\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 247, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/247\hfil Persistence and extinction]
{Persistence and extinction for stochastic logistic model with L\'evy
noise and impulsive perturbation}

\author[C. Lu, Q. Ma, X. H. Ding \hfil EJDE-2015/247\hfilneg]
{Chun Lu, Qiang Ma, Xiaohua Ding}

\address{Chun Lu \newline
Department of Mathematics,
Qingdao Technological University,
Qingdao 266520, China}
\email{mathlc@163.com}

\address{Qiang Ma\newline
Department of Mathematics,
Harbin Institute of Technology,
Weihai 264209, China}
\email{hitmaqiang@hotmail.com}

\address{Xiaohua Ding \newline
Department of Mathematics,
Harbin Institute of Technology,
Weihai 264209, China}
\email{mathlc@126.com}

\thanks{Submitted September 5, 2014. Published September 23, 2015.}
\subjclass[2010]{64H10, 60J75, 35R12}
\keywords{Logistic equation; L\'evy noise; impulsive perturbation;
\hfill\break\indent stochastic permanence}

\begin{abstract}
 This article investigates a stochastic logistic model with L\'evy
 noise and impulsive perturbation. In the model, the impulsive perturbation
 and L\'evy noise are taken into account simultaneously.
 This model is new and more feasible and more accordance with the actual.
 The definition of solution to a stochastic differential equation with
 L\'evy noise and impulsive perturbation is established. Based on this
 definition, we show that our model has a unique global positive solution
 and obtains its explicit expression. Sufficient conditions for extinction
 are established as well as nonpersistence in the mean, weak persistence and
 stochastic permanence. The threshold between weak persistence and extinction
 is obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}\label{sec:1}

Persistence and extinction of logistic model is one of the important topics
in mathematical biology. Many scholars have investigated the topic for
the classical stochastic logistic model with L\'evy noise
(see \cite{Bao11,Bao12,Liu14,Zou14}):
\begin{equation} \label{e1.1}
dx(t)=x(t)(r(t)-a(t)x(t))dt+\sigma(t)
x(t)dB(t)+x(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du),
\end{equation}
where $x(t)$ is the population size, $B(t)$ is a standard
Brownian motion, $x(t^{-})=\lim_{s\uparrow t}x(s)$, $N(dt,
du)$ is a real-valued Poisson counting measure with characteristic
measure $\lambda$ on a measurable subset $\mathbb{Y}$ of $\mathbb{R}_{+}=[0,\infty)$
with $\lambda(\mathbb{Y})<\infty, \tilde{N}(dt, du)=N(dt,du)-\lambda(du)dt$
and $\gamma(u)>-1$. There is an important and interesting literature
about stochastic differential equation with jumps
(see \cite{Applebaum88,Applebaum09,Applebaum10,Ikeda81,Situ05}).
To simulate the phenomena well in reality, e.g.,
epidemics, earthquakes, hurricanes, ocean red tide and so on, Lots of
authors have introduced the L\'evy noise into biological model
(see \cite{Liu13c,Lu14a,Lu14d,Lu14b,Ma14,Wu14}).

However, in the real world, owing to some natural and man-made factors,
such as fire, drought, crop-dusting, deforestation,
hunting, harvesting, etc., the growth of species often undergoes some
discrete changes of relatively short time interval at some fixed times.
These phenomena cannot be considered continually, so in this case,
system \eqref{e1.1} cannot describe these phenomena. Introducing the
impulsive effects, which can not boil down to L\'evy noise from its definition,
into the model may describe such phenomena well,
see \cite{Bainov93,Lakshmikantham89}.

Recently, several authors have incorporated the impulsive perturbation into
the stochastic population dynamics and some results on dynamical behavior
for such systems have been reported
(see \cite{Liu12,Liu13a,Liu13b,Lu14c,Wu14b}) and the references therein.
However, so far as we know, there are no papers published which consider the
impulsive perturbation in stochastic population model with L\'evy noise.
Motivated by these arguments presented above, we will consider the following
stochastic logistic model with L\'evy noise and impulsive perturbation:
\begin{equation} \label{e1.2}
 \begin{gathered}
 dx(t)=x(t)(r(t)-a(t)x(t))dt+\sigma(t) x(t)dB(t)+x(t^{-})
\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du),\\
 t\neq t_{k}, \quad K\in \mathbb{N}\\
 x(t_{k}^{+})-x(t_{k})=h_{k}x(t_{k}),\quad k\in \mathbb{N}
 \end{gathered}
\end{equation}
where $N$ denotes the set of positive integers,
$0<t_{1}<t_{2}\dots ,\lim_{k\to\infty}t_{k}=\infty$, $r(t), a(t)$ and
$\sigma(t)$ are continuous and boundeded function on $\mathbb{R}_{+}$ and
$\inf_{t\in \mathbb{R}_{+}}a(t)>0$. Here, we assume that $B(t)$ is independent of
$N(dt,du)$.
Other parameters are defined and required as before.

The main contributions of this paper are listed as follows:

(1) The model includes two types of environmental noise and impulsive
perturbation which is more grounded in the real world. We establish the
definition of solution to a stochastic differential equation with L\'evy
noise and impulsive perturbation. The explicit solution for the model is
 given in Theorem \ref{thm2.1};

(2) We give sufficient conditions for extinction, nonpersistence in the mean,
weak persistence and stochastic permanence of the solution.
In addition, the threshold between weak persistence and extinction is obtained.

(3) The effects of the impulsive perturbation on the population are
investigated in detail, see Remark \ref{rmk3.5}, examples and figures. Our
results imply that the impulsive perturbation has great impacts on the model.


For model \eqref{e1.2} we assume the following conditions:
\begin{itemize}
\item[(A1)] As far as biological meanings is concerned, we
consider $1+h_{k}>0,~k\in \mathbb{N}$. When $h_{k}>0$, is satisfied, the
perturbation turn to be the description process of planting of
species and harvesting if not $h_{k}<0$.

\item[(A2)] For each $m>0$ there exists $L_{m}$ such that
$\int_{\mathbb{Y}}|H(x,u)-H(y,u)|^{2}\lambda(du)\leq L_{m}|x-y|^{2}$ where $H(x,u)=\gamma(u)x(t^{-})$ with $|x|\vee |y|\leq m$.


