t)b(t)ˇx1ˇx2f(t)ˉx11+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx2=f(t)(ˉx1ˇx1)+a(t)b(t)ˉx1ˇx1(ˉx2ˇx2)+b(t)ˇx1(ˉx2ˇx2)b(t)ˇx2(ˉx1ˇx1)(1+a(t)ˇx1+b(t)ˇx2+a(t)b(t)ˇx1ˇx2)(1+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx2)2f(t)(ˉx1ˇx1),

    then

    lnˇx2(t)lnx2(0)t1t[0<τk<tln(1+ρ2k)+t0r2(s)ds]h(t)ˇx2(t)+N2(t)t+f(t)ˉx11+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx22f(t)(ˉx1ˇx1). (4.12)

    Construct the Lyapunov function V3(t)=|lnˉx1(t)lnˇx1(t)|, where V3(t) is a positive function on R+. By virtue of Itˆo's formula and Eq (4.9), we achieve the following expression:

    D+V3(t)sgn(ˉx1ˇx1){k(ˉx1ˇx1)+ω(t)ˇx21+a(t)ˇx1+b(t)ˇx2+a(t)b(t)ˇx1ˇx2}. (4.13)

    Moreover, integrating the above inequality from 0 to t and dividing by t on both sides of the above inequality result in V3(t)V3(0)tωuˇx2(t)kl|ˉx1ˇx1|. Then we achieve

    ˉx1ˇx1ωuklˇx2(t).

    By substituting the above inequality into Eq (4.12) and taking the superior limit of the inequality, we obtain

    [lnˇx2(t)lnx2(0)t]ˆr2huˇx2(t)+N2(t)t+f(t)ˉx11+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx22fuωuklˇx2(t)ˆr2+f(t)ˉx11+a(t)ˉx1+b(t)ˉx2+a(t)b(t)ˉx1ˉx22(fuωukl+hu+1)ε1>0, (4.14)

    Equation (4.14) contradicts Lemma 4.1, therefore x2(t)>0 a.s. The proof is hence completed.

    (d). By the second equation of (4.3), we obtain the following equation

    lnx2(t)lnx2(0)tˆr2+fahlx2(t)+N2(t)t. (4.15)

    Moreover, from the definition of superior limit and Eq (4.2), for the given positive number ε, there is a T2>0 satisfying 1t[0<τk<tln(1+ρ2k)+t0r2(s)ds]<ˆr2+ε3,fa<fa+ε3,N2(t)t<ε3, for all t>T2. In accordance with Lemma 2.1 and the arbitrariness of ε, we easily achieve

    x2(t)ˆr2+fahlMy.

    The desired result is obtained.

    (e). If x2>0 a.s is false, let Ω={x2=0}, then P(Ω)>0. For an arbitrary νΩ, we have limtx2(t,ν)=0. From the second equation of (4.3) and by virtue of Eq (4.2), we show that [lnx2(t,ν)t]=^r2>0 a.s. Then we follow that P{[lnx2(t,ν)t]>0}>0, which contradicts with Lemma 4.1. The result is then concluded.

    Remark 1. By the proof of Theorem 4.1.2, we observe that if ˆr1<0, then kˆr2+fˆr1<0. Thus, if the prey species is extinct, then the predator species will also be extinct. This notion is consistent with the reality. Moreover, if ˆr1>0 and kˆr2+fˆr1<0, then even if the prey population is persistent, the predators end in extinction because of an excessively large diffusion coefficient σ22.

    Remark 2. According to conclusion (5) of Theorem 4.1.2, with the effect of impulsive perturbations despite the regression of the prey population to extinction, the predator may remain weakly persistent.

    Theorem 4.2.1 If (max{σu1,σu2})2+2(ωubl+gu)<2min{rl,flau} holds, then Eq (1.2) is stochastically permanent.

    Proof. The whole proof is divided into two parts. First, we must prove that for arbitrary ε>0, there is a constant δ>0 satisfying P{|X1(t)|δ}1ε, where X1(t)=(x1(t),x2(t)).

