VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA173 10.15559/21-VMSTA173 Research Article Long-time behavior of a nonautonomous stochastic predator–prey model with jumps BorysenkoOlgaolga_borisenko@ukr.neta https://orcid.org/0000-0002-6670-8605 BorysenkoOleksandrodb@univ.kiev.uab∗ Department of Mathematical Physics, National Technical University of Ukraine, 37, Prosp.Peremohy, Kyiv, 03056, Ukraine Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine, 64 Volodymyrska Str., Kyiv, 01601 Ukraine Corresponding author. 2021 832021811739 20112020 1222021 2122021 © 2021 The Author(s). Published by VTeX2021 Open access article under the CC BY license.

The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic predator–prey model with a modified version of the Leslie–Gower term and Holling-type II functional response disturbed by white noise, centered and noncentered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of a solution to the considered system.

 Stochastic predator–prey model Leslie–Gower and Holling-type II functional response global solution stochastic ultimate boundedness stochastic permanence extinction nonpersistence in the mean weak persistence in the mean 92D25 60H10 60H30 Introduction

The deterministic predator–prey model with modified version of Leslie–Gower term and Holling-type II functional response is studied in [1]. This model has the form dx1(t)=x1(t)a−bx1(t)−cx2(t)m1+x1(t)dt,dx2(t)=x2(t)r−fx2(t)m2+x1(t)dt, where x1(t) and x2(t) are the prey and predator population densities at time t, respectively. The positive constants a, b, c, r, f, m1 , m2 are defined as follows: a is the growth rate of prey x1 ; b measures the strength of competition among individuals of species x1 ; c is the maximum value of the per capita reduction rate of x1 due to x2 ; m1 and m2 measure the extent to which the environment provides protection to the prey x1 and to the predator x2 , respectively; r is the growth rate of the predator x2 , and f has a similar meaning to c. In [1] the authors study boundedness and global stability of the positive equilibrium of the model (1).

In the papers [6, 7, 9] the stochastic version of model (1) is considered in the form dx1(t)=x1(t)a−bx1(t)−cx2(t)m1+x1(t)dt+αx1(t)dw1(t),dx2(t)=x2(t)r−fx2(t)m2+x1(t)dt+βx2(t)dw2(t), where w1(t) and w2(t) are mutually independent Wiener processes in [6, 7], and processes w1(t) , w2(t) are correlated in [9]. In [6] the authors proved that there is a unique positive solution to the system (2), obtaining the sufficient conditions for extinction and persistence in the mean of predator and prey. In [7] it is shown that, under appropriate conditions, there is a stationary distribution of the solution to the system (2) which is ergodic. In [9] the authors prove that the densities of the distributions of the solution to the system (2) can converge in L1 to an invariant density or can converge weakly to a singular measure under appropriate conditions.

Population systems may suffer abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. So it is natural to introduce Poisson noises into the population model for describing such discontinuous systems.

In this paper, we consider the nonautonomous predator–prey model with modified version of the Leslie–Gower term and Holling-type II functional response, disturbed by white noise and jumps generated by centered and noncentered Poisson measures. So, we take into account not only “small” jumps, corresponding to the centered Poisson measure, but also the “large” jumps, corresponding to the noncentered Poisson measure. This model is driven by the system of stochastic differential equations dxi(t)=xi(t)ai(t)−bi(t)xi(t)−ci(t)x2(t)m(t)+x1(t)dt+σi(t)xi(t)dwi(t)+∫Rγi(t,z)xi(t)ν˜1(dt,dz)+∫Rδi(t,z)xi(t)ν2(dt,dz),xi(0)=xi0>0,i=1,2, 0,\hspace{2.5pt}i=1,2,\end{array}\]]]> where x1(t) and x2(t) are the prey and predator population densities at time t, respectively, b2(t)≡0 , wi(t),i=1,2 , are independent standard one-dimensional Wiener processes, νi(t,A),i=1,2 , are independent Poisson measures, which are independent on wi(t),i=1,2 , ν˜1(t,A)=ν1(t,A)−tΠ1(A) , E[νi(t,A)]=tΠi(A) , i=1,2 , Πi(A),i=1,2 , are finite measures on the Borel sets A in R .

To the best of our knowledge, there have been no papers devoted to the dynamical properties of the stochastic predator–prey model (3), even in the case of centered Poisson noise. It is worth noting that the impact of centered and noncentered Poisson noises to the stochastic nonautonomous logistic model and to the stochastic two-species mutualism model is studied in the papers [2–4].

In the following we will use the notations X(t)=(x1(t),x2(t)) , X0=(x10,x20) , |X(t)|=x12(t)+x22(t) , R+2={X∈R2:x1>0,x2>0} 0,{x_{2}}>0\}$]]>, αi(t)=ai(t)+∫Rδi(t,z)Π2(dz),βi(t)=σi2(t)2+∫R[γi(t,z)−ln(1+γi(t,z))]Π1(dz)−∫Rln(1+δi(t,z))Π2(dz), i=1,2 . For bounded, continuous functions fi(t),t∈[0,+∞) , i=1,2 , let us denote fisup=supt≥0fi(t),fiinf=inft≥0fi(t),i=1,2,fmax=max{f1sup,f2sup},fmin=min{f1inf,f2inf}.

We prove that the system (3) has a unique, positive, global (no explosion in a finite time) solution for any positive initial value, and that this solution is stochastically ultimately bounded. The sufficient conditions for stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of solution are derived.

The rest of this paper is organized as follows. In Section 2, we prove the existence of the unique global positive solution to the system (3) and derive some auxiliary results. In Section 3, we prove the stochastic ultimate boundedness of the solution to the system (3), obtainig conditions under which the solution is stochastically permanent. The sufficient conditions for nonpersistence in the mean, weak persistence in the mean and extinction of the solution are derived.

