Stochastic two-species mutualism model with jumps
Olga Borysenko Oleksandr Borysenko
math.PR



Research Article




a]Olg.Olga Borysenko[label=e1]olga_borisenko@ukr.net
b]O.Oleksandr Borysenkocor1[label=e2]odb@univ.kiev.ua
[type=corresp,id=cor1]Corresponding author.
[a]Department of Mathematical Physics,
National Technical University of Ukraine, 37, Prosp. Peremohy, Kyiv, 03056, Ukraine

[b]Department of Probability Theory, Statistics and Actuarial
Mathematics, Taras Shevchenko National University of Kyiv, Ukraine,
64 Volodymyrska Str., Kyiv, 01601 Ukraine











Olg. Borysenko, O. BorysenkoStochastic two-species mutualism model with jumps



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


[MSC2010]
92D25
60H10
60H30



3 12 2019
21 2 2020
21 2 2020
3 3 2020







§ INTRODUCTION



The construction of the mutualism model and its properties are presented
in K. Gopalsamy <cit.>. Mutualism occurs when one species provides
some benefit in exchange for another benefit. A deterministic two-species
mutualism model is described by the system


    dN_1(t)/dt=r_1(t)N_1(t) [
    K_1(t)+α _1(t)N_2(t)/1+N_2(t)-N_1(t) ],
    
    dN_2(t)/dt=r_2(t)N_2(t) [
    K_2(t)+α _2(t)N_1(t)/1+N_1(t)-N_2(t) ],


where N_1(t) and N_2(t) denote the population densities of each species
at time t, r_i(t)>0, i=1,2, denotes the intrinsic growth rate of species
N_i,i=1,2, and α _i(t)>K_i(t)>0, i=1,2. The carrying capacity
of species N_i(t) is K_i(t), i=1,2, in the absence of other species.
In the paper by Hong Qiu, Jingliang Lv and Ke Wang <cit.>
the stochastic mutualism modelstochastic ! mutualism model of the form



    dx(t)=x(t) [a_1(t)+a_2(t)y(t)/1+y(t)-c_1(t)x(t)
     ]+σ _1(t)x(t)dw_1(t),
    
    
    dy(t)=y(t) [b_1(t)+b_2(t)x(t)/1+x(t)-c_2(t)y(t)
     ]+σ _2(t)y(t)dw_2(t)


is considered, where a_i(t), b_i(t), c_i(t), σ _i(t), i =1,2, are all positive,
continuous and bounded functions on [0,+∞ ), and
w_1(t), w_2(t) are independent Wiener processes. The authors show
that the stochastic system  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 permanencestochastic ! permanence and persistence in the mean of the solution to
the system  are obtained.

Population systems may suffer abrupt environmental perturbations,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 <cit.> the authors
consider the stochastic mutualism modelstochastic ! mutualism model with the white and centered Poisson
noises:


    dx(t)   =x(t^-) [0pt20pt (r_1(t)-
    b_1(t)x(t)/K_1(t)+y(t)-ε _1(t)x(t) )dt
     .
    
       +α _1(t)dw_1(t)+ .∫ _𝕐γ _1(t,z)
    ν̃(dt,dz) ],
    
    
    dy(t)   =y(t^-) [0pt20pt (r_2(t)-
    b_2(t)y(t)/K_2(t)+x(t)-ε _2(t)y(t) )dt
     .
    
       +α _2(t)dw_2(t)+ .∫ _𝕐γ _2(t,z)
    ν̃(dt,dz) ],


where x(t^-), y(t^-) are the left limit of x(t) and y(t) respectively,
r_i(t), b_i(t), K_i(t), α _i(t), i=1,2, are all positive,
continuous and bounded functions, 𝕐 is measurable subset of
(0,+∞ ), w_i(t), i=1,2, are independent standardindependent standard one-dimensional
Wiener processes, ν̃(t,A)=ν (t,A)-tΠ (A) is the centered
Poisson measure independent on w_i(t), i=1,2,
E[ν (t,A)]=tΠ (A), Π (𝕐)<∞,
γ _i(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,stochastic ! permanence persistencepersistence in the mean and extinctionextinction of the solution
are obtained.