    At this point, we show that for any initial value ¯X(0) =(y1(0),y2(0))R2+, the solution ¯X(t)=(y1(t),y2(t)) holds the property that

    lim suptE(1|¯X(t)|θ)M0,

    where θ>0 is a sufficiently small constant, such that

    min{rl,flau}>(ωubl+gu)+(θ+1)2(max{σu1,σu2})2. (4.16)

    By virtue of Eq (4.16), there is an arbitrary constant p>0 such that

    min{rl,flau}(θ+1)2(max{σu1,σu2})2(ωubl+gu)p>0. (4.17)

    Define V(y1,y2)=y1+y2, then

    dV(y1,y2)=y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)dt+y1σ1(t)dB1(t)+y2(g(t)h(t)0<ρ2k<t(1+ρ1k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1)dt+y2σ2(t)dB2(t).

    Let U(y1,y2)=1V(y1,y2), according to the Itˆo's formula, we obtain

    dU(¯X)=U2(¯X)[y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)dt]U2(¯X)y2(g(t)h(t)0<ρ2k<t(1+ρ1k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1)dt+U3(¯X)[y21σ21(t)+y22σ22(t)]dtU2(¯X)σ1(t)y1dB1(t)+σ2(t)y2dB2(t)=LU(¯X)dtU2(¯X)[σ1(t)y1dB1(t)+σ2(t)y2dB2(t)].

    Under the condition of this theorem, a positive constant θ can be chosen to satisfy Eq (4.16). By the Itô formula,

    L(1+U(¯X))θ=θ(1+U(¯X))θ1LU(¯X)+12θ(θ1)(1+U(¯X))θ2U4(¯X)+(y21σ21(t)+y22σ22(t)).

    Then we choose p>0 to be sufficiently small such that the term satisfies Eq (4.17). We define W(¯X)=ept(1+U(¯X))θ and consequently achieve

    LW(¯X)=pept(1+U(¯X))θ+eptL(1+U(¯X))θ=ept(1+U(¯X))θ2{p(1+U(¯X))2θU2(¯X)y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)θU2(¯X)y2(g(t)h(t)0<τk<t(1+ρ2k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1)θU3(¯X)y2(g(t)h(t)0<τk<t(1+ρ2k)y2)θU3(¯X)[y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)+f(t)λ(t)0<τk<t(1+ρ1k)y1]+θU3(¯X)[y21σ21(t)+y22σ22(t)]+θ(θ+1)2U4(¯X)[y21σ21(t)+y22σ22(t)]}ept(1+U(¯X))θ2{(p+θmax{kuM,hu˜M})+[2pθmin{rl,flau}+θmax{kuM,hu˜M}]U(¯X)+[p+θ(ωubl+gu)θmin{rl,flau}+θ(θ+1)2(max{σu1,δu1})2]U2(¯X)}.

    By Eq (4.17), a positive constant S satisfying LW(¯X)Sept is easily noted. Consequently, E[ept(1+U(¯X))θ](1+U(0))θ+S(ept1)p and then

    lim suptE[Uθ(¯X(t))]lim suptE[(1+U(¯X(t))θ]Sp.

    Notably, (y1+y2)θ2θ(y21+y22)θ2=2θ|¯X|θ, where ¯X=(y1,y2)R2+. Then, we obtain

    lim suptE[1|¯X(t)|θ]2θlim suptEUθ(¯X)2θSp:=M0,

    and

    lim suptE[1|X1(t)|θ]¯mθlim suptEUθ(¯X)¯mθM0:=¯M0,

    where ¯m=min{m,˜m}. Therefore, for arbitrary ε>0, we let δ=(ε¯M0)1θ in accordance with Chebyshev's inequality, thereby yielding

    P{|X1(t)|<δ}=P{|X1(t)|θ>δθ}E[|X1(t)|θ]/δθ=δθE[|X1(t)|θ],

    thus, P{|X1(t)|δ}1ε.

    In the following relations, we prove that for any ε>0, there exists χ>0 satisfying P{|X1(t)|χ}1ε. Define V4(¯X)=yq1+yq2, herein 0<q<1 and ¯X=(y1,y2)R2+, then by virtue of Itˆo's formula, we obtain the expression

    dV4(¯X(t))=qyq1[r(t)k(t)0<τk<t(1+ρ1k)y1+q12σ21(t)ω(t)λ(t)0<τk<t(1+ρ2k)y2]dt+qyq2[g(t)h(t)0<τk<t(1+ρ2k)+q12σ22(t)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1]dt+qyq1σ1(t)dB1(t)+qyq2σ2(t)dB2(t).

