MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA150 10.15559/20-MSTA150 Research Article Stochastic two-species mutualism model with jumps BorysenkoOlgaolga_borisenko@ukr.neta 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. 2020 33202071115 3122019 2122020 2122020 © 2020 The Author(s). Published by VTeX2020 Open access article under the CC BY license.

The existence and uniqueness are proved for the global positive solution to the system of stochastic differential equations describing a two-species mutualism model disturbed by the white noise, the centered and non-centered Poisson noises. We obtain sufficient conditions for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, strong persistence in the mean and extinction of the solution to the considered system.

 Stochastic mutualism model global solution stochastic ultimate boundedness stochastic permanence extinction nonpersistence in the mean strong persistence in the mean 92D25 60H10 60H30 Introduction

The construction of the mutualism model and its properties are presented in K. Gopalsamy [4]. Mutualism occurs when one species provides some benefit in exchange for another benefit. A deterministic two-species mutualism model is described by the system dN1(t)dt=r1(t)N1(t)K1(t)+α1(t)N2(t)1+N2(t)−N1(t),dN2(t)dt=r2(t)N2(t)K2(t)+α2(t)N1(t)1+N1(t)−N2(t), where N1(t) and N2(t) denote the population densities of each species at time t, ri(t)>0 0$]]>, i=1,2 , denotes the intrinsic growth rate of species Ni,i=1,2 , and αi(t)>Ki(t)>0 {K_{i}}(t)>0$]]>, i=1,2 . The carrying capacity of species Ni(t) is Ki(t) , i=1,2 , in the absence of other species. In the paper by Hong Qiu, Jingliang Lv and Ke Wang [9] the stochastic mutualism model of the form dx(t)=x(t)a1(t)+a2(t)y(t)1+y(t)−c1(t)x(t)+σ1(t)x(t)dw1(t),dy(t)=y(t)b1(t)+b2(t)x(t)1+x(t)−c2(t)y(t)+σ2(t)y(t)dw2(t) is considered, where ai(t),bi(t),ci(t),σi(t),i=1,2 , are all positive, continuous and bounded functions on [0,+∞) , and w1(t),w2(t) are independent Wiener processes. The authors show that the stochastic system (1) has a unique global (no explosion in a finite time) solution for any positive initial value and that this stochastic model is stochastically ultimately bounded. The sufficient conditions for stochastic permanence and persistence in the mean of the solution to the system (1) are obtained.

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 the paper by Mei Li, Hongjun Gao and Binjun Wang [5] the authors consider the stochastic mutualism model with the white and centered Poisson noises: dx(t)=x(t−)r1(t)−b1(t)x(t)K1(t)+y(t)−ε1(t)x(t)dt+α1(t)dw1(t)+∫Yγ1(t,z)ν˜(dt,dz),dy(t)=y(t−)r2(t)−b2(t)y(t)K2(t)+x(t)−ε2(t)y(t)dt+α2(t)dw2(t)+∫Yγ2(t,z)ν˜(dt,dz), where x(t−) , y(t−) are the left limit of x(t) and y(t) respectively, ri(t) , bi(t) , Ki(t) , αi(t) , i=1,2 , are all positive, continuous and bounded functions, Y is measurable subset of (0,+∞) , wi(t) , i=1,2 , are independent standard one-dimensional Wiener processes, ν˜(t,A)=ν(t,A)−tΠ(A) is the centered Poisson measure independent on wi(t) , i=1,2 , E[ν(t,A)]=tΠ(A) , Π(Y)<∞ , γi(t,z),i=1,2 , are random, measurable, bounded, continuous in t. The global existence and uniqueness of the positive solution to this problem are proved. The sufficient conditions of stochastic boundedness, stochastic permanence, persistence in the mean and extinction of the solution are obtained.