 Existence of global solution and some auxiliary lemmas

Let (Ω,F,P) be a probability space, wi(t),i=1 , 2,t≥0 , are independent standard one-dimensional Wiener processes on (Ω,F,P) , and νi(t,A),i=1,2 , are independent Poisson measures defined on (Ω,F,P) independent on wi(t),i=1,2 . Here E[νi(t,A)]=tΠi(A) , i=1,2 , ν˜i(t,A)=νi(t,A)−tΠi(A) , i=1,2 , Πi(·),i=1,2 , are finite measures on the Borel sets in R . On the probability space (Ω,F,P) we consider an increasing, right continuous family of complete sub-σ-algebras {Ft}t≥0 , where Ft=σ{wi(s),νi(s,A),s≤t,i=1,2} .

We need the following assumption.

It is assumed that ai(t) , b1(t) , ci(t),σi(t),γi(t,z),δi(t,z),i=1,2 , m(t) are bounded, continuous on t functions, ai(t)>0 0$]]>, i=1,2 , b1inf>0 0$]]>, ciinf>0 0$]]>, i=1,2 , minf=inft≥0m(t)>0 0$]]>, and ln(1+γi(t,z)),ln(1+δi(t,z)),i=1,2 , are bounded, Πi(R)<∞ , i=1,2 .

In what follows we will assume that Assumption 1 holds.

There exists a unique global solution X(t) to the system (3) for any initial value X(0)=X0∈R+2 , and P{X(t)∈R+2}=1 .

Let us consider the system of stochastic differential equations dξi(t)=ai(t)−bi(t)exp{ξi(t)}−ci(t)exp{ξ2(t)}m(t)+exp{ξ1(t)}−βi(t)dt+σi(t)dwi(t)+∫Rln(1+γi(t,z))ν˜1(dt,dz)+∫Rln(1+δi(t,z))ν˜2(dt,dz),vi(0)=lnxi0,i=1,2. The coefficients of the system (4) are locally Lipschitz continuous. So, for any initial value (ξ1(0),ξ2(0)) there exists a unique local solution Ξ(t)=(ξ1(t),ξ2(t)) on [0,τe) , where supt<τe|Ξ(t)|=+∞ (cf. Theorem 6, p. 246, [5]). Therefore, from the Itô formula we derive that the process X(t)=(exp{ξ1(t)},exp{ξ2(t)}) is a unique, positive local solution to the system (3). To show this solution is global, we need to show that τe=+∞ a.s. Let n0∈N be sufficiently large for xi0∈[1/n0,n0] , i=1,2 . For any n≥n0 we define the stopping time τn=inft∈[0,τe):X(t)∉1n,n×1n,n. It is easy to see that τn is increasing as n→+∞ . Denote τ∞=limn→∞τn , whence τ∞≤τe a.s. If we prove that τ∞=∞ a.s., then τe=∞ a.s. and X(t)∈R+2 a.s. for all t∈[0,+∞) . So we need to show that τ∞=∞ a.s. If it is not true, there are constants T>0 0$]]> and ε∈(0,1) , such that P{τ∞<T}>ε \varepsilon $]]>. Hence, there is n1≥n0 such that P{τn<T}>ε,∀n≥n1. \varepsilon ,\hspace{1em}\forall n\ge {n_{1}}.\]]]> For the nonnegative function V(X)= ∑i=12ki(xi−1−lnxi) , X=(x1,x2) , xi>0 0$]]>, ki>0 0$]]>, i=1,2 , by the Itô formula we obtain dV(X(t))= ∑i=12ki{(xi(t)−1)ai(t)−bi(t)xi(t)−ci(t)x2(t)m(t)+x1(t)+βi(t)+∫Rδi(t,z)xi(t)Π2(dz)}dt+ ∑i=12ki{(xi(t)−1)σi(t)dwi(t)+∫R[γi(t,z)xi(t)−ln(1+γi(t,z))]ν˜1(dt,dz)+∫R[δi(t,z)xi(t)−ln(1+δi(t,z))]ν˜2(dt,dz)}. Let us consider the function f(t,x1,x2)=ϕ(t,x1)+ψ(t,x1,x2) , x1>0 0$]]>, x2>0 0$]]>, where ϕ(t,x1)=−k1b1(t)x12+k1(α1(t)+b1(t))x1+k1β1(t)+k2β2(t)−k1a1(t)−k2a2(t),ψ(t,x1,x2)=(m(t)+x1)−1[−k2c2(t)x22+(k2α2(t)−k1c1(t))x1x2+(k2α2(t)m(t)+k1c1(t)+k2c2(t))x2]. Under Assumption 1 there is a constant L1(k1,k2)>0 0$]]>, such that ϕ(t,x1)≤k1−b1infx12+α1sup+b1supx1+βmax(k1+k2)≤L1(k1,k2). If α2sup≤0 , then for the function ψ(t,x1,x2) we have ψ(t,x1,x2)≤−k2c2infx22+(k1+k2)cmaxx2m(t)+x1≤L2(k1,k2). If α2sup>0 0$]]>, then for k2=k1c1infα2sup there is a constant L3(k1,k2)>0 0$]]>, such that ψ(t,x1,x2)≤{−k2c2infx22+(k2α2sup−k1c1inf)x1x2+[k2α2supmsup+(k1+k2)cmax]x2}(m(t)+x1)−1=k1m(t)+x1{−c1infc2infα2supx22+[c1infmsup+1+c1infα2supcmax]x2}≤L3(k1,k2). Therefore, there is a constant L(k1,k2)>0 0$]]>, such that f(t,x1,x2)≤L(k1,k2) . So, from (6) we obtain by integrating V(X(T∧τn))≤V(X0)+L(k1,k2)(T∧τn)+ ∑i=12ki{ ∫0T∧τn(xi(t)−1)σi(t)dwi(t)+ ∫0T∧τn∫R[γi(t,z)xi(t)−ln(1+γi(t,z))]ν˜1(dt,dz)+ ∫0T∧τn∫R[δi(t,z)xi(t)−ln(1+δi(t,z))]ν˜2(dt,dz)}. Taking expectations we derive from (7) EV(X(T∧τn))≤V(X0)+L(k1,k2)T. Set Ωn={τn≤T} for n≥n1 . Then by (5), P(Ωn)=P{τn≤T}>ε \varepsilon $]]>, ∀n≥n1 . Note that for every ω∈Ωn at least one of x1(τn,ω) and x2(τn,ω) equals either n or 1/n . So V(X(τn))≥min{k1,k2}min{n−1−lnn,1n−1+lnn}. From (8) it follows V(X0)+L(k1,k2)T≥E[1ΩnV(X(τn))]≥εmin{k1,k2}min{n−1−lnn,1n−1+lnn}, where 1Ωn is the indicator function of Ωn . Letting n→∞ leads to the contradiction ∞>V(X0)+L(k1,k2)T=∞ V({X_{0}})+L({k_{1}},{k_{2}})T=\infty $]]>. This completes the proof of the theorem.  □

