\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 230, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/230\hfil Persistence and extinction]
{Persistence and extinction of a stochastic single-species population
model in a polluted environment with impulsive toxicant input}
\author[M. Liu, K. Wang \hfil EJDE-2013/230\hfilneg]
{Meng Liu, Ke Wang} % in alphabetical order
\address{Meng Liu \newline
School of Mathematical Science, Huaiyin Normal University,
Huaian 223300, China}
\email{liumeng0557@sina.com}
\address{Ke Wang \newline
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China}
\email{w\_k@hotmail.com}
\thanks{Submitted September 17, 2013. Published October 16, 2013.}
\subjclass[2000]{92D25, 92D40, 60H30}
\keywords{Pollution; white noise; extinction; persistence;
stochastic permanence}
\begin{abstract}
A stochastic single-species population system in a polluted environment
with impulsive toxicant input is proposed and studied.
Sufficient conditions for extinction, non-persistence in the mean,
strong persistence in the mean and stochastic permanence of the population
are established. The threshold between strong persistence in the mean
and extinction is obtained. Some simulation figures are introduced
to illustrate the main results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
In the world today, a large quantity of toxicants and contaminants
are emitted into ecosystems by various industries and other
activities of human. Uncontrolled contribution of toxicants to the
environment has led many populations to extinction and several
others to be on the verge of extinction. This motivates scholars to
investigate the effect of toxins on the populations and to find a
theoretical threshold value which determines extinction or
persistence of a species or community. These investigations are
becoming more and more important.
Recently, Hallam et al \cite{Hallam83a,Hallam83b,Hallam84} proposed
some deterministic models to study the effect of toxicants on a
single species initially. From then on, many important deterministic
models were proposed and studied, see e.g.
\cite{Buonomo,Cai,Freedman,Hallamma,He07,He09},\cite{Jiao}-\cite{Liu91},\cite{Ma90,Ma89,Ma97},\cite{X.Z.Meng10}-\cite{Zhao}
and the references cited therein. Particularly, Liu, Chen and Zhang
\cite{B.Liu} proposed a single-species population model in a
polluted environment with impulsive toxicant input. The authors
obtained the survival threshold and investigated the globally
asymptotical stability of the positive periodic solution of the
model.
In the real world, population systems are inevitably affected by the
environmental noises. Taking the random perturbations into account,
Gard \cite{Gard92} proposed and studied the following stochastic
model
\begin{equation}\label{1} dx=\frac{x}{g(x)}\Big[(r_0-r_1C_0(t)-f(x)x)dt+\alpha_1dB_1(t)\Big],\end{equation}
where $x=x(t)$ is the population size at time $t$; $r_0$ stands for
the intrinsic growth rate of the population without toxicant; $r_1$
denotes the population response to the pollutant present in the
organism; $C_0(t)$ is the concentration of toxicant in the organism
at time $t$; The functions $f$ and $g$ in equation \eqref{1}
represent intra specific density dependent interactions and are
assumed to be in $C[\mathbb{R}_+,\mathbb{R}_+]$ and
$C[\mathbb{R}_+,\mathbb{R}_+-\{0\}]$ respectively. $B_1(t)$ is a
standard Brownian motion and $\alpha_1$ is a constant representing
the intensity of the white noise. Under the assumption that the
toxicant stress $C_0(t)$ in equation \eqref{1} was a constant, the
author obtained the conditions for the existence of an invariant
distribution on $(0,+\infty)$. Then Liu and Wang
\cite{Liu1}-\cite{Liu4} studied the persistence and extinction of
system \eqref{1} and its generalized forms without the constant
assumption. Recently, Liu and Wang \cite{Liu31,Liu32,Liu33} investigated the
stochastic Lotka-Volterra competitive model and
cooperation model in a polluted environment respectively. For each
species, the authors established the survival threshold. However, in
all of these stochastic models, it is supposed that the exogenous
input of toxicant is continuous. In fact, in practical situation,
many toxicants are emitted in regular pulses. For example,
pesticides can be sprayed instantaneously and regularly. Chemical
plant and artificial industry often termly let sewage or other
pollutant into rivers, soil and air. So far as our knowledge is
concerned, there are no studies of the stochastic system in a
polluted environment with impulsive toxicant input. So in this
paper, we try to study this problem. The contributions of this paper
are therefore clear.
To obtain our results, we need the following widely used
concepts (see e.g.
\cite{Hallamma}-\cite{He09,Liu91}-\cite{Ma89}, \cite{Ma97,Ma20,Yang}).
\begin{definition} \rm \quad
\begin{itemize}
\item[(I)] The population, $x(t)$, is said to go to extinction if
$\lim_{t\to +\infty}x(t)=0$.