In this paper, we consider the stochastic mutualism model with 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 “large” jumps, corresponding to the noncentered Poisson measure. This model is driven by the system of stochastic differential equations dxi(t)=xi(t)ai1(t)+ai2(t)x3−i(t)1+x3−i(t)−ci(t)xi(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{1em}i=1,2,\end{array}\]]]> where wi(t),i=1,2 , are independent standard one-dimensional Wiener processes, ν˜1(t,A)=ν1(t,A)−tΠ1(A) , νi(t,A),i=1,2 , are independent Poisson measures, which are independent on wi(t),i=1,2 , 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 are no papers devoted to the dynamical properties of the stochastic mutualism model (2), even in the case of the centered Poisson noise. It is worth noting that the impact of the centered and noncentered Poisson noises to the stochastic nonautonomous logistic model is studied in the papers by O.D. Borysenko and D.O. Borysenko [1, 2].

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\}$]]>, 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 the 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 will prove that system (2) has a unique, positive, global (no explosion in a finite time) solution for any positive initial value, and that this solution is stochastically ultimate bounded. The sufficient conditions for stochastic permanence, nonpersistence in the mean, strong 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 (2). In Section 3, we prove the stochastic ultimate boundedness of the solution to the system (2). In Section 4, we obtain conditions under which the solution to the system (2) is stochastically permanent, and in Section 5 the sufficient conditions for nonpersistence in the mean, strong persistence in the mean and extinction of the solution are obtained.

 Existence of the global solution

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 aij(t),i,j=1,2 , ci(t),σi(t),i=1,2 , are bounded, continuous in t functions, aij(t)>0 0$]]>, i,j=1,2 , cmin>0 0$]]>, γi(t,z),δi(t,z) , i=1,2 , are continuous in t functions and ln(1+γi(t,z)),ln(1+δi(t,z)) , i=1,2 , are bounded, Πi(R)<∞ , i=1,2 .

Let Assumption 1 be fulfilled. Then there exists a unique global solution X(t) to the system (2) for any initial value X(0)=X0>0 0$]]>, and P{X(t)∈R+2}=1 .

Let us consider the system of stochastic differential equations dvi(t)=ai1(t)+ai2(t)exp{v3−i(t)}1+exp{v3−i(t)}−ci(t)exp{vi(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 equation (3) are locally Lipschitz continuous. Therefore, for any initial value (v1(0),v2(0)) there exists a unique local solution V(t)=(v1(t),v2(t)) on [0,τe) , where supt<τe|V(t)|=+∞ (cf. Theorem 6, p. 246, [3]). So, from the Itô formula we derive that the process X(t)=(exp{v1(t)},exp{v2(t)}) is a unique, positive local solution to the system (2). 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 clear that τn is increasing as n→+∞ . Set 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 this statement is false, 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=12(xi−1−lnxi) , xi>0 0$]]>, i=1,2 , by the Itô formula we have V(X(T∧τn))=V(X0)+ ∑i=12 ∫0T∧τn(xi(t)−1)ai1(t)+ai2(t)x3−i(t)1+x3−i(t)−ci(t)xi(t)+βi(t)+∫Rδi(t,z)xi(t)Π2(dz)dt+ ∫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).

Under the conditions of the theorem there is a constant K>0 0$]]> such that ∑i=12(xi−1)ai1(t)+ai2(t)x3−i1+x3−i−ci(t)xi+βi(t)+∫Rδi(t,z)xiΠ2(dz)≤ ∑i=12(amax+cmax)xi−cminxi2+βmax+xiδmaxΠ2(R)≤K, where amax=maxi,j=1,2{supt≥0aij(t)} . From (5) and (6) we have V(X(T∧τn))≤V(X0)+K(T∧τn)+ ∑i=12 ∫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).

Whence, taking expectations yields EV(X(T∧τn))≤V(X0)+KT.

Set Ωn={τn≤T} for n≥n1 . Then by (4), P(Ωn)=P{τn≤t}>ε \varepsilon $]]>, ∀n≥n1 . Note that for every ω∈Ωn there is some i such that xi(τn,ω) equals either n or 1/n . So V(X(τn))≥min{n−1−lnn,1n−1+lnn}.

It then follows from (7) that V(X0)+KT≥E[1ΩnV(X(τn))]≥εmin{n−1−lnn,1n−1+lnn}, where 1Ωn is the indicator function of Ωn . Letting n→∞ leads to the contradiction ∞>V(X0)+KT=∞ V({X_{0}})+KT=\infty $]]>. This completes the proof of the theorem.  □

 Stochastically ultimate boundedness

([6]) The solution X(t) to the system (2) 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 this solution has the property lim supt→∞P{|X(t)|>χ}<ε. \chi \}<\varepsilon .\]]]>

Under Assumption 1 the solution X(t) to the system (2) is stochastically ultimately bounded for any initial value X0∈R+2 .

