# Long-time dynamics of a stochastic density dependent predator-prey model
with Holling II functional response and jumps

Olg.Olga<EMAIL_ADDRESS>[
O.Oleksandr<EMAIL_ADDRESS>[ 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

(2022)

###### Abstract

The existence and uniqueness of a global positive solution is proven for the
system of stochastic differential equations describing a nonautonomous
stochastic density dependent predator-prey model with Holling-type II
functional response disturbed by white noise, centered and non-centered
Poisson noises. Sufficient conditions are obtained for stochastic ultimate
boundedness, stochastic permanence, non-persistence in the mean, weak
persistence in the mean and extinction of a population densities in the
considered stochastic predator-prey model.

Stochastic Predator-Prey Model,

Predator Density Dependence,

Holling-type II functional response,

Global Solution,

Stochastic Ultimate Boundedness,

Stochastic Permanence,

Extinction,

Non-Persistence in the Mean,

Weak Persistence in the Mean,

92D25,

60H10,

60H30,

###### keywords:

###### keywords:

[MSC2010]

††volume: 0††issue: 0††articletype: research-article

Research Article

[id=cor1]Corresponding author.

##  1 Introduction

The deterministic autonomous Rosenzweig-Mac’Arthur model ([1]) is a
generalization of Voltera-Verhulst model in that the linear functional
response is replaced by Holling II functional response. This model has a form

$\displaystyle
dx_{1}(t)=x_{1}(t)\left(a_{1}-bx_{1}(t)-\frac{cx_{2}(t)}{1+mx_{1}(t)}\right)dt,$
$\displaystyle dx_{2}(t)=x_{2}(t)\left(-a_{2}+\frac{\kappa
cx_{1}(t)}{1+mx_{1}(t)}\right)dt,$ (1)

where $x_{1}(t)$ and $x_{2}(t)$ are the prey and predator population densities
at time $t$, respectively; $a_{1}>0$ is the growth rate of prey $x_{1}$; $b>0$
measures the strength of competition among individuals of species $x_{1}$; $c$
is the maximum ingestion rate; $m>0$ is the half-saturation; $a_{2}>0$ is the
death rate of predator $x_{2}$, and $\kappa>0$ is the conversion factor.

In the paper [2] the stochastic version of model $(\ref{eq1})$ is considered
in the following form

$\displaystyle
dx_{1}(t)=x_{1}(t)\left(a_{1}-bx_{1}(t)-\frac{cx_{2}(t)}{1+mx_{1}(t)}\right)dt+\sigma_{1}x_{1}(t)dw_{1}(t),$
$\displaystyle dx_{2}(t)=x_{2}(t)\left(-a_{2}+\frac{\kappa
cx_{1}(t)}{1+mx_{1}(t)}\right)dt+\sigma_{2}x_{2}(t)dw_{2}(t),$ (2)

where $w_{1}(t)$ and $w_{2}(t)$ are mutually independent Wiener processes. In
[2] the authors proved that there is a unique positive solution to the system
$(\ref{eq2})$, deduced the conditions that there is a stationary distribution
of the system, which implies that the system is permanent, and obtained the
sufficient conditions for the system that is going to be extinct.

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.
There are considerable evidences in nature that predator species may be
density dependent. So we need to take into account levels of predator density
dependence.

In this paper, we consider the non-autonomous density dependent predator-prey
model with Holling-type II functional response, disturbed by white noise and
jumps generated by centered and non-centered 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 non-centered Poisson
measure. This model is driven by the system of stochastic differential
equations

$\displaystyle
dx_{i}(t)=x_{i}(t)\left[(-1)^{i-1}\left(a_{i}(t)-\frac{c_{i}(t)x_{3-i}(t)}{1+m(t)x_{1}(t)}\right)-b_{i}(t)x_{i}(t)\right]dt$
$\displaystyle+\sigma_{i}(t)x_{i}(t)dw_{i}(t)\\!\\!+\\!\\!\int\limits_{\mathbb{R}}\gamma_{i}(t,z)x_{i}(t^{-})\tilde{\nu}_{1}(dt,dz)\\!+\\!\\!\int\limits_{\mathbb{R}}\delta_{i}(t,z)x_{i}(t^{-})\nu_{2}(dt,dz),$
$\displaystyle x_{i}(0)=x_{i0}>0,\ i=1,2.$ (3)

where $x_{1}(t)$ and $x_{2}(t)$ are the prey and predator population densities
at time $t$, respectively, $b_{2}(t)>0$ is the predator density dependence
rate, $c_{2}(t)=\kappa c_{1}(t)$, $w_{i}(t),i=1,2$ are independent standard
one-dimensional Wiener processes, $\nu_{i}(t,A),i=1,2$ are independent Poisson
measures, which are independent on $w_{i}(t),i=1,2$,
$\tilde{\nu}_{1}(t,A)=\nu_{1}(t,A)-t\Pi_{1}(A)$,
$E[\nu_{i}(t,A)]=t\Pi_{i}(A),i=1,2$, $\Pi_{i}(A),i=1,2$ are finite measures on
the Borel sets $A$ in $\mathbb{R}$.

To the best of our knowledge, there have been no papers devoted to the
dynamical properties of the stochastic predator-prey model $(\ref{eq3})$, even
in the case of centered Poisson noise. It is worth noting that the impact of
centered and non-centered Poisson noises to the stochastic non-autonomous
logistic model, to the stochastic two-species mutualism model and to the
stochastic predator-prey model with modified version of Leslie-Gower term and
Holling-type II functional response is studied in the papers [3] – [6].

In the following we will use the notations $X(t)=(x_{1}(t),x_{2}(t))$,
$X_{0}=(x_{10},x_{20})$, $|X(t)|=\sqrt{x_{1}^{2}(t)+x_{2}^{2}(t)}$,
$\mathbb{R}^{2}_{+}=\\{X\in\mathbb{R}^{2}:\ x_{1}>0,x_{2}>0\\}$,

$\displaystyle\alpha_{i}(t)=a_{i}(t)+\int_{\mathbb{R}}\delta_{i}(t,z)\Pi_{2}(dz),$
$\displaystyle\beta_{i}(t)\\!=\\!\frac{\sigma^{2}_{i}(t)}{2}\\!+\\!\\!\\!\int\limits_{\mathbb{R}}[\gamma_{i}(t,z)\\!-\\!\ln(1\\!+\\!\gamma_{i}(t,z))]\Pi_{1}(dz)\\!-\\!\\!\\!\int\limits_{\mathbb{R}}\ln(1\\!+\\!\delta_{i}(t,z))\Pi_{2}(dz),$

$i=1,2$. For the bounded, continuous functions
$f(t),f_{i}(t),t\in[0,+\infty),i=1,2$, let us denote

$\displaystyle f_{\sup}=\sup_{t\geq 0}f(t),f_{\inf}=\inf_{t\geq
0}f(t),f_{i\sup}=\sup_{t\geq 0}f_{i}(t),f_{i\inf}=\inf_{t\geq
0}f_{i}(t),i=1,2,$ $\displaystyle
f_{\max}=\max\\{f_{1\sup},f_{2\sup}\\},f_{\min}=\min\\{f_{1\inf},f_{2\inf}\\}.$

We prove that the system $(\ref{eq3})$ 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, non-persistence 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 $(\ref{eq3})$
and derive some auxiliary results. In Section 3, we prove the stochastic
ultimate boundedness of the solution to the system $(\ref{eq3})$, obtainig
conditions under which the solution is stochastically permanent. The
sufficient conditions for non-persistence in the mean, weak persistence in the
mean and extinction of the solution are derived.