In this paper, we consider the stochastic mutualism modelstochastic ! 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
equationsstochastic ! differential equations



    dx_i(t)=x_i(t) [
    a_i1(t)+a_i2(t)x_3-i(t)/1+x_3-i(t)-c_i(t)x_i(t)
     ]dt+σ _i(t)x_i(t)dw_i(t)
    
    
    +∫ _ℝγ _i(t,z)x_i(t)ν̃_1(dt,dz)+
    ∫ _ℝδ _i(t,z)x_i(t)ν _2(dt,dz),
    
    
    x_i(0)=x_i0>0,    i=1,2,.


where w_i(t), i=1,2, are independent standardindependent standard one-dimensional Wiener
processes, ν̃_1(t,A)=ν _1(t,A)-tΠ _1(A),
ν _i(t,A), i=1,2, are independent Poisson measures,independent Poisson measures which are independent
on w_i(t),i=1,2, E[ν _i(t,A)]=tΠ _i(A), i=1,2,
Π _i(A), i=1,2, are  finite measuresfinite 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 modelstochastic ! mutualism model , 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
<cit.>.

In the following we will use the notations
X(t)=(x_1(t),x_2(t)), X_0=(x_10,x_20),
|X(t)|=√(x_1^2(t)+x_2^2(t)),
ℝ^2_+={X∈ℝ^2: x_1>0,x_2>0},



    β _i(t)=σ ^2_i(t)/2+∫ _ℝ[
    γ _i(t,z)-ln (1+γ _i(t,z))]Π _1(dz)-∫ _ℝln (1+δ _i(t,z))]Π _2(dz),


i=1,2. For the bounded, continuous functions
f_i(t), t∈ [0,+∞ ), i=1,2, let us denote


    f_isup=sup _t≥ 0f_i(t),      f_iinf=inf _t≥ 0f_i(t),    i=1,2,
    
    
    f_max=max{f_1sup,f_2sup},      f_min=min{f_1inf,f_2
    inf}.


We will prove that system  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,stochastic ! permanence nonpersistence in the mean, strong persistence
in the mean and extinctionextinction of solution are derived.

The rest of this paper is organized as follows. In Section <ref>, we prove
the existence of the unique global positive solution to the system
. In Section <ref>, we prove the stochastic ultimate boundedness
of the solution to the system . In Section <ref>, we obtain
conditions under which the solution to the system  is stochastically
permanent,stochastically permanent and in Section <ref> the sufficient conditions for nonpersistence
in the mean, strong persistence in the mean and extinctionextinction of the solution
are obtained.




§ EXISTENCE OF THE GLOBAL SOLUTION



Let (Ω ,ℱ,P) be a probability space,
w_i(t), i=1, 2, t≥ 0, are independent standardindependent standard one-dimensional Wiener
processes on (Ω ,ℱ,P), and
ν _i(t,A), i=1,2, are independent Poisson measuresindependent Poisson measures defined on
(Ω ,ℱ,P) independent on w_i(t), i=1,2. Here
E[ν _i(t,A)]=tΠ _i(A), i=1,2,
ν̃_i(t,A)=ν _i(t,A)-tΠ _i(A), i=1,2,
Π _i(· ), i=1,2, are finite measuresfinite 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
{ℱ_t}_t≥ 0, where
ℱ_t=σ{w_i(s),ν _i(s,A), s≤ t,i=1,2}.

We need the following assumption.




It is assumed that a_ij(t),i,j=1,2,
c_i(t), σ _i(t), i=1,2, are bounded, continuous in t functions,
a_ij(t)>0, i,j=1,2, c_min>0,
γ _i(t,z), δ _i(t,z), i=1,2, are continuous in t functions
and ln (1+γ _i(t,z)),ln (1+δ _i(t,z)), i=1,2, are bounded,
Π _i(ℝ)<∞, i=1,2.





Let Assumption <ref> be fulfilled. Then there exists a unique global
solution X(t) to the system  for any initial value
X(0)=X_0>0, and P{X(t)∈ℝ^2_+}= 1.



Let us consider the system of stochastic differential equationsstochastic ! differential equations



    dv_i(t)= [
    a_i1(t)+a_i2(t)exp{v_3-i(t)}/1+exp{v_3-i(t)}-c_i(t)
    exp{v_i(t)}-β _i(t) ]dt
    
    
    +σ _i(t)dw_i(t) +∫ _ℝln (1+γ _i(t,z))
    ν̃_1(dt,dz)+∫ _ℝln (1+δ _i(t,z))
    ν̃_2(dt,dz),
    
    
    v_i(0)=ln x_i0,    i=1,2.