\item[(II)] $x(t)$ is said to be non-persistent in the
mean if $\lim_{t\to +\infty}\langle x(t)\rangle=0$,
where $\langle f(t)\rangle =t^{-1}\int_{0}^{t}f(s)ds$.
\item[(III)] $x(t)$ is said to be strongly persistent
in the mean if $\langle x(t)\rangle_\ast>0$, where
$f_\ast=\liminf_{t\to+\infty}f(t),$ $f^\ast=\limsup_{t\to+\infty}f(t)$.
\item[(IV)] $x(t)$ is said to be stochastically permanent if for
arbitrary $\varepsilon>0$, there are two positive constants~
$\sigma$ and $\delta$ such that
$$
\mathcal {P}_\ast\{x(t)\geq \sigma\}\geq 1-\varepsilon,\quad
\mathcal{P}_\ast\{x(t)\leq \delta\}\geq 1-\varepsilon.
$$
\end{itemize}\end{definition}
The rest of the paper is arranged as follows. In Section 2,
we propose our stochastic model. In Section 3, we carry out the
survival analysis for our system. Sufficient conditions for
extinction, non-persistence in the mean, strong persistence in the mean and stochastic permanence are
established. The threshold between strong persistence in the mean and
extinction is obtained. In Section 4, we work out some figures to
illustrate our main results. We close the paper with conclusions in
Section 5.
\section{Model Formulation}
The model we consider is based on the following single species model
with pulse toxicant input at fixed moment (Liu, Chen and Zhang
\cite{B.Liu})
\begin{equation}\label{1.21}
\begin{gathered}
\left. \begin{gathered}
\frac{dx(t)}{dt}=x(t)[r_0-r_1C_0(t)-fx]\\
\frac{dC_0(t)}{dt}=kC_e(t)-(g+m)C_0(t)\\
\frac{dC_e(t)}{dt}=-hC_e(t)
\end{gathered} \right\} \quad t\neq n\tau,\; n\in Z^+. \\
\Delta x(t)=0,\quad \Delta C_0(t)=0,\quad \Delta C_e(t)=b,\quad
t=n\tau, \quad n\in Z^+.
\end{gathered}
\end{equation}
Here, the state variable $x(t)$ is the population size, $C_0(t)$
is the concentration of toxicant in the organism and $C_e(t)$
is the concentration of toxicant in the environment,
$x(0)>0$, $0\leq C_0(0)\leq1$, $0\leq C_e(0)\leq1$,
$\Delta x(t)=x(t^+)-x(t)$, $\Delta C_0(t)=C_0(t^+)-C_0(t)$,
$\Delta C_e(t)=C_e(t^+)-C_e(t)$,
$r_0,r_1,f,k,g,m,h,b$ are positive constants and
$Z^+=\{1,2,\dots \}$; $\tau$ is the period of the impulsive effect
about the exogenous input of toxicant and $b$ is the toxicant input
amount at every time; $kC_e(t)$ stands for the organism's net uptake
of toxicant from the environment, and $gC_0(t)$ and $mC_0(t)$
represent the egestion and depuration rates of the toxicant in the
organism, respectively; $hC_e(t)$ represents the toxicant loss from
the environment itself by volatilization and so on.
As said above, population system is inevitably affected by
environmental noises. In fact, it has been noted that (see e.g.
\cite{Bandyopadhyay}) if systems assume that parameters are
deterministic, they would have some limitations in mathematical
modeling of ecological systems, besides they would be difficult to
fit data perfectly and to predict the future dynamics of the system accurately.
May \cite{May01} has also pointed out that due to
environmental noise, the birth rate should
exhibit random fluctuation.
Recall that the parameter $r_{0}$ denotes the intrinsic growth rate of the
population. In practice we usually estimate it by an average growth
rate plus an error term. If we still use $r_0$ to denote the average
growth rate, then the growth rate becomes
$$
r_0+\text{error}.
$$
Let us consider a small subsequent time interval $dt$, during
which $x(t)$ becomes $x(t)+dx(t)$. Therefore the first equation
in \eqref{1.21} becomes
$$
\frac{dx(t)}{dt}=x(t)[r_0+\text{error}-r_1C_0(t)-fx].
$$
By the well-known central
limit theorem, the error term follows a normal distribution. Thus we can approximate it by a normal distribution with mean
zero and variance $\gamma^2$; that is,
$$
\text{error}dt \backsim N(0,\gamma^2dt).
$$
Taking into account that the random noise in the environment may not be single,
thus we may describe them by an $n$-dimensional
Brownian motion $B(t) = (B_1(t), \dots, B_n(t))^T$ as follows
$$
\text{error}\,dt=\sum_{i=1}^n\alpha_idB_i(t),
$$
where $\sum_{i=1}^n\alpha_i^2=\gamma$.