The density of the prey population x1(t) obeys lim supt→∞ln(m+x1(t))t≤0,∀m>0,a.s. 0,\hspace{2em}\textit{a.s.}\]]]>

By the Itô formula for the process etln(m+x1(t)) we have etln(m+x1(t))−ln(m+x10)=∫0tes{ln(m+x1(s))+x1(s)m+x1(s)[a1(s)−b1(s)x1(s)−c1(s)x2(s)m(s)+x1(s)]−σ12(s)x12(s)2(m+x1(s))2+∫R[ln(1+γ1(s,z)x1(s)m+x1(s))−γ1(s,z)x1(s)m+x1(s)]Π1(dz)}ds+∫0tesσ1(s)x1(s)m+x1(s)dw1(s)+ ∫0t∫Resln(1+γ1(s,z)x1(s)m+x1(s))ν˜1(ds,dz)+ ∫0t∫Resln(1+δ1(s,z)x1(s)m+x1(s))ν2(ds,dz). For 0<κ≤1 , let us denote the process ζκ(t)=∫0tesσ1(s)x1(s)m+x1(s)dw1(s)+ ∫0t∫Resln(1+γ1(s,z)x1(s)m+x1(s))ν˜1(ds,dz)+ ∫0t∫Resln(1+δ1(s,z)x1(s)m+x1(s))ν2(ds,dz)−κ2∫0te2sσ12(s)x1(s)m+x1(s)2ds−1κ ∫0t∫R1+γ1(s,z)x1(s)m+x1(s)κes−1−κesln1+γ1(s,z)x1(s)m+x1(s)Π1(dz)ds−1κ ∫0t∫R1+δ1(s,z)x1(s)m+x1(s)κes−1Π2(dz)ds. By virtue of the exponential inequality ([3], Lemma 2.2) for any T>0 0$]]>, 0<κ≤1 , β>0 0$]]>, we have P{sup0≤t≤Tζκ(t)>β}≤e−κβ. \beta \}\le {e^{-\kappa \beta }}.\]]]> Choosing T=kτ , k∈N , τ>0 0$]]>, κ=εe−kτ , β=θekτε−1lnk , 0<ε<1 , θ>1 1$]]>, we get P{sup0≤t≤Tζκ(t)>θekτε−1lnk}≤1kθ. \theta {e^{k\tau }}{\varepsilon ^{-1}}\ln k\}\le \frac{1}{{k^{\theta }}}.\]]]> By the Borel–Cantelli lemma, for almost all ω∈Ω , there is a random integer k0(ω) , such that, for ∀k≥k0(ω) and 0≤t≤kτ , we have ∫0tesσ1(s)x1(s)m+x1(s)dw1(s)+ ∫0t∫Resln(1+γ1(s,z)x1(s)m+x1(s))ν˜1(ds,dz)+ ∫0t∫Resln(1+δ1(s,z)x1(s)m+x1(s))ν2(ds,dz)≤ε2ekτ∫0te2sσ1(s)x1(s)m+x1(s)2ds+ekτε ∫0t∫R[1+γ1(s,z)x1(s)m+x1(s)εes−kτ−1−εes−kτln1+γ1(s,z)x1(s)m+x1(s)]Π1(dz)ds+ekτε ∫0t∫R1+δ1(s,z)x1(s)m+x1(s)εes−kτ−1Π2(dz)ds+θekτlnkε. By using the inequality xr≤1+r(x−1) , ∀x≥0 , 0≤r≤1 , with x=1+γ1(s,z)x1(s)m+x1(s) , r=εes−kτ , and then with x=1+δ1(s,z)x1(s)m+x1(s) , r=εes−kτ , we derive the estimates ekτε ∫0t∫R[1+γ1(s,z)x1(s)m+x1(s)εes−kτ−1−εes−kτln1+γ1(s,z)x1(s)m+x1(s)]Π1(dz)ds≤ ∫0t∫Resγ1(s,z)x1(s)m+x1(s)−ln1+γ1(s,z)x1(s)m+x1(s)Π1(dz)ds, ekτε ∫0t∫R1+δ1(s,z)x1(s)m+x1(s)εes−kτ−1Π2(dz)ds≤ ∫0t∫Resδ1(s,z)x1(s)m+x1(s)Π2(dz)ds. From (10), by using (12)–(14) we get etln(m+x1(t))≤ln(m+x10)+∫0tes{ln(m+x1(s))+x1(s)m+x1(s)[a1(s)−b1(s)x1(s)−c1(s)x2(s)m(s)+x1(s)]−σ12(s)x12(s)2(m+x1(s))2×1−εes−kτ+∫Rδ1(s,z)x1(s)m+x1(s)Π2(dz)}ds+θekτlnkε,a.s. It is easy to see that, under Assumption 1, for any x>0 0$]]> there exists a constant L>0 0$]]> independent on k, s and x, such that ln(m+x)−x2b1(s)m+x+xα1(s)m+x≤L. So, from (15) for any (k−1)τ≤t≤kτ we have (a.s.) ln(m+x1(t))lnt≤e−tln(m+x10)lnt+Llnt(1−e−t)+θekτlnkεe(k−1)τln(k−1)τ. Therefore, lim supt→∞ln(m+x1(t))lnt≤θeτε,∀θ>1,τ>0,ε∈(0,1),a.s. 1,\tau >0,\varepsilon \in (0,1),\hspace{1em}\text{a.s.}\]]]> If θ↓1 , τ↓0 , ε↑1 , we obtain lim supt→∞ln(m+x1(t))lnt≤1,a.s. So, lim supt→∞ln(m+x1(t))t≤0,a.s.  □