\item[(A3)] There exists a constant $c>0$ such that
$\int_{\mathbb{Y}}(\ln(1+\gamma(u))^{2}\lambda(du)\leq c$.
\end{itemize}
For simplicity, we define the  notation:
\[
\langle f(t)\rangle=\frac{1}{t}\int_{0}^{t}f(s)ds,\quad
f_{*}=\liminf_{t\to\infty}f(t),\quad f^{*}=\limsup_{t\to\infty}f(t).
\]
If $\nu(t)$ is a continuous bounded function on $\mathbb{R}_{+}$, define
$\hat{\nu}=\sup_{t\in \mathbb{R}_{+}}\nu(t)$ and 
$\check{\nu}=\inf_{t\in\mathbb{R}_{+}}\nu(t)$.
The following definitions are commonly used
and we list them here.

1. The population $x(t)$ is said to be extinct if $\lim_{t\to\infty}x(t)=0$.

2. The population $x(t)$ is said to be nonpersistence in the mean \cite{Ma90}
if  \\ $\limsup_{t\to\infty}\langle x(t)\rangle=0$.

3. The population $x(t)$ is said to be weak persistence \cite{Hallam86}
if $\limsup_{t\to\infty}x(t)>0$.

4. The population $x(t)$ is said to be stochastic permanence \cite{Liu14} if
for an arbitrary $\varepsilon>0$, there are constants $\beta>0,\alpha>0$
such that
\[
\liminf_{t\to \infty}\mathcal {P}\{x(t)\geq\beta\}\geq 1-\varepsilon
\quad\text{and}\quad
\liminf_{t\to \infty}\mathcal {P}\{x(t)\leq \alpha\}\geq 1-\varepsilon.
\]


\section{Positive and global solutions}

Throughout this paper, let
$(\Omega,\mathscr{F},\{\mathscr{F}_{t}\}_{t\geq 0},\mathcal {P})$ be
a complete probability space with a filtration
$\{\mathscr{F}_{t}\}_{t\geq 0\mathbf{}}$ satisfying the usual
conditions and $B(t)$ denotes a standard Brownian motion defined
on this probability space.

\begin{definition} \label{def1} \rm
Consider the stochastic differential equation
with L\'evy noise and impulsive perturbation:
\begin{equation} \label{e2.1}
 \begin{gathered}
dx(t)=f(t,x(t),\omega)dt+g(t,x(t),\omega)dB(t)+\int_{\mathbb{Y}}
\gamma(t,x(t^{-}),u,\omega)\tilde{N}(dt,du),\\
 t\neq t_k, k\in \mathbb{N}\\
x(t_k^+)-x(t_k)=h_k x(t_k), \quad k\in \mathbb{N}
 \end{gathered}
\end{equation}
with initial condition $x(0)=x_{0}$. Here,
$x(t^{-})=\lim_{s\uparrow t}x(s)$, $N(dt, du)$ is a
real-valued Poisson counting measure with characteristic measure
$\lambda$ on a measurable subset $\mathbb{Y}$ of $\mathbb{R}_{+}$ with
$\lambda(\mathbb{Y})<\infty, \tilde{N}(dt, du)=N(dt,du)-\lambda(du)dt$ and
$B(t)$ is independent of $N$. A stochastic
process $x(t)$, $t\in \mathbb{R}_{+}$, is said to be a solution of \eqref{e2.1} if
\begin{itemize}
\item[(i)] $x(t)$ is $\mathscr{F}_{t}$-adapted on $(0,t_1)$ and each
interval $(t_{k},t_{k+1})\in \mathbb{R}_{+},k\in \mathbb{N}$;
$f(t,x):\mathbb{R}_{+}\times R\times\Omega\to R$,
$g(t,x): \mathbb{R}_{+}\times R\times\Omega\to R$ and
$\gamma:\mathbb{R}_{+}\times R\times\mathbb{Y}\times\Omega\to R$
are jointly measurable and $\mathscr{F}_{t}$-adapted where,
furthermore, $\gamma$ is $\mathscr{F}_{t}$-predictable;

\item[(ii)] For each $t_k,k\in \mathbb{N},x(t_k^{+})
=\lim_{t\to t_{k}^{+}}x(t)$ and
$x(t_{k}^{-})=\lim_{t\to t_{k}^{-}}x(t)$ exist and
 $x(t_{k})=x(t_{k}^{-})$ with probability one;

\item[(iii)] For almost all $t\in [0,t_{1}]$ and $k\in \mathbb{N},x(t)$, $x(t)$
satisfies the integral equation
\begin{equation} \label{e2.2}
\begin{split}
x(t)&=x(0)+\int_{0}^{t}f(s,x(s),\omega)+\int_{0}^{t}g(s,x(s),\omega)dB(s)\\
&\quad +\int_{0}^{t}\int_{\mathbb{Y}}\gamma(s,x(s^{-}),u,\omega)\tilde{N}(ds,du).
\end{split}
\end{equation}
And for almost all $t\in (t_{k},t_{k+1}], k\in \mathbb{N}, x(t)$ satisfies
\begin{equation} \label{e2.3}
 \begin{split}
x(t)&=x(t_{k}^{+})+\int_{t_{k}}^{t}f(s,x(s),\omega)
 +\int_{t_{k}}^{t}g(s,x(s),\omega)dB(s)\\
&\quad +\int_{t_{k}}^{t}\int_{\mathbb{Y}}\gamma(s,x(s^{-}),u,\omega)
\tilde{N}(ds,du).
\end{split}
 \end{equation}
Moreover, $x(t)$ satisfies the impulsive conditions at each
$t=t_{k}, k\in \mathbb{N}$ with probability one.
\end{itemize}
\end{definition}

\begin{remark} \label{rmk1.1} \rm
 Now let us clarify the derivation procedure of Definition 1.
Firstly, noticing that the stochastic differential equation
with jumps and impulsive perturbation \eqref{e2.1} becomes the following
stochastic differential equation with jumps:
$$
dx(t)=f(t,x(t),\omega)dt+g(t, x(t),\omega)dB(t)
+\int_{\mathbb{Y}}\gamma(t,x(t^{-}),u,\omega)\tilde{N}(dt,du)
$$
on interval $[0,t_1]$ and each interval
$(t_{k},t_{k+1}]\in \mathbb{R}_{+},k\in \mathbb{N}$.
In the light of the classical definition of a solution of stochastic
differential equation with jumps (see  \cite[page 76]{Situ05}),
condition (i), Equations \eqref{e2.2} and \eqref{e2.3} should be satisfied. Second,
since there exists impulsive perturbation in \eqref{e2.1}, then the condition
(ii) and (iii) should be satisfied. According to the two facts
above, the Definition 1 is proposed.
\end{remark}