##  2 Existence of global solution and some auxiliary lemmas

Let $(\Omega,{\cal F},\mathrm{P})$ be a probability space,
$w_{i}(t),i=1,2,t\geq 0$ are independent standard one-dimensional Wiener
processes on $(\Omega,{\cal F},\mathrm{P})$, and $\nu_{i}(t,A),i=1,2$ are
independent Poisson measures defined on $(\Omega,{\cal F},\mathrm{P})$
independent on $w_{i}(t),i=1,2$. Here
$\mathrm{E}[\nu_{i}(t,A)]=t\Pi_{i}(A),i=1,2$,
$\tilde{\nu}_{i}(t,A)=\nu_{i}(t,A)-t\Pi_{i}(A),i=1,2$, $\Pi_{i}(\cdot),i=1,2$
are finite measures on the Borel sets in $\mathbb{R}$. On the probability
space $(\Omega,{\cal F},\mathrm{P})$ we consider an increasing, right
continuous family of complete sub-$\sigma$-algebras $\\{{\cal F}_{t}\\}_{t\geq
0}$, where ${\cal F}_{t}=\sigma\\{w_{i}(s),\nu_{i}(s,A),s\leq t,i=1,2\\}$.

We need the following assumption.

######  Assumption 1.

It is assumed, that $a_{i}(t),b_{i}(t)$,
$c_{i}(t),\sigma_{i}(t),\gamma_{i}(t,z),\delta_{i}(t,z),i=1,2$, $m(t)$ are
bounded, continuous on $t$ functions, $a_{i}(t)>0,i=1,2$, $b_{i\inf}>0$,
$c_{i\inf}>0,i=1,2$, $m_{\inf}>0$, and
$\ln(1+\gamma_{i}(t,z)),\ln(1+\delta_{i}(t,z)),i=1,2$ are bounded,
$\Pi_{i}(\mathbb{R})<\infty,i=1,2$.

In what follows we will assume that Assumption 1 holds.

######  Theorem 1.

There exists a unique global solution $X(t)$ to the system $(\ref{eq3})$ for
any initial value $X(0)=X_{0}\in\mathbb{R}^{2}_{+}$, and
$\mathrm{P}\\{X(t)\in\mathbb{R}^{2}_{+}\\}=1$.

###### Proof.

Let us consider the system of stochastic differential equations

$\displaystyle
d\xi_{i}(t)=\left[(-1)^{i-1}\left(a_{i}(t)-\frac{c_{i}(t)\exp\\{\xi_{3-i}(t)\\}}{1+m(t)\exp\\{\xi_{1}(t)\\}}\right)-b_{i}(t)\exp\\{\xi_{i}(t)\\}\right.$
$\displaystyle\left.-\beta_{i}(t)\rule{0.0pt}{14.0pt}\right]dt+\sigma_{i}(t)dw_{i}(t)+\int\limits_{\mathbb{R}}\ln(1+\gamma_{i}(t,z))\tilde{\nu}_{1}(dt,dz)$
$\displaystyle+\int\limits_{\mathbb{R}}\ln(1+\delta_{i}(t,z))\tilde{\nu}_{2}(dt,dz),\quad\xi_{i}(0)=\ln
x_{i0},\ i=1,2.$ (4)

The coefficients of the system $(\ref{eq4})$ are local Lipschitz continuous.
So, for any initial value $(\xi_{1}(0),\xi_{2}(0))$ there exists a unique
local solution $\Xi(t)=(\xi_{1}(t),\xi_{2}(t))$ on $[0,\tau_{e})$, where
$\sup_{t<\tau_{e}}|\Xi(t)|=+\infty$ (cf. Theorem 6, p.246, [7]). Therefore,
from the Itô’s formula we derive that the process
$X(t)=(\exp\\{\xi_{1}(t)\\},\linebreak\exp\\{\xi_{2}(t)\\})$ is a unique,
positive local solution to the system (1). To show this solution is global, we
need to show that $\tau_{e}=+\infty$ a.s. Let $n_{0}\in\mathbb{N}$ be
sufficiently large for $x_{i0}\in[1/n_{0},n_{0}],i=1,2$. For any $n\geq n_{0}$
we define the stopping time

$\displaystyle\tau_{n}=\inf\left\\{t\in[0,\tau_{e}):\
X(t)\notin\left(\frac{1}{n},n\right)\times\left(\frac{1}{n},n\right)\right\\}.$

It is easy to see that $\tau_{n}$ is increasing as $n\to+\infty$. Denote
$\tau_{\infty}=\lim_{n\to\infty}\tau_{n}$, whence $\tau_{\infty}\leq\tau_{e}$
a.s. If we prove that $\tau_{\infty}=\infty$ a.s., then $\tau_{e}=\infty$ a.s.
and $X(t)\in\mathbb{R}^{2}_{+}$ a.s. for all $t\in[0,+\infty)$. So we need to
show that $\tau_{\infty}=\infty$ a.s. If it is not true, there are constants
$T>0$ and $\varepsilon\in(0,1)$, such that
$\mathrm{P}\\{\tau_{\infty}<T\\}>\varepsilon$. Hence, there is $n_{1}\geq
n_{0}$ such that

$\displaystyle\mathrm{P}\\{\tau_{n}<T\\}>\varepsilon,\quad\forall n\geq
n_{1}.$ (5)

For the non-negative function $V(X)=\sum\limits_{i=1}^{2}k_{i}(x_{i}-1-\ln
x_{i})$, $X=(x_{1},x_{2})$, $x_{i}>0$, $k_{i}>0$, $i=1,2$ by the Itô’s formula
we obtain

$\displaystyle
dV(X(t))=\sum_{i=1}^{2}k_{i}\left\\{\rule{0.0pt}{20.0pt}(-1)^{i-1}(x_{i}(t)-1)\left(a_{i}(t)-\frac{c_{i}(t)x_{3-i}(t)}{1+m(t)x_{1}(t)}\right)\right.$
$\displaystyle\left.-b_{i}(t)x_{i}(t)(x_{i}(t)-1)+\beta_{i}(t)+\int\limits_{\mathbb{R}}\delta_{i}(t,z)x_{i}(t)\Pi_{2}(dz)\right\\}dt$
$\displaystyle\\!+\\!\sum_{i=1}^{2}k_{i}\\!\left\\{\\!\rule{0.0pt}{20.0pt}(x_{i}(t)\\!-\\!1)\sigma_{i}(t)dw_{i}(t)\right.\\!\\!+\\!\\!\int\limits_{\mathbb{R}}\\![\gamma_{i}(t,z)x_{i}(t^{-})\\!-\\!\ln(1\\!+\\!\gamma_{i}(t,z))]\tilde{\nu}_{1}(dt,dz)$
$\displaystyle\left.\\!+\\!\int\limits_{\mathbb{R}}[\delta_{i}(t,z)x_{i}(t^{-})\\!-\\!\ln(1\\!+\\!\delta_{i}(t,z))]\tilde{\nu}_{2}(dt,dz)\right\\}.$
(6)

Let us consider the function

$\displaystyle
f(t,x_{1},x_{2})=-k_{1}b_{1}(t)x_{1}^{2}+k_{1}\left(\rule{0.0pt}{12.0pt}\alpha_{1}(t)+b_{1}(t)\right)x_{1}+k_{1}(\beta_{1}(t)-a_{1}(t))$
$\displaystyle-
k_{2}b_{2}(t)x_{2}^{2}+k_{2}\left(-a_{2}(t)+b_{2}(t)+\int_{\mathbb{R}}\delta_{2}(t,z)\Pi_{2}(dz)\right)x_{2}+k_{2}(\beta_{2}(t)+a_{2}(t))$
$\displaystyle+\frac{k_{2}(x_{2}-1)c_{2}(t)x_{1}-k_{1}(x_{1}-1)c_{1}(t)x_{2}}{1+m(t)x_{1}},x_{i}>0,i=1,2.$

