\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx, amssymb} \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). \begin{thebibliography}{99} \bibitem{Bandyopadhyay} M. Bandyopadhyay, J. Chattopadhyay; \emph{Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability}, {Nonlinearity} \textbf{18} (2005), 913--936. \bibitem{Buonomo} B. Buonomo, A. D. Liddo, I. Sgura; \emph{A diffusive-convective model for the dynamics of population-toxicant intentions: Some analytical and numerical results}, {Math. Biosci.} \textbf{157} (1999), 37--64. \bibitem{Cai} S.Cai; \emph{A stage-structured single species model with pulse input in a polluted environment}. {Nonlinear Dynam.} \textbf{57} (2009), 375-382. \bibitem{Freedman} H. I. Freedman, J. B. Shukla; \emph{Models for the effect of toxicant in single-species and predator-prey systems}. {J. Math. Biol.} \textbf{30} (1991), 15--30. \bibitem{Gard92} T. C. Gard; \emph{Stochastic models for toxicant-stressed populations}. Bull. Math. Biol. \textbf{54} (1992), 827--837. \bibitem{Hallam83a} T. G. Hallam, C. E. Clark, R. R. Lassider; \emph{Effects of toxicant on population: a qualitative approach I. Equilibrium environmental exposure}, {Ecol. Model.} \textbf{8} (1983), 291--304. \bibitem{Hallam83b} T. G. Hallam, C. E. Clar, G. S. Jordan; \emph{Effects of toxicant on population: a qualitative approach II. First Order Kinetics}, {J. Math. Biol.} \textbf{109} (1983), 411--429. \bibitem{Hallam84} T. G. Hallam, J. L. Deluna; \emph{Effects of toxicant on populations: a qualitative approach III. Environmental and food chain pathways}, {J. Theor. Biol.} \textbf{109} (1984), 411--429. \bibitem{Hallamma} T. G. Hallam, Z. Ma; \emph{Persistence in population models with demographic fluctuations}, {J. Math. Biol.} \textbf{24} (1986), 327--339. \bibitem{He07} J. He, K. Wang; \emph{The survival analysis for a single-species population model in a polluted environment}, {Appl. Math. Model.} \textbf{31} (2007), 2227--2238. \bibitem{He09} J. He, K. Wang; \emph{The survival analysis for a population in a polluted environment}, {Nonlinear Anal. Real World Appl.} \textbf{10} (2009), 1555--1571. \bibitem{Higham} D. J. Higham; \emph{An algorithmic introduction to numerical simulation of stochastic diffrential equations}, {SIAM Rev.} \textbf{43} (2001), 525-546. \bibitem{Jiao} J. Jiao, L. Chen; \emph{Dynamical analysis of a chemostat model with delayed response in growth and pulse input in polluted environment}, {J. Math. Chem.} \textbf{46} (2009), 502-513. \bibitem{Jiao09} J. Jiao, W. Long, L. Chen; \emph{A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin}. {Nonlinear Anal. Real World Appl.} 10 (2009), 3073-3081. \bibitem{Jiao11} J. Jiao, K. Ye, L. Chen; \emph{Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant}. {Chaos Solitons Fractals} \textbf{44} (2011), 17-27. \bibitem{B.Liu} B. Liu, L. Chen, Y. Zhang; \emph{The effects of impulsive toxicant input on a population in a polluted environment}. {J. Biol. Syst.} \textbf{11} (2003), 265-274. \bibitem{B.Liu09} B. Liu, L. Zhang; \emph{Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input}, {Appl. Math. Comput.} \textbf{214} (2009), 155-162. \bibitem{Liu91} H. Liu, Z. Ma; \emph{The threshold of survival for system of two species in a polluted environment}, {J. Math. Biol.} \textbf{30} (1991), 49--51. \bibitem{Liu1} M. Liu, K. Wang; \emph{Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II.} {J. Theoret. Biol.