The coefficients of  equation  are locally Lipschitz continuous.locally Lipschitz continuous
Therefore, for any initial value (v_1(0),v_2(0)) there exists a unique
local solution V(t)=(v_1(t), v_2(t)) on [0,τ _e), where
sup _t<τ _e|V(t)|=+∞ (cf. Theorem 6, p. 246, <cit.>).
So, from the Itô formulaItô formula we derive that the process
X(t)=(exp{v_1(t)},exp{v_2(t)}) is a unique, positive local
solution to the system (<ref>). To show this solution is global,
we need to show that τ _e=+∞ a.s. Let
n_0∈ℕ be sufficiently large for
x_i0∈ [1/n_0,n_0], i=1,2. For any n≥ n_0 we define the
stopping time


    τ _n=inf{  t∈ [0,τ _e): X(t)∉ (
    1/n,n )× (1/n,n )}  .


It is clear that τ _n is increasing as n→ +∞. Set
τ _∞=lim _n→∞τ _n, whence
τ _∞≤τ _e a.s. If we prove that
τ _∞=∞ a.s., then τ _e=∞ 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 and ε∈ (0,1), such that
P{τ _∞<T}>ε. Hence, there is
n_1≥ n_0 such that



    P{τ _n<T}>ε ,  ∀ n≥ n_1.


For the nonnegative function
V(X)=∑ _i=1^2(x_i-1-ln x_i), x_i>0,
i=1,2, by the Itô formulaItô formula we have



    V(X(T∧τ _n))=V(X_0)+∑ _i=1^2{∫ _0^T
    ∧τ _n [0pt20pt(x_i(t)-1) (
    a_i1(t)+a_i2(t)x_3-i(t)/1+x_3-i(t) . .
     .
    
     . .-c_i(t)x_i(t)0pt14pt )+ β _i(t)+
    ∫ _ℝδ _i(t,z)x_i(t)Π _2(dz)
     ]dt
    
    
    +∫ _0^T∧τ _n(x_i(t)-1)
    σ _i(t)dw_i(t) +∫ _0^T∧τ _n∫ _ℝ[γ _i(t,z)x_i(t)-ln (1+
    γ _i(t,z))]ν̃_1(dt,dz)
    
     .+ ∫ _0^T∧τ _n∫ _ℝ[δ _i(t,z)x_i(t)-ln (1+δ _i(t,z))]
    ν̃_2(dt,dz)}  .


Under the conditions of the theorem there is a constant K>0 such that



    ∑ _i=1^2 [(x_i-1) (
    a_i1(t)+a_i2(t)x_3-i/1+x_3-i-c_i(t)x_i )+
    β _i(t) .
    
     .+∫ _ℝδ _i(t,z)x_iΠ _2(dz)
     ]≤∑ _i=1^2 [(a_max+c_max)x_i-c_minx_i^2+
    β _max .
    
     .+ x_iδ _maxΠ _2(ℝ) ]≤ K,


where a_max=max _i,j=1,2{sup _t≥ 0a_ij(t)}. From
 and  we have


    V(X(T∧τ _n))≤ V(X_0)+K(T∧τ _n)+∑ _i=1^2{∫ _0^T∧τ _n(x_i(t)-1)σ _i(t)dw_i(t)
     .
    
    
    + . ∫ _0^T∧τ _n∫ _ℝ[γ _i(t,z)x_i(t)-ln (1+γ _i(t,z))]
    ν̃_1(dt,dz) .
    
    
    + . ∫ _0^T∧τ _n∫ _ℝ[δ _i(t,z)x_i(t)-ln (1+δ _i(t,z))]
    ν̃_2(dt,dz)}  .


Whence, taking expectations yields



    E [V(X(T∧τ _n)) ]≤ V(X_0)+KT.


Set Ω _n={τ _n≤ T} for n≥ n_1. Then by
,
P(Ω _n)=P{τ _n≤ t}>ε,
∀ n≥ n_1. Note that for every ω∈Ω _n there
is some i such that x_i(τ _n,ω ) equals either n or
1/n. So


    V(X(τ _n))≥min{n-1-ln n,1/n-1+ln n}.


It then follows from  that


    V(X_0)+KT≥E[1_Ω _nV(X(τ _n))]
    ≥εmin{n-1-ln n,1/n-1+ln n},


where 1_Ω _n is the indicator function of
Ω _n. Letting n→∞ leads to the contradiction
∞ >V(X_0)+KT=∞. This completes the proof of the theorem.