Hence we obtain the following stochastic system
\begin{equation}\label{1.2}
\begin{gathered}
\left. \begin{gathered}
dx(t)=x(t)[r_0-r_1C_0(t)-fx(t)]dt+\sum_{i=1}^n\alpha_ix(t)dB_i(t)\\
\frac{dC_0(t)}{dt}=kC_e(t)-(g+m)C_0(t)\\
\frac{dC_e(t)}{dt}=-hC_e(t)
\end{gathered}\right\} \quad t\neq n\tau,\; n\in Z^+. \\
\Delta x(t)=0,\quad \Delta C_0(t)=0,\quad \Delta C_e(t)=b,\quad
t=n\tau,\quad n\in Z^+.\\
x(0)>0,\quad 0\leq C_0(0)\leq1,\quad 0\leq C_e(0)\leq1.
\end{gathered}
\end{equation}
There are many methods to analysis deterministic system, such as
Lyapunov functions, coincidence degree theory, Jacobian matrix and
so on. But there is lack of mathematical machinery available to
analyze stochastic system. One of current approaches for studying
stochastic system is to make use of Fokker-Planck equation (see e.g.
\cite{Mao06}). However, \eqref{1.2} is a non-autonomous stochastic
system, whose corresponding Fokker-Planck equation is not an
ordinary differential equations but a partial differential equation.
Moreover, the ultimate boundedness of $x(t)$ in deterministic model
\eqref{1.21} is destroyed in model \eqref{1.2} by stochastic
disturbance. It is well-known that boundedness is a very important
property in the proof. In this work, we mainly use It\^o's
formula, theory of stochastic differential equation and Lyapunov
function to analyze the properties of system \eqref{1.2}.
Since Each of $C_0(t)$ and $C_e(t)$ is a concentration, thus the
inequalities $0\leq C_0(t)\leq1$ and $0\leq C_e(t)\leq1$ must be
satisfied. Now let us prepare an important lemma to close this
section.
\begin{lemma}[\cite{B.Liu}]
For system \eqref{1.2}, if
$k\leq g+m,~b\leq 1-e^{-h\tau}$,
then $0\leq C_0(t)\leq1$ and $0\leq C_e(t)\leq1$ for all $t\geq0$.
\end{lemma}
For the rest of this article, we impose $k\leq g+m$ and $b\leq 1-e^{-h\tau}$
in this paper.
\section{Persistence and extinction}
To begin with, let us give some basic properties of the following
subsystem of \eqref{1.2}:
\begin{equation}\label{1.23}
\begin{gathered}
\left.\begin{gathered}
\frac{dC_0(t)}{dt}=kC_e(t)-(g+m)C_0(t)\\
\frac{dC_e(t)}{dt}=-hC_e(t)
\end{gathered} \right\} \quad t\neq n\tau,\; n\in Z^+. \\
\Delta C_0(t)=0,\quad \Delta C_e(t)=b,\quad t=n\tau, \quad n\in Z^+.\\
0\leq C_0(0)\leq1,\quad 0\leq C_e(0)\leq1.
\end{gathered}
\end{equation}
\begin{lemma}[\cite{B.Liu}] \label{lem3.1b}
System \eqref{1.23} has a unique positive $\tau$-periodic solution\\
$(\tilde{C}_0(t),\tilde{C}_e(t))^T$ and for every solution
$(C_0(t),C_e(t))^T$ of \eqref{1.23}, $C_0(t)\to \tilde{C}_0(t)$ and
$C_e(t)\to \tilde{C}_e(t)$ as $t\to\infty$. Moreover,
$C_0(t)>\tilde{C}_0(t)$ and
$C_e(t)>\tilde{C}_e(t)$ for all $t\geq0$ if $C_0(0)>\tilde{C}_0(0)$
and $C_e(0)>\tilde{C}_e(0)$, where
\begin{gather*}
\tilde{C}_0(t)=\tilde{C}_0(0)e^{-(g+m)(t-n\tau)}
+\frac{kb(e^{-(g+m)(t-n\tau)}-e^{-h(t-n\tau)})}{(h-g-m)(1-e^{-h\tau})},\\
\tilde{C}_e(t)=\frac{be^{-h(t-n\tau)}}{1-e^{-h\tau}},\\
\tilde{C}_0(0)=\frac{kb(e^{-(g+m)\tau}-e^{-h\tau})}{(h-g-m)(1-e^{-(g+m)\tau})(1-e^{-h\tau})},\\
\tilde{C}_e(0)=\frac{b}{1-e^{-h\tau}}
\end{gather*}
for $t\in(n\tau,(n+1)\tau]$ and $n\in Z^+$. In addition,
\begin{equation}\label{3.2}
\lim_{t\to +\infty}t^{-1}\int_{0}^{t}\tilde{C}_0(s)ds
=\frac{kb}{h(g+m)\tau}.