We have estimate

$\displaystyle
f(t,x_{1},x_{2})\leq\sum_{i=1}^{2}k_{i}(\rule{0.0pt}{14.0pt}-b_{i\inf}x_{i}^{2}+(\alpha_{i\sup}+b_{i\sup})x_{i})+k_{1}c_{1\sup}x_{2}$
$\displaystyle+k_{1}(\beta_{1\sup}-a_{1\inf})+k_{2}(\beta_{2\sup}+a_{2\sup})+\frac{c_{1}(t)x_{1}x_{2}(k_{2}\kappa-
k_{1})}{1+m(t)x_{1}}$

So for $k_{2}=k_{1}/\kappa$ there is a constant $L(k_{1},k_{2})>0$, such that
$f(t,x_{1},x_{2})\leq L(k_{1},k_{2})$. From $(\ref{eq6})$ we obtain by
integrating

$\displaystyle V(X(T\wedge\tau_{n}))\leq
V(X_{0})+L(k_{1},k_{2})(T\wedge\tau_{n})$
$\displaystyle+\sum_{i=1}^{2}k_{i}\left\\{\int\limits_{0}^{T\wedge\tau_{n}}(x_{i}(t)-1)\sigma_{i}(t)dw_{i}(t)+\int\limits_{0}^{T\wedge\tau_{n}}\\!\\!\\!\int\limits_{\mathbb{R}}\left[\gamma_{i}(t,z)x_{i}(t^{-})\\!-\\!\ln(1\right.\right.$
$\displaystyle\left.\left.+\gamma_{i}(t,z))\right]\tilde{\nu}_{1}(dt,dz)+\int\limits_{0}^{T\wedge\tau_{n}}\\!\\!\\!\int\limits_{\mathbb{R}}[\delta_{i}(t,z)x_{i}(t^{-})\\!-\\!\ln(1\\!+\\!\delta_{i}(t,z))]\tilde{\nu}_{2}(dt,dz)\right\\}.$
(7)

Taking expectations we derive from $(\ref{eq7})$

$\displaystyle\mathrm{E}\left[V(X(T\wedge\tau_{n}))\right]\leq
V(X_{0})+L(k_{1},k_{2})T.$ (8)

Set $\Omega_{n}=\\{\tau_{n}\leq T\\}$ for $n\geq n_{1}$. Then by
$(\ref{eq5})$, $\mathrm{P}(\Omega_{n})=\mathrm{P}\\{\tau_{n}\leq
T\\}>\varepsilon$, $\forall n\geq n_{1}$. Note that for every
$\omega\in\Omega_{n}$ at least one of $x_{1}(\tau_{n},\omega)$ and
$x_{2}(\tau_{n},\omega)$ equals either $n$ or $1/n$. So

$\displaystyle V(X(\tau_{n}))\geq\min\\{k_{1},k_{2}\\}\min\left\\{n-1-\ln
n,\frac{1}{n}-1+\ln n\right\\}.$

From $(\ref{eq8})$ it follows

$\displaystyle
V(X_{0})+L(k_{1},k_{2})T\geq\mathrm{E}[\mathbf{1}_{\Omega_{n}}V(X(\tau_{n}))]$
$\displaystyle\geq\varepsilon\min\\{k_{1},k_{2}\\}\min\left\\{n-1-\ln
n,\frac{1}{n}-1+\ln n\right\\},$

where $\mathbf{1}_{\Omega_{n}}$ is the indicator function of $\Omega_{n}$.
Letting $n\to\infty$ leads to the contradiction
$\infty>V(X_{0})+L(k_{1},k_{2})T=\infty$. This completes the proof of the
theorem. ∎

######  Lemma 1.

The density of the population $x_{i}(t),i=1,2$ obeys

$\displaystyle\limsup_{t\to\infty}\frac{\ln x_{i}(t)}{t}\leq 0,\
i=1,2\qquad\hbox{a.s.}$

###### Proof.

By the Itô’s formula we have for $i=1,2$

$\displaystyle e^{t}\ln x_{i}(t)-\ln
x_{i0}=\int\limits_{0}^{t}e^{s}\left\\{\ln
x_{i}(s)+(-1)^{i-1}\left[a_{i}(s)-\frac{c_{i}(s)x_{3-i}(s)}{1+m(s)x_{1}(s)}\right]\right.$
$\displaystyle\left.-b_{i}(s)x_{i}(s)-\frac{\sigma_{i}^{2}(s)}{2}+\int\limits_{\mathbb{R}}[\ln(1+\gamma_{i}(s,z))-\gamma_{i}(s,z)]\Pi_{1}(dz)\right\\}ds+\psi_{i}(t),$
(9)

where

$\displaystyle\psi_{i}(t)=\int\limits_{0}^{t}e^{s}\sigma_{i}(s)dw_{i}(s)+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}e^{s}\ln(1+\gamma_{i}(s,z))\tilde{\nu}_{1}(ds,dz)$
$\displaystyle+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}e^{s}\ln(1+\delta_{i}(s,z))\nu_{2}(ds,dz),\
i=1,2.$

By virtue of the exponential inequality ([3], Lemma 2.2) we have

$\displaystyle\mathrm{P}\left\\{\sup_{0\leq t\leq
T}\zeta_{i}(\mu,t)>\beta\right\\}\leq e^{-\mu\beta},\ \forall 0<\mu\leq 1,\
\beta>0,\ i=1,2$

where

$\displaystyle\zeta_{i}(\mu,t)=\psi_{i}(t)-\frac{\mu}{2}\int\limits_{0}^{t}e^{2s}\sigma_{i}^{2}(s)ds-\frac{1}{\mu}\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\left[(1+\gamma_{i}(s,z))^{\mu
e^{s}}-1\right.$ $\displaystyle\left.-\mu
e^{s}\ln(1\\!+\\!\gamma_{i}(s,z))\rule{0.0pt}{14.0pt}\right]\Pi_{1}(dz)ds\\!-\\!\frac{1}{\mu}\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\left[(1\\!+\\!\delta_{i}(s,z))^{\mu
e^{s}}\\!-\\!1\right]\Pi_{2}(dz)ds,$

$i=1,2$. Choosing $T=k\tau,k\in\mathbb{N},\tau>0,\mu=e^{-k\tau},\beta=\theta
e^{k\tau}\ln k$, $\theta>1$ we get

$\displaystyle\mathrm{P}\left\\{\sup_{0\leq t\leq
k\tau}\zeta_{i}(\mu,t)>\theta e^{k\tau}\ln
k\right\\}\leq\frac{1}{k^{\theta}},\ i=1,2.$

By the Borel-Cantelli lemma for almost all $\omega\in\Omega$, there is a
random integer $k_{0}(\omega)$, such that for $\forall k\geq k_{0}(\omega)$
and $0\leq t\leq k\tau$

$\displaystyle\psi_{i}(t)\leq\frac{1}{2e^{k\tau}}\int\limits_{0}^{t}e^{2s}\sigma_{i}^{2}(s)ds+e^{k\tau}\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\left[(1+\gamma_{i}(s,z))^{e^{s-k\tau}}-1\right.$
$\displaystyle\left.-e^{s-k\tau}\ln(1+\gamma_{i}(s,z))\right]\Pi_{1}(dz)ds+e^{k\tau}\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\left[(1+\delta_{i}(s,z))^{e^{s-k\tau}}\right.$
$\displaystyle\left.-1\rule{0.0pt}{14.0pt}\right]\Pi_{2}(dz)ds+\theta
e^{k\tau}\ln k,\ i=1,2.$ (10)