§ STOCHASTICALLY ULTIMATE BOUNDEDNESS






(<cit.>) The solution X(t) to the system  is said
to be stochastically ultimately bounded, if for any
ε∈ (0,1) there is a positive constant
χ =χ (ε )>0 such that for any initial value
X_0∈ℝ^2_+

this solution has the property


    lim sup _t→∞P{|X(t)|>χ}<ε .







Under Assumption <ref> the solution X(t) to the system
 is stochastically ultimately bounded for any initial value
X_0∈ℝ^2_+.



Let τ _n be the stopping time defined in Theorem
<ref>. Applying the Itô formulaItô formula to the process
V(t,x_i(t))=e^tx_i^p(t), i=1,2, p>0, we obtain for
i=1,2 that



    V(t∧τ _n,x_i(t∧τ _n))=x_i0^p+∫ _0^t∧τ _ne^sx_i^p(s){0pt18pt1+p [
    a_i1(s)+a_i2(s)x_3-i(s)/1+x_3-i(s) . .
    
     . .- c_i(s)x_i(s) ]+p(p-1)σ _i^2(s)/2+∫ _ℝ [(1+γ _i(s,z))^p-1-pγ _i(s,z) ]
    Π _1(dz) .
    
     . +∫ _ℝ [(1+δ _i(s,z))^p-1 ]Π _2(dz) }  ds +∫ _0^t∧τ _np e^sx_i^p(s)σ _i(s)dw_i(s)
    
    
    +∫ _0^t∧τ _n∫ _ℝe^sx_i^p(s) [(1+γ _i(s,z))^p-1 ]
    ν̃_1(ds,dz)
    
    
    +∫ _0^t∧τ _n∫ _ℝe^sx_i^p(s) [(1+δ _i(s,z))^p-1 ]
    ν̃_2(ds,dz).


Under Assumption <ref> there is a constant K_i(p)>0 such that



    e^sx_i^p{0pt18pt1+p [
    a_i1(s)+a_i2(s)x_3-i/1+x_3-i- c_i(s)x_i ]+
    p(p-1)σ _i^2(s)/2+ .
    
     .+∫ _ℝ [(1+γ _i(s,z))^p-1-pγ _i(s,z) ]Π _1(dz) +∫ _ℝ [(1+δ _i(s,z))^p-1 ]Π _2(dz)
    }
    ≤ K_i(p)e^s.


From  and , taking expectations, we obtain


    E[V(t∧τ _n,x_i(t∧τ _n))]≤ x_i0^p+K(p)
    E[e^t∧τ _n]≤ x_i0^p+K_i(p)e^t.


Letting n→∞ leads to the estimate



    e^tE[x_i^p(t)]≤ x_i0^p+e^tK_i(p).


So we have for i=1,2 that



    limsup _t→∞E[x_i^p(t)]≤ K_i(p).


For X=(x_1,x_2)∈ℝ^2_+ we have
|X|^p≤ 2^p/2(x_1^p+x_2^p), therefore, from

limsup _t→∞E[|X(t)|^p]≤ L(p)=2^p/2(K_1(p)+K_2(p)).
Let χ >(L(p)/ε )^1/p, p>0,
∀ε∈ (0,1). Then applying the Chebyshev inequality
yields


    lim sup _t→∞P{|X(t)|>χ}≤1/χ ^plimsup _t→∞E[|X(t)|^p]≤L(p)/χ ^p<ε .







§ STOCHASTIC PERMANENCESTOCHASTIC ! PERMANENCE



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



(<cit.>) The solution X(t) to the system  is said to
be stochastically permanentstochastically permanent if for any ε >0 there are positive
constants H=H(ε ), h=h(ε ) such that


    lim inf _t→∞P{x_i(t)≤ H}≥ 1-
    ε ,    lim inf _t→∞P{x_i(t)
    ≥ h}≥ 1-ε ,


for i=1,2 and for any inial value X_0∈ℝ^2_+.




Let Assumption <ref> be fulfilled. If


    min _i=1,2inf _t≥ 0(a_imin(t)-β _i(t))>0,  where a_imin(t)=min _j=1,2a_ij(t),    i=1,2,


then the solution X(t) to the system  with the initial condition
X_0∈ℝ^2_+ is stochastically permanent.stochastically permanent




For the process U_i(t)=1/x_i(t), i=1,2, by the Itô formulaItô formula we
have


    U_i(t)=U_i(0)+∫ _0^t U_i(s)  [
    0pt20pt-a_i1(s)+a_i2(s)x_3-i(s)/1+x_3-i(s)+c_i(s)x_i(s)+
    σ _i^2(s) .
    
