\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 04, pp. 1--8.\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-2013/04\hfil Global dynamics]
{Global dynamics of a predator-prey model incorporating a constant prey refuge}

\author[X. Yu, F. Sun \hfil EJDE-2013/04\hfilneg]
{Xiaolei Yu, Fuqin Sun} 

\address{Xiaolei Yu \newline
School of Science,
Tianjin University of Technology and Education,
Tianjin 300222, China}
\email{13752383154@126.com, yuxioalei12@126.com}

\address{Fuqin Sun \newline
School of Science,
Tianjin University of Technology and Education,
Tianjin 300222, China}
\email{sfqwell@163.com}

\thanks{Submitted May 8, 2012. Published January 7, 2013.}
\subjclass[2000]{34D05, 34D20, 92D25}
\keywords{Predator-prey model; prey refuge; global stability}

\begin{abstract}
 In this article, a general predator-prey model incorporating
 a constant prey refuge with Hassell-Varley type functional response
 is studied. Sufficient conditions for the stability of the equilibria
 are obtained. It is shown that the positive equilibrium exists if
 predator death rate multiplied by a constants is smaller than its
 growth rate multiplied by capturing rate. Moreover, by constructing
 a Lyapunov function, it is shown that the positive equilibrium is
 globally stable.
\end{abstract}

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

\newcommand{\p}{\varphi}

\section{Introduction}

 In 1980, Freedman \cite{Freedman} proposed a predator-prey model
with Holling type II functional response:
\begin{equation} \label{e1}
\begin{gathered}
 x'(t) =ax(1-x/K)-bxy/(\beta+x),\\
 y'(t) =y(ex/(\beta+x)-D),\\
 x(0)>0,\quad y(0)>0,
\end{gathered}
\end{equation}
where $x$ and $y$ denote the prey and predator populations,
respectively at any time $t$; $a>0$ represents the intrinsic growth
rate of the prey; $K>0$ is the carrying capacity of the prey in the
absence of predator; $b>0$ is the conversion factor denoting the
number of newly born predators for each captured prey; $D>0$ is the
death rate of the predator; $e>0$ is the intrinsic growth rate of
the predator; $\beta$ represents half saturation constant. The model
exhibits the well-known but highly controversial paradox of
enrichinent observed by Hairston et al \cite{Hairston} (1960) and by
Rosenzweig \cite{Rosenzweig} (1969) which is rarely reported in
nature. To address this problem, Arditi and Ginzburg
\cite{Arditi} (1989) proposed the following predator-prey model with
ratio-dependent type functional response:
\begin{equation} \label{e2}
 \begin{gathered}
 x'(t) =ax(1-x/K)-bxy/(\beta y+x),\\
 y'(t) =y(ex/(\beta y+x)-D),\\
 x(0)>0, \quad y(0)>0.
 \end{gathered}
\end{equation}
It is well known system \eqref{e2} can display richer and more plausible
dynamics than system \eqref{e1}.

A general predator-prey model with Hassell-Varley type functional
response may take the following form (Hsu 2008) \cite{Hsu}:
\begin{equation} \label{e3}
 \begin{gathered}
 x'(t) =ax(1-x/K)-bxy/(\beta y^\gamma+x),\\
 y'(t) =y(ex/(\beta y^\gamma+x)-D),\\
 x(0)>0,\quad y(0)>0,
 \end{gathered}
\end{equation}
where the constant $\gamma>0$ is called the Hassel-Varley constant.
A unified mechanistic approach was provided by Cosner et al
\cite{Cosner} where the functional response in system \eqref{e3} was
derived. In a typical predator-prey interaction where predators do
not form groups, one can assume that $\gamma=1$, producing the so-called
ratio-dependent predator-prey dynamics. For terrestrial predators
that form a fixed number of tight groups,
it is often reasonable to assume that $\gamma=1/2$.
For aquatic predators that form a fixed number of tight groups,
$\gamma=1/3$ maybe more appropriate.
Since most predators do not form a fixed number of tight groups,
it can be argued that for most realistic
predator-prey interactions,$\gamma\in[0,1]$.