Let τn be the stopping time defined in Theorem 1. Applying the Itô formula to the process V(t,xi(t))=etxip(t) , i=1,2 , p>0 0$]]>, we obtain for i=1,2 that V(t∧τn,xi(t∧τn))=xi0p+ ∫0t∧τnesxip(s)1+pai1(s)+ai2(s)x3−i(s)1+x3−i(s)−ci(s)xi(s)+p(p−1)σi2(s)2+∫R(1+γi(s,z))p−1−pγi(s,z)Π1(dz)+∫R(1+δi(s,z))p−1Π2(dz)ds+ ∫0t∧τnpesxip(s)σi(s)dwi(s)+ ∫0t∧τn∫Resxip(s)(1+γi(s,z))p−1ν˜1(ds,dz)+ ∫0t∧τn∫Resxip(s)(1+δi(s,z))p−1ν˜2(ds,dz).

Under Assumption 1 there is a constant Ki(p)>0 0$]]> such that esxip1+pai1(s)+ai2(s)x3−i1+x3−i−ci(s)xi+p(p−1)σi2(s)2++∫R(1+γi(s,z))p−1−pγi(s,z)Π1(dz)+∫R(1+δi(s,z))p−1Π2(dz)≤Ki(p)es. From (8) and (9), taking expectations, we obtain E[V(t∧τn,xi(t∧τn))]≤xi0p+K(p)E[et∧τn]≤xi0p+Ki(p)et.

Letting n→∞ leads to the estimate etE[xip(t)]≤xi0p+etKi(p). So we have for i=1,2 that limsupt→∞E[xip(t)]≤Ki(p). For X=(x1,x2)∈R+2 we have |X|p≤2p/2(x1p+x2p) , therefore, from (11) limsupt→∞E[|X(t)|p]≤L(p)=2p/2(K1(p)+K2(p)) . Let χ>(L(p)/ε)1/p {(L(p)/\varepsilon )^{1/p}}$]]>, p>0 0$]]>, ∀ε∈(0,1) . Then applying the Chebyshev inequality yields lim supt→∞P{|X(t)|>χ}≤1χplimsupt→∞E[|X(t)|p]≤L(p)χp<ε. \chi \}\le \frac{1}{{\chi ^{p}}}\lim \underset{t\to \infty }{\sup }E[|X(t){|^{p}}]\le \frac{L(p)}{{\chi ^{p}}}<\varepsilon .\]]]>  □

 Stochastic permanence

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

([5]) The solution X(t) to the system (2) 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{xi(t)≤H}≥1−ε,lim inft→∞P{xi(t)≥h}≥1−ε, for i=1,2 and for any inial value X0∈R+2 .

Let Assumption 1 be fulfilled. If mini=1,2inft≥0(aimin(t)−βi(t))>0,whereaimin(t)=minj=1,2aij(t),i=1,2, 0,\hspace{1em}\textit{where}\hspace{2.5pt}{a_{i\min }}(t)=\underset{j=1,2}{\min }{a_{ij}}(t),\hspace{1em}i=1,2,\]]]> then the solution X(t) to the system (2) with the initial condition X0∈R+2 is stochastically permanent.

For the process Ui(t)=1/xi(t) , i=1,2 , by the Itô formula we have Ui(t)=Ui(0)+ ∫0tUi(s)−ai1(s)+ai2(s)x3−i(s)1+x3−i(s)+ci(s)xi(s)+σi2(s)+∫Rγi2(s,z)1+γi(s,z)Π1(dz)ds− ∫0tUi(s)σi(s)dwi(s)− ∫0t∫RUi(s)γi(s,z)1+γi(s,z)ν˜1(ds,dz)− ∫0t∫RUi(s)δi(s,z)1+δi(s,z)ν2(ds,dz).