Appling the inequality $x^{r}\leq 1+r(x-1)$, $\forall x\geq 0$, $0\leq r\leq
1$ with $x=1+\gamma_{i}(s,z)$, $r=e^{s-k\tau}$, then with
$x=1+\delta_{i}(s,z)$, $r=e^{s-k\tau}$, we derive from $(\ref{eq10})$ the
estimates

$\displaystyle\psi_{i}(t)\\!\leq\\!\frac{1}{2e^{k\tau}}\int\limits_{0}^{t}e^{2s}\sigma_{i}^{2}(s)ds\\!+\\!\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}e^{s}\left[\gamma_{i}(s,z)\\!-\\!\ln(1\\!+\\!\gamma_{i}(s,z))\right]\Pi_{1}(dz)ds$
$\displaystyle+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}e^{s}\delta_{i}(s,z)\Pi_{2}(dz)ds+\theta
e^{k\tau}\ln k,\ i=1,2.$ (11)

So from $(\ref{eq9})$ and $(\ref{eq11})$ we get for $i=1,2$

$\displaystyle e^{t}\ln x_{i}(t)\leq\ln
x_{i0}+\int\limits_{0}^{t}e^{s}\left\\{\ln
x_{i}(s)+(-1)^{i-1}\left[a_{i}(s)-\frac{c_{i}(s)x_{3-i}(s)}{1+m(s)x_{1}(s)}\right]\right.$
$\displaystyle\left.-b_{i}(s)x_{i}(s)-\frac{\sigma_{i}^{2}(s)}{2}\left(1-e^{s-k\tau}\right)+\int\limits_{\mathbb{R}}\delta_{i}(s,z)\Pi_{2}(dz)\right\\}ds+\theta
e^{k\tau}\ln k$ $\displaystyle\leq\ln x_{i0}+\int\limits_{0}^{t}e^{s}[\ln
x_{i}(s)-b_{i\inf}x_{i}(s)+K_{i}]ds+\theta e^{k\tau}\ln k$
$\displaystyle\leq\ln x_{i0}+L(e^{t}-1)+\theta e^{k\tau}\ln k,\forall k\geq
k_{0}(\omega),0\leq t\leq k\tau,$

for some constant $L>0$, where $K_{1}=\alpha_{1\sup},K_{2}=\sup_{t\geq
0}\int_{\mathbb{R}}\delta_{2}(t,z)\Pi_{2}(dz)+c_{2\sup}/m_{\inf}$.

So for any $(k-1)\tau\leq t\leq k\tau$, $\forall k\geq k_{0}(\omega)$ we have

$\displaystyle\frac{\ln x_{i}(t)}{\ln t}\leq e^{-t}\frac{\ln x_{i0}}{\ln
t}+\frac{L}{\ln t}(1-e^{-t})+\frac{\theta e^{k\tau}\ln
k}{e^{(k-1)\tau}\ln(k-1)\tau},i=1,2\quad\hbox{a.s.}$

Therefore

$\displaystyle\limsup_{t\to\infty}\frac{\ln x_{i}(t)}{\ln t}\leq\theta
e^{\tau},\ i=1,2,\ \forall\theta>1,\ \forall\tau>0,\quad\hbox{a.s.}$

If $\theta\downarrow 1$, $\tau\downarrow 0$, then we obtain

$\displaystyle\limsup_{t\to\infty}\frac{\ln x_{i}(t)}{\ln t}\leq 1,\
i=1,2\quad\hbox{a.s.}$

So

$\displaystyle\limsup_{t\to\infty}\frac{\ln x_{i}(t)}{t}\leq 0,\
i=1,2\quad\hbox{a.s.}$

∎

######  Lemma 2.

Let $p>0$.Then for any initial value $x_{i0}>0,i=1,2$ we have

$\displaystyle\limsup_{t\to\infty}\mathrm{E}\left[x_{i}^{p}(t)\right]\leq
K_{i}(p),\ i=1,2,$

where $K_{i}(p)>0,i=1,2$ are some constants depending on $p$.

###### Proof.

Let $\tau_{n}$ be the stopping time defined in Theorem 1. Applying the Itô’s
formula to the process $V(t,x_{i}(t))=e^{t}x_{i}^{p}(t)$, $i=1,2,p>0$, we
obtain for $i=1,2$

$\displaystyle V(t\wedge\tau_{n},x_{i}(t\wedge\tau_{n}))=x_{i0}^{p}\\!$
$\displaystyle+\\!\\!\\!\\!\\!\int\limits_{0}^{t\wedge\tau_{n}}\\!\\!\\!\\!e^{s}x_{i}^{p}(s)\\!\\!\left\\{\\!\rule{0.0pt}{18.0pt}1\\!+\\!p\left[(-1)^{i-1}\rule{0.0pt}{16.0pt}\\!\\!\left(a_{i}(s)\\!-\\!\frac{c_{i}(s)x_{3-i}(s)}{1+m(s)x_{1}(s)}\right)\left.\rule{0.0pt}{16.0pt}-b_{i}(s)x_{i}(s)\right]\right.\right.$
$\displaystyle+\frac{p(p-1)\sigma_{i}^{2}(s)}{2}+\int\limits_{\mathbb{R}}\left[(1+\gamma_{i}(s,z))^{p}-1-p\gamma_{i}(s,z)\right]\Pi_{1}(dz)$
$\displaystyle\left.+\\!\\!\\!\int\limits_{\mathbb{R}}\\!\left[(1\\!+\\!\delta_{i}(s,z))^{p}\\!-\\!1\right]\Pi_{2}(dz)\right\\}ds\\!+\\!\\!\\!\\!\int\limits_{0}^{t\wedge\tau_{n}}\\!\\!\\!\\!pe^{s}x_{i}^{p}(s)\sigma_{i}(s)dw_{i}(s)\\!$
$\displaystyle+\\!\\!\\!\\!\int\limits_{0}^{t\wedge\tau_{n}}\\!\\!\\!\\!\int\limits_{\mathbb{R}}\\!\\!e^{s}x_{i}^{p}(s^{-})\left[(1\\!+\\!\gamma_{i}(s,z))^{p}\\!-\\!1\right]\tilde{\nu}_{1}(ds,dz)$
$\displaystyle+\int\limits_{0}^{t\wedge\tau_{n}}\\!\\!\\!\int\limits_{\mathbb{R}}e^{s}x_{i}^{p}(s^{-})\left[(1+\delta_{i}(s,z))^{p}-1\right]\tilde{\nu}_{2}(ds,dz).$
(12)

Under Assumption 1 there are a constants $K_{i}(p)>0,i=1,2$, such that

$\displaystyle
e^{s}x_{i}^{p}(s)\left\\{\rule{0.0pt}{18.0pt}1\\!+\\!p\left[(-1)^{i-1}\rule{0.0pt}{16.0pt}\left(a_{i}(s)\\!-\\!\frac{c_{i}(s)x_{3-i}(s)}{1+m(s)x_{1}(s)}\right)-b_{i}(s)x_{i}(s)\right]\right.$
$\displaystyle\left.+\frac{p(p-1)\sigma_{i}^{2}(s)}{2}+\int\limits_{\mathbb{R}}\left[(1+\gamma_{i}(s,z))^{p}-1-p\gamma_{i}(s,z)\right]\Pi_{1}(dz)\right.$
$\displaystyle\left.+\\!\\!\int\limits_{\mathbb{R}}\\!\left[(1\\!+\\!\delta_{i}(s,z))^{p}\\!-\\!1\right]\Pi_{2}(dz)\right\\}\leq
e^{s}K_{i}(p)$ (13)

From $(\ref{eq12})$ and $(\ref{eq13})$, taking the expectation, we obtain

$\displaystyle\mathrm{E}\left[V(t\wedge\tau_{n},x_{i}(t\wedge\tau_{n}))\right]\leq
x_{i0}^{p}+K_{i}(p)e^{t},i=1,2.$