 Because species compete, evolve and disperse
often simply for the purpose of seeking resources to sustain
their struggle for their very existence. Their extinctions are often
the results of their failure in obtaining the minimum level of
resources needed for their subsistence. Thus, prey refuge are widely
believed to prevent prey extinction and damp predator-prey
oscillations. For example, Gonzalez-Olivares and Ramos-Jiliberto \cite{Gonzalez}
studied the dynamic consequences of the following predator-prey systems with
constant number of prey using refuges, which protects of prey $m$
from predation
\begin{equation} \label{e4}
\begin{gathered}
 x'(t) =ax(1-x/K)-\frac{b(x-m)y}{1+c(x-m)},\\
 y'(t) =-Dy+\frac{be(x-m)y}{1+c(x-m)},\\
 \end{gathered}
\end{equation}
$m>0$ is a constant number of prey using refuges, which protects $m$
of prey from predation; $c>0$ is a constant.

Inspired by [4] and [10], we will consider a more general
predator-prey model incorporating a constant prey refuge with
Hassell-Varley type functional response.
\begin{equation} \label{e5}
 \begin{gathered}
 x'(t) =ax(1-x/K)-\frac{b(x-m)y}{y^\gamma+c(x-m)},\\
 y'(t) =-Dy+\frac{be(x-m)y}{y^\gamma+c(x-m)},\\
 x(0)>m,\quad y(0)>0.
\end{gathered}
\end{equation}
Mathematically, system \eqref{e1} or \eqref{e2} can be viewed as
limiting cases of system \eqref{e3} if one chooses $\gamma=0$ or 1
in system \eqref{e3}.
 System \eqref{e4} can be viewed as limiting case of system \eqref{e5}
if one chooses $\gamma=0$ in system \eqref{e5}. In this paper we
take $\gamma\in(0,1)$.

In this article, we will find sufficient conditions of the
stability properties to the equilibria of \eqref{e5}. We will
show the positive equilibrium exists if predator death rate
multiplied by a constant is smaller than its growth rate multiplied
by capturing rate. We construct a Lyapunov function to show the
global stability of the positive equilibrium.

\section{Preliminary analysis}

In this section, we present the basic results
 on the boundedness of positive solutions and the local stabilities
of nonnegative equilibria in \eqref{e5}.

Let $\Omega_0=\{(x,y):x>m,\; y>0\}$, for practical biological
meaning, we simply study system \eqref{e5} in $\Omega_0$.

By the scaling: $t\to at$, $x\to x/m$, $y\to\alpha y$.
System \eqref{e5} turns into
\begin{equation} \label{e6}
\begin{gathered}
x'(t) =x\Big(1-\frac{m}{K}x\Big)-\frac{s(x-1)y}{y^\gamma+x-1}
 \equiv F(x,y),\\
y'(t) =\delta y\Big(-d+\frac{x-1}{y^\gamma+x-1}\Big) \equiv G(x,y),\\
 x(0)>1,\quad y(0)>0,
\end{gathered} 
\end{equation}
where $s=\frac{b}{a}(cm)^{\frac{1}{\gamma}-1}$,
$\delta=\frac{be}{ac}$, $d=\frac{Dc}{be}$. we thus define that
$F(1,0)=G(1,0)=0$. Clearly with this assumption, both $F$ and $G$
are continuous on the closure of $\Omega$, where
$\Omega=\{(x,y)|x>1,y>0\}$.