\begin{theorem} \label{thm2.1}
Under assumptions {\rm (A1)--(A2)}, for any initial value $x(0)=x_{0}>0$,
there is a unique solution $x(t)$ to \eqref{e1.2} a.s., which is global and
represented by
$$
x(t)=\frac{\prod_{0<t_{k}<t}(1+h_{k})\phi(t)}{\frac{1}{x_{0}}
+\int_{0}^{t}\prod_{0<t_{k}<s}(1+h_{k})a(s)\phi(s)ds},
$$
where
\begin{align*}
\phi(t)
&=\exp\Big(\int_{0}^{t}\Big[r(\zeta)-\frac{1}{2}\sigma^{2}(\zeta)
 +\int_{\mathbb{Y}}(\ln(1+\gamma(u))-\gamma(u))\lambda(du)\Big]d\zeta \\
&\quad +\int_{0}^{t}\sigma(\zeta)dB(\zeta)
 +\int_{0}^{t}\int_{\mathbb{Y}}\ln(1+\gamma(u))\tilde{N}(d\zeta,du)\Big).
\end{align*}
\end{theorem}

\begin{proof}
Consider the  stochastic differential equation with jumps
\begin{equation} \label{e2.4}
 \begin{split}
dy(t)
&=y(t)\Big[r(t)-\prod_{0<t_{k}<t}(1+h_{k})a(t)y(t)\Big]+\sigma(t)y(t)dB(t)\\
&\quad +y(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du)
\end{split}
\end{equation}
with initial value $y(0)=x_{0}$. Then \eqref{e2.4} has the explicit
solution  \cite[Lemma 4.2]{Bao11}
$$
y(t)=\frac{\phi(t)}{\frac{1}{x_{0}}
+\int_{0}^{t}\prod_{0<t_{k}<s}(1+h_{k})a(s)\phi(s)ds},
$$
where
\begin{align*}
\phi(t)&=\exp\Big(\int_{0}^{t}\Big[r(\zeta)-\frac{1}{2}\sigma^{2}(\zeta)
 +\int_{\mathbb{Y}}(\ln(1+\gamma(u))-\gamma(u))\lambda(du)\Big]d\zeta\\
&\quad +\int_{0}^{t}\sigma(\zeta)dB(\zeta)
+\int_{0}^{t}\int_{\mathbb{Y}}\ln(1+\gamma(u))\tilde{N}(d\zeta,du)\Big).
\end{align*}
Now let
$$
x(t)=\prod_{0<t_{k}<t}(1+h_{k})y(t).
$$
We show that $x(t)$ is the solution \eqref{e1.2}. On the interval $[0,t_1)$ 
and each interval $(t_{k},t_{k+1})\in \mathbb{R}_{+},k\in \mathbb{N}$, we obtain
\begin{align*}
dx(t)&=d\Big[\prod_{0<t_{k}<t}(1+h_{k})y(t)\Big]\\
&=\prod_{0<t_{k}<t}(1+h_{k})dy(t)\\
&=\prod_{0<t_{k}<t}(1+h_{k})y(t)\Big[r(t)-\prod_{0<t_{k}<t}(1+h_{k})a(t)y(t)\Big]\\
&\quad +\prod_{0<t_{k}<t}(1+h_{k})\sigma(t)y(t)dB(t)
 +\prod_{0<t_{k}<t}(1+h_{k})y(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du)\\
&=\prod_{0<t_{k}<t}(1+h_{k})y(t)\Big[r(t)-\prod_{0<t_{k}<t}(1+h_{k})a(t)y(t)\Big]\\
&\quad +\prod_{0<t_{k}<t}(1+h_{k})\sigma(t)y(t)dB(t)
 +\prod_{0<t_{k}<t^{-}}(1+h_{k})y(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du)\\
&=x(t)(r(t)-a(t)x(t))dt+\sigma(t)x(t)dB(t)
 +x(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du).
\end{align*}
Moreover, for every $k\in \mathbb{N}$ and $t_{k}\in [0,\infty)$,
\begin{align*}
x(t_{k}^{+})
&=\lim_{t\to t_{k}^{+}}\prod_{0<t_{j}<t}(1+h_{j})y(t)
 =\prod_{0<t_{j}\leq t_{k}}(1+h_{j})y(t_{k}^{+})\\
&=(1+h_{k})\prod_{0<t_{j}<t_{k}}(1+h_{j})y(t_{k})
 =(1+h_{k})x(t_{k}).
\end{align*}
In addition,
\begin{align*}
x(t_{k}^{-})
&=\lim_{t\to t_{k}^{-}}\prod_{0<t_{j}<t}(1+h_{j})y(t)
=\prod_{0<t_{j}<t_{k}}(1+h_{j})y(t_{k}^{-})\\
&=\prod_{0<t_{j}<t_{k}}(1+h_{j})y(t_{k})=x(t_{k}).
\end{align*}
Now let us prove the uniqueness of the solution. 
For $t\in [0,t_{1}]$, equation \eqref{e1.2} becomes the  classical equation
\begin{equation} \label{e2.5}
dx(t)=x(t)(r(t)-a(t)x(t))dt+\sigma(t)
x(t)dB(t)+x(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du),
\end{equation}
for $t\in (0,t_{1})$.
Since the coefficients of \eqref{e2.5} are locally
Lipschitz continuous, by the theory of stochastic differential
equation with jumps  \cite{Applebaum09,Ikeda81}, the solution of  \eqref{e2.5} is
unique. For $t\in (t_{k},t_{k+1}], k\in \mathbb{N}$, equation \eqref{e1.2} becomes
\begin{equation} \label{e2.6}
dx(t)=x(t)(r(t)-a(t)x(t))dt+\sigma(t)
x(t)dB(t)+x(t^{-})\int_{\mathbb{Y}}\gamma(u)\tilde{N}(dt,du)
\end{equation}
for $t\in (t_{k},t_{k+1}]$.
Note that the coefficients of  \eqref{e2.6} are also locally Lipschitz
continuous; then the solution of \eqref{e2.6} is also unique.
Therefore, the solution of model \eqref{e1.2} is unique.
\end{proof}