    Let n0 be a sufficiently large constant, such that y1(0), y2(0) remain within the internal [1n0,n0]. For each integer nn0, we define the stopping time tn=inf{t0:y1(t)(1/n,n)ory2(t)(1/n,n)}. Obviously, tn increases as n+. Using Itˆo's formula again for exp{t}V4(¯X) and accounting for the expectations on both sides, we show that

    E[exp{ttn}¯Xq(ttn)]¯Xq(0)qEttn0exp{s}yq1(s)[1+q(r(s)k(s)0<τk<s(1+ρ1k)y1(s)1q2σ21(s))]ds+qEttn0exp{s}φq(s)[1+q(g(s)h(s)0<τk<s(1+ρ2k)y2(s)+f(s)a(s)1q2σ22(s))]dsEttn0(K1+K2)exp{s}ds(K1+K2)(exp{t}1),

    where K1,K2 are positive constants. Letting n+ yields

    exp{t}E[¯Xq(t)]¯Xq(0)+(K1+K2)(exp{t}1).

    Then, we achieve lim supt+E[¯Xq(t)]K1+K2 and lim supt+E[Xq1(t)]¯Mq(K1+K2), where ¯M=max{M,˜M}. Therefore, at any given ε>0, we let χ=¯M(K1+K2)1/qε1/q, by virtue of the Chebyshev inequality, we easily show that

    P{|X1(t)|>χ}=P{|X1(t)|q>χq}E[|X1(t)|q]/χq.

    Consequently, P{|X1(t)|χ}1ε.

    Theorem 4.2.1 is proven.

    Remark 3. From the conditions of Theorem 4.2.1, we find that although the stochastic disturbance greatly influence the dynamical property of the system, the bounded impulsive perturbations do not affect the stochastic permanence of the model.

    Remark 4. We should point out that the definition of stochastically permanent which requires that all species have positive upper bounds and at least one species has a positive lower bound, cannot demonstrate the permanence of all species. It has some limitations and deficiency. If there is only one species having a positive lower bound and all the other species go extinction, the system is still permanent. A new definition of stochastic permanence [31] may be more appropriate.

    In this section, we provide some sufficient criteria to ensure the global attractiveness of the Equation (1.2).

    Theorem 5.1. For any initial value (y1(0),y2(0))R+2, (y1(t),y2(t)) is a solution of Eq (3.1) on [0,+). Then almost every sample path of (y1(t),y2(t)) is uniformly continuous.

    The proof of Theorem 5.1 is given in Appendix A.

    Theorem 5.2. Suppose that constants μi>0(i=1,2) satisfying lim inftAi(t)>0 exist, where

    A1(t)=μ1[mk(t)2ω(t)a(t)b(t)]2μ2f(t),A2(t)=μ2[˜mh(t)2f(t)b(t)a(t)]2μ1ω(t), (5.1)

    then Eq (1.2) is globally attractive.

    Proof. Let (y1(t),y2(t)), (˜y1(t),˜y2(t)) be two arbitrary solutions of Eq (3.1) with initial value (y1(0),y2(0)), (˜y1(0),˜y2(0))R2+. Denote ˜λ(t)=1+a(t)0<τk<t(1+ρ1k)˜y1+b(t)0<τk<t(1+ρ2k)˜y2+a(t)b(t)0<τk<t(1+ρ1k)0<τk<t(1+ρ2k)˜y1˜y2 and define a Lyapunov function as follows:

    V(t)=μ1|lny1(t)ln˜y1(t)|+μ2|lny2(t)ln˜y2(t)|.

    Then,

    D+(V(t))=μ1sgn(y1˜y1)([r(t)k(t)0<τk<t(1+ρ1k)y10.5σ21(t)ω(t)λ(t)0<τk<t(1+ρ2k)y2]dt[r(t)k(t)0<τk<t(1+ρ1k)˜y10.5σ21(t)ω(t)˜λ(t)0<τk<t(1+ρ2k)˜y2]dt+μ2sgn(y2˜y2)([g(t)h(t)0<τk<t(1+ρ2k)y20.5σ22(t)+f(t)λ(t)0<τk<t(1+ρ1k)y1]dt[g(t)h(t)0<τk<t(1+ρ2k)˜y20.5σ22(t)+f(t)˜λ(t)0<τk<t(1+ρ1k)˜y1]dt)μ1sgn(y1˜y1)(k(t)0<τk<t(1+ρ1k)(y1˜y1)+ω(t)[0<τk<t(1+ρ2k)˜y2˜λ(t)0<τk<t(1+ρ2k)y2λ(t)])dt+μ2sgn(y2˜y2)(h(t)0<τk<t(1+ρ2k)(y2˜y2)+f(t)[0<τk<t(1+ρ1k)y1λ(t)0<τk<t(1+ρ1k)˜y1˜λ(t)])[μ1(mk(t)2ω(t)a(t)b(t))2μ2f(t)]|y1˜y1|[μ2(˜mh(t)2f(t)b(t)a(t))2μ1ω(t)]|y2˜y2|.