If $n\to\infty$, then we get

$\displaystyle e^{t}\mathrm{E}\left[x_{i}^{p}(t)\right]\leq
x_{i0}^{p}+K_{i}(p)e^{t},\ i=1,2.$

Hence $\limsup_{t\to\infty}\mathrm{E}\left[x_{i}^{p}(t)\right]\leq K_{i}(p),\
i=1,2$. ∎

######  Lemma 3.

If $p_{2\inf}>0$, where $p_{2}(t)=-a_{2}(t)-\beta_{2}(t)$, then for any
initial value $x_{20}>0$, the predator population density $x_{2}(t)$ has the
property that

$\displaystyle\limsup_{t\to\infty}\mathrm{E}\left[\left(\frac{1}{x_{2}(t)}\right)^{\theta}\right]\leq
K(\theta),\ 0<\theta<1,$ (14)

###### Proof.

For the process $U(t)=1/x_{2}(t)$ by the Itô’s formula we have

$\displaystyle
U(t)=U(0)+\int\limits_{0}^{t}U(s)\left[\rule{0.0pt}{20.0pt}a_{2}(s)-\frac{c_{2}(s)x_{1}(s)}{1+m(s)x_{1}(s)}+b_{2}(s)x_{2}(s)+\sigma_{2}^{2}(s)\right.$
$\displaystyle\left.+\int\limits_{\mathbb{R}}\frac{\gamma_{2}^{2}(s,z)}{1+\gamma_{2}(s,z)}\Pi_{1}(dz)\right]ds-\int\limits_{0}^{t}U(s)\sigma_{2}(s)dw_{2}(s)$
$\displaystyle-\int\limits_{0}^{t}\\!\\!\\!\int\limits_{\mathbb{R}}U(s^{-})\frac{\gamma_{2}(s,z)}{1+\gamma_{2}(s,z)}\tilde{\nu}_{1}(ds,dz)-\int\limits_{0}^{t}\\!\\!\\!\int\limits_{\mathbb{R}}U(s^{-})\frac{\delta_{2}(s,z)}{1+\delta_{2}(s,z)}\nu_{2}(ds,dz).$

Then by the Itô’s formula we derive for $0<\theta<1$

$\displaystyle(1+U(t))^{\theta}=(1+U(0))^{\theta}+\int\limits_{0}^{t}\theta(1+U(s))^{\theta-2}\left\\{\rule{0.0pt}{20.0pt}(1+U(s))U(s)\right.$
$\displaystyle\times\left[a_{2}(s)-\frac{c_{2}(s)x_{1}(s)}{1+m(s)x_{1}(s)}+b_{2}(s)x_{2}(s)\sigma_{2}^{2}(s)\\!+\\!\int\limits_{\mathbb{R}}\frac{\gamma_{2}^{2}(s,z)}{1+\gamma_{2}(s,z)}\Pi_{1}(dz)\right]$

$\displaystyle+\frac{\theta-1}{2}U^{2}(s)\sigma_{2}^{2}(s)$
$\displaystyle+\frac{1}{\theta}\int\limits_{\mathbb{R}}\left[(1+U(s))^{2}\left(\left(\frac{1+U(s)+\gamma_{2}(s,z)}{(1+\gamma_{2}(s,z))(1+U(s))}\right)^{\theta}-1\right)\right.$
$\displaystyle+\left.\theta(1+U(s))\frac{U(s)\gamma_{2}(s,z)}{1+\gamma_{2}(s,z)}\right]\Pi_{1}(dz)$

$\displaystyle\left.+\frac{1}{\theta}\int\limits_{\mathbb{R}}(1+U(s))^{2}\left[\left(\frac{1+U(s)+\delta_{2}(s,z)}{(1+\delta_{2}(s,z))(1+U(s))}\right)^{\theta}-1\right]\Pi_{2}(dz)\right\\}ds$
$\displaystyle-\int\limits_{0}^{t}\theta(1+U(s))^{\theta-1}U(s)\sigma_{2}(s)dw_{2}(s)$
$\displaystyle+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\left[\left(1+\frac{U(s^{-})}{1+\gamma_{2}(s,z)}\right)^{\theta}\\!-\\!(1\\!+\\!U(s^{-}))^{\theta}\right]\tilde{\nu}_{1}(ds,dz)$

$\displaystyle+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\left[\left(1+\frac{U(s^{-})}{1\\!+\\!\delta_{2}(s,z)}\right)^{\theta}\\!-\\!(1\\!+\\!U(s^{-}))^{\theta}\right]\tilde{\nu}_{2}(ds,dz)=(1+U(0))^{\theta}$
$\displaystyle+\int\limits_{0}^{t}\theta(1+U(s))^{\theta-2}J(s)ds-
I_{1,stoch}(t)+I_{2,stoch}(t)+I_{3,stoch}(t),$ (15)

where $I_{j,stoch}(t),j=\overline{1,3}$ are corresponding stochastic integrals
in $(\ref{eq15})$. Under Assumption 1 there exist constants
$|K_{1}(\theta)|<\infty$, $|K_{2}(\theta)|<\infty$ such, that for the process
$J(t)$ we have the estimate

$\displaystyle
J(t)\leq(1+U(t))U(t)\left[a_{2}(t)+b_{2}(t)U^{-1}(t)+\sigma_{2}^{2}(t)\\!\\!\phantom{\int\limits_{\mathbb{R}}}\right.$
$\displaystyle\left.+\int\limits_{\mathbb{R}}\gamma_{2}(s,z)\Pi_{1}(dz)\right]+\frac{\theta-1}{2}U^{2}(s)\sigma_{2}^{2}(s)$
$\displaystyle+\frac{1}{\theta}\int\limits_{\mathbb{R}}(1+U(s))^{2}\left[\left(\frac{1}{1+\gamma_{2}(s,z)}+\frac{1}{1+U(s)}\right)^{\theta}-1\right]\Pi_{1}(dz)$
$\displaystyle+\frac{1}{\theta}\int\limits_{\mathbb{R}}(1+U(s))^{2}\left[\left(\frac{1}{1+\delta_{2}(s,z)}+\frac{1}{1+U(s)}\right)^{\theta}-1\right]\Pi_{2}(dz)$
$\displaystyle\leq
U^{2}(t)\left[a_{2}(t)+\frac{\sigma_{2}^{2}(t)}{2}+\int\limits_{\mathbb{R}}\gamma_{2}(t,z)\Pi_{1}(dz)+\frac{\theta}{2}\sigma_{2}^{2}(t)\right.$
$\displaystyle\left.+\frac{1}{\theta}\int\limits_{\mathbb{R}}[(1+\gamma_{2}(t,z))^{-\theta}-1]\Pi_{1}(dz)+\frac{1}{\theta}\int\limits_{\mathbb{R}}[(1+\delta_{2}(t,z))^{-\theta}-1]\Pi_{2}(dz)\right]$
$\displaystyle+K_{1}(\theta)U(t)+K_{2}(\theta)=-K_{0}(t,\theta)U^{2}(t)+K_{1}(\theta)U(t)+K_{2}(\theta).$

Here we used the inequality $(x+y)^{\theta}\leq x^{\theta}+\theta
x^{\theta-1}y$, $0<\theta<1$, $x,y>0$, in the first integral term for
$x=(1+\gamma_{2}(s,z))^{-1}$, $y=(1+U(s))^{-1}$, and then in the second
integral term for $x=(1+\delta_{2}(s,z))^{-1}$, $y=(1+U(s))^{-1}$. Due to