\section{Persistence and extinction for model \eqref{e1.2}}

 For later applications, we introduce the following lemmas.

\begin{lemma} \label{lem3.1}
Let $M(t)$, $t\geq 0$ be a local martingale at time 0 and define
$$
\rho_{M}(t)=\int_{0}^{t}\frac{d\langle M\rangle(s)}{(1+s)^{2}}, \quad t\geq 0,
$$
where $\langle M\rangle(t)=\langle M, M\rangle(t)$ is Meyer's angle
bracket process. Then
$\lim_{t\to\infty}\frac{M(t)}{t}=0$ a.s.
provided that $\lim_{t\to\infty}\rho_{M}(t)<\infty$
a.s.  \cite{Lipster80}.
\end{lemma}

\begin{theorem} \label{thm3.2}
Let  assumptions {\rm (A1)--(A3)} hold. 
Suppose that $x(t)$ is a solution of \eqref{e1.2}, then
\begin{align*}
\limsup_{t\to\infty}t^{-1}\ln x(t)
&\leq\limsup_{t\to\infty}t^{-1}\Big[\sum_{0<t_{k}<t}\ln(1+h_{k})
 +\int_{0}^{t}(r(s)-0.5\sigma^{2}(s))ds\Big] \\
&\quad -\int_{\mathbb{Y}}(\gamma(u)-\ln(1+\gamma(u)))\lambda(du)=g^{*},\quad
\text{a.s.},
\end{align*}
where
\[
g(t)=t^{-1}\Big[\sum_{0<t_{k}<t}\ln(1+h_{k})
 +\int_{0}^{t}(r(s)-0.5\sigma^{2}(s))ds\Big]
 -\int_{\mathbb{Y}}(\gamma(u)-\ln(1+\gamma(u)))\lambda(du).
\]
In particular, if $g^{*}<0$, then
$\lim_{t\to\infty}x(t)=0$  a.s.
\end{theorem}

\begin{proof}
By  It\^{o}'s formula \cite{Kunita10}, we derive from \eqref{e2.4} that
\begin{align*}
d\ln y(t)
&=\Big[r(t)-\frac{\sigma^{2}(t)}{2}-\prod_{0<t_{k}<t}(1+h_{k})a(t)y(t)
 +\int_{\mathbb{Y}}(\ln(1+\gamma(u))\\
&\quad -\gamma(u))\lambda(du)\Big]dt
 +\sigma(t)dB(t)+\int_{\mathbb{Y}}\ln(1+\gamma(u))\tilde{N}(dt,du)\\
&=\Big[r(t)-\frac{\sigma^{2}(t)}{2}-a(t)x(t)+\int_{\mathbb{Y}}
 (\ln(1+\gamma(u))-\gamma(u))\lambda(du)\Big]dt\\
&\quad +\sigma(t)dB(t)+\int_{\mathbb{Y}}\ln(1+\gamma(u))\tilde{N}(dt,du).
\end{align*}
Integrating both sides from $0$ to $t$, we have
\begin{equation} \label{e3.1}
 \begin{split}
\ln y(t)-\ln y(0)
&= \int_{0}^{t}\Big[r(s)-0.5\sigma^{2}(s)-a(s)x(s)\Big]ds
 +t\int_{\mathbb{Y}}(\ln(1+\gamma(u))\\
&\quad -\gamma(u))\lambda(du)+M_{1}(t)+M_{2}(t)
\end{split}
\end{equation}
where 
\[
M_{1}(t)=\int_{0}^{t}\sigma(s)dB(s), \quad
M_{2}(t)=\int_{0}^{t}\int_{\mathbb{Y}}\ln(1+\gamma(u))\tilde{N}(ds,du).
\]
The quadratic variation of $M_{1}(t)$ is $\langle
M_{1}(t),M_{1}(t)\rangle=\int_{0}^{t}\sigma^{2}(s)ds\leq\hat{\sigma^{2}}t$.
By the strong law of large numbers for martingales leads to
\begin{equation} \label{e3.2}
\lim_{t\to\infty}\frac{M_{1}(t)}{t}=0,\quad  a.s.
\end{equation}
Under  assumption (A2), 
$\langle M_{2}\rangle(t)=\int_{0}^{t}\int_{\mathbb{Y}}
(\ln(1+\gamma(u)))^{2}\lambda(du)ds\leq ct$. We derive
$$
\int_{0}^{t}\frac{1}{(1+s)^{2}}ds=\frac{t}{t+1}<\infty.
$$
By Lemma \ref{lem3.1}, we then obtain
\begin{equation} \label{e3.3}
\lim_{t\to\infty}\frac{M_{2}(t)}{t}=0,\quad  \text{a.s.}
 \end{equation}
On the other hand, it follows from \eqref{e3.1} that
\begin{align*}
&\sum_{0<t_{k}<t}\ln(1+h_{k})+\ln y(t)-\ln y(0)\\
&= \sum_{0<t_{k}<t}\ln(1+h_{k})
+\int_{0}^{t}\Big[r(s)-0.5\sigma^{2}(s)-a(s)x(s)\Big]ds\\
&\quad +t\int_{\mathbb{Y}}(\ln(1+\gamma(u))-\gamma(u))\lambda(du)
+M_{1}(t)+M_{2}(t).
\end{align*}
Thus
\begin{equation} \label{e3.4}
\begin{split}
\ln x(t)-\ln
x(0)
&= \sum_{0<t_{k}<t}\ln(1+h_{k})
 +\int_{0}^{t}\Big[r(s)-0.5\sigma^{2}(s)-a(s)x(s)\Big]ds\\
&\quad -t\int_{\mathbb{Y}}(\gamma(u)
 -\ln(1+\gamma(u)))\lambda(du)+M_{1}(t)+M_{2}(t).
\end{split}
 \end{equation}
Using \eqref{e3.2} and \eqref{e3.3}, we immediately obtain the desired assertion.
\end{proof}