The density of the prey population x1(t) obeys lim supt→∞lnx1(t)t≤0,a.s.

The density of the predator population x2(t) has the property that lim supt→∞lnx2(t)t≤0,a.s.

Making use of the Itô formula we get etlnx2(t)−lnx20=∫0tes{lnx2(s)+a2(s)−c2(s)x2(s)m(s)+x1(s)−σ22(s)2+∫R[ln(1+γ2(s,z))−γ2(s,z)]Π1(dz)}ds+ψ(t), where ψ(t)=∫0tesσ2(s)dw2(s)+ ∫0t∫Resln(1+γ2(s,z))ν˜1(ds,dz)+ ∫0t∫Resln(1+δ2(s,z))ν2(ds,dz). By virtue of the exponential inequality (11) we have P{sup0≤t≤Tζκ(t)>β}≤e−κβ,∀0<κ≤1,β>0, \beta \}\le {e^{-\kappa \beta }},\forall 0<\kappa \le 1,\beta >0,\]]]> where ζκ(t)=ψ(t)−κ2 ∫0te2sσ22(s)ds−1κ ∫0t∫R[(1+γ2(s,z))κes−1−κesln(1+γ2(s,z))]Π1(dz)ds−1κ ∫0t∫R(1+δ2(s,z))κes−1Π2(dz)ds. Choosing T=kτ , k∈N , τ>0 0$]]>, κ=e−kτ , β=θekτlnk , θ>1 1$]]>, we get P{sup0≤t≤Tζκ(t)>θekτlnk}≤1kθ. \theta {e^{k\tau }}\ln k\}\le \frac{1}{{k^{\theta }}}.\]]]> By the same arguments as in the proof of Lemma 1, using the Borel–Cantelli lemma, we derive from (16) etlnx2(t)≤lnx20+∫0tes{lnx2(s)+a2(s)−c2(s)x2(s)m(s)+x1(s)−σ22(s)21−es−kτ+∫Rδ2(s,z)Π2(dz)}ds+θekτlnk,a.s., for all sufficiently large k≥k0(ω) and 0≤t≤kτ .

Using inequality lnx−cx≤−lnc−1 , ∀x≥0 , c>0 0$]]>, with x=x2(s) , c=c2(s)m(s)+x1(s) , we derive from (17) the estimate etlnx2(t)≤lnx20+ ∫0tesln(msup+x1(s))ds+L(et−1)+θekτlnk, for some constant L>0 0$]]>.

So, for (k−1)τ≤t≤kτ , k≥k0(ω) , we have lim supt→∞lnx2(t)t≤lim supt→∞1t ∫0tes−tln(msup+x1(s))ds≤0, by virtue of Lemma 1.  □

Let p>0 0$]]>. Then for any initial value x10>0 0$]]>, the pth-moment of the prey population density x1(t) obeys lim supt→∞Ex1p(t)≤K1(p), where K1(p)>0 0$]]> is independent of x10 .

For any initial value x20>0 0$]]>, the expectation of the predator population density x2(t) obeys lim supt→∞Ex2(t)≤K2, where K2>0 0$]]> is independent of x20 .

Let τn be the stopping time defined in Theorem 1. Applying the Itô formula to the process V(t,x1(t))=etx1p(t) , p>0 0$]]>, we obtain V(t∧τn,x1(t∧τn))=x10p+ ∫0t∧τnesx1p(s){1+p[a1(s)−b1(s)x1(s)−c1(s)x2(s)m(s)+x1(s)]+p(p−1)σ12(s)2+∫R(1+γ1(s,z))p−1−pγ1(s,z)Π1(dz)+∫R(1+δ1(s,z))p−1Π2(dz)}ds+ ∫0t∧τnpesx1p(s)σ1(s)dw1(s)+ ∫0t∧τn∫Resx1p(s)(1+γ1(s,z))p−1ν˜1(ds,dz)+ ∫0t∧τn∫Resx1p(s)(1+δ1(s,z))p−1ν˜2(ds,dz).

Under Assumption 1 there is a constant K1(p)>0 0$]]>, such that esx1p{1+pa1(s)−b1(s)x1−c1(s)x2m(s)+x1+p(p−1)σ12(s)2++∫R(1+γ1(s,z))p−1−pγ1(s,z)Π1(dz)+∫R(1+δ1(s,z))p−1Π2(dz)}≤K1(p)es. From (20) and (21), taking expectations, we obtain E[V(t∧τn,x1(t∧τn))]≤x10p+K1(p)et.