The variational matrix of the system \eqref{e6} is 
\begin{equation*}
A(x,y)= \begin{pmatrix}
 1-\frac{2m}{K}x-\frac{sy}{y^\gamma+x-1}
+\frac{s(x-1)y}{(y^\gamma+x-1)^2} 
& \frac{-s(x-1)}{(y^\gamma+x-1)^2}[(x-1)+(1-\gamma)y^\gamma] \\
 \frac{\delta y^{\gamma+1}}{(y^\gamma+x-1)^2} 
& \delta(\frac{x-1}{y^\gamma+x-1}-\frac{\gamma(x-1)y^\gamma}{(y^\gamma+x-1)^2}-d) 
 \end{pmatrix}.
\end{equation*}

\begin{proposition} \label{prop1}
Let $(x(t),y(t))$ be any solution of \eqref{e6} with $(x(0),y(0))\in\Omega$.
Then
\begin{equation*}
\lim\sup_{t\to\infty}(x(t)+\frac{s}{\delta}y(t))\leq
\frac{(1+d\delta)^2K}{4dm\delta}.
\end{equation*}
\end{proposition}

\begin{proof}
 It follows immediately from the existence and uniqueness of
solutions for ordinary differential equations with initial
conditions  that the solution is positive on its domain of definition. 
Let $V(t)=x(t)+\frac{s}{\delta}y(t)$ and differentiating $V$ once yields
\begin{equation*}
V'(t)=-\frac{m}{K}x^2+(1+d\delta)x-d\delta V(t)
\leq\frac{(1+d\delta)^2}{4m/K}-d\delta V.
\end{equation*}
Hence we have 
\[
0<V(t)\leq \frac{(1+d\delta)^2K}{4dm\delta}+(V(0)
-\frac{(1+d\delta)^2}{4d\delta m/K})e^{-dt\delta}.
\]
This gives the desired result. 
\end{proof}

System \eqref{e6} has three equilibria. They are $E_0=(0,0)$ which is not
in $\Omega$, $E_1=(\frac{K}{m},0)$ and $E_*=(x_*,y_*)$, where
$x_*>1$, $y_*>0$ and
\begin{gather*} 
 x_*\left(1-\frac{m}{K}x_*\right)-\frac{s(x_*-1)y_*}{y_*^\gamma+x_*-1}=0, \\
 \frac{x_*-1}{y_*^\gamma+x_*-1}=d,
 \end{gather*} 
if $d\in(0,1)$.

At $E_1$, we have
\begin{equation*}
A(\frac{K}{m},0)= \begin{pmatrix}
 -1 & s \\
 0 & \delta(1-d)
 \end{pmatrix}.
  \end{equation*}
For the rest of this article, we always assume that $0<d<1$. 
With this assumption,  it is easy to know that $E_1$ is a saddle point.
 At $E_*$, we have
 \begin{gather*}
\begin{aligned}
&A(x_*,y_*)\\
&=\begin{pmatrix}
 {1-\frac{2m}{K}x_*-\frac{sy_*}{y_*^\gamma+x_*-1}
 +\frac{s(x_*-1)y_*}{(y_*^\gamma+x_*-1)^2}} 
& \frac{-s(x_*-1)}{(y_*^\gamma+
 x_*-1)^2}[(x_*-1)+(1-\gamma)y_*^\gamma] \\
 \frac{\delta y_*^{\gamma+1}}{(y_*^\gamma+x_*-1)^2}
& -\frac{\delta\gamma(x_*-1)y_*^\gamma}{(y_*^\gamma+x_*-1)^2} 
 \end{pmatrix},
\end{aligned}\\
\begin{aligned}
\det A(x_*,y_*
&={\frac{-\delta\gamma y_*^\gamma(x_*-1)
 (1-\frac{2m}{K}x_*)}{(y_*^\gamma+x_*-1)^2}
+\frac{s\gamma\delta(x_*-1)y_*^{\gamma+1}}{(y_*^\gamma+x_*-1)^3}}\\
&\quad +{\frac{s\delta(x_*-1)y_*^{\gamma+1}[(1-\gamma)(x_*-1)
 +(1-\gamma)y_*^\gamma]}{(y_*^\gamma+x_*-1)^4}},
\end{aligned} 
\\
\operatorname{tr} A(x_*,y_*)=1-\frac{2m}{K}x_*-\frac{sy_*}{y_*^\gamma+x_*-1}
+\frac{(x_*-1)y_*(s-\delta\gamma y_*^{\gamma-1})}{(y_*^\gamma+x_*-1)^2}.
\end{gather*}
Hence the stability of $E_*$ is determined by the sign of $\det A(x_*,y_*)$ 
and $\operatorname{tr} A(x_*,y_*)$.