\begin{theorem} \label{thm3.3}
Under  assumptions {\rm (A1)--(A3)}, if $g^{*}=0$, then the population model
\eqref{e1.2} is nonpersistence in the mean a.s.
\end{theorem}

\begin{proof}
In view of $g^{*}=0$, there exists a constant $T$ such
that for all $\varepsilon>0$,
\begin{align*}
&t^{-1}\Big[\sum_{0<t_{k}<t}\ln(1+b_{k})
 +\int_{0}^{t}b(s)ds\Big]
 -\int_{\mathbb{Y}}[\gamma(u)-\ln(1+\gamma(u))]\lambda(du)\\
&+\frac{M_{1}(t)}{t}+\frac{M_{2}(t)}{t}<\varepsilon, \quad t>T.
\end{align*}
Substituting this inequality into \eqref{e3.4} yields
\begin{align*}
\ln x(t)
&\leq \ln x(0)+\sum_{0<t_{k}<t}\ln(1+h_{k})
 +\int_{0}^{t}(b(s)-a(s)x(s))ds\\
&\quad -\int_{0}^{t}\int_{\mathbb{Y}}[\gamma(u)
-\ln(1+\gamma(u))]\lambda(du)ds+M_{1}(t)+M_{2}(t)\\
&<\varepsilon t-\int_{0}^{t}a(s)x(s)ds
\end{align*}
for all $t>T$.
The rest of proof is similar to \cite[Theorem 3]{Liu2011d} 
and hence is omitted.
\end{proof}

\begin{theorem} \label{thm3.4}
Under the assumptions {\rm (A1)--(A3)}, if $g^{*}>0$, then the population 
$x(t)$ modeled by \eqref{e1.2} is weak persistence a.s.
\end{theorem}

\begin{proof}
If this assertion is not true, let
$F=\{\limsup_{t\to\infty}x(t)=0\}$ and suppose
$\mathcal {P}(F)>0$. In the light of \eqref{e3.4},
\begin{equation} \label{e3.5}
 \begin{split}
&t^{-1}\Big[\ln x(t)-\ln x(0)\Big]\\
&= t^{-1}\Big[\sum_{0<t_{k}<t}\ln(1+b_{k})
 +\int_{0}^{t}(b(s)-a(s)x(s))ds\\
&\quad -\int_{0}^{t}\int_{\mathbb{Y}}[\gamma(u)
 -\ln(1+\gamma(u))]\lambda(du)ds\Big]
+\frac{M_{1}(t)}{t}+\frac{M_{2}(t)}{t}.
\end{split}
 \end{equation}
On the other hand, for for all $\omega\in F$, we have
$\lim_{t\to\infty}x(t,\omega)=0$.
Therefore, substituting \eqref{e3.2} and \eqref{e3.3} into \eqref{e3.5}, 
one can deduce the contradiction
$$
0\geq\limsup_{t\to\infty}[t^{-1}\ln x(t,\omega)]=g^{*}>0.
$$
\end{proof}

\begin{remark} \label{rmk3.1}\rm
Theorems \ref{thm3.2}--\ref{thm3.4} have a direct biological explanation. 
It is obvious to see that the extinction and
persistence of population $x(t)$ modeled by \eqref{e1.2} largely rely on
$g^{*}$. Under the assumption (A1)--(A3), 
if $g^{*}>0$, the population $x(t)$ will be weakly persistent;
 Under the assumption  (A1)--(A3), if $g^{*}<0$, the
population $x(t)$ will go to extinction. That is to say, under
the assumption  (A1)--(A3), $g^{*}$ is the threshold between weak persistence
and extinction for the population $x(t)$.
\end{remark}

When it comes to the study of population system, the role of
stochastic permanence indicating the eternal existence of the
population, can never be ignorant with its theoretical and practical
significance. And its importance has catched the eyes of scientists
all over the world.
 So now let us show that $x(t)$ modeled by \eqref{e1.2} is
stochastic permanent in some cases.
We define the assumption
\begin{itemize}
\item[(A4)] There are two positive constants $m$ and $M$
such that $m\leq\prod_{0<t_{k}<t}(1+h_{k})\leq M$ for all
$t>0$.
\end{itemize}

\begin{remark} \label{rmk3.2} \rm
Assumption (A4) is easy to be satisfied. For example, if
$h_{k}=e^{\frac{(-1)^{k+1}}{k}}-1$, then
$e^{0.5}<\prod_{0<t_{k}<t}(1+h_{k})<e$ for all $t>0$. Thus
$1\leq\prod_{0<t_{k}<t}(1+h_{k})\leq e$ for all $t>0$.
\end{remark}

\begin{theorem} \label{thm3.5}
Under assumptions {\rm (A1), (A2), (A4)}. If
\[
\big(r(t)-0.5\sigma^{2}(t)\big)_{*}-\int_{\mathbb{Y}}\gamma(u)\lambda(du)>0
\]
and $\gamma(u)\geq 0$, then the population $x(t)$ represented by 
\eqref{e1.2} will be stochastic permanence.
\end{theorem}

\begin{proof}
First, we claim that for arbitrary $\varepsilon>0$, there is
constant $\beta>0$ such that 
$\liminf_{t\to \infty}\mathcal {P}\{x(t)\geq\beta\}\geq 1-\varepsilon$.

Define $V_{1}(y)=1/y$ for $y>0$. Applying It\^{o}'s formula
to \eqref{e2.4} we can obtain that
\begin{align*}
dV_{1}(y)
&= -V_{1}(y)\Big[r(t)-\prod_{0<t_{k}<t}(1+h_{k})a(t)y(t)\Big]dt\\
&\quad +V_{1}(y)\int_{\mathbb{Y}}\Big(\frac{1}{1+\gamma(u)}-1
  +\gamma(u)\Big)\lambda(du)dt
 +V_{1}(y)\sigma^{2}(t)dt\\
&\quad -V_{1}(y)\sigma(t)dw(t)
 +V_{1}(y)\int_{\mathbb{Y}}\Big(\frac{1}{1+\gamma(u)}-1\Big)\tilde{N}(dt,du).
\end{align*}
Since
$\big(r(t)-0.5\sigma^{2}(t)\big)_{*}-\int_{\mathbb{Y}}\gamma(u)\lambda(du)>0$,
we can choose a sufficient small constant $0<\kappa<1$ such that
$r(t)-0.5\sigma^{2}(t)-\int_{\mathbb{Y}}\gamma(u)\lambda(du)-0.5\kappa\sigma^{2}(t)>0$.