    By the condition lim inftAi(t)>0(i=1,2), constants α>0 and T0>0 satisfying Ai(t)α(i=1,2) exist for all tT0. Moreover, we obtain the following relation

    D+(V(t))α(|y1˜y1|+|y2˜y2|), (5.2)

    for all tT0. By integrating Eq (5.2) from T0 to t, we achieve

    V(t)V(T0)αtT0(|y1(s)˜y1(s)|+|y2(s)˜y2(s)|)ds.

    Consequently,

    V(t)+αtT0(|y1(s)˜y1(s)|+|y2(s)˜y2(s)|)dsV(T0)<+. (5.3)

    Then by V(t)0, we note that |y1(t)˜y1(t)|L1[0,+),|y2(t)˜y2(t)|L1[0,+). Thus,

    limt+|y1(t)˜y1(t)|=0,limt+|y2(t)˜y2(t)|=0.

    Next,

    limt+|x1(t)˜x1(t)|=limt+0<τk<t(1+ρ1k)|y1(t)˜y1(t)|Mlimt+|y1(t)˜y1(t)|=0,a.s.limt+|x2(t)˜x2(t)|=limt+0<τk<t(1+ρ2k)|y2(t)˜y2(t)|˜Mlimt+|y2(t)˜y2(t)|=0.a.s.

    Therefore, the desired assertion is obtained by Theorem 5.1 and Lemma 2.2.

    In this section, some numerical simulations and examples are given to illustrate and augment our theoretical findings of Eq (1.2) by means of the Milstein method mentioned in Higham [35]. Moreover, the effects of impulsive and stochastic perturbations on population dynamics are discussed.

    Example 1.

    {dx1(t)=x1[r(t)(0.7+0.01sint)x1(0.1+0.02sint)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x1σ1(t)dB1(t),dx2(t)=x2[(0.2+0.05sint)(0.2+0.01sint)x2+(0.2+0.02sint)x11+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x2σ2(t)dB2(t),tτk,x1(τ+k)=(1+ρ1k)x1(τk),x2(τ+k)=(1+ρ2k)x2(τk),t=τk,k=1,2,3, (6.1)

    We set r(t)=0.2+0.01sint, a(t)=0.1+0.04sint, b(t)=0.5+0.05sint, σ21(t)2=σ22(t)2=0.3+0.02sint, ρ1k=ρ2k=e(1)k+11k1 and τk=k, then it can be obtained that ˆr1=0.1<0. By Theorems 4.1.1 and 4.1.2, both prey and predator populations (x1 and x2, respectively) regress to extinction, which is also further confirmed by Figure 1.

    Figure 1.  Solutions of Eq (1.2) with σ21(t)2=σ22(t)2=0.3+0.02sint, ρ1k=ρ2k=e(1)k+11k1, τk=k. Both prey population x1 and predator population x2 go to extinction.

    We then choose σ21(t)2=0.3+0.02sint, σ22(t)2=0.4+0.02sint, ρ1k=ρ2k=e(1)k+11k1, and r(t)=0.4+0.01sint. The other parameters are the same as that in example 1, then ˆr1=0.1>0, ˆr2=0.6<0, and kˆr2+fˆr1=0.116<0. In Figure 2, although the prey population x1 is weakly persistent in the mean, the predator population x2 end in extinction because of the effects of the white noises, which are of great importance in maintaining the coexistence of populations.

    Figure 2.  σ21(t)2=0.3+0.02sint, σ22(t)2=0.4+0.02sint, ρ1k=ρ2k=e(1)k+11k1, τk=k. The prey population is weakly persistent in the mean and predator population is extinct.

    Example 2.