$\displaystyle\lim_{\theta\to
0+}\left[\frac{\theta}{2}\sigma_{i}^{2}(t)+\frac{1}{\theta}\int\limits_{\mathbb{R}}[(1+\gamma_{i}(t,z))^{-\theta}-1]\Pi_{1}(dz)\right.$
$\displaystyle\left.+\frac{1}{\theta}\int\limits_{\mathbb{R}}[(1+\delta_{i}(t,z))^{-\theta}-1]\Pi_{2}(dz)+\int\limits_{\mathbb{R}}\ln(1+\gamma_{i}(t,z))\Pi_{1}(dz)\right.$
$\displaystyle\left.+\int\limits_{\mathbb{R}}\ln(1+\delta_{i}(t,z))\Pi_{2}(dz)\right]=\lim_{\theta\to
0+}\Delta(\theta,t)=0,$

and the condition $p_{2\inf}>0$ we can choose a sufficiently small
$0<\theta<1$ so that

$\displaystyle K_{0}(\theta)=\inf_{t\geq 0}K_{0}(t,\theta)=\inf_{t\geq
0}[p_{2}(t)-\Delta(\theta,t)]>0$

is satisfied. So from $(\ref{eq15})$ and the estimate for $J(t)$ we derive

$\displaystyle
d\left[(1+U(t))^{\theta}\right]\leq\theta(1+U(t))^{\theta-2}[-K_{0}(\theta)U^{2}(t)+K_{1}(\theta)U(t)+K_{2}(\theta)]dt$
$\displaystyle-\theta(1+U(t))^{\theta-1}U(t)\sigma_{2}(t)dw_{2}(t)+\int\limits_{\mathbb{R}}\left[\left(1+\frac{U(t^{-})}{1+\gamma_{2}(t,z)}\right)^{\theta}\right.$
$\displaystyle\left.\\!-(1\\!+\\!\\!U(t^{-}))^{\theta}\rule{0.0pt}{18.0pt}\right]\\!\tilde{\nu}_{1}(dt,dz)\\!\\!+\\!\\!\\!\int\limits_{\mathbb{R}}\\!\left[\left(1\\!+\\!\frac{U(t^{-})}{1\\!+\\!\delta_{2}(t,z)}\right)^{\theta}\\!\\!\\!\\!-\\!(1\\!+U(t^{-}))^{\theta}\right]\\!\tilde{\nu}_{2}(dt,dz).$
(16)

By the Itô’s formula and $(\ref{eq16})$ we have

$\displaystyle d\left[e^{\lambda t}(1+U(t))^{\theta}\right]=\lambda e^{\lambda
t}(1+U(t))^{\theta}dt+e^{\lambda t}d\left[(1+U(t))^{\theta}\right]$
$\displaystyle\leq e^{\lambda
t}\theta(1+U(t))^{\theta-2}\left[-\left(K_{0}(\theta)-\frac{\lambda}{\theta}\right)U^{2}(t)+\left(K_{1}(\theta)+\frac{2\lambda}{\theta}\right)U(t)\right.$
$\displaystyle\left.+K_{2}(\theta)+\frac{\lambda}{\theta}\right]dt-\theta
e^{\lambda t}(1+U(t))^{\theta-1}U(t)\sigma_{2}(t)dw_{2}(t)$
$\displaystyle+e^{\lambda
t}\int\limits_{\mathbb{R}}\left[\left(1+\frac{U(t^{-})}{1+\gamma_{2}(t,z)}\right)^{\theta}-(1+U(t^{-}))^{\theta}\right]\tilde{\nu}_{1}(dt,dz)$
$\displaystyle\displaystyle+e^{\lambda
t}\int\limits_{\mathbb{R}}\left[\left(1+\frac{U(t^{-})}{1+\delta_{2}(t,z)}\right)^{\theta}-(1+U(t^{-}))^{\theta}\right]\tilde{\nu}_{2}(dt,dz).$
(17)

Let us choose $\lambda=\lambda(\theta)>0$ such, that
$K_{0}(\theta)-\lambda/\theta>0$. Then there is a constant $K>0$, such that

$\displaystyle(1+U(t))^{\theta-2}\left[-\left(K_{0}(\theta)-\frac{\lambda}{\theta}\right)U^{2}(t)\right.$
$\displaystyle\left.+\left(K_{1}(\theta)+\frac{2\lambda}{\theta}\right)U(t)+K_{2}(\theta)+\frac{\lambda}{\theta}\right]\leq
K.$ (18)

Let $\tau_{n}$ be the stopping time defined in the Theorem 1. Then by
integrating $(\ref{eq17})$, using $(\ref{eq18})$ and taking the expectation we
obtain

$\displaystyle\mathrm{E}\left[e^{\lambda(t\wedge\tau_{n})}(1+U(t\wedge\tau_{n}))^{\theta}\right]\leq\left(1+\frac{1}{x_{20}}\right)^{\theta}+\frac{\theta}{\lambda}K\left(e^{\lambda
t}-1\right).$

Letting $n\to\infty$ leads to the estimate

$\displaystyle
e^{t}\mathrm{E}\left[(1+U(t))^{\theta}\right]\leq\left(1+\frac{1}{x_{20}}\right)^{\theta}+\frac{\theta}{\lambda}K\left(e^{\lambda
t}-1\right).$ (19)

From $(\ref{eq19})$ we obtain

$\displaystyle\limsup_{t\to\infty}\mathrm{E}\left[\left(\frac{1}{x_{2}(t)}\right)^{\theta}\right]=\limsup_{t\to\infty}\mathrm{E}\left[U^{\theta}(t)\right]$
$\displaystyle\leq\limsup_{t\to\infty}\mathrm{E}\left[(1+U(t))^{\theta}\right]\leq\frac{\theta}{\lambda(\theta)}K,$

this implies $(\ref{eq14})$. ∎

##  3 The long time behaviour

######  Definition 1.

([8]) The solution $X(t)$ to the system $(\ref{eq3})$ will be said
stochastically ultimately bounded, if for any $\varepsilon\in(0,1)$, there is
a positive constant $\chi=\chi(\varepsilon)>0$, such that for any initial
value $X_{0}\in\mathbb{R}^{2}_{+}$, the solution to the system $(\ref{eq3})$
has the property that

$\displaystyle\limsup_{t\to\infty}\mathrm{P}\left\\{|X(t)|>\chi\right\\}<\varepsilon.$

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

######  Theorem 2.

The solution $X(t)$ to the system $(\ref{eq3})$ is stochastically ultimately
bounded for any initial value $X_{0}\in\mathbb{R}^{2}_{+}$.

###### Proof.

From Lemma 2 we have estimate

$\displaystyle\limsup_{t\to\infty}E[x_{i}(t)]\leq K_{i},\ i=1,2.$ (20)

For $X=(x_{1},x_{2})\in\mathbb{R}^{2}_{+}$ we have $|X|\leq x_{1}+x_{2}$,
therefore, from $(\ref{eq32})$$\limsup_{t\to\infty}E[|X(t)|]\leq
L=K_{1}+K_{2}$. Let $\chi>L/\varepsilon$, $\forall\varepsilon\in(0,1)$. Then
applying the Chebyshev inequality yields

$\displaystyle\limsup_{t\to\infty}\mathrm{P}\\{|X(t)|>\chi\\}\leq\frac{1}{\chi}\lim\sup_{t\to\infty}E[|X(t)|]\leq\frac{L}{\chi}<\varepsilon.$

∎

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

######  Definition 2.

The population density $x(t)$ will be said stochastically permanent if for any
$\varepsilon>0$, there are positive constants $H=H(\varepsilon)$,
$h=h(\varepsilon)$ such that

$\displaystyle\liminf\limits_{t\to\infty}\mathrm{P}\\{x(t)\leq H\\}\geq
1-\varepsilon,\quad\liminf\limits_{t\to\infty}\mathrm{P}\\{x(t)\geq h\\}\geq
1-\varepsilon,$

for any inial value $x(0)=x_{0}>0$.