\end{equation}
\end{lemma}
Now we can give our main results.
\begin{theorem}\label{thm1}
For model \eqref{1.2}, if
$$
r_0<\frac{r_1kb}{h(g+m)\tau}+0.5\sum_{i=1}^n\alpha_i^2,
$$
then the population goes to extinction almost surely (a.s.).
\end{theorem}
\begin{proof}
From Lemma 3.1, for for all $\varepsilon>0$,
there exists a constant $T>0$ such that
$$
\tilde{C}_0(t)-\varepsilon\leq
C_0(t)\leq\tilde{C}_0(t)+\varepsilon,\quad t>T.
$$
Then for $t>T$,
\begin{align*}
t^{-1}\int_{T}^{t}(\tilde{C}_0(s)-\varepsilon)ds
&\leq t^{-1}\int_{0}^{t}C_0(s)ds=t^{-1}
\Big[\int_{0}^{T}C_0(s)ds+\int_{T}^{t}C_0(s)ds\Big]\\
&\leq t^{-1}\Big[T+\int_{T}^{t}(\tilde{C}_0(s)+\varepsilon)ds\Big].
\end{align*}
By \eqref{3.2},
\begin{equation}\label{liu3.2}
\lim_{t\to +\infty}t^{-1}\int_{0}^{t}C_0(s)ds
=\frac{kb}{h(g+m)\tau}.
\end{equation}
Applying
It\^{o}'s formula to the first equation of \eqref{1.2} gives
$$
d\ln x=\frac{dx}{x}-\frac{(dx)^2}{2x^2}=\Big[r_0-r_1C_0(t)
-fx-0.5\sum_{i=1}^n\alpha_i^2\Big]{dt}+\sum_{i=1}^n\alpha_idB_i(t).
$$
In other words, we have shown that
$$
\ln (x(t)/x_0)=\int_0^t\Big[r_0-r_1C_0(s)-fx(s)-0.5\sum_{i=1}^n\alpha_i^2\Big]ds
+\sum_{i=1}^n\alpha_iB_i(t)
$$
Then for sufficiently
large $t$, we have
\begin{equation}\label{3.3}
t^{-1}\ln (x(t)/x_0)= r_0-0.5\sum_{i=1}^n\alpha_i^2-r_1\langle
C_0(t)\rangle-f\langle x(t)\rangle+\sum_{i=1}^n\alpha_iB_i(t)/t.
\end{equation}
Using the strong law of large numbers for martingales (see
e.g. Mao \cite{Mao06} on page 16) leads to
\begin{equation}\label{3.5}
\lim_{t\to+\infty}B_i(t)/t=0,\quad i=1,\dots ,n.
\end{equation}
Substituting \eqref{liu3.2} and \eqref{3.5}
into \eqref{3.3} and then using the arbitrariness of
$\varepsilon$, one can obtain
$$
\big[t^{-1}\ln \frac{x(t)}{x_0}\big]^\ast
\leq r_0-\frac{r_1kb}{h(g+m)\tau}-0.5\sum_{i=1}^n\alpha_i^2<0.
$$
Thus $\lim_{t\to+\infty}x(t)=0$, a.s.
\end{proof}
\begin{theorem}\label{thm2}
For model \eqref{1.2}, if $r_0=\frac{r_1kb}{h(g+m)\tau}+0.5\sum_{i=1}^n\alpha_i^2$,
then the population is nonpersistent in the mean a.s..
\end{theorem}
\begin{proof}
For arbitrarily fixed $\varepsilon>0$, there is a constant $T$ such
that
$$
r_1t^{-1}\int_0^tC_0(s)ds>\frac{r_1kb}{h(g+m)\tau}-\varepsilon/2,\quad
\sum_{i=1}^n\alpha_iB_i(t)/t<\varepsilon/2
$$
for $t>T$. Substituting these inequalities into
\eqref{3.3} results in
\begin{align*}
\ln (x(t)/x_0)
&=[r_0-0.5\sum_{i=1}^n\alpha_i^2]t-r_1\int_0^tC_0(s)ds-f\int_0^tx(s)ds
+\sum_{i=1}^n\alpha_iB_i(t)\\
&<\Big[r_0-0.5\sum_{i=1}^n\alpha_i^2-\frac{r_1kb}{h(g+m)\tau}
+\varepsilon\Big]t-f\int_0^tx(s)ds
\end{align*}
for all $t\geq T$ almost surely. Set $h(t)=\int_{0}^{t}x(s)ds$,
$\psi=r_0-0.5\sum_{i=1}^n\alpha_i^2-\frac{r_1kb}{h(g+m)\tau}+\varepsilon$,
then we obtain
$$
\ln(dh/dt)<\psi t-fh(t)+\ln~x_0.
