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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 108, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/108\hfil Persistence and extinction]
{Persistence and extinction of single population in a polluted
environment}

\author[Z. Li, Z. Shuai, K. Wang\hfil EJDE-2004/108\hfilneg]
{Zhan Li, Zhisheng Shuai, Ke Wang}  % in alphabetical order

\address{Zhan Li\hfill\break
Department of Mathematics, Key Laboratory for Vegetation
Ecology of Education Ministry\\
Northeast Normal University \\
Changchun 130024, China}
\email{lizhan1125@hotmail.com \quad Tel: 00 86 631 5689621}

\address{Zhisheng Shuai\hfill\break
Department of Mathematics, Key Laboratory for Vegetation
Ecology of Education Ministry\\
Northeast Normal University \\
Changchun 130024, China}
\email{shuaizs@hotmail.com}


\address{Ke Wang\hfill\break
  Department of Mathematics, Harbin Institute of Technology,
   Weihai 264209,   Shandong, China}
\email{w\_k@hotmail.com}


\date{}
\thanks{Submitted January 9, 2004. Published Sept 12, 2004.}
\thanks{Supported by the National Natural Science Foundation of China
 (No.10171010), and by the \hfill\break\indent
 Key Project on Science
 and Technology of the Education Ministry of China (No. Key 01061).}

\subjclass[2000]{93C15}
\keywords{Population dynamics; environment pollution; persistence}


\begin{abstract}
 In this paper, we consider the ODE system corresponding
 to a diffusive-convective model for the dynamics of a population
 living in a polluted environment. Sufficient criteria on persistence
 and extinction of the population are derived.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

Today, the most threatening problem to the society is the change
in environment caused by pollution, affecting the long term
survival of species, human life style and biodiversity of the
habitat. Therefore the study of the effects of toxicant on the
population and the assessment of the risk to populations is
becoming more important.

In the early eighties a deterministic modelling approach to the
problem of assessing the effects of a pollutant on an ecological
system was proposed by Hallam and his co-workers\cite{h1,h2,h3}.  Since
then, such models have been the subject of many investigations and
improvements.  Population-toxicant coupling has been applied in
several contexts, including Lotka-Volterra and chemostat-like
environments, resulting in ordinary, integro-differential and
stochastic models.  Usually a qualitative analysis was performed
which focuses on the survival or extinction of populations
\cite{m1,m2}.
All these studies rely on the hypothesis of a complete spatially
homogeneous environment.

Recently, a first attempt to consider a spatial structure has been
carried out in \cite{b2,b3} where a reaction-diffusion model is
proposed to describe the dynamics of a living population
interacting with a toxicant present in the environment(external
toxicant) through the amount of toxicant stored into the bodies of
the living organisms(internal toxicant).  However, as the authors
pointed out, even if the resulting model presents many features
which make stimulating its study, such a modelling approach is a
rough approximation to the biological phenomena at hand. In 1999,
Buonomo et.al. viewed the internal toxicant as drifted by the
living population and then, by balance arguments, they derived a
PDE system consisting into two reaction diffusion equations
coupled with a first order convection equation, and the
corresponding ODE system was obtained as well \cite{b1}. This model is
the most realistic by now but the analysis of it is so difficult
that they only used some analytic and numerical approaches.
Obviously, the more clear work is deserved to do.

In this paper, we use some new methods to investigate the model
made by Buonomo et al. and the conditions of survival and
extinction are obtained.

\section{The Model}

We utilize a modified logistic equation \cite{f1} to model the effect of
toxin on single species.  We take

$n(t)$: concentration of the population biomass

$c(t)$: concentration of the toxicant in the environment

$z(t)$: concentration of the toxicant in the population.

We assume that there is a given(external) toxicant in the
environment, and the living organisms absorb into their bodies
part of this toxicant so that the dynamics of the population is
affected by this(internal) toxicant.  Concerning the growth rate
of the population we assume that the birth rate is $b(n)=b_0-fn$
and the death rate is $d(n,c)=d_0+\alpha c$, where $b_0$,$d_0$ and
$\alpha$ are positive constants.  $f$ is assumed to be a
non-negative constant.  Therefore we assume, in absence of
toxicant, a malthusian($f=0$) or a logistic growth rate($f>0$). We
can see that if $b_0-d_0-\alpha c\leq 0$, $n(t)$ will be extinct
in the end, so we suppose
\begin{equation}
c<\frac{b_0-d_0}{\alpha}:=c_1.\label{e2.1}
\end{equation}
We propose the following model governing the system
\begin{equation} \label{eM}
\begin{gathered}
 \frac{dn}{dt}=n(b_0-d_0-\alpha c-fn)\\
\frac{dc}{dt}=kz-(r+m+b_0-fn)c\\
\frac{dz}{dt}=-kzn+(r+d_0+\alpha c)cn-hz+u(t).
\end{gathered}
\end{equation}
with initial data
$$
n(0)=n_0\geq0;\quad c(0)=c_0\geq0; \quad z(0)=z_0\geq 0.
$$
Here
$\alpha$ is the depletion rate coefficient of the population due
to organismal pollutant concentration.
\\
$k$ is the depletion rate of toxicant in the environment due to its
intake made by the population.
\\
$r$ is the depletion rate of toxicant in the population due to
egestion.
\\
$m$ is the depletion rate of toxicant in the population due to
metabolization processes.
\\
$h$ is the depletion rate of the toxicant in the environment.
\\
$u$ is the exogen+ous toxicant input rate which is assumed to be a
smooth bounded non-negative function of $t$.