######  Theorem 3.

If $p_{2\inf}>0$, where $p_{2}(t)=-a_{2}(t)-\beta_{2}(t)$, then for any
initial value $x_{20}>0$, the predator population density $x_{2}(t)$ is
stochastically permanent.

###### Proof.

From Lemma 2 we have estimate

$\displaystyle\limsup_{t\to\infty}E[x_{2}(t)]\leq K.$

Thus for any given $\varepsilon>0$, let $H=K/\varepsilon$, by virtue of
Chebyshev’s inequality, we can derive that

$\displaystyle\limsup\limits_{t\to\infty}\mathrm{P}\\{x_{2}(t)\geq
H\\}\leq\frac{1}{H}\limsup_{t\to\infty}E[x_{2}(t)]\leq\varepsilon.$

Consequently $\liminf\limits_{t\to\infty}\mathrm{P}\\{x_{2}(t)\leq H\\}\geq
1-\varepsilon$.

From Lemma 3 we have the estimate

$\displaystyle\limsup_{t\to\infty}\mathrm{E}\left[\left(\frac{1}{x_{2}(t)}\right)^{\theta}\right]\leq
K(\theta),\ 0<\theta<1.$

For any given $\varepsilon>0$, let $h=(\varepsilon/K(\theta))^{1/\theta}$,
then by Chebyshev’s inequality, we have

$\displaystyle\limsup\limits_{t\to\infty}\mathrm{P}\\{x_{2}(t)<h\\}\leq\limsup\limits_{t\to\infty}\mathrm{P}\left\\{\left(\frac{1}{x_{2}(t)}\right)^{\theta}>h^{-\theta}\right\\}$
$\displaystyle\leq
h^{\theta}\limsup\limits_{t\to\infty}\mathrm{E}\left[\left(\frac{1}{x_{2}(t)}\right)^{\theta}\right]\leq\varepsilon.$

Consequently $\liminf\limits_{t\to\infty}\mathrm{P}\\{x_{2}(t)\geq h\\}\geq
1-\varepsilon$. ∎

If the predator is absent, i.e. $x_{2}(t)\equiv 0$ a.s., then the equation for
prey population is the non-autonomous stochastic logistic differential
equation. The sufficient conditions for the stochastic permanence of the
solution to the stochastic logistic differential equation are obtained in [6]:

######  Theorem 4.

If the predator is absent, i.e. $x_{2}(t)\equiv 0$ a.s., and $p_{1\inf}>0$,
where $p_{1}(t)=a_{1}(t)-\beta_{1}(t)$, then for any initial value $x_{10}>0$,
the prey population density $x_{1}(t)$ is stochastically permanent.

######  Definition 3.

The solution $X(t)=(x_{1}(t),x_{2}(t))$, $t\geq 0$ to the system $(\ref{eq3})$
will be said extinct if for every initial data $X_{0}\in\mathbb{R}^{2}_{+}$,
we have $\lim_{t\to\infty}x_{i}(t)=0$ a.s., $i=1,2$.

######  Theorem 5.

If
${\bar{q}}^{*}_{i}=\limsup_{t\to\infty}\frac{1}{t}\int\limits_{0}^{t}q_{i}(s)ds<0$,
where

$\displaystyle q_{1}(t)=a_{1}(t)-\beta_{1}(t),\
q_{2}(t)=-a_{2}(t)+\frac{c_{2}(t)}{m(t)}-\beta_{2}(t)$

then the solution $X(t)$ to the system $(\ref{eq3})$ with the initial
condition $X_{0}\in\mathbb{R}^{2}_{+}$ will be extinct.

###### Proof.

By the Itô’s formula, we have

$\displaystyle\ln x_{i}(t)=\ln
x_{i0}+\int_{0}^{t}\left\\{(-1)^{i-1}\left[a_{i}(s)-\frac{c_{i}(s)x_{3-i}(s)}{1+m(s)x_{1}(s)}\right]\right.$
$\displaystyle\left.-\beta_{i}(s)-b_{i}(s)x_{i}(s)\rule{0.0pt}{14.0pt}\right\\}ds\\!+\\!M_{i}(t)\leq\ln
x_{i0}\\!+\\!\int_{0}^{t}q_{i}(s)ds\\!+\\!M_{i}(t),i=1,2$ (21)

where the martingale

$\displaystyle
M_{i}(t)=\int\limits_{0}^{t}\sigma_{i}(s)dw_{i}(s)+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\ln(1+\gamma_{i}(s,z))\tilde{\nu}_{1}(ds,dz)$
$\displaystyle+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\ln(1+\delta_{i}(s,z))\tilde{\nu}_{2}(ds,dz),\
i=1,2,$ (22)

has quadratic variation (Meyer’s angle bracket process)

$\displaystyle\langle
M_{i},M_{i}\rangle(t)=\int\limits_{0}^{t}\sigma^{2}_{i}(s)ds+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\ln^{2}(1+\gamma_{i}(s,z))\Pi_{1}(dz)ds$
$\displaystyle+\int\limits_{0}^{t}\\!\\!\int\limits_{\mathbb{R}}\ln^{2}(1+\delta_{i}(s,z))\Pi_{2}(dz)ds\leq
Kt,\ i=1,2.$

Then the strong law of large numbers for local martingales ([9]) yields
$\lim_{t\to\infty}M_{i}(t)/t=0,i=1,2$ a.s. Therefore, from $(\ref{eq21})$ we
obtain

$\limsup_{t\to\infty}\frac{\ln
x_{i}(t)}{t}\leq\limsup_{t\to\infty}\frac{1}{t}\int\limits_{0}^{t}q_{i}(s)ds<0,\quad\hbox{a.s.}$

So $\lim_{t\to\infty}x_{i}(t)=0,i=1,2$ a.s. ∎

######  Definition 4.

([10]) The solution $X(t)=(x_{1}(t),x_{2}(t))$, $t\geq 0$ to the system
$(\ref{eq3})$ will be said non-persistent in the mean if for every initial
data $X_{0}\in\mathbb{R}^{2}_{+}$, we have
$\lim_{t\to\infty}\frac{1}{t}\int\limits_{0}^{t}x_{i}(s)ds=0$ a.s., $i=1,2$.

######  Theorem 6.

If ${\bar{q}}_{i}^{*}=0$, $i=1,2$, then the solution
$X(t)=(x_{1}(t),x_{2}(t))$ to the system $(\ref{eq3})$ with the initial value
$X_{0}\in\mathbb{R}^{2}_{+}$ will be non-persistent in the mean.