    {dx1(t)=x1[(0.6+0.1sint))(0.7+0.01sint)x1(0.1+0.02sint)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x1σ1(t)dB1(t),dx2(t)=x2[(0.2+0.05sint)(0.2+0.01sint)x2+(0.62+0.02sint)x11+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x2σ2(t)dB2(t),tτk,x1(τ+k)=(1+ρ1k)x1(τk),x2(τ+k)=(1+ρ2k)x2(τk),t=τk,k=1,2,3, (6.2)

    We let σ21(t)2=0.1+0.05sint, σ22(t)2=0.1+0.02sint, a(t)=0.1+0.04sint, b(t)=0.5+0.05sint ρ1k=ρ2k=e1k21, and τk=k, then ˆr1=0.5>0, ˆr2=0.3<0. Both the prey and predator populations (x1 and x2, respectively) are weakly persistent in the mean of Figure 3.

    Figure 3.  Both populations are weakly persistent in the mean for Eq (1.2).

    Example 3.

    {dx1(t)=x1[(1.21+0.01sint))(0.7+0.01sint)x1(0.1+0.02sint)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x1σ1(t)dB1(t),dx2(t)=x2[(0.2+0.05sint)(0.2+0.01sint)x2+(0.62+0.02sint)x11+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x2σ2(t)dB2(t),tτk,x1(τ+k)=(1+ρ1k)x1(τk),x2(τ+k)=(1+ρ2k)x2(τk),t=τk,k=1,2,3, (6.3)

    In Figure 4, we choose σ1(t)=0.2+0.02sint, σ2(t)=0.3+0.02sint, a(t)=0.1+0.04sint, b(t)=0.5+0.05sint, and the impulsive perturbations are (a) ρ1k=ρ2k=e(1)k+1k1, (b)ρ1k=ρ2k=e1k21, (c)ρ1k=ρ2k=0. The conditions of Theorem 4.2.1 are satisfied in all of those cases and the Eq (1.2) displays stochastic permanence in Figure 4. Moreover, in Figure 4(a)-4(c), the bounded impulsive perturbations do not affect the stochastic permanence of the model.

    Figure 4.  σ1(t)=0.2+0.02sint, σ2(t)=0.3+0.02sint, the only difference between these graphs is the impulses: (a) ρ1k=ρ2k=e(1)k+1k1, (b)ρ1k=ρ2k=e1k21, (c)ρ1k=ρ2k=0. The system is stochastic permanence.
    Figure 5.  The figures show the attractivity of Eq (1.2).
    Figure 6.  σ1(t)=0.2+0.02sint, σ2(t)=0.3+0.02sint, and the different impulses chosen for the graphs are : (a) ρ1k=ρ2k=0, (b)ρ1k=e1.921, ρ2k=e0.021. The negative impulses do not benefit species coexistence.
    Figure 7.  σ21(t)2=σ22(t)2=0.3+0.02sint, and the impulsive perturbations are: (a) ρ1k=ρ2k=0, (b)ρ1k=e1k21, ρ2k=e0.81, (c)ρ1k=ρ2k=e0.81. Positive impulses are advantageous for the coexistence of ecosystems.

    Example 4.

    {dx1(t)=x1[(1.8+0.01sint))(0.7+0.01sint)x1(0.1+0.02sint)x21+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x1σ1(t)dB1(t),dx2(t)=x2[(0.2+0.05sint)(0.8+0.01sint)x2+(1.12+0.02sint)x11+a(t)x1+b(t)x2+a(t)b(t)x1x2]dt+x2σ2(t)dB2(t),tτk,x1(τ+k)=(1+ρ1k)x1(τk),x2(τ+k)=(1+ρ2k)x2(τk),t=τk,k=1,2,3, (6.4)

    Set σ21(t)2=0.1+0.05sint, σ22(t)2=0.6+0.02sint, x1(0)=2, x2(0)=3, ˜x1(0)=0.5, and ˜x2(0)=0.3. We then observe from Figure 5 that Eq (1.2) is globally attractive.