$$
In other words, for $t\geq T$, we have
$$
e^{fh(t)}(dh/dt)\frac{r_1kb}{h(g+m)\tau}+0.5\sum_{i=1}^n\alpha_i^2,
$$
then the population $x(t)$
represented by system \eqref{1.2} is strongly persistent in the
mean. Moreover,
$$
\lim_{t\to +\infty}\langle x(t)\rangle
= \Big[r_0-\frac{r_1kb}{h(g+m)\tau}-0.5\sum_{i=1}^n\alpha_i^2\Big]/f.
$$
\end{theorem}
\begin{proof}
For all $\varepsilon>0$, there exists a constant $T$ such that
$$
\sum_{i=1}^n\alpha_iB_i(t)/t>-\varepsilon/2,\quad
t^{-1}\int_0^tr_1C_0(s)ds<\frac{r_1kb}{h(g+m)\tau}+\varepsilon/2
$$
for all $t>T$. Substituting these inequalities into \eqref{3.3} results in
$$
\ln (x(t))-\ln x_0>\rho t-f\int_0^t x(s)ds;\quad t>T,
$$
where
$$
\rho=r_0-\frac{r_1kb}{h(g+m)\tau}-0.5\sum_{i=1}^n\alpha_i^2-\varepsilon.
$$
Let $g(t)=\int_{0}^{t}x(s)ds$, then we have
$$
\ln \frac{dg}{dt}>\rho t-fg(t)+\ln x_0;\quad t>T.
$$
In other words, for $t\geq T$, one can see that
$$
e^{fg(t)}(dg/dt)>x_0e^{\rho t}.
$$
Integrating this inequality from $T$ to $t$ yields
$$
f^{-1}[e^{fg(t)}-e^{fg(T)}] >x_0\rho^{-1}[e^{\rho t}-e^{\rho T}].
$$
Rewriting this inequality one can observe that
$$
e^{fg(t)}>e^{fg(T)}+x_0f\rho^{-1}e^{\rho t}-x_0f\rho ^{-1}e^{\rho T}.
$$
Taking the logarithm of both sides leads to
$$
g(t)>f^{-1}\ln\{x_0f\rho^{-1}e^{\rho t}+e^{fg(T)}-x_0f\rho ^{-1}e^{\rho T}\}.$$ In other words, we have
already shown that
$$
\Big\{t^{-1}\int_{0}^{t}x(s)ds\Big\}_\ast\geq f^{-1}
\Big\{t^{-1}\ln\Big\{x_0f\rho^{-1}e^{\rho t}
+e^{fg(T)}-x_0f\rho ^{-1}e^{\rho T}\Big\}\Big\}_\ast.
$$
An application of the L'Hospital's rule gives
$$
\langle x(t)\rangle_\ast\geq f^{-1}\Big\{t^{-1}
\ln\Big[x_0f\rho^{-1}e^{\rho t}\Big]\Big\}_\ast=\rho/f.
$$
Thus it follows from the arbitrariness of $\varepsilon$ that
$$
\langle x\rangle_\ast
\geq \Big[r_0-\frac{r_1kb}{h(g+m)\tau}-0.5\sum_{i=1}^n\alpha_i^2\Big]/f.
$$
This, togethers with \eqref{liu1}, complete the proof.
\end{proof}
It follows from Theorems \ref{thm1}-\ref{thm4} that
$$
r_0-\frac{r_1kb}{h(g+m)\tau}-0.5\sum_{i=1}^n\alpha_i^2
$$
is the threshold between strong
persistence in the mean and extinction for $x(t)$.
Now let us turn to establishing the stochastic permanence of model
\eqref{1.2}. To this end, we need the following useful lemma.
\begin{lemma} \label{lem3.1}
For all $p>0$, there exists a positive
constant $K=K(p)$ such that
$$
\limsup_{t\to+\infty}E[x^p(t)]\leq K(p).
$$
\end{lemma}
The proof of the above lemma is a modification of Liu and Wang
\cite{Liu1} (the second part of Theorem 4.5) and hence is omitted.
\begin{theorem}\label{thm5}
If
$$
r_0> (r_1\tilde{C}_0(t))^\ast+0.5\sum_{i=1}^n\alpha_i^2,
$$
then the population $x(t)$ is stochastically permanent.