Letting n→∞ leads to the estimate etE[x1p(t)]≤x10p+etK1(p). So from (22) we derive (18).

Let us prove the estimate (19). Applying the Itô formula to the process U(t,X(t))=et[k1x1(t)+k2x2(t)] , ki>0 0$]]>, i=1,2 , we obtain dU(t,X(t))=et{k1x1(t)+k2x2(t)+k1[a1(t)x1(t)−b1(t)x12(t)−c1(t)x1(t)x2(t)m(t)+x1(t)]+k2[a2(t)x2(t)−c2(t)x22(t)m(t)+x1(t)]+ ∑i=12ki∫Rxi(t)δi(t,z)Π2(dz)}dt+et{ ∑i=12ki[xi(t)σi(t)dwi(t)+∫Rxi(t)γi(t,z)ν˜1(dt,dz)+∫Rxi(t)δi(t,z)ν˜2(dt,dz)]}. For the function f(t,x1,x2)=1m(t)+x1{k1[−b1(t)x13+(1+a1(t)+δ¯1(t)−b1(t)m(t))x12+m(t)(1+a1(t)+δ¯1(t))x1]+[k2(1+a2(t)+δ¯2(t))−k1c1(t)]x1x2+k2[−c2(t)x22+m(t)(1+a2(t)+δ¯2(t))x2]},whereδ¯i(t)=∫Rδi(t,z)Π2(dz),i=1,2, we have f(t,x1,x2)≤ϕ1(x1,x2)+ϕ2(x2)m(t)+x1, where ϕ1(x1,x2)=k1−b1infx13+(d1−b1infminf)x12+msupd1x1+[k2d2−k1c1inf]x1x2,ϕ2(x2)=k2−c2infx22+msupd2x2,di=1+aisup+|δ¯i|sup,i=1,2. For k2=k1c1inf/d2 there is a constant L′>0 0$]]>, such that ϕ1(x1,x2)≤L′k1 and ϕ2(x2)≤L′k1 . So, there is a constant L>0 0$]]>, such that f(t,x1,x2)≤Lk1. From (23) and (24) by integrating and taking expectation, we derive E[U(t∧τn,X(t∧τn))]≤k1x10+c1infd2x20+Let. Letting n→∞ leads to the estimate etEx1(t)+c1infd2x2(t)≤x10+c1infd2x20+Let. So, E[x2(t)]≤d2c1infx10+x20e−t+d2c1infL. From (25) we have (19).  □

If p2inf>0 0$]]>, where p2(t)=a2(t)−β2(t) , then for any initial value x20>0 0$]]>, the predator population density x2(t) satisfies lim supt→∞E1x2(t)θ≤K(θ),0<θ<1,

For the process U(t)=1/x2(t) by the Itô formula we derive U(t)=U(0)+ ∫0tU(s)[c2(s)x2(s)m(s)+x1(s)−a2(s)+σ22(s)+∫Rγ22(s,z)1+γ2(s,z)Π1(dz)]ds− ∫0tU(s)σ2(s)dw2(s)− ∫0t∫RU(s)γ2(s,z)1+γ2(s,z)ν˜1(ds,dz)− ∫0t∫RU(s)δ2(s,z)1+δ2(s,z)ν2(ds,dz). Then, by applying the Itô formula, we derive, for 0<θ<1 , (1+U(t))θ=(1+U(0))θ+ ∫0tθ(1+U(s))θ−2{(1+U(s))U(s)×[c2(s)x2(s)m(s)+x1(s)−a2(s)+σ22(s)+∫Rγ22(s,z)1+γ2(s,z)Π1(dz)]+θ−12U2(s)σ22(s)+1θ∫R[(1+U(s))2((1+U(s)+γ2(s,z)(1+γ2(s,z))(1+U(s)))θ−1)+θ(1+U(s))U(s)γ2(s,z)1+γ2(s,z)]Π1(dz)+1θ∫R(1+U(s))2[(1+U(s)+δ2(s,z)(1+δ2(s,z))(1+U(s)))θ−1]Π2(dz)}ds− ∫0tθ(1+U(s))θ−1U(s)σ2(s)dw2(s)+ ∫0t∫R[(1+U(s)1+γ2(s,z))θ−(1+U(s))θ]ν˜1(ds,dz)+ ∫0t∫R[(1+U(s)1+δ2(s,z))θ−(1+U(s))θ]ν˜2(ds,dz)=(1+U(0))θ+ ∫0tθ(1+U(s))θ−2J(s)ds−I1,stoch(t)+I2,stoch(t)+I3,stoch(t), where Ij,stoch(t),j=1,3‾ , are the corresponding stochastic integrals in (27). Under Assumption 1 there exist constants |K1(θ)|<∞ , |K2(θ)|<∞ , such that for the process J(t) we have the estimate J(t)≤(1+U(t))U(t)[−a2(t)+c2supU−1(t)minf+σ22(t)∫R+∫Rγ22(s,z)1+γ2(s,z)Π1(dz)]+θ−12U2(s)σ22(s)+1θ∫R[(1+U(s))2((11+γ2(s,z)+11+U(s))θ−1)+θ(1+U(s))U(s)γ2(s,z)1+γ2(s,z)]Π1(dz)+1θ∫R(1+U(s))2[(11+δ2(s,z)+11+U(s))θ−1]Π2(dz)≤U2(t)[−a2(t)+σ22(t)2+∫Rγ2(t,z)Π1(dz)+θ2σ22(t)+1θ∫R[(1+γ2(t,z))−θ−1]Π1(dz)+1θ∫R[(1+δ2(t,z))−θ−1]Π2(dz)]+K1(θ)U(t)+K2(θ)=−K0(t,θ)U2(t)+K1(θ)U(t)+K2(θ), where we used the inequality (x+y)θ≤xθ+θxθ−1y , 0<θ<1 , x,y>0 0$]]>. Due to limθ→0+[θ2σ22(t)+1θ∫R[(1+γ2(t,z))−θ−1]Π1(dz)+1θ∫R[(1+δ2(t,z))−θ−1]Π2(dz)+∫Rln(1+γ2(t,z))Π1(dz)+∫Rln(1+δ2(t,z))Π2(dz)]=limθ→0+Δ(θ)=0, and the condition p2inf>0 0$]]> we can choose a sufficiently small 0<θ<1 so that K0(θ)=inft≥0K0(t,θ)=inft≥0[p2(t)−Δ(θ)]=p2inf−Δ(θ)>0 0\]]]> is satisfied. So, from (27) and the estimate for J(t) we derive d[(1+U(t))θ]≤θ(1+U(t))θ−2[−K0(θ)U2(t)+K1(θ)U(t)+K2(θ)]dt−θ(1+U(t))θ−1U(t)σ2(t)dw2(t)+∫R[(1+U(t)1+γ2(t,z))θ−(1+U(t))θ]ν˜1(dt,dz)+∫R[(1+U(t)1+δ2(t,z))θ−(1+U(t))θ]ν˜2(dt,dz). By the Itô formula and (28) we have deλt(1+U(t))θ=λeλt(1+U(t))θdt+eλtd(1+U(t))θ≤eλtθ(1+U(t))θ−2[−(K0(θ)−λθ)U2(t)+(K1(θ)+2λθ)U(t)+K2(θ)+λθ]dt−θeλt(1+U(t))θ−1U(t)σ2(t)dw2(t)+eλt∫R[(1+U(t)1+γ2(t,z))θ−(1+U(t))θ]ν˜1(dt,dz)+eλt∫R[(1+U(t)1+δ2(t,z))θ−(1+U(t))θ]ν˜2(dt,dz).