    In the following instance, the effects of negative and positive impulses on the species are investigated. We let ρ1k=e1.921, ρ2k=e0.021, and τk=k. The other parameters are the same as those in Example 3, then we obtain that ˆr1=0.03<0 and ˆr2=0.36<0. On the basis of Theorems 4.1.1 and 4.1.2, both the prey and predator populations end in extinction, which is further confirmed by Figure 6(b). By comparing Figure 6(a) with Figure 6(b), we observe that the negative impulses do not benefit species coexistence. Moreover, considering the impulsive perturbations are (a) ρ1k=ρ2k=0, (b) ρ1k=e1k21, ρ2k=e0.81, (c) ρ1k=ρ2k=e0.81 and the other parameters are the same as those in Example 1. Hence, the system can be altered from extinction to persistence with the effects of positive impulsive perturbations (Figure 7(a)-7(c)). Herein, persistence can be divided into two cases as follows: first, the predator population x2(t) is weakly persistent and the prey population x1(t) proceeds to extinction and second, both of the populations are persistent. Therefore, positive impulses are advantageous for the coexistence of ecosystems. Moreover, comparing figures 3 and 6, figures 1 and 7, we can derive that if the impulsive perturbations are unbounded, some properties may be changed significantly.

    In this paper, we propose a stochastic non-autonomous predator-prey system with impulsive perturbations and investigate the qualitative dynamic properties of the model. Under some sufficient conditions, we present the extinction and a series of persistence in the mean of the system, including non-persistence, weak persistence and strong persistence in the mean. Furthermore, we obtain the global attractivity of the model. From the assumptions of Theorems 4.1.1, 4.1.2 and 4.2.1, we demonstrate that the stochastic and impulsive disturbances greatly influence the extinction and persistence of the system. Positive impulses are advantageous for the coexistence of ecosystem, whereas negative impulses are not beneficial for species coexistence. Moreover, the results show that the bounded impulsive perturbations do not affect all the properties, such as the stochastic permanence of the model. However, if the impulsive perturbations are unbounded, some properties may be changed significantly.

    Some interesting topics require further investigations. If we also consider the effects of time delays and telephone noise [36,37,38] on Eq (1.2) to propose a more realistic model, then how will the properties change? We leave it for future investigation.

    This work is supported by The National Natural Science Foundation of China (11901110, 11961003), and the National Natural Science Foundation of Jiangxi (20192BAB211003, 20192ACBL20004).

    The authors declare that they have no competing of interests regarding the publication of this paper.

    Proof of Theorem 3.1. From Lemma 3.1, it suffices to show that Eq (3.1) has a unique solution, (y1(t),y2(t)), for all t0 and will remain in R2+ with probability one. The proof of this theorem is standard.

    Let n0>0 be sufficiently large such that y1(0), y2(0) lie within the interval [1/n0,n0]. For each integer n>n0, define the stopping times tn=inf{t[0,te]:y1(t)(1/n,n)ory2(t)(1/n,n)}. Obviously, tn is increasing as n+. Denote t+=limn+tn, thus t+te. To complete the proof, it only needs to show t+=+ a.s. If the statement is not true, there exist two constants T>0 and ε(0,1), such that P{t+<+}>ε. Therefore, there is an integer n1n0 satisfying P{tnT}ε, for all n>n1.

    Define a C2-function V:R2+R+ by V(y1,y2)=(y11lny1)+(y21lny2), then we obtain that V(y1,y2) is nonnegative. By virtue of Itˆo's formula, we achieve that

    dV(y1,y2)=Vy1dy1+0.5Vy1y1(dy1)2+Vy2dy2+0.5Vy2y2(dy2)2=[(11/y1)y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)+(11/y2)y2(g(t)h(t)0<τk<t(1+ρ2k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1)]dt+0.5(σ21(t)+σ22(t))dt+(11/y1)y1σ1(t)dB1(t)+(11/y2)y2σ2(t)dB2(t)[(ru+kuM)y1klmy21rl+ωubl+gu+(˜Mhu+fualgl)y2˜mhly22]dt+0.5[(σu1)2+(σu2)2]dt+(y11)σ1(t)dB1(t)+(y21)σ2(t)dB2(t).

    According to the negative coefficients of the quadratic terms, there is a positive number G satisfying

    dV(y1,y2)Gdt+(y11)σ1(t)dB1(t)+(y21)σ2(t)dB2(t).

    Thus

    tnT0dV(y1,y2)tnT0Gdt+tnT0[(y1(t)1)σ1(t)dB1(t)+(y2(t)1)σ2(t)dB2(t)],

    where tnT=min{tn,T}. Taking expectation yields that

    EV(y1(tnT),y2(tnT))V(y1(0),y2(0))+GE(tnT)V(y1(0),y2(0))+GT.