\section{The conditions of survival and extinction for the
population when $u(t)=Q>0$}

We now recall the definitions of persistence and extinction.  A
component $n(t)$ of a given ODE system is said to be persistent if
for any $n(0)>0$ it follows that $n(t)>0, t>0$ and
$\liminf_{t\to\infty} n(t)>0$.  If there exists $\delta>0$
(independent of $n(0)$) such that $n(t)$ is persistent and is
bounded and $\liminf_{t\to\infty} n(t)\geq\delta$, then
$n(t)$ is said to be uniformly strongly persistent.  If there
exists $\delta>0$ (independent of $n(0)$) such that
$\limsup_{t\to\infty} n(t)\geq\delta$, then $n(t)$ is said
to be uniformly weakly persistent.  If
$\limsup_{t\to\infty} n(t)\leq 0$, then $n(t)$ is said to
be extinct.


\begin{theorem} \label{thm1}
The system \eqref{eM} is uniformly strongly
persistent if and only if
\[
\alpha kQ<h( b_{0}-d_{0}) ( r+m+b_{0}) .
\]
\end{theorem}

\begin{proof}
First, we deduce  that the system is
uniformly weakly persistent.
Assume that the system is not uniformly weakly persistent then
there exists a sequence of initial value
$(n_{k}(0),c_{k}(0),z_{k}(0))\in ( 0,+\infty ) \times R_{+}^{2}$,
such that
\[
\limsup_{t\to\infty} n_{k}(t)=\varepsilon _{k}\to 0,\quad \mbox{as }
k\to +\infty .
\]
Then there exist $T_{k}>0$, such that
\begin{equation}
n_{k}(t)<2\varepsilon _{k} \quad\mbox{for } t\geq T_{k}.  \label{e3.1}
\end{equation}
From \eqref{e2.1} and \eqref{e3.1} we have
$$
\dot{z_{k}}\leq (r+d_0+\alpha c)cn_{k}-hz_{k}+Q\leq(r+b_0)c_12\varepsilon
_{k}-hz_{k}+Q \quad\mbox{for } t\geq T_{k}\,.
$$
Using the Comparison Theorem we have
$$
\limsup_{t\to\infty} z_{k}(t)\leq \frac{c_1(r+b_0)2\varepsilon _{k}+Q}{h}.
$$
 So for all $\varepsilon>0$, there exists $H_{k}>T_{k}>0$, such that
\begin{equation}
z_{k}(t)\leq \frac{c_1(r+b_0)2\varepsilon _{k}+Q}{h}+\varepsilon
=:\overline{z}_{k} \quad\mbox{for } t\geq H_{k} .  \label{e3.2}
\end{equation}
 From \eqref{e3.1} and \eqref{e3.2} we
see that
\begin{align*}
\dot{c_{k}} &\leq k\overline{z}_{k}-(r+m+b_0)c_{k}+f2\varepsilon _{k}c_{k}\\
&= k\overline{z}_{k}-(r+m+b_0-f2\varepsilon_{k})c_{k}~~~for~~
t\geq H_{k}\,.
\end{align*}
Similarly by the Comparison Theorem and let
$\varepsilon\to 0$, we have
 $$
\limsup_{t\to\infty} c_{k}(t)\leq \frac{kc_1(r+b_0)2\varepsilon
_{k}+kQ}{h(r+m+b_0-f2\varepsilon _{k})}\,.
 $$
Then, for $t\geq H_{k}$,
\begin{equation}
c_{k}(t)\leq \frac{kc_1(r+b_0)2\varepsilon _{k}+kQ}{h(r+m+b_0-f2\varepsilon
_{k})}:=\beta(\varepsilon_{k}).  \label{e3.3}
\end{equation}
 Now we consider the
first equation of the model \eqref{eM}, from \eqref{e3.3} it is easy to see
there exists $S_{k}>H_{k}>0$, such that
$$
\dot{n}_{k} \geq n_{k}(b_0-d_0-\alpha \beta(\varepsilon_{k})-fn_{k})~~~for~~
t\geq S_{k}.
$$
Using the Comparison Theorem again we have
\begin{equation}
\liminf_{t\to\infty} n_{k}(t)\geq
\frac{b_0-d_0-\alpha\beta(\varepsilon_{k})}{f}.  \label{e3.4}
\end{equation}
By
\eqref{e3.4} and the assumption
$\limsup_{t\to\infty}n_{k}(t)=\varepsilon _{k}$, we obtain
$$
\limsup_{t\to\infty}
n_{k}(t)=\varepsilon _{k}\geq \liminf_{t\to\infty}
n_{k}(t)\geq \frac{b_0-d_0-\alpha\beta(\varepsilon_{k})}{f}.
$$
Let $k\to +\infty$, it follows that $\varepsilon _{k}\to 0$ and
$\beta(\varepsilon_{k})\to \frac{kQ}{h(r+m+b_0)}$. Hence,
$$
0\geq \frac{b_0-d_0-\alpha\beta(\varepsilon_{k})}{f}\to
\frac{h(b_0-d_0)(r+m+b_0)-\alpha kQ}{fh(r+m+b_0)}\,.
$$
That is,
$\alpha kQ\geq h(b_0-d_0)(r+m+b_0)$. So there is uniformly
weakly persistent if
$$ \alpha k Q<h( b_{0}-d_{0})( r+m+b_{0}).
$$
Using the well known result which says that uniform weak
persistence implies uniform strong persistence \cite[Section 2]{h4},
then the proof is completed.
\end{proof}