Then by the Itô formula we derive for 0<θ<1 : (1+Ui(t))θ=(1+Ui(0))θ+ ∫0tθ(1+Ui(s))θ−2(1+Ui(s))Ui(s)×−ai1(s)+ai2(s)x3−i(s)1+x3−i(s)+ci(s)xi(s)+σi2(s)∫R+∫Rγi2(s,z)1+γi(s,z)Π1(dz)+θ−12Ui2(s)σi2(s)+1θ∫R(1+Ui(s))21+Ui(s)+γi(s,z)(1+γi(s,z))(1+Ui(s))θ−1+θ(1+Ui(s))Ui(s)γi(s,z)1+γi(s,z)Π1(dz) +1θ∫R(1+Ui(s))21+Ui(s)+δi(s,z)(1+δi(s,z))(1+Ui(s))θ−1Π2(dz)ds− ∫0tθ(1+Ui(s))θ−1Ui(s)σi(s)dwi(s)+ ∫0t∫R1+Ui(s)1+γi(s,z)θ−(1+Ui(s))θν˜1(ds,dz)+ ∫0t∫R1+Ui(s)1+δi(s,z)θ−(1+Ui(s))θν˜2(ds,dz)=(1+Ui(0))θ+ ∫0tθ(1+Ui(s))θ−2J(s)ds−I1,stoch(t)+I2,stoch(t)+I3,stoch(t), where Ii,stoch(t),i=1,3‾ , are corresponding stochastic integrals in (12). Under the Assumption 1 there exist constants |K1(θ)|<∞ , |K2(θ)|<∞ such that for the process J(t) we have the estimate J(t)≤(1+Ui(t))Ui(t)−aimin(t)+cmaxUi−1(t)+σi2(t)∫R+∫Rγi2(s,z)1+γi(s,z)Π1(dz)+θ−12Ui2(s)σi2(s)+1θ∫R(1+Ui(s))211+γi(s,z)+11+Ui(s)θ−1+θ(1+Ui(s))Ui(s)γi(s,z)1+γi(s,z)Π1(dz)+1θ∫R(1+Ui(s))211+δi(s,z)+11+Ui(s)θ−1Π2(dz)≤Ui2(t)−aimin(t)+σi2(t)2+∫Rγi(t,z)Π1(dz)+θ2σi2(t)+1θ∫R[(1+γi(t,z))−θ−1]Π1(dz)+1θ∫R[(1+δi(t,z))−θ−1]Π2(dz)+K1(θ)Ui(t)+K2(θ). Here we used the inequality (x+y)θ≤xθ+θxθ−1y , 0<θ<1 , x,y>0 0$]]>. Due to limθ→0+θ2σi2(t)+1θ∫R[(1+γi(t,z))−θ−1]Π1(dz)+1θ∫R[(1+δi(t,z))−θ−1]Π2(dz)=−∫Rln(1+γi(t,z))Π1(dz)−∫Rln(1+δi(t,z))Π2(dz), and the condition mini=1,2inft≥0(aimin(t)−βi(t))>0 0$]]> we can choose a sufficiently small 0<θ<1 to satisfy K0(θ)=mini=1,2inft≥0aimin(t)−σi2(t)2−∫Rγi(t,z)Π1(dz)−θ2σi2(t)−1θ∫R[(1+γi(t,z))−θ−1]Π1(dz)−1θ∫R[(1+δi(t,z))−θ−1]Π2(dz)>0. 0.\end{array}\]]]>

So from (12) and the estimate for J(t) we derive d(1+Ui(t))θ≤θ(1+Ui(t))θ−2[−K0(θ)Ui2(t)+K1(θ)Ui(t)+K2(θ)]dt−θ(1+Ui(t))θ−1Ui(t)σi(t)dwi(t)+∫R1+Ui(t)1+γi(t,z)θ−(1+Ui(t))θν˜1(dt,dz)+∫R1+Ui(t)1+δi(t,z)θ−(1+Ui(t))θν˜2(dt,dz). By the Itô formula and (13) we have deλt(1+Ui(t))θ=λeλt(1+Ui(t))θdt+eλtd(1+Ui(t))θ≤eλtθ(1+Ui(t))θ−2−K0(θ)−λθUi2(t)+K1(θ)+2λθUi(t)+K2(θ)+λθdt−θeλt(1+Ui(t))θ−1Ui(t)σi(t)dwi(t)+eλt∫R1+Ui(t)1+γi(t,z)θ−(1+Ui(t))θν˜1(dt,dz)+eλt∫R1+Ui(t)1+δi(t,z)θ−(1+Ui(t))θν˜2(dt,dz).