    Let Ωn={tnT}, then P(Ωn)ε. For any wΩn, y1(tn,w) or y2(tn,w) equals either n or 1/n, thus

    V(y1(tn,w),y2(tn,w))min{n1lnn,1/n1+lnn}.

    Therefore it can be shown that

    V(y1(0),y2(0))+G1TE[1Ωn(w)V(y1(tn,w),y2(tn,w))]εmin{n1lnn,1/n1+lnn},

    where 1Ωn is the indicator function of Ωn. Letting n+ leads to the contradiction.

    So we obtain that t+=+ a.s. This completes the proof.

    Proof of Theorem 3.2. Define V1(y1)=yp1 and V2(y2)=yp2, respectively, for (y1,y2)R2+ and p>1. According to the Itˆo's formula, we have

    d(yp1)=pyp11dy1+0.5p(p1)yp21(dy1)2=pyp1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2+0.5(p1)σ21(t))dt+pyp1σ1(t)dB1(t)pyp1[rumkly1+0.5(p1)(σu1)2]dt+pyp1σ1(t)dB1(t).

    and

    d(yp2)=pyp2(g(t)h(t)0<τk<t(1+ρ2k)y2+f(t)λ(t)0<τk<t(1+ρ1k)y1+0.5(p1)σ22(t))dt+pyp2σ2(t)dB2(t)pyp2[fual+0.5p(σu2)2˜mhly2]dt+pyp2σ2(t)dB2(t).

    Taking expectation and then

    dE[yp1(t)]dtp{[ru+0.5(p1)(σu1)2]E[yp1(t)]mklE[yp+11(t)]}p{[ru+0.5(p1)(σu1)2]E[yp1(t)]mkl[E(yp1(t))]1+1p}pE[yp1(t)]{[ru+0.5p(σu1)2]mkl[E(yp1(t))]1p} (A.1)

    and

    dE[yp2(t)]dtpE[yp2(t)]{[fual+0.5p(σu2)2]˜mhl[E(yp2(t))]1p}. (A.2)

    For Eq (A.1), considering the following equation

    dv(t)dt=pv(t){[ru+0.5p(σu1)2]mklv1p(t)}

    with initial value v(0)=v0. Obviously, we can obtain that

    v(t)=(v1/p0(ru+0.5p(σu1)2)[ru+0.5p(σu1)2]e(ru+0.5p(σu1)2)t+mklv1/p0(1e(ru+0.5p(σu1)2)t))p.

    Let t and thus limt+v(t)=(ru+0.5p(σu1)2mkl)p. Using the comparison theorem yields that lim supt+E[yp1(t)]G1<+, where G1=(ru+0.5p(σu1)2mkl)p. In the same way, we can achieve lim supt+E[yp2(t)]G2<+ and G2=(fu/al+0.5p(σu2)2˜mhl)p.

    Consequently, for a given constant ε>0, there is a T>0 satisfying E[yp1(t)]G1+ε, E[yp2(t)]G2+ε, for all t>T. Considering the continuity of E[yp1(t)] and E[yp2(t)], there exist ¯G1(p),¯G2(p)>0 such that E[yp1(t)]¯G1(p) and E[yp2(t)]¯G2(p) for tT. Denote M1(p)=max{¯G1(p),G1+ε}, M2(p)=max{¯G2(p),G2+ε}, then for all tR+,

    E[yp1(t)]M1(p),E[yp2(t)]M2(p)

    and

    E[xp1(t)]=E[(0<τk<t(1+ρ1k)y1(t))p]MpM1(p),E[xp2(t)]=E[(0<τk<t(1+ρ2k)y2(t))p]˜MpM1(p).

    Thus, for X1(t)=(x1(t),x2(t))R2+, it is obvious that |X1(t)|p2p2[xp1(t)+xp2(t)]. Therefore,

    E|X1(t)|pMp<+,

    herein, Mp=2p2(MpM1(p)+˜MpM2(p)). By virtue of the Chebyshev inequality, the proof is completed.