Moreover, if we look at the system restricted to
$\{0\}\times R_{+}^{2}$, then there is a unique
equilibrium
$$
\overline{X}=( 0,\overline{c},\overline{z})
$$
with
$$
\overline{z}=\frac{Q}{h}\quad\mbox{and}\quad
\overline{c}=\frac{k\overline{z}}{(r+m+b_{0}) }
$$
 Then an easy investigation
of the linearized equation at $\overline{X}$ shows that when
$\alpha kQ>h( b_{0}-d_{0}) ( r+m+b_{0}) $,
$\overline{X}$ is locally asymptotically stable.
In particular the system is not
uniformly persistent anymore.


\begin{theorem} \label{thm2}
Consider the system \eqref{eM}. If
$Q>\frac{(r+m+b_0)(b_0-d_0)(fh+k(b_0-d_0))}{\alpha kf}$, then the
population is extinct.
\end{theorem}

\begin{proof}  From \eqref{e3.1} we know $n(t)\leq n_1$, for $t>t_1$.
 So from the last equation of the model \eqref{eM} , we can obtain
$$
\frac{dz}{dt}>Q-hz-kn_1z \quad\mbox{for } t>t_1.
$$
Now we use the Comparison Theorem again, then we have
 $$
\liminf_{t\to +\infty} z(t) \geq \frac{Q}{h+kn_1}.
$$
 Let $\varepsilon \to 0$, we have
$$
\liminf_{t \to +\infty}z(t)\geq=\frac{fQ}{fh+k(b_0-d_0)}\\:= m_z.
$$
That is to say for all $\varepsilon >0,\exists T_1$, such that
$z(t)>m_z-\varepsilon$,  for all $t>T_1$.
 From the second equation of the model \eqref{eM} , if $t>T_1$, it is easy
to see
$$ \frac{dc}{dt}>k(m_z-\varepsilon)-(r+m+b_0)c.  $$
Similarly by the Comparison Theorem,
$$
\liminf_{t\to +\infty} c(t)
\geq\frac{km_z-\varepsilon)}{r+m+b_0}.$$
Let  $\varepsilon \to 0$. Then we have
 \[
\liminf_{t \to +\infty} c(t) \geq  \frac{k(m_z}{r+m+b_0}
= \frac{kfQ}{(r+m+b_0)(fh+k(b_0-d_0))}\\
=: m_c.
\]
Then for all $\varepsilon >0$ there exists $T_2>T_1$, such that
$c(t)>m_c-\varepsilon$ for all $t>T_2$.
Obviously, from the first equation of the model \eqref{eM}, if $t>T_2$,
we have
$$
\frac{dn}{dt}<n(b_0-d_0-\alpha(m_c-\varepsilon)-fn).
$$
By the Comparison Theorem, we have
$$
\limsup_{t\to+\infty} n(t) \leq \frac{b_0-d_0-\alpha(m_c-\varepsilon)}{f}.
$$
Let $\varepsilon \to 0$. Then we obtain
 \[
\limsup_{t \to +\infty} n(t) \leq
\frac{b_0-d_0-\alpha m_c}{f}
=: M_n\,.
\]
Clearly,
$$
\limsup_{t \to +\infty} n(t) \leq M_n \leq \frac{b_0-d_0}{f}.
$$
If $M_n<0$, that is
$Q>\frac{(r+m+b_0)(b_0-d_0)(fh+k(b_0-d_0))}{\alpha kf}$, the
population will be extinct.
\end{proof}

\subsection*{Acknowledgements}
We thank the anonymous referee for his/her remarks that helped us
improving Theorem \ref{thm1}.

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\end{thebibliography}
\end{document}