\end{theorem}
\begin{proof}
Firstly, let us demonstrate that for given $\varepsilon>0$, there is
a positive constant $\sigma$ such that $\mathcal {P}_\ast\{x(t)\geq
\sigma\}\geq 1-\varepsilon$. Define
$V_1(x)=1/x^2$ for $x\in R_+$.
Applying It\^o's formula to the first equation of \eqref{1.2}
we have
\begin{align*}
dV_1(x)&=-2x^{-3}dx+3x^{-4}(dx)^2\\
&=2V_1(x)[fx-r_0+r_1C_0(t)]dt+3V_1(x)
\sum_{i=1}^n\alpha_i^2dt-2V_1(x)\sum_{i=1}^n\alpha_idB_i(t)\\
&=2V_1(x)\Big[fx-r_0+r_1C_0(t)+1.5\sum_{i=1}^n\alpha_i^2\Big]dt
-2V_1(x)\sum_{i=1}^n\alpha_idB_i(t),
\end{align*}
For arbitrarily small $\varepsilon$ satisfying
$r_0-0.5\sum_{i=1}^n\alpha^2_i-(r_1C_0(t))^\ast>\varepsilon>0$,
we can choose a positive constant $\theta$ such
that
$$
r_0-0.5\sum_{i=1}^n\alpha^2_i-(r_1C_0(t))^\ast
-\varepsilon-\theta\sum_{i=1}^n\alpha^2_i >0.
$$
Define
$$
V_2(x)=(1+V_1(x))^\theta.
$$
An application of It\^o's formula gives
\begin{align*}
dV_2(x)
&=\theta(1+V_1(x))^{\theta-1}dV_1+0.5\theta(\theta-1)(1+V_1(x))^{\theta-2}(dV_1)^2\\&
=\theta(1+V_1(x))^{\theta-2}\Big\{(1+V_1(x))2V_1(x)\\
&\quad \times\Big[fx-r_0+r_1C_0(t)+1.5\sum_{i=1}^n\alpha_i^2\Big]
+2(\theta-1)V_1^2(x)\sum_{i=1}^n\alpha_i^2\Big\}dt\\
&\quad -2\theta(1+V_1(x))^{\theta-1}V_1(x)\sum_{i=1}^n\alpha_idB_i(t)
\\
&=\theta(1+V_1(x))^{\theta-2}\Big\{-2\Big[r_0-r_1C_0(t)
-0.5\sum_{i=1}^n\alpha^2_i
-\theta\sum_{i=1}^n\alpha_i^2\Big]V_1^2(x)\\
&\quad +2fV_1^{1.5}(x)+\Big[-2r_0+2r_1C_0(t)
+3\sum_{i=1}^n\alpha_i^2\Big]V_1(x)+2fV^{0.5}_1(x)\Big\}dt
\\
&\quad -2\theta(1+V_1(x))^{\theta-1}V_1(x)\sum_{i=1}^n\alpha_idB_i(t)
\\
&\leq\theta(1+V_1(x))^{\theta-2}\Big\{-2\Big[r_0-0.5\sum_{i=1}^n\alpha^2_i
-(r_1\tilde{C}_0(t))^\ast-\theta\sum_{i=1}^n\alpha_i^2-\varepsilon\Big]V_1^2(x)
\\
&\quad +2fV_1^{1.5}(x)+\Big[2r_1+3\sum_{i=1}^n\alpha_i^2\Big]V_1(x)
+2fV^{0.5}_1(x)\Big\}dt\\
&-2\theta(1+V_1(x))^{\theta-1}V_1(x)\sum_{i=1}^n\alpha_idB_i(t)
\end{align*}
for sufficiently large $t$.
Now, choose $\eta>0$ sufficiently small to satisfy
\begin{equation}\label{333}
0<\eta/\theta<2\Big[r_0-0.5\sum_{i=1}^n\alpha^2_i-(r_1\tilde{C}_0(t))^\ast
-\theta\sum_{i=1}^n\alpha^2_i
-\varepsilon\Big].