Let us choose λ=λ(θ)>0 0$]]>, such that K0(θ)−λ/θ>0 0$]]>. Then there is a constant K>0 0$]]>, such that (1+U(t))θ−2[−(K0(θ)−λθ)U2(t)+(K1(θ)+2λθ)U(t)+K2(θ)+λθ]≤K. Let τn be the stopping time defined in Theorem 1. Then by integrating (29), using (30) and taking the expectation we obtain Eeλ(t∧τn)(1+U(t∧τn))θ≤1+1x20θ+θλKeλt−1. Letting n→∞ leads to the estimate etE(1+U(t))θ≤1+1x20θ+θλKeλt−1. From (31) we obtain lim supt→∞E1x2(t)θ=lim supt→∞EUθ(t)≤lim supt→∞E(1+U(t))θ≤θλ(θ)K, this implies (26).  □

 The long-time behaviour ([<xref ref-type="bibr" rid="j_vmsta173_ref_008">8</xref>]).

The solution X(t) to the system (3) is said to be stochastically ultimately bounded, if for any ε∈(0,1) , there is a positive constant χ=χ(ε)>0 0$]]>, such that for any initial value X0∈R+2 , the solution to the system (3) has the property that lim supt→∞P|X(t)|>χ<ε. \chi \right\}<\varepsilon .\]]]>

In what follows in this section we will assume that Assumption 1 holds.

The solution X(t) to the system (3) with the initial value X0∈R+2 is stochastically ultimately bounded.

From Lemma 3 we have the estimate lim supt→∞E[xi(t)]≤Ki,i=1,2. For X=(x1,x2)∈R+2 we have |X|≤x1+x2 , therefore, from (32) lim supt→∞E[|X(t)|]≤L=K1+K2 . Let χ>L/ε L/\varepsilon $]]>, ∀ε∈(0,1) . Then applying the Chebyshev inequality yields lim supt→∞P{|X(t)|>χ}≤1χlimsupt→∞E[|X(t)|]≤Lχ<ε. \chi \}\le \frac{1}{\chi }\lim \underset{t\to \infty }{\sup }E[|X(t)|]\le \frac{L}{\chi }<\varepsilon .\]]]>  □

The property of stochastic permanence is important since it means the long-time survival in a population dynamics.

The population density x(t) is said to be stochastically permanent if for any ε>0 0$]]>, there are positive constants H=H(ε) , h=h(ε) such that lim inft→∞P{x(t)≤H}≥1−ε,lim inft→∞P{x(t)≥h}≥1−ε, for any inial value x0>0 0$]]>.

If p2inf>0 0$]]>, where p2(t)=a2(t)−β2(t) , then for any initial value x20>0 0$]]>, the predator population density x2(t) is stochastically permanent.

From Lemma 3 we have estimate lim supt→∞E[x2(t)]≤K. Thus for any given ε>0 0$]]>, let H=K/ε , by virtue of Chebyshev’s inequality, we can derive that lim supt→∞P{x2(t)≥H}≤1Hlim supt→∞E[x2(t)]≤ε. Consequently, lim inft→∞P{x2(t)≤H}≥1−ε .

From Lemma 4 we have the estimate lim supt→∞E1x2(t)θ≤K(θ),0<θ<1. For any given ε>0 0$]]>, let h=(ε/K(θ))1/θ , then by Chebyshev’s inequality, we have lim supt→∞P{x2(t)<h}≤lim supt→∞P1x2(t)θ>h−θ≤hθlim supt→∞E1x2(t)θ≤ε. {h^{-\theta }}\right\}\\ {} \displaystyle \le {h^{\theta }}\underset{t\to \infty }{\limsup }\mathrm{E}\left[{\left(\frac{1}{{x_{2}}(t)}\right)^{\theta }}\right]\le \varepsilon .\end{array}\]]]> Consequently, lim inft→∞P{x2(t)≥h}≥1−ε .  □

If the predator is absent, i.e. x2(t)=0 a.s., and p1inf>0 0$]]>, where p1(t)=a1(t)−β1(t) , then for any initial value x10>0 0$]]>, the prey population density x1(t) is stochastically permanent.