    Proof of Theorem 4.2. From Eq (3.1), we have

    dy1 y1(r(t)k(t)0<τk<t(1+ρ1k)y1)dt+y1σ1(t)dB1(t),dy2y2(h(t)0<τk<t(1+ρ2k)y2+f(t)a(t))dt+y2σ2(t)dB2(t). (A.3)

    Construct the comparison equation

    dˉy1=ˉy1(r(t)k(t)0<τk<t(1+ρ1k)ˉy1)dt+ˉy1σ1(t)dB1(t),dˉy2=ˉy2(h(t)0<τk<t(1+ρ2k)ˉy2+f(t)a(t))dt+ˉy2σ2(t)dB2(t), (A.4)

    where (ˉy1(t),ˉy2(t)) is a solution of Eq (A.4) with initial value (y1(0),y2(0))R+2. According to the comparison theorem for stochastic differential equations ([33]) and Theorem 4.1 ([34]), we can obtain that lim suptlny1(t)lntlim suptlnˉy1(t)lnt1, lim suptlny2(t)lntlim suptlnˉy2(t)lnt1, a.s. Then,

    lim suptlny1(t)tlim suptlny1(t)lnt.lim suptlnttlim suptlntt=0.

    Similarly, we have that lim suptlny2(t)t0. Therefore,

    lim suptlnx1(t)t=lim supt0<τk<tln(1+ρ1k)+lny1(t)t=lim suptlny1(t)t+lim supt0<τk<tln(1+ρ1k)lnt.lntt0,
    lim suptlnx2(t)t=lim supt0<τk<tln(1+ρ2k)+lny2(t)t0. (A.5)

    The proof is completed.

    Proof of Theorem 5.1. The first equation of Eq (3.1) is equivalent to the following stochastic integral equation

    y1(t)=y1(0)+t0y1(s)(r(s)k(s)0<τk<s(1+ρ1k)y1(s)ω(s)λ(s)0<τk<s(1+ρ2k)y2(s))ds+t0y1(s)σ1(s)dB1(s),

    Denote f1(s)=y1(s)(r(s)k(s)0<τk<s(1+ρ1k)y1(s)ω(s)λ(s)0<τk<s(1+ρ2k)y2(s)), f2(s)=y1(s)σ1(s), then

    E|f1(t)|p=E|y1(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)|p=E[|y1|p|(r(t)k(t)0<τk<t(1+ρ1k)y1ω(t)λ(t)0<τk<t(1+ρ2k)y2)|p]12E|y1|2p+12E|ru+Mkuy1+˜Mωuy2|2p12E|y1|2p+1232p1[(ru)2p+(Mku)2pE|y1|2p+(˜Mωu)2pE|y2|2p]12M1(2p)+32p12[(ru)2p+(Mku)2pM1(2p)+(˜Mωu)2pM2(2p)]F1(p), (A.6)
    E|f2(t)|p=E|y1σ1(t)|pσu1E|y1|pσu1M1(p)F2(p). (A.7)

    By virtue of the moment inequality for stochastic integrals, we can show that for 0t1t2 and p>2,

    E|t2t1f2(s)dB1(s)|p(p(p1)2)p2(t2t1)p22t2t1E|f2(s)|pds(p(p1)2)p2(t2t1)p2F2(p).

    Thus, for 0<t1<t2<, t2t11, 1p+1q=1 and by Eqs (A.6) and (A.7), we achieve

    E|y1(t2)y1(t1)|p=E|t2t1f1(s)ds+t2t1f2(s)dB1(s)|p2p1E|t2t1f1(s)ds|p+2p1E|t2t1f2(s)dB1(s)|p
    2p1(t2t1)pqE[t2t1|f1(s)|pds]+2p1(p(p1)2)p2(t2t1)p2F2(p)2p1(t2t1)pqF1(p)(t2t1)+2p1(p(p1)2)p2(t2t1)p2F2(p)=2p1(t2t1)pF1(p)+2p1(p(p1)2)p2(t2t1)p2F2(p)=2p1(t2t1)p2{(t2t1)p2F1(p)+(p(p1)2)p2F2(p)}2p1(t2t1)p2{1+(p(p1)2)p2}F(p),

    where F(p)=max{F1(p),F2(p)}. By Lemma 5 in [11] and the definition in [30], we obtain that almost every sample path of y1(t) is locally but uniformly H¨older-continuous with exponent υ for every υ(0,p22p). Similarly, it can be proved that almost every sample path of y2(t) is also uniformly continuous on t0. The proof is completed.



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