Let us choose λ>0 0$]]> such that K0(θ)−λ/θ>0 0$]]>. Then the function (1+Ui(t))θ−2−K0(θ)−λθUi2(t)+K1(θ)+2λθUi(t)+K2(θ)+λθ is bounded from above by some constant K>0 0$]]>. So by integrating (14) and taking the expectation we obtain eλtE(1+Ui(t))θ≤1+1xi0θ+λθKeλt−1, because the expectation of stochastic integrals are equal zero by (10) and E ∫0te2λs(1+Ui(s))2θ−2Ui2(s)σi2(s)ds=E ∫0te2λsσi2(s)(1+xi−1(s))2−2θxi2(s)ds≤ ∫0te2λsσi2(s)E[xi2θ]ds<∞,E ∫0t∫Re2λs1+Ui(s)1+γi(s,z)θ−(1+Ui(s))θ2Π1(dz)ds≤L1 ∫0te2λsExi2θ(s)ds<∞,E ∫0t∫Re2λs1+Ui(s)1+δi(s,z)θ−(1+Ui(s))θ2Π2(dz)ds≤L2 ∫0te2λsExi2θ(s)ds<∞, where L1=θ2max(γ2)max(1+γmin)2,(γ2)max(1+γmin)2θΠ1(R)<∞,L2=θ2max(δ2)max(1+δmin)2,(δ2)max(1+δmin)2θΠ2(R)<∞.

From (15) we obtain lim supt→∞E1xi(t)θ=lim supt→∞EUiθ(t)≤lim supt→∞E(1+Ui(t))θ≤θKλ,i=1,2.

From (11) and (16) by the Chebyshev inequality we can derive that for an arbitrary ε∈(0,1) there are positive constants H=H(ε) and h=h(ε) such that lim inft→∞P{xi(t)≤H}≥1−ε,lim inft→∞P{xi(t)≥h}≥1−ε,i=1,2.  □

 Extinction, nonpersistence and strong persistence in the mean

The property of extinction in the stochastic models of population dynamics means that every species will become extinct with probability 1.

The solution X(t)=(x1(t),x2(t)) , t≥0 , to the system (2) is said to be extinct if for every initial data X0>0 0$]]> we have limt→∞xi(t)=0 almost surely (a.s.), i=1,2 .

Let Assumption 1 be fulfilled. If p¯i∗=lim supt→∞1t ∫0tpimax(s)ds<0,wherepimax(s)=aimax(s)−βi(s), aimax(t)=maxj=1,2aij(t) , i=1,2 , then the solution X(t) to the system (2) with the initial condition X0∈R+2 will be extinct.

By the Itô formula, we have lnxi(t)=lnxi0+ ∫0t∫Rai1(s)+ai2(s)x3−i(s)1+x3−i(s)−ci(s)xi(s)−σi2(s)2+∫R[ln(1+γi(s,z))−γi(s,z)]Π1(dz)+∫Rln(1+δi(s,z))Π2(dz)ds+ ∫0tσi(s)dwi(s)+ ∫0t∫Rln(1+γi(s,z))ν˜1(ds,dz)+ ∫0t∫Rln(1+δi(s,z))ν˜2(ds,dz)≤lnxi0+ ∫0tpimax(s)ds+M(t), where the martingale M(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) has quadratic variation ⟨M,M⟩(t)= ∫0tσi2(s)ds+ ∫0t∫Rln2(1+γi(s,z))Π1(dz)ds+ ∫0t∫Rln2(1+δi(s,z))Π2(dz)ds≤Kt.

Then the strong law of large numbers for local martingales ([7]) yields limt→∞M(t)/t=0 a.s. Therefore, from (17) we have lim supt→∞lnxi(t)t≤lim supt→∞1t ∫0tpimax(s)ds<0,a.s. So limt→∞xi(t)=0 , i=1,2 , a.s.  □

([8]) The solution X(t)=(x1(t),x2(t)) , t≥0 , to the system (2) is said to be nonpersistence in the mean if for every initial data X0>0 0$]]> we have limt→∞1t ∫0txi(s)ds=0 a.s., i=1,2 .