From Lemma 3 we have the estimate lim supt→∞E[x1(t)]≤K. Thus for any given ε>0 0$]]>, let H=K/ε , by virtue of Chebyshev’s inequality, we can derive that lim supt→∞P{x1(t)≥H}≤1Hlim supt→∞E[x1(t)]≤ε. Consequently, lim inft→∞P{x1(t)≤H}≥1−ε .

For the process U(t)=1/x1(t) , by the Itô formula we have U(t)=U(0)+ ∫0tU(s)[b1(s)x1(s)−a1(s)+σ12(s)+∫Rγ12(s,z)1+γ1(s,z)Π1(dz)]ds− ∫0tU(s)σ1(s)dw1(s)− ∫0t∫RU(s)γ1(s,z)1+γ1(s,z)ν˜1(ds,dz)− ∫0t∫RU(s)δ1(s,z)1+δ1(s,z)ν2(ds,dz). Then, using the same arguments as in the proof of Lemma 4 we can derive the estimate lim supt→∞E1x1(t)θ≤K(θ),0<θ<1. For any given ε>0 0$]]>, let h=(ε/K(θ))1/θ . Then by Chebyshev’s inequality, we have lim supt→∞P{x1(t)<h}=lim supt→∞P1x1(t)θ>h−θ≤hθlim supt→∞E1x1(t)θ≤ε. {h^{-\theta }}\right\}\\ {} \displaystyle \le {h^{\theta }}\underset{t\to \infty }{\limsup }\mathrm{E}\left[{\left(\frac{1}{{x_{1}}(t)}\right)^{\theta }}\right]\le \varepsilon .\end{array}\]]]> Consequently, lim inft→∞P{x1(t)≥h}≥1−ε .  □

If the predator is absent, i.e. x2(t)=0 a.s., then the equation for the prey x1(t) has the logistic form. So, Theorem 4 gives us the sufficient conditions for the stochastic permanence of the solution to the stochastic nonautonomous logistic equation disturbed by white noise, centered and noncentered Poisson noises.

The solution X(t)=(x1(t),x2(t)) , t≥0 , to equation (3) will be said extinct if for every initial data X0∈R+2 , we have limt→∞xi(t)=0 almost surely (a.s.), i=1,2 .

If p¯i∗=lim supt→∞1t ∫0tpi(s)ds<0,wherepi(t)=ai(t)−βi(t),i=1,2, then the solution X(t) to equation (3) with the initial condition X0∈R+2 will be extinct.

By the Itô formula, we have dlnxi(t)=ai(t)−bi(t)xi(t)−ci(t)x2(t)m(t)+x1(t)−βi(t)dt+dMi(t)≤[ai(t)−βi(t)]dt+dMi(t),i=1,2, where the martingale Mi(t)= ∫0tσi(s)dwi(s)+ ∫0t∫Rln(1+γi(s,z))ν˜1(ds,dz)+ ∫0t∫Rln(1+δi(s,z))ν˜2(ds,dz),i=1,2, has quadratic variation ⟨Mi,Mi⟩(t)= ∫0tσi2(s)ds+ ∫0t∫Rln2(1+γi(s,z))Π1(dz)ds+ ∫0t∫Rln2(1+δi(s,z))Π2(dz)ds≤Kt,i=1,2.

Then the strong law of large numbers for local martingales ([10]) yields limt→∞Mi(t)/t=0 , i=1,2 , a.s. Therefore, from (33) we obtain lim supt→∞lnxi(t)t≤lim supt→∞1t ∫0tpi(s)ds<0,a.s. So, limt→∞xi(t)=0 , i=1,2 , a.s.  □

 ([<xref ref-type="bibr" rid="j_vmsta173_ref_011">11</xref>]).

The population density x(t) will be said nonpersistent in the mean if limt→∞1t∫0tx(s)ds=0a.s.

If p¯1∗=0 , then the prey population density x1(t) with the initial condition x10>0 0$]]> will be nonpersistent in the mean.

From the first equality in (33) for i=1 we have lnx1(t)≤lnx10+ ∫0tp1(s)ds−b1inf ∫0tx1(s)ds+M1(t), where the martingale M1(t) is defined in (34). From the definition of p¯1∗ and the strong law of large numbers for M1(t) it follows, that ∀ε>0 0$]]>, ∃t0≥0 , and ∃Ωε⊂Ω , with P(Ωε)≥1−ε , such that 1t ∫0tp1(s)ds≤p¯1∗+ε2,M1(t)t≤ε2,∀t≥t0,ω∈Ωε. So, from (35) we derive lnx1(t)−lnx10≤t(p¯1∗+ε)−b1inf ∫0tx1(s)ds=tε−b1inf ∫0tx1(s)ds,∀t≥t0,ω∈Ωε. Let y1(t)=∫0tx1(s)ds , then from (36) we have lndy1(t)dt≤εt−b1infy1(t)+lnx10⇒eb1infy1(t)dy1(t)dt≤x10eεt,∀t≥t0,ω∈Ωε. By integrating the last inequality from t0 to t we obtain eb1infy1(t)≤b1infx10εeεt−eεt0+eb1infy1(t0),∀t≥t0,ω∈Ωε. So, y1(t)≤1b1inflneb1infy1(t0)+b1infx10εeεt−eεt0,∀t≥t0,ω∈Ωε, and therefore lim supt→∞1t ∫0tx1(s)ds≤εb1inf,∀ω∈Ωε. Since ε>0 0$]]> is arbitrary and x1(t)>0 0$]]> a.s., we have limt→∞1t ∫0tx1(s)ds=0a.s.  □

If p¯2∗=0 and p¯1∗<0 , then the predator population density x2(t) with the initial condition x20>0 0$]]> will be nonpersistent in the mean.