Define $V_{2}(y)=(1+V_{1}(y))^{\kappa}$.
Using  It\^{o}'s formula again leads to
\begin{align*}
&dV_{2}\\
&=\kappa(1+V_{1}(y))^{\kappa-1}dV_{1}+0.5\kappa(\kappa-1)(1+V_{1}(y))^{\kappa-2}
 V_{1}^{2}(y)\sigma^{2}(t)dt\\
&\quad+\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)+V_{1}(y)\Big(\frac{1}{(1+\gamma(u))}
 -1\Big)\Big)^{\kappa}-(1+V_{1}(y))^{\kappa}-\kappa(1+V_{1}(y))^{\kappa-1}\\
&\quad \times V_{1}(y)\Big(\frac{1}{1+\gamma(u)}-1\Big)\Big]\lambda(du)dt
 -\kappa(1+V_{1}(y))^{\kappa-1}V_{1}(y)\sigma(t)dw(t)\\
&\quad+\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)+V_{1}(y)\Big(\frac{1}{1+\gamma(u)}
 -1\Big)\Big)^{\kappa}-(1+V_{1}(y))^{\kappa}\Big]\tilde{N}(dt,du)\\
&\leq\kappa(1+V_{1}(y))^{\kappa-2}\Big\{-(1+V_{1}(y))V_{1}{(y)}
 \Big[r(t)-Ma(t)y(t)\Big]+(1+V_{1}(y))V_{1}(y)\\
&\times\int_{\mathbb{Y}}\Big(\frac{1}{1+\gamma(u)}-1+\gamma(u)\Big)
 \lambda(du)+(1+V_{1}(y))V_{1}(y)\sigma^{2}(t)+0.5(\kappa-1)V_{1}^{2}(y)
 \sigma^{2}(t)\\
&-(1+V_{1}(y))V_{1}(y)\int_{\mathbb{Y}}\Big(\frac{1}{1+\gamma(u)}-1\Big)
 \lambda(du)\Big\}dt-\kappa(1+V_{1}(y))^{\kappa-1}V_{1}(y)\sigma(t)dw(t)\\
&+\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)+V_{1}(y)
 \Big(\frac{1}{1+\gamma(u)}-1\Big)\Big)^{\kappa}-(1+V_{1}(y))^{\kappa}\Big]
 \tilde{N}(dt,du)\\
&= \kappa(1+V_{1}(y))^{\kappa-2}\Big\{-V_{1}^{2}(y)\Big[r(t)-0.5\sigma^{2}(t)
 -\int_{\mathbb{Y}}\gamma(u)\lambda(du)-0.5\kappa\sigma^{2}(t)\Big]\\
&\quad +V_{1}(y)\Big[Ma(t)-r(t)+\sigma^{2}(t)
 +\int_{\mathbb{Y}}\gamma(u)\lambda(du)\Big]+Ma(t)\Big\}dt\\
&+\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)+V_{1}(y)
 \Big(\frac{1}{1+\gamma(u)}-1\Big)\Big)^{\kappa}-(1+V_{1}(y))^{\kappa}\Big]
 \tilde{N}(dt,du)\\
&\quad -\kappa(1+V_{1}(y))^{\kappa-1}V_{1}(y)\sigma(t)dw(t)
\end{align*}
for sufficiently large $t\geq T$. The first inequity follows from
$\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(x)+V_{1}(x)
\Big(\frac{1}{(1+\gamma(u))^{2}}-1\Big)\Big)^{\kappa}-(1+V_{1}(x))^{\kappa}\Big]
\lambda(du)\leq 0$ for $\gamma(u)\geq 0$. Now, let $\eta>0$ be sufficiently
small satisfy
$$
0<\eta/\kappa<r(t)-0.5\sigma^{2}(t)-\int_{\mathbb{Y}}\gamma(u)\lambda(du)
-0.5\kappa\sigma^{2}(t).
$$
Define $V_{3}(y)=e^{\eta t}V_{2}(y)$. By It\^{o}'s formula
\begin{align*}
&dV_{3}(y(t))\\
&=\eta e^{\eta t}V_{2}(y)+e^{\eta t}dV_{2}(y)\\
&\leq\kappa e^{\eta t}(1+V_{1}(y(t)))^{\kappa-2}
 \Big\{\eta(1+V_{1}(y))^{2}/\kappa-V_{1}^{2}(y)\Big[r(t)-0.5\sigma^{2}(t)\\
&\quad -\int_{\mathbb{Y}}\gamma(u)\lambda(du)
 -0.5\kappa\sigma^{2}(t)\Big]+V_{1}(y)
 \Big[Ma(t)-r(t)+\sigma^{2}(t)\\
&\quad +\int_{\mathbb{Y}}\gamma(u)\lambda(du)\Big]+Ma(t)\Big\}dt
 -e^{\eta t}\kappa(1+V_{1}(y))^{\kappa-1}V_{1}(y)\sigma(t)dw(t)\\
&\quad +e^{\eta t}\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)+V_{1}(y)
 \Big(\frac{1}{1+\gamma(u)}-1\Big)\Big)^{\kappa}
 -(1+V_{1}(y))^{\kappa}\Big]\tilde{N}(dt,du)\\
&\leq \kappa e^{\eta t}(1+V_{1}(y(t)))^{\kappa-2}
 \Big\{-V_{1}^{2}(y)\Big[\Big(r(t)-0.5\sigma^{2}(t)\Big)_{*}\\
&\quad -\int_{\mathbb{Y}}\gamma(u)\lambda(du)-\frac{\eta}{\kappa}
 -0.5\kappa\sigma^{2}(t)\Big]+V_{1}(y)\Big[\frac{2\eta}{\kappa}
 +M\hat{a}-\check{r}+\hat{\sigma^{2}}\\
&\quad +\int_{\mathbb{Y}}\gamma(u)\lambda(du)\Big]+M\hat{a}
 +\frac{\eta}{\kappa}\Big\}dt
 +e^{\eta t}\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)\\
&\quad +V_{1}(y)
 \Big(\frac{1}{1+\gamma(u)}-1\Big)\Big)^{\kappa}
 -(1+V_{1}(y))^{\kappa}\Big]\tilde{N}(dt,du)\\
&= e^{\eta t}H(y)dt-e^{\eta t}\kappa(1+V_{1}(y))^{\kappa-1}V_{1}(y)\sigma(t)dw(t)
 +e^{\eta t}\int_{\mathbb{Y}}\Big[\Big(1+V_{1}(y)\\
&\quad +V_{1}(y)\Big(\frac{1}{1+\gamma(u)}-1\Big)\Big)^{\kappa}
 -(1+V_{1}(y))^{\kappa}\Big]\tilde{N}(dt,du)
\end{align*}
for $t\geq T$. Note that $H(y)$ is upper bounded in $\mathbb{R}_{+}$, namely
$H=\sup_{y\in \mathbb{R}_{+}}H(y)<\infty$. Consequently,
\begin{align*}
&dV_{3}(y(t))\\
&=H e^{\eta t}dt-e^{\eta t}\kappa(1+V_{1}(y))^{\kappa-1}V_{1}(y)\sigma(t)dw(t)\\
&\quad +e^{\eta t}\int_{\mathbb{Y}}
 \Big[\Big(1+V_{1}(y)+V_{1}(y)\Big(\frac{1}{1+\gamma(u)}-1\Big)\Big)^{\kappa}
 -(1+V_{1}(y))^{\kappa}\Big]\tilde{N}(dt,du)
\end{align*}
for sufficiently large $t$. Integrating both sides of the above
inequality and then taking eypectations gives
$$
E[V_{3}(y(t))]=E[e^{\eta t}(1+V_{1}(y(t)))^{\kappa}]
\leq e^{\eta T}(1+V_{1}(y(T)))^{\kappa}+\frac{H}{\eta}\Big(e^{\eta t}
-e^{\eta T}\Big).
$$
That is to say
$$\limsup_{t\to\infty}E[V_{1}^{\kappa}(y(t))]\leq\limsup_{t\to\infty}E[(1+V_{1}(y(t)))^{\kappa}]<\frac{H}{\eta}.$$
In other words, we have already shown that
$$
\limsup_{t\to\infty}E\Big[\frac{1}{y^{\kappa}(t)}\Big]\leq \frac{H}{\eta}.
$$
Then
$$
\limsup_{t\to\infty}E[1/x^{\kappa}(t)]
=\limsup_{t\to\infty}\Big[\prod_{0<t_{k}<t}(1+h_{k})\Big]^{-\kappa}
E[1/y^{\kappa}(t)]\leq m^{-\kappa}\frac{H}{\eta}=H_{1}.
$$
So for $\varepsilon>0$, we set
$\beta=\varepsilon^{1/\kappa}/H_{1}^{1/\kappa}$, by Chebyshev's
inequality, one can derive that
$$
\mathcal{P}\{x(t)<\beta\}
=\mathcal {P}\Big\{\frac{1}{x^{\kappa}(t)}>\frac{1}{\beta^{\kappa}}\Big\}
\leq\frac{E[1/x^{\kappa}(t)]}{1/\beta^{\kappa}}.
$$
This is to say
$$
\limsup_{t\to\infty}\{x(t)<\beta\}\leq \beta^{\kappa}H_{1}=\varepsilon.
$$
Consequently
$$
\liminf_{t\to\infty}\{x(t)\geq\beta\}\geq 1-\varepsilon.
$$
Next, we prove that for arbitrary $\varepsilon>0$, there are
constants $\alpha>0$ such that 
$\liminf_{t\to \infty}\mathcal {P}\{x(t)\leq\alpha\}\geq 1-\varepsilon$.