\end{equation}
Define
$$
V_3(x)=e^{\eta t}V_2(x)=e^{\eta t}(1+V_1(x))^\theta.
$$
An application of It\^o's formula yields
\begin{align*}
dV_3(x(t))
&=\eta e^{\eta t}V_2(x)dt+e^{\eta t}dV_2(x)\\
&\leq\theta e^{\eta t}(1+V_1(x))^{\theta-2}\Big\{\eta(1+V_1(x))^2/\theta\\
&\quad -2\Big[r_0-0.5\sum_{i=1}^n\alpha^2_i-(r_1\tilde{C}_0(t))^\ast
-\theta\sum_{i=1}^n\alpha_i^2-\varepsilon\Big]V_1^2(x)\\
&\quad +2fV_1^{1.5}(x)+\Big[2r_1+3\sum_{i=1}^n\alpha_i^2\Big]V_1(x)
+2fV^{0.5}_1(x)\Big\}dt\\
&\quad -2e^{\eta t}\theta(1+V_1(x))^{\theta-1}V_1(x)
\sum_{i=1}^n\alpha_idB_i(t)\\
&=\theta e^{\eta t}(1+V_1(x))^{\theta-2}
\Big\{-2\Big[r_0-0.5\sum_{i=1}^n\alpha^2_i-(r_1\tilde{C}_0(t))^\ast\\
&\quad -\theta\sum_{i=1}^n\alpha_i^2-\varepsilon-0.5\eta/\theta\Big]V_1^2(x)\\
&\quad +2fV_1^{1.5}(x)+\Big[2r_1+3\sum_{i=1}^n\alpha_i^2+2\eta/\theta\Big]
V_1(x)+2fV^{0.5}_1(x)+\eta/\theta\Big\}dt\\
&\quad -2e^{\eta t}\theta(1+V_1(x))^{\theta-1}V_1(x)\sum_{i=1}^n\alpha_idB_i(t)\\
&=:e^{\eta t}F(x)dt-2e^{\eta t}\theta(1+V_1(x))^{\theta-1}V_1(x)
\sum_{i=1}^n\alpha_idB_i(t)
\end{align*}
for sufficiently large $t$, where
\begin{align*}
F(x)&=\theta (1+V_1(x))^{\theta-2}\Big\{-2\Big[r_0
-0.5\sum_{i=1}^n\alpha^2_i-(r_1\tilde{C}_0(t))^\ast
-\theta\sum_{i=1}^n\alpha_i^2-\varepsilon\\
&\quad -0.5\eta/\theta\Big]V_1^2(x)
+2fV_1^{1.5}(x)+\Big[2r_1+3\sum_{i=1}^n\alpha_i^2
+2\eta/\theta\Big]V_1(x) \\
&\quad t\neq n\tau,\; n\in Z^+. +2fV^{0.5}_1(x)+\eta/\theta\Big\}.
\end{align*}
It follows from \eqref{333} that $F(x)$ is bounded form above in $R_+$, namely
$F_1:=\sup_{x\in R_+}F(x)<+\infty$. Thus,
$$
dV_3(x(t))\leq F_1e^{\eta t}dt-2e^{\eta t}\theta(1+V_1(x))^{\theta-1}V_1(x)
\sum_{i=1}^n\alpha_idB_i(t).
$$
Integrating both sides and then taking expectations, one can see
that
$$
E\Big[e^{\eta t}\Big(1+V_1(x(t))\Big)^\theta\Big]
\leq \Big(1+V_1(x(0))\Big)^\theta +F_1(e^{\eta t}-1)/\eta.
$$
In other words, we have already shown
that
$$
\limsup_{t\to+\infty}E[V_1^\theta(x(t))]\leq
\limsup_{t\to+\infty}E\Big[\Big(1+V_1(x(t))\Big)^\theta\Big]\leq
F_1/\eta.
$$
That is to say
$$
\limsup_{t\to+\infty}E[x^{-2\theta}(t)]\leq F_1/\eta=:F_2.
$$
Thus for any given $\varepsilon>0$, denote
$\sigma=\varepsilon^{0.5/\theta}/F_2^{0.5/\theta}$. By
Chebyshev's inequality (see e.g. Mao \cite[page 7]{Mao06}), one
can derive that
$$
\mathcal {P}\{x(t)<\sigma\}=\mathcal {P}\{x^{-2\theta}(t)>\sigma^{-2\theta}\}
\leq E[x^{-2\theta}(t)]/\sigma^{-2\theta}=\sigma^{2\theta}
E[x^{-2\theta}(t)],
$$
which is to say $\mathcal {P}^\ast\{x(t)<
\sigma\}\leq\sigma^{2\theta}F_2=\varepsilon$. Thus
$\mathcal {P}_\ast\{x(t)\geq \sigma\}\geq 1-\varepsilon$.
Next we prove that for arbitrary fixed $\varepsilon>0$, there exists
a positive constant $\delta$ such that
$\mathcal {P}_\ast(x(t)\leq \delta)\geq 1-\varepsilon$.
The proof follows from Lemma 3.1 and Chebyshev's inequality immediately.