Let Assumption 1 be fulfilled. If p¯i∗=0 , i=1,2 , then the solution X(t) to the system (2) with the initial condition X0∈R+2 will be nonpersistence in the mean.

From the first equality in (17) we have lnxi(t)≤lnxi0+ ∫0tpimax(s)ds−cmin ∫0txi(s)ds+M(t), where the martingale M(t) is defined in (18). From the definition of p¯i∗ and the strong law of large numbers for M(t) it follows that ∀ε>0 0$]]>, ∃t0≥0 such that 1t ∫0tpimax(s)ds≤p¯i∗+ε2,M(t)t≤ε2,∀t≥t0,a.s. So, from (19) we derive lnxi(t)−lnxi0≤t(p¯i∗+ε)−cmin ∫0txi(s)ds=tε−cmin ∫0txi(s)ds. Let yi(t)=∫0txi(s)ds , then from (20) we have lndyi(t)dt≤εt−cminyi(t)+lnxi0⇒ecminyi(t)dyi(t)dt≤xi0eεt. Integrating the last inequality from t0 to t yields ecminyi(t)≤cminxi0εeεt−eεt0+ecminyi(t0),∀t≥t0,a.s. So yi(t)≤1cminlnecminyi(t0)+cminxi0εeεt−eεt0,∀t≥t0,a.s., and therefore lim supt→∞1t ∫0txi(s)ds≤εcmin,a.s. Since ε>0 0$]]> is arbitrary and X(t)∈R+2 , we have limt→∞1t ∫0sxi(s)ds=0,i=1,2,a.s.  □

([8]) The solution X(t)=(x1(t),x2(t)) , t≥0 , to the system (2) is said to be strongly persistence in the mean if for every initial data X0>0 0$]]> we have lim inft→∞1t ∫0txi(s)ds>0 0$]]> a.s., i=1,2 .

Let Assumption 1 be fulfilled. If p¯i∗=lim inft→∞1t ∫0tpimin(s)ds>0 0$]]>, where pimin(s)=aimin(s)−βi(s) , aimin(t)=minj=1,2aij(t) , i=1,2 , then lim inft→∞1t ∫0txi(s)ds≥p¯i∗cisup. Therefore, the solution X(t) to the system (2) with the initial condition X0∈R+2 will be strongly persistence in the mean.

From the first equality in (17) we have lnxi(t)≥lnxi0+ ∫0tpimin(s)ds−cisup ∫0txi(s)ds+M(t), where the martingale M(t) is defined in (18). From the definition of p¯i∗ and the strong law of large numbers for M(t) it follows that ∀ε>0 0$]]>, ∃t0≥0 such that 1t ∫0tpimin(s)ds≥p¯i∗−ε2 , M(t)t≥−ε2 , ∀t≥t0 , a.s. So, from (21) we obtain lnxi(t)≥lnxi0+t(p¯i∗−ε)−cisup ∫0txi(s)ds. Let us choose sufficiently small ε>0 0$]]> such that p¯i∗−ε>0 0$]]>.

Let yi(t)=∫0txi(s)ds , then from (22) we have lndyi(t)dt≥(p¯i∗−ε)t−cisupyi(t)+lnxi0. Hence ecisupyi(t)dyi(t)dt≥xi0e(p¯i∗−ε)t . Integrating the last inequality from t0 to t yields ecisupyi(t)≥cisupxi0p¯i∗−εe(p¯i∗−ε)t−e(p¯i∗−ε)t0+ecisupyi(t0),∀t≥t0,a.s. So yi(t)≥1cisuplnecisupyi(t0)+cisupxi0p¯i∗−εe(p¯i∗−ε)t−e(p¯i∗−ε)t0,a.s., ∀t≥t0 , and therefore lim inft→∞1t ∫0txi(s)ds≥(p¯i∗−ε)cisup,a.s. Using the arbitrariness of ε>0 0$]]> we get the assertion of the theorem.  □

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