Let $0<p<1$, and compute
\begin{align*}
&dy^{p}(t)\\
&= py^{p-1}(t)\Big[y(t)\Big(r(t)-\prod_{0<t_{k}<t}(1+h_{k})a(t)y(t)\Big)dt
 +\sigma(t)y(t)dB(t)\Big]\\
&\quad+\frac{1}{2}p(p-1)\sigma^{2}(t)y^{p}(t)dt
 +\int_{\mathbb{Y}}[(1+\gamma(u))^{p}-1-p\gamma(u)]x^{p}(t)\lambda(du)dt\\
&\quad+\int_{\mathbb{Y}}[(1+\gamma(u))^{p}-1]\tilde{N}(dt,du)x^{p}(t)\\
&\leq \Big(-pma(t)y^{p+1}(t)+pr(t)y^{p}(t)+\frac{1}{2}p(p-1)\sigma^{2}(t)y^{p}(t)
 \Big)dt+p\sigma(t)y^{p}(t)dB(t)\\
&\quad +\int_{\mathbb{Y}}[(1+\gamma(u))^{p}-1-p\gamma(u)]x^{p}(t)\lambda(du)dt
 +\int_{\mathbb{Y}}[(1+\gamma(u))^{p}-1]\tilde{N}(dt,du)x^{p}(t).
\end{align*}
Using  the inequality $x^{q}\leq 1+q(x-1)$, $x\geq 0$, $0\leq q\leq 1$, we have
$\int_{\mathbb{Y}}[(1+\gamma(u))^{p}-1-p\gamma(u)]\lambda(du)<0$.
And there exists $K>0$ such that
$-pma(t)y^{p+1}(t)+\Big(pr(t)+\frac{1}{2}p(p-1)\sigma^{2}(t)\Big)y^{p}(t)\leq
K$. Then,
$$
dy^{p}(t)\leq
Kdt+p\sigma(t)y^{p}(t)dB(t)+\int_{\mathbb{Y}}[(1+\gamma(u))^{p}-1]
\tilde{N}(dt,du)x^{p}(t).
$$
Therefore
$$
E(e^{t}x^{p}(t))\leq x^{p}_{0}+\int_{0}^{t}e^{s}ds=x^{p}_{0}+K(e^{t}-1).
$$
This immediately implies that
$\limsup_{t\to\infty}E(y^{p}(t))\leq K$.
Consequently,
$$
\limsup_{t\to\infty}E(x^{p}(t))
=\limsup_{t\to\infty}\Big[\prod_{0<t_{k}<t}(1+h_{k})\Big]^{p}E(x^{p}(t))
\leq M^{p}K=\alpha
.$$
Then the desired assertion follows from the Chebyshev inequality.
 This completes the proof.
\end{proof}