From the first equality in (33) with i=2 we have, for c=c2inf/msup , lnx2(t)≤lnx20+ ∫0tp2(s)ds−c2inf ∫0tx2(s)m(s)+x1(s)ds+M2(t)=lnx20+ ∫0tp2(s)ds−c2inf ∫0t1m(s)x2(s)−x1(s)x2(s)m(s)+x1(s)ds+M2(t)≤lnx20+ ∫0tp2(s)ds−c ∫0tx2(s)ds+c ∫0tx1(s)x2(s)msup+x1(s)ds+M2(t), where the martingale M2(t) is defined in (34). From Theorem 5, the definition of p¯2∗ and the strong law of large numbers for M2(t) it follows, that ∀ε>0 0$]]>, ∃t0≥0 , and ∃Ωε⊂Ω with P(Ωε)≥1−ε , such that 1t ∫0tp2(s)ds≤p¯2∗+ε2,M2(t)t≤ε2,x1(t)msup+x1(t)≤ε,∀t≥t0,ω∈Ωε. So, from (37) we derive lnx2(t)−lnx20≤t(p¯2∗+ε)−c(1−ε) ∫t0tx2(s)ds=tε−c(1−ε) ∫t0tx2(s)ds,∀t≥t0,ω∈Ωε. Let y2(t)=∫t0tx2(s)ds . Then from (38) we have lndy2(t)dt≤εt−c(1−ε)y2(t)+lnx20⇒ec(1−ε)y2(t)dy2(t)dt≤x20eεt,∀t≥t0,ω∈Ωε. By integrating the last inequality from t0 to t we obtain ec(1−ε)y2(t)≤c(1−ε)x20εeεt−eεt0+1,∀t≥t0,ω∈Ωε. So, y2(t)≤1c(1−ε)ln1+c(1−ε)x20εeεt−eεt0,∀t≥t0,ω∈Ωε, and therefore lim supt→∞1t ∫0tx2(s)ds≤εc(1−ε),∀ω∈Ωε. Since ε>0 0$]]> is arbitrary and x2(t)>0 0$]]> a.s., we have limt→∞1t ∫0tx2(s)ds=0a.s.  □

 ([<xref ref-type="bibr" rid="j_vmsta173_ref_011">11</xref>]).

The population density x(t) will be said weakly persistent in the mean if x¯∗=lim supt→∞1t∫0tx(s)ds>0a.s. 0\hspace{2.5pt}\text{a.s.}\]]]>

If p¯2∗>0 0$]]>, then the predator population density x2(t) with the initial condition x20>0 0$]]> will be weakly persistent in the mean.

If the assertion of theorem is not true, then P{x¯2∗=0}>0 0$]]>. From the first equality in (33) we get 1t(lnx2(t)−lnx20)=1t∫0tp2(s)ds−1t∫0tc2(s)x2(s)m(s)+x1(s)ds+M2(t)t≥1t∫0tp2(s)ds−c2supminft∫0tx2(s)ds+M2(t)t, where the martingale M2(t) is defined in (34). For ∀ω∈{ω∈Ω|x¯2∗=0} in virtue of the strong law of large numbers for the martingale M2(t) we have lim supt→∞lnx2(t)t≥p¯2∗>0. 0.\]]]> Therefore, Pω∈Ω|lim supt→∞lnx2(t)t>0>0. 0\right\}>0.\]]]> But from Lemma 2 we have Pω∈Ω|lim supt→∞lnx2(t)t≤0=1. This is a contradiction.  □

If p¯1∗>0 0$]]> and p¯2∗<0 , then the prey population density x1(t) with the initial condition x10>0 0$]]> will be weakly persistent in the mean.

Let P{x¯1∗=0}>0 0$]]>. From the first equality in (33) with i=1 we get 1t(lnx1(t)−lnx10)=1t∫0tp1(s)ds−1t∫0tb1(s)x1(s)ds−1t∫0tc1(s)x2(s)m(s)+x1(s)ds+M1(t)t≥1t∫0tp1(s)ds−b1supt∫0tx1(s)ds−c1supminft∫0tx2(s)ds+M1(t)t where the martingale M1(t) is defined in (34). From the definition of p¯1∗ , the strong law of large numbers for the martingale M1(t) and Theorem 2 for x2(t) , we have ∀ε>0 0$]]>, ∃t0≥0 , ∃Ωε⊂Ω with P(Ωε)≥1−ε , such that 1t ∫0tp1(s)ds≥p¯1∗−ε3,M1(t)t≥−ε3,1t∫0tx2(s)ds≤εminf3c1sup,∀t≥t0,ω∈Ωε. So, from (39) we get for ω∈{ω∈Ω|x¯1∗=0}∩Ωε lim supt→∞lnx1(t)t≥p¯1∗−ε>0 0\]]]> for a sufficiently small ε>0 0$]]>. Therefore, Pω∈Ω|lim supt→∞lnx1(t)t>0>0. 0\right\}>0.\]]]> But from Corollary 1 Pω∈Ω|lim supt→∞lnx1(t)t≤0=1. Therefore we have a contradiction.  □

 Acknowledgments

The authors wish to thank the Referees for their very careful reading of the manuscript and very useful comments which allowed us to improve the presentation of the paper.

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