###### Proof.

From the first equality in $(\ref{eq21})$ we have

$\displaystyle\ln x_{i}(t)\leq\ln x_{i0}+\int\limits_{0}^{t}q_{i}(s)ds-
b_{i\inf}\int\limits_{0}^{t}x_{i}(s)ds+M_{i}(t),\ i=1,2,$ (23)

where martingales $M_{i}(t),i=1,2$ are defined in $(\ref{eq22})$. From the
definition of ${\bar{q}}^{*}_{i},i=1,2$ and the strong law of large numbers
for $M_{i}(t),i=1,2$ it follows, that $\forall\varepsilon>0$, $\exists
t_{0}\geq 0$, and $\exists\Omega_{\varepsilon}\subset\Omega$,
$\mathrm{P}(\Omega_{\varepsilon})\geq 1-\varepsilon$ such that

$\displaystyle\frac{1}{t}\int\limits_{0}^{t}q_{i}(s)ds\leq{\bar{q}}^{*}_{i}+\frac{\varepsilon}{2},\
\frac{M_{i}(t)}{t}\leq\frac{\varepsilon}{2},\ i=1,2,\ \forall t\geq t_{0},\
\omega\in\Omega_{\varepsilon}.$

So, from $(\ref{eq23})$ we derive

$\displaystyle\ln x_{i}(t)-\ln x_{i0}\leq
t({\bar{q}}^{*}_{i}+\varepsilon)-b_{i\inf}\int\limits_{0}^{t}x_{i}(s)ds$
$\displaystyle=t\varepsilon-b_{i\inf}\int\limits_{0}^{t}x_{i}(s)ds,\ i=1,2,\
\forall t\geq t_{0},\ \omega\in\Omega_{\varepsilon}.$ (24)

Let $y_{i}(t)=\int_{0}^{t}x_{i}(s)ds,i=1,2$, then from $(\ref{eq24})$ we have
for $i=1,2$

$\displaystyle\ln\left(\frac{dy_{i}(t)}{dt}\right)\leq\varepsilon
t-b_{i\inf}y_{i}(t)+\ln x_{i0},\ \forall t\geq t_{0},\
\omega\in\Omega_{\varepsilon}.$

Therefore

$\displaystyle e^{b_{i\inf}y_{i}(t)}\frac{dy_{i}(t)}{dt}\leq
x_{i0}e^{\varepsilon t},\ i=1,2,\ \forall t\geq t_{0},\
\omega\in\Omega_{\varepsilon}.$

By integrating the last inequality from $t_{0}$ to $t$ we obtain

$\displaystyle
e^{b_{i\inf}y_{i}(t)}\leq\frac{b_{i\inf}x_{i0}}{\varepsilon}\left(e^{\varepsilon
t}-e^{\varepsilon t_{0}}\right)+e^{b_{i\inf}y_{i}(t_{0})},\ i=1,2,\ \forall
t\geq t_{0},\ \omega\in\Omega_{\varepsilon}.$

So

$\displaystyle
y_{i}(t)\leq\frac{1}{b_{i\inf}}\ln\left[e^{b_{i\inf}y_{i}(t_{0})}+\frac{b_{i\inf}x_{i0}}{\varepsilon}\left(e^{\varepsilon
t}-e^{\varepsilon t_{0}}\right)\right],\ i=1,2,\ \forall t\geq t_{0},\
\omega\in\Omega_{\varepsilon},$

and therefore

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\int\limits_{0}^{t}x_{i}(s)ds\leq\frac{\varepsilon}{b_{i\inf}},\
i=1,2,\ \forall\omega\in\Omega_{\varepsilon}.$

Since $\varepsilon>0$ is arbitrary and $x_{i}(t)>0,i=1,2$ a.s., we have for
$i=1,2$

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\int\limits_{0}^{t}x_{i}(s)ds=0\
a.s.$

∎

######  Definition 5.

([10]) The population $x_{i}(t),i=1,2$ will be said weakly persistent in the
mean if for every initial data $x_{i0}>0,i=1,2$, we have

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\int\limits_{0}^{t}x_{i}(s)ds>0,\
a.s.,\ i=1,2$

######  Theorem 7.

If ${\bar{p}}_{2}^{*}>0$, where $p_{2}(t)=-a_{2}(t)-\beta_{2}(t)$, then the
predator population density $x_{2}(t)$ with the initial condition $x_{20}>0$
will be weakly persistence in the mean

$\displaystyle{\bar{x}}_{2}^{*}=\limsup_{t\to\infty}\frac{1}{t}\int_{0}^{t}x_{2}(s)ds>0\
\hbox{a.s.}$

###### Proof.

If the assertion of theorem is not true, then $\mathrm{P}\\{\omega\in\Omega\
|\ {\bar{x}}_{2}^{*}=0\\}>0$. From the first equality in $(\ref{eq21})$ we get

$\displaystyle\frac{1}{t}(\ln x_{2}(t)-\ln
x_{20})+\frac{1}{t}\int_{0}^{t}b_{2}(s)x_{2}(s)ds=\frac{1}{t}\int_{0}^{t}p_{2}(s)ds$
$\displaystyle+\frac{1}{t}\int_{0}^{t}\frac{c_{2}(s)x_{1}(s)}{1+m(s)x_{1}(s)}ds+\frac{M_{2}(t)}{t}\geq\frac{1}{t}\int_{0}^{t}p_{2}(s)ds+\frac{M_{2}(t)}{t},$

where martingale $M_{2}(t)$ is defined in $(\ref{eq22})$. For
$\forall\omega\in\\{\omega\in\Omega|\ {\bar{x}}_{2}^{*}=0\\}$ in virtue of the
strong law of large numbers for martingale $M_{2}(t)$ we have

$\displaystyle\limsup_{t\to\infty}\frac{\ln
x_{2}(t)}{t}\geq{\bar{p}}_{2}^{*}>0.$

Therefore

$\displaystyle\mathrm{P}\left\\{\omega\in\Omega|\
\limsup_{t\to\infty}\frac{\ln x_{2}(t)}{t}>0\right\\}>0.$

But from Lemma 1 we have

$\displaystyle\mathrm{P}\left\\{\omega\in\Omega\ |\
\limsup_{t\to\infty}\frac{\ln x_{2}(t)}{t}\leq 0\right\\}=1.$

This is a contradiction. ∎

######  Theorem 8.

If ${\bar{p}}_{1}^{*}>0$ and ${\bar{q}}_{2}^{*}<0$, then the prey population
density $x_{1}(t)$ with initial condition $x_{10}>0$ will be weakly
persistence in the mean

$\displaystyle{\bar{x}}_{1}^{*}=\limsup_{t\to\infty}\frac{1}{t}\int_{0}^{t}x_{1}(s)ds>0\
\hbox{a.s.}$

###### Proof.

Let $\mathrm{P}\\{{\bar{x}}_{1}^{*}=0\\}>0$. From the first equality in
$(\ref{eq21})$ we get

$\displaystyle\frac{1}{t}(\ln x_{1}(t)-\ln
x_{10})+\frac{1}{t}\int_{0}^{t}b_{1}(s)x_{1}(s)ds=\frac{1}{t}\int_{0}^{t}p_{1}(s)ds$
$\displaystyle-\frac{1}{t}\int_{0}^{t}\frac{c_{1}(s)x_{2}(s)}{m(s)+x_{1}(s)}ds+\frac{M_{1}(t)}{t}$
$\displaystyle\geq\frac{1}{t}\int_{0}^{t}p_{1}(s)ds-\frac{c_{1\sup}}{m_{\inf}t}\int_{0}^{t}x_{2}(s)ds+\frac{M_{1}(t)}{t}$
(25)

where martingale $M_{1}(t)$ is defined in $(\ref{eq22})$. In virtue of the
strong law of large numbers for martingale $M_{1}(t)$ and the result of
Theorem 5 for the predator population density $x_{2}(t)$ we obtain from
$(\ref{eq25})$

$\displaystyle\limsup_{t\to\infty}\frac{\ln
x_{1}(t)}{t}\geq{\bar{p}}_{1}^{*}>0$

for $\forall\omega\in\\{\omega\in\Omega|\ {\bar{x}}_{2}^{*}=0\\}$. Therefore

$\displaystyle\mathrm{P}\left\\{\omega\in\Omega|\
\limsup_{t\to\infty}\frac{\ln x_{1}(t)}{t}>0\right\\}>0.$

But from Lemma 1 we have

$\displaystyle\mathrm{P}\left\\{\omega\in\Omega|\
\limsup_{t\to\infty}\frac{\ln x_{1}(t)}{t}\leq 0\right\\}=1.$

Therefore we have a contradiction. ∎

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