\begin{remark} \label{rmk3.3} \rm
Generally speaking, as the biology has implied,
Theorem \ref{thm3.2} reveals that the population probably will go to an end
in the worst cases, while Theorem \ref{thm3.3} shows that the living chances
are considerably rare. From Theorem \ref{thm3.4} we can easily find that the
population size is limited to zero with the time permitted, however,
the opportunity of the survival of it still exists. Theorem \ref{thm3.5} means
that if the time is sufficiently large, the population size will be neither 
too small nor too large with large probability. That is
to say, the population will stably exist, which is the best result. 
In other words, the survival conditions of Theorem \ref{thm3.5} are better than 
Theorems \ref{thm3.2}--\ref{thm3.4}. 
This can well explain why the conditions are gradually 
stronger from Theorem \ref{thm3.2} to Theorem \ref{thm3.5}.
\end{remark}

\begin{remark} \label{rmk3.4}\rm
 When the jump coefficient $\gamma(u)$ degenerates to zero, our results in
 Theorems \ref{thm2.1} and \ref{thm3.2} coincide with 
\cite[Theorems 1 and 2]{Liu12}. Therefore, our results generalize the
 work of \cite{Liu12}.
\end{remark}

\begin{remark} \label{rmk3.5} \rm
In view of 
\begin{align*}
g^{*}&=\limsup_{t\to\infty}t^{-1}
\Big[\sum_{0<t_{k}<t}\ln(1+h_{k})+\int_{0}^{t}(r(s)-0.5\sigma^{2}(s))ds\Big]\\
&\quad -\int_{\mathbb{Y}}(\gamma(u)-\ln(1+\gamma(u)))\lambda(du)
\end{align*}
in Theorems \ref{thm3.2}--\ref{thm3.5}, we can find that the impulse does
 not affect the persistence and extinction if the impulsive perturbations
satisfy assumption (A4). If the impulsive perturbations do not
satisfy assumption (A4), it can effect the population: the positive 
impulses $h_{k}$ are advantageous for the population and the negative 
impulses $h_{k}$ not favorable to the population.
\end{remark}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=1\textwidth]{fig1}
\end{center}
 \caption{The horizontal axis and the vertical axis in this and following 
figures represent the time $t$ and the populations size $x(t)$
 (step size $\Delta t=0.001$).} \label{fig1} 
\end{figure}

\section{Examples and numerical simulations}

 In this section, we shall use the Euler scheme \cite{Protter97}
to illustrate the analytical findings.

In Figure \ref{fig1} (a), (b) (c), 
we choose $r(t)=0.28+0.05\sin t$, $a(t)=0.2+0.01\cos t$,
$\sigma^{2}(t)=0.3$, $\mathbb{Y}=(0,\infty)$, $\lambda(\mathbb{Y})=1$,
$\gamma(u)=0.63$, $x_{0}=0.3$ and step size 
$\Delta t= 0.001$ and $t_{k}=100k$ for $k\in \mathbb{N}$. 
The only difference in Figure \ref{fig1} (a) (b) and (c) is that the
 representations of $h_{k}$ are different. In Figure \ref{fig1}(a), we
choose $h_{k}=0$, then $g^{*}=-0.01<0$. From Theorem \ref{thm3.2},
the population $x(t)$ will go to extinction. In Figure \ref{fig1}(b), we
choose $t_{k}=10k$ and
$h_{k}=e^{0.1}-1$, then $g^{*}=0$. In view of
Theorem \ref{thm3.3}, population $x(t)$ will be nonpersistence in the
mean. In Figure \ref{fig1}(c),
we consider $t_{k}=10k$ and
$h_{k}=e^{0.2}-1$, then $g^{*}=0.01>0$. 
By Theorem \ref{thm3.4}, population $x(t)$ will be weak persistence.
 By the numerical simulations above, we can find
that the impulsive perturbation can change the properties of the population
models significantly.

In Figure \ref{fig1}(d), we consider 
$r(t)=0.43+0.06\sin t$, $a(t)=0.2+0.01\cos t$, $\gamma(u)=0.24$, 
$\sigma^{2}(t)=0.3$, $\mathbb{Y}=(0,\infty)$, $\lambda(\mathbb{Y})=1$, 
$x_{0}=0.3$, step size $\Delta t=0.001$, $t_{k}=10k$ and
$h_{k}=e^{\frac{(-1)^{k+1}}{k}}-1$, then $g^{*}=0.04>0$.
Using  Theorem \ref{thm3.5}, the population $x(t)$ will be
stochastic permanence.

\subsection*{Conclusions and future directions}

In this article, we considered a stochastic logistic model with L\'evy
noise and impulsive perturbation. From the conclusions we know that the 
impulsive perturbation can have an impact on the population in some degree.
 Generally speaking, the impulsive perturbation is small when compared 
with L\'evy jumps. Yet it may represent human factor to protect the
population even if it suffer sudden environmental shocks that can be 
modelled by L\'evy noise. When the population will be extinct, we
should take measure, i.e. positive impulsive perturbation, to avoid the 
case as far as possible. In contrast, we may go into action, i.e. 
negative impulsive perturbation, to take precautions against population 
explosion ahead of schedule.

Some interesting and significant topics deserve our further engagement. 
One may put forward a more realistic and sophisticated model to integrate the
colored noise into the model \cite{Li13,Li12,Mao06}. 
Another significant problem is that one should incorporate L\'evy noise
and impulsive perturbation into multidimensional stochastic model with time 
delay or without time delay 
\cite{Liu2015,Wang2013a,Liu2013b,Wang2015,Zhang2015}, and such 
investigations are to be done in future.


\subsection*{Acknowledgments} 
The authors would like to thank the editor and the referee for their
important and valuable comments. 
This work was supported by the National Natural Science Foundation of China 
(Nos. 11501150 and 11271101), the NNSF of Shandong Province in China
(ZR2010AQ021), the Key Project of Science and Technology of Weihai 
(No. 2014DXGJ14), the Natural Scientific Research Innovation Foundation 
in Harbin Institute of Technology (No. HIT. NSRIF. 2016081),
 the Disciplinary Construction Guide Foundation of Harbin Institute 
of Technology at Weihai (No. WH20140206) and the Scientific Research 
Foundation of Harbin Institute of Technology at Weihai (No. HIT(WH)201319).

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