     .+∫ _ℝγ _i^2(s,z)/1+γ _i(s,z)Π _1(dz) ]ds -
    ∫ _0^tU_i(s)σ _i(s)dw_i(s)
    
    
    -∫ _0^t∫ _ℝU_i(s)
    γ _i(s,z)/1+γ _i(s,z)ν̃_1(ds,dz)-
    ∫ _0^t∫ _ℝU_i(s)
    δ _i(s,z)/1+δ _i(s,z)ν _2(ds,dz).


Then by the Itô formulaItô formula we derive for 0<θ <1:


    (1+U_i(t))^θ=(1+U_i(0))^θ+∫ _0^tθ (1+U_i(s))^θ -2{0pt20pt(1+U_i(s))U_i(s)
     .
    
    × [-a_i1(s)+a_i2(s)x_3-i(s)/1+x_3-i(s)+c_i(s)x_i(s)+
    σ _i^2(s)
    ∫ _ℝ .
    
     . +∫ _ℝγ _i^2(s,z)/1+γ _i(s,z)Π _1(dz) ] +
    θ -1/2U_i^2(s)σ _i^2(s)
    
    
    + 1/θ∫ _ℝ [(1+U_i(s))^2 ( (
    1+U_i(s)+γ _i(s,z)/(1+γ _i(s,z))(1+U_i(s)) )^θ-1 ) .
    
    
    + . θ (1+U_i(s))
    U_i(s)γ _i(s,z)/1+γ _i(s,z) ]Π _1(dz)




    . +1/θ∫ _ℝ(1+U_i(s))^2 [ (
    1+U_i(s)+δ _i(s,z)/(1+δ _i(s,z))(1+U_i(s)) )^θ-1 ]Π _2(dz)}  ds
    
    
    -∫ _0^tθ (1+U_i(s))^θ -1U_i(s)σ _i(s)dw_i(s)
    
    
    + ∫ _0^t∫ _ℝ [ (1+
    U_i(s)/1+γ _i(s,z) )^θ-(1+U_i(s))^θ ]ν̃_1(ds,dz)
    
    
    + ∫ _0^t∫ _ℝ [ (1+
    U_i(s)/1+δ _i(s,z) )^θ-(1+U_i(s))^θ ]ν̃_2(ds,dz)= (1+U_i(0))^θ
    
    +∫ _0^tθ (1+U_i(s))^θ -2J(s)ds-I_1,stoch(t)+I_2,stoch(t)+I_3,stoch(t),


where I_i,stoch(t), i=1,3, are corresponding stochastic integralsstochastic ! integrals
in . Under the Assumption <ref> there exist constants
|K_1(θ )|<∞, |K_2(θ )|<∞ such that for the
process J(t) we have the estimate


    J(t)≤ (1+U_i(t))U_i(t) [-a_imin(t)+c_maxU_i^-1(t)
    +σ _i^2(t)∫ _ℝ .
    
     .+∫ _ℝγ _i^2(s,z)/1+γ _i(s,z)Π _1(dz) ] +
    θ -1/2U_i^2(s)σ _i^2(s)
    
    
    +1/θ∫ _ℝ [(1+U_i(s))^2 ( (1/1+γ _i(s,z)+ 1/1+U_i(s) )^θ-1 ) .
    
    
    + . θ (1+U_i(s))
    U_i(s)γ _i(s,z)/1+γ _i(s,z) ]Π _1(dz)
    
    
    +1/θ∫ _ℝ(1+U_i(s))^2 [
     (1/1+δ _i(s,z)+ 1/1+U_i(s) )^θ-1 ]Π _2(dz)
    
    ≤ U_i^2(t) [-a_imin(t)+σ _i^2(t)/2 +
    ∫ _ℝγ _i(t,z)Π _1(dz)+
    θ/2σ _i^2(t) .
    
     .+1/θ∫ _ℝ[(1+γ _i(t,z))^-
    θ-1]Π _1(dz) +1/θ∫ _ℝ[(1+
    δ _i(t,z))^-θ-1]Π _2(dz) ]
    
    
    +K_1(θ )U_i(t)+K_2(θ ).