\end{proof}
\section{Numerical simulations}
In this section we will introduce some figures to illustrate our
main results.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig1a}
\includegraphics[width=0.45\textwidth]{fig1b}\\
\includegraphics[width=0.45\textwidth]{fig1c}
\includegraphics[width=0.45\textwidth]{fig1d}
\end{center}
\caption{Solutions of system \eqref{1.2} for $r_0=0.75$,
$r_1=k=h=g=m=\tau=1$, $b=0.1$, $f=0.18$, $n=2$, $\alpha_2^2/2=0.1$, $x(0)=0.4$,
$C_0(0)=0.5$ and $C_e(0)=0.5$. The horizontal axis represents the
time $t$. (a) is with $\alpha_1^2/2=0.605$;
(b) is with $\alpha_1^2/2=0.6$;
(c) is with $\alpha_1^2/2=0.55$;
(d) is with $\alpha_1^2/2=0.51$}
\label{fig1}
\end{figure}
In Figure \ref{fig1}, we choose $r_0=0.75$,
$r_1=k=h=g=m=\tau=1$, $b=0.1$, $f=0.18$, $n=2$,
$\alpha_2^2/2=0.1$. The only
difference between conditions of Figure \ref{fig1} are
the values of $\alpha_1^2/2$.
In Figure \ref{fig1}(a), we choose $\alpha_1^2/2=0.605$. It then
follows from Theorem \ref{thm1} that population goes to extinction.
Figure \ref{fig1}(a) confirms this. In Figure \ref{fig1}(b), we choose
$\alpha_1^2/2=0.6$. Making use of Theorem \ref{thm2} yields that
population $x$ is nonpersistent in the mean. See Figure \ref{fig1}(b).
In Figure \ref{fig1}(c), we choose
$\alpha_1^2/2=0.55$. By Theorem \ref{thm4}, population $x$ is
strongly persistent in the mean and
$$
\limsup_{t\to+\infty}\langle x(t)\rangle=\Big[r_0-\alpha_1^2/2-\alpha_2^2/2-
\frac{r_1kb}{h(g+m)\tau}\Big]/f=0.2778.
$$ See Figure \ref{fig1}(c). In Figure \ref{fig1}(d), we
choose $\alpha_1^2/2=0.41$. By virtue of Theorem \ref{thm5}, we can
observe that population $x$ is stochastically permanent. See
Figure \ref{fig1}(d).
\section{Conclusions and further research}
In this paper, a stochastic single-species population system in a
polluted environment with impulsive toxicant input is proposed and
studied. Owing to its theoretical and practical significance,
population model in a polluted environment with pulse toxicant input
has deserved a lot of attention (see e.g.
\cite{Cai,Jiao,Jiao09,Jiao11,B.Liu,B.Liu09,X.Z.Meng10,Tao,Yang,Zhao}),
but mainly in deterministic case. The present paper is the first
attempt, up to our knowledge, of such a study in a stochastic
setting.
It is shown that
\begin{itemize}
\item[(A)] If $ r_0-0.5\sum_{i=1}^n\alpha_i^2-
\frac{r_1kb}{h(g+m)\tau}<0$, then the population is extinctive
with probability one;
\item[(B)] If $ r_0-0.5\sum_{i=1}^n\alpha_i^2-
\frac{r_1kb}{h(g+m)\tau}=0$, then the population is non-persistent in the
mean with probability one;
\item[(C)] If $ r_0-0.5\sum_{i=1}^n\alpha_i^2-
\frac{r_1kb}{h(g+m)\tau}>0$, then the population is strongly persistent in the
mean with probability one and
$$
\lim_{t\to+\infty}\langle x(t)\rangle=\Big[r_0-0.5\sum_{i=1}^n\alpha_i^2-
\frac{r_1kb}{h(g+m)\tau}\Big]/f,\text{ a.s.};
$$
\item[(D)] If $ r_0>
(r_1\tilde{C}_0(t))^\ast+0.5\sum_{i=1}^n\alpha_i^2$,
then the population is stochastically permanent.
\end{itemize}
From our results, we can see that both the white noises
(i.e., $\alpha_i^2$) and the impulsive effect period (i.e., $\tau$)
play very important roles in determining the
extinction and persistence of the population.
Since many population models are inevitably affected by some
stochastic noises, so the studies of these stochastic models are
important and useful for better understanding of the real world.
This paper devotes to studying a stochastic single-species model in
a polluted environment with impulsive toxicant input which is basic
and important, and the methods developed in this paper can be
referred when one further studies other stochastic models, for
example, Lotka-Volterra system with stochastic disturbance.
\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation of
China (grants 11301207, 11171081, 11301112, 11171056),
Natural Science Foundation of Jiangsu Province (grant BK20130411),
Natural Science Research Project of Ordinary Universities in Jiangsu Province
(grant 13KJB110002).
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\end{document}