} \textbf{267} (2010), 283-291. \bibitem{Liu33} M. Liu, K. Wang; \emph{Survival analysis of a stochastic cooperation system in a polluted environment}, {J. Biol. Syst.} \textbf{2} (2011), 1-22. \bibitem{Liu31} M. Liu, K. Wang, Q. Wu; \emph{Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle}, {Bull. Math. Biol.} 73 (2011), 1969-2012. \bibitem{Liu21} M. Liu, K. Wang; \emph{Persistence and extinction of a single-species population system in a polluted environment with random perturbations and impulsive toxicant input}, {Chaos Solitons Fractals} \textbf{45} (2012), 1541-1550. \bibitem{Liu22} M. Liu, K. Wang; \emph{Dynamics of a two-prey one-predator system in random environments}, {J. Nonlinear Sci.} \textbf{23} (2013), 751-775. \bibitem{Liu32} M. Liu, K. Wang; \emph{Population dynamical behavior of Lotka--Volterra cooperative systems with random perturbations}, {Discrete Contin. Dyn. Syst.} \textbf{33} (2013), 2495-2522. \bibitem{Liu13a} M. Liu, K. Wang; \emph{A note on delay Lotka-Volterra competitive system with random perturbations}, {Appl. Math. Lett.} \textbf{26} (2013), 589-594. \bibitem{Liu2} M. Liu, K. Wang; \emph{Dynamics of a Leslie-Gower Holling-type II predator-prey system with L\'{e}vy jumps}. {Nonlinear Anal. } \textbf{85} (2013), 204-213. \bibitem{Liu4} M. Liu, K. Wang; \emph{Stochastic Lotka-Volterra systems with L\'{e}vy noise}, {J. Math. Anal. Appl.} \textbf{410} (2014), 750-763. \bibitem{Ma90} Z. Ma, G. Cui, W. Wang; \emph{Persistence and extinction of a population in a polluted environment.} {Math. Biosci.} \textbf{101} (1990), 75--97. \bibitem{Ma89} Z. Ma, B. Song, T. G. Hallam; \emph{The threshold of survival for systems in a fluctuating environment}, {Bull. Math. Biol.} \textbf{51} (1989), 311-323. \bibitem{May01} R. M. May; \emph{Stability and Complexity in Model Ecosystems}, (Princeton University Press, NJ, 2001.) \bibitem{Ma97} Z. Ma, W. Zong, Z. Luo; \emph{The thresholds of survival for an n-dimensional food chain model in a polluted environment}, {J. Math. Anal. Appl.} \textbf{210} (1997), 440--458. \bibitem{Mao06} X. Mao, C. Yuan; \emph{Stochastic Differential Equations with Markovian Switching}, Imperial College Press, London, UK, 2006. \bibitem{X.Z.Meng10} X. Z. Meng, Z. Zhao, J. J. Nieto; \emph{Dynamic analysis of Michaelis-Menten chemostat-type competition models with time delay and pulse in a polluted environment}, {J. Math. Chem.} \textbf{47} (2010), 123-144. \bibitem{Ma20}{\it J.Pan, Z.Jin, Z.Ma}: Thresholds of survival for an n-dimensional Volterra mutualistic system in a polluted environment. {J. Math. Anal. Appl.} \textbf{252} (2000), 519-531. \bibitem{Tao} F. Tao, B. Liu; \emph{Dynamic behaviors of a single-species population model with birth pulses in a polluted environment}, {Rocky Mountain J. Math.} \textbf{38} (2008), 1663-1684. \bibitem{Thomas} D. M. Thomas, T. W. Snell, S. M. Joffer; \emph{ A control problem in a polluted environment}, {Math. Biosci.} \textbf{133} (1996), 139-163. \bibitem{Wang} W. Wang, Z. Ma; \emph{Permanence of a nonautomonous population model}, Acta Math. Appl. Sin. Engl. Ser. \textbf{1} (1998), 86--95. \bibitem{Yang}X. Yang, Z. Jin, Y. Xue; \emph{Weak average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input}, {Chaos Solitons Fractals} \textbf{31} (2007), 726-735. \bibitem{Zhao} Z. Zhao, L. Chen, X. Song; \emph{Extinction and permanence of chemostat model with pulsed input in a polluted environment}, {Commun. Nonlinear Sci. Numer. Simul.} \textbf{14} (2009), 1737-1745. \end{thebibliography} \end{document}