Here we used the inequality
(x+y)^θ≤ x^θ+θ x^θ -1y,
0<θ <1, x,y>0. Due to


    lim _θ→ 0+ [θ/2σ _i^2(t)+
    1/θ∫ _ℝ[(1+γ _i(t,z))^-
    θ-1]Π _1(dz) .
    
     . +1/θ∫ _ℝ[(1+δ _i(t,z))^-
    θ-1]Π _2(dz) ]
    
    
    =-∫ _ℝln (1+γ _i(t,z))Π _1(dz)-
    ∫ _ℝln (1+δ _i(t,z))Π _2(dz),


and the condition
min _i=1,2inf _t≥ 0(a_imin(t)-β _i(t))>0 we can choose
a sufficiently small 0<θ <1 to satisfy


    K_0(θ )=min _i=1,2inf _t≥ 0{  a_imin(t)-
    σ _i^2(t)/2-∫ _ℝγ _i(t,z)
    Π _1(dz)-θ/2σ _i^2(t) .
    
     . - 1/θ∫ _ℝ[(1+γ _i(t,z))^-
    θ-1]Π _1(dz) -1/θ∫ _ℝ[(1+
    δ _i(t,z))^-θ-1]Π _2(dz)}  >0.


So from  and the estimate for J(t) we derive



    d [(1+U_i(t))^θ ]≤θ (1+U_i(t))^θ -2[-K_0(
    θ )U_i^2(t)+K_1(θ )U_i(t)+K_2(θ )]dt
    
    
    -θ (1+U_i(t))^θ -1U_i(t)σ _i(t)dw_i(t)+
    ∫ _ℝ [ (1+
    U_i(t)/1+γ _i(t,z) )^θ .
    
     . -(1 +U_i(t))^θ0pt18pt ]ν̃_1(dt,dz)
    + ∫ _ℝ [ (1+
    U_i(t)/1+δ _i(t,z) )^θ -(1+U_i(t))^θ ]ν̃_2(dt,dz).


By the Itô formulaItô formula and  we have



    d [e^λ t(1+U_i(t))^θ ]=λ e^λ t(1+U_i(t))^θdt+e^λ t d [(1+U_i(t))^θ ]
    
    ≤ e^λ tθ (1+U_i(t))^θ -2 [- (K_0(
    θ )-λ/θ )U_i^2(t)+ (K_1(
    θ )+2λ/θ )U_i(t) .
    
     . +K_2(θ )+λ/θ ]dt-θ e^λ t(1+U_i(t))^θ -1U_i(t)σ _i(t)dw_i(t)
    
    
    +e^λ t∫ _ℝ [ (1+
    U_i(t)/1+γ _i(t,z) )^θ-(1+U_i(t))^θ ]ν̃_1(dt,dz)
    
     + e^λ t∫ _ℝ [ (1+
    U_i(t)/1+δ _i(t,z) )^θ-(1+U_i(t))^θ ]ν̃_2(dt,dz).


Let us choose λ >0 such that
K_0(θ )-λ /θ >0. Then the function


    (1+U_i(t))^θ -2 [- (K_0(θ )-
    λ/θ )U_i^2(t)+ (K_1(θ )+
    2λ/θ )U_i(t) +K_2(θ )+
    λ/θ ]


is bounded from above by some constant K>0. So by integrating
 and taking the expectation we obtain



    e^λ tE [(1+U_i(t))^θ ]≤ (1+
    1/x_i0 )^θ+λ/θK (e^λ t-1 ),


because the expectation of stochastic integralsstochastic ! integrals are equal zero by
 and


    E [∫ _0^t e^2λ s(1+U_i(s))^2
    θ -2U^2_i(s)σ _i^2(s)ds ]=E [
    ∫ _0^te^2λ sσ _i^2(s)/(1+x^-1_i(s))^2-2θx^2_i(s)ds
     ]
    
    ≤∫ _0^t e^2λ sσ _i^2(s)E[x_i^2
    θ]ds<∞ ,
    
    E [∫ _0^t∫ _ℝ
    e^2λ s [ (1+U_i(s)/1+γ _i(s,z) )^θ-(1+U_i(s))^θ ]^2Π _1(dz)ds
     ]
    
    ≤ L_1∫ _0^te^2λ sE [x_i^2
    θ(s) ]ds<∞ ,
    
    E [∫ _0^t∫ _ℝ
    e^2λ s [ (1+U_i(s)/1+δ _i(s,z) )^θ-(1+U_i(s))^θ ]^2Π _2(dz)ds
     ]
    
    ≤ L_2∫ _0^te^2λ sE [x_i^2
    θ(s) ]ds<∞ ,


where


    L_1=θ ^2max{(γ ^2)_max/(1+γ _min)^2,
    (γ ^2)_max/(1+γ _min)^2θ}Π _1(ℝ)<∞ ,
    
    
    L_2=θ ^2max{(δ ^2)_max/(1+δ _min)^2,
    (δ ^2)_max/(1+δ _min)^2θ}Π _2(ℝ)<∞ .


From  we obtain



    lim sup _t→∞E [ (1/x_i(t) )^θ ]=lim sup _t→∞E [U_i^θ(t) ]
    
    ≤lim sup _t→∞E [(1+U_i(t))^θ ]≤θ K/λ,    i=1,2.


From  and  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 inf _t→∞P{x_i(t)≤ H}≥ 1-
    ε ,    lim inf _t→∞P{x_i(t)
    ≥ h}≥ 1-ε ,    i=1,2.








§ EXTINCTION,EXTINCTION NONPERSISTENCE AND STRONG PERSISTENCE IN THE MEAN



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



The solution X(t)=(x_1(t),x_2(t)), t≥ 0, to the system
 is said to be extinctextinct if for every initial data
X_0>0 we have lim _t→∞x_i(t)=0 almost surely (a.s.),
i=1,2.




Let Assumption <ref> be fulfilled. If


    p̅^*_i=lim sup _t→∞1/t∫ _0^tp_i
    max(s)ds<0,  where p_imax(s)=a_imax(s)-β _i(s),


a_imax(t)=max _j=1,2a_ij(t), i=1,2, then the solution
X(t) to the system  with the initial condition
X_0∈ℝ^2_+ will be extinct.extinct



By the Itô formula,Itô formula we have



    ln x_i(t)=ln x_i0+∫ _0^t [∫ _ℝa_i1(s)+a_i2(s)x_3-i(s)/1+x_3-i(s)-c_i(s)x_i(s)-
    σ _i^2(s)/2 .
    
     .+ ∫ _ℝ[ln (1+γ _i(s,z))-γ _i(s,z)]
    Π _1(dz)+∫ _ℝln (1+δ _i(s,z))Π _2(dz)
     ]ds
    
    
    + ∫ _0^tσ _i(s)dw_i(s)+∫ _0^t∫ _ℝln (1+γ _i(s,z))ν̃_1(ds,dz)
    
    
    + ∫ _0^t∫ _ℝln (1+δ _i(s,z))
    ν̃_2(ds,dz)≤ln x_i0+∫ _0^tp_imax(s)ds+M(t),


where the martingale



    M(t)=∫ _0^tσ _i(s)dw_i(s)+∫ _0^t∫ _ℝln (1+γ _i(s,z))ν̃_1(ds,dz)
    
    
    + ∫ _0^t∫ _ℝln (1+δ _i(s,z))
    ν̃_2(ds,dz)


has quadratic variation


    ⟨ M,M⟩ (t)=∫ _0^tσ ^2_i(s)ds+
    ∫ _0^t∫ _ℝln ^2(1+
    γ _i(s,z))Π _1(dz)ds
    
    
    + ∫ _0^t∫ _ℝln ^2(1+
    δ _i(s,z))Π _2(dz)ds≤ Kt.


Then the strong law of large numbers for local martingales (<cit.>)
yieldslim _t→∞M(t)/t=0 a.s. Therefore, from
 we have


    lim sup _t→∞ln x_i(t)/t≤lim sup _t→∞1/t∫ _0^t p_imax(s)ds<0,   a.s.


So lim _t→∞x_i(t)=0, i=1,2, a.s.




(<cit.>) The solution X(t)=(x_1(t),x_2(t)), t≥ 0, to the system
 is said to be nonpersistence in the mean if for every initial
data X_0>0 we have
lim _t→∞1/t∫ _0^tx_i(s)ds=0 a.s.,
i=1,2.




Let Assumption <ref> be fulfilled. If
p̅_i^*=0, i=1,2, then the solution X(t) to the system
 with the initial condition X_0∈ℝ^2_+ will
be nonpersistence in the mean.



From the first equality in  we have



    ln x_i(t)≤ln x_i0+∫ _0^t p_imax(s)ds -c_min∫ _0^tx_i(s)ds+M(t),


where the martingale M(t)martingale is defined in . From the definition
of p̅^*_i and the strong law of large numbers for M(t) it
follows that ∀ε >0, ∃ t_0≥ 0 such that


    1/t∫ _0^t p_imax(s)ds≤p̅^*_i+
    ε/2, M(t)/t≤ε/2,  ∀ t≥ t_0, a.s.


So, from  we derive



    ln x_i(t)- ln x_i0≤ t(p̅^*_i+ε )-c_min∫ _0^tx_i(s)ds=tε -c_min∫ _0^tx_i(s)ds.


Let y_i(t) =∫ _0^tx_i(s)ds, then from  we have


    ln (dy_i(t)/dt )≤ε t-c_min y_i(t)+
    ln x_i0⇒ e^c_miny_i(t)dy_i(t)/dt≤ x_i0e^ε t.


Integrating the last inequality from t_0 to t yields


    e^c_miny_i(t)≤c_minx_i0/ε (e^ε t-e^ε t_0 ) +e^c_miny_i(t_0),
      ∀ t≥ t_0, a.s.


So


    y_i(t)≤1/c_minln [e^c_miny_i(t_0)+
    c_minx_i0/ε (e^ε t-e^ε t_0 )  ],   ∀ t≥ t_0, a.s.,


and therefore


    lim sup _t→∞1/t∫ _0^tx_i(s)ds≤ε/c_min,   a.s.


Since ε >0 is arbitrary and
X(t)∈ℝ^2_+, we have


    lim _t→∞1/t∫ _0^sx_i(s)ds=0,    i=1,2,
     a.s.






(<cit.>) The solution X(t)=(x_1(t),x_2(t)), t≥ 0, to the system
 is said to be strongly persistencepersistence in the mean if for every
initial data X_0>0 we have
lim inf _t→∞1/t∫ _0^tx_i(s)ds>0 a.s.,
i=1,2.




Let Assumption <ref> be fulfilled. If
p̅_i*=lim inf _t→∞1/t∫ _0^tp_i
min(s)ds>0, where p_imin(s)=a_imin(s)-β _i(s),
a_imin(t)=min _j=1,2a_ij(t), i=1,2, then


    lim inf _t→∞1/t∫ _0^tx_i(s)ds≥p̅_i*/c_isup.


Therefore, the solution X(t) to the system  with the initial
condition X_0∈ℝ^2_+ will be strongly persistencepersistence in
the mean.



From the first equality in  we have



    ln x_i(t)≥ln x_i0+∫ _0^t p_imin(s)ds -c_i
    sup∫ _0^tx_i(s)ds+M(t),


where the martingale M(t)martingale is defined in . From the definition
of p̅_i* and the strong law of large numbers for M(t) it follows
that ∀ε >0, ∃ t_0≥ 0 such that
1/t∫ _0^t p_imin(s)ds≥p̅_i*-
ε/2, M(t)/t≥ -ε/2, ∀ t≥ t_0, a.s. So, from  we obtain



    ln x_i(t)≥ln x_i0+ t(p̅_i*-ε )-c_isup∫ _0^tx_i(s)ds.


Let us choose sufficiently small ε >0 such that
p̅_i*-ε >0.

Let y_i(t) =∫ _0^tx_i(s)ds, then from  we have


    ln (dy_i(t)/dt )≥ (p̅_i*-
    ε ) t-c_isup y_i(t)+ln x_i0.


Hence
e^c_isupy_i(t)dy_i(t)/dt≥ x_i0e^( p̅_i*-
ε ) t. Integrating the last inequality from t_0 to t yields


    e^c_isupy_i(t)≥c_isupx_i0/p̅_i*-ε (e^(p̅_i*-
    ε ) t-e^(p̅_i*-ε ) t_0 ) +e^c_i
    supy_i(t_0),  ∀ t≥ t_0,  a.s.


So


    y_i(t)≥1/c_isupln [e^c_isupy_i(t_0)+
    c_isupx_i0/p̅_i*-ε (e^(p̅_i*-
    ε ) t-e^(p̅_i*-ε ) t_0 )
     ], a.s.,


∀ t≥ t_0, and therefore


    lim inf _t→∞1/t∫ _0^tx_i(s)ds≥(p̅_i*-ε )/c_isup,   a.s.


Using the arbitrariness of ε >0 we get the assertion of the
theorem.














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