\begin{proposition} \label{prop2}
For system \eqref{e6}, the following statements hold:
\begin{enumerate}
 \item $E_1$ is a saddle point;
 
\item when $det A(x_*,y_*)>0$, then $E_*$ is locally asymptotically stable 
if $\operatorname{tr} A(x_*,y_*)<0$;
$E_*$ is unstable if $\operatorname{tr} A(x_*,y_*)>0$;

\item when $\det A(x_*,y_*)<0$, then $E_*$ is a saddle point.
\end{enumerate}
\end{proposition}

\section{Uniform persistence}

The objective of this section is to present conditions ensuring the
system \eqref{e6} is uniformly persistent. To this end, we make the change
of variables $(x,y)\to(u,z)$ in system \eqref{e6}, where
$u=\frac{x-1}{y^\gamma}$, $z=y^\sigma$ and $\sigma$ will be chosen
later. This reduces  it to the system
\begin{equation} \label{e7}
\begin{gathered}
 u'(t) =g(u)-\varphi_1(u)z^{\sigma_1}-\varphi_2(u)z^{\sigma_2}+(1-m/K)z^{-1} 
 \equiv f_1(u,z), \\
 z'(t) =\psi(u)z \equiv f_2(u,z),\\
 u(0)>0,\quad z(0)>0,
 \end{gathered}
 \end{equation}
where
\begin{gather*} 
 g(u) =\frac{u}{1+u}[1+\gamma\delta d-2m/K+(1+\gamma\delta d-\gamma\delta-2m/K)u], \\
 \varphi_1(u) =2m/Ku^2,\quad 
 \varphi_2(u) =\frac{su}{1+u},\\
 \psi(u) =\sigma\delta\Big(-d+\frac{u}{1+u}\Big),
 \end{gather*} 
and $\sigma_1=\gamma/\sigma$ and
$\sigma_2=(1-\gamma)/\sigma$. Now let $\sigma=\gamma$ if
$\gamma\in(0,1/2)$
 and $\sigma=1-\gamma$ if $\gamma\in[1/2,1)$, then
\begin{equation*} \label{eq:7}
\sigma_1 = \begin{cases}
 1 & \text{if } \gamma\in(0,\frac1{2}), \\
 \gamma/(1-\gamma) &\text{if } \gamma\in[\frac1{2},1)
 \end{cases} 
 \end{equation*}
and
\begin{equation*} 
\sigma_2 = \begin{cases}
 (1-\gamma)/{\gamma} &\text{if } \gamma\in(0,1/2), \\
 1 &\text{if }\gamma\in[1/2,1).
 \end{cases}
 \end{equation*}
Hence, $\sigma_i\geq1,i=1,2$ and the vector field $(f_1,f_2)$ is
$C^1$ smooth on the set $\mathbb{R}_+^2$ where
$\mathbb{R}_+^2=\{(u,z):,u>0,\,z>0\}$. 
Since $0<d<1$, we have $\psi(u_*)=0$ where
$u_*=d/(1-d)$ and
$$
\psi(u)=\frac{\sigma\delta(1-d)(u-u_*)}{1+u}.
$$
Moreover, $g(u)>0$ on $\mathbb{R}_+^2$ if $\gamma\delta\leq\frac{1-2m/K}{1-d}$ 
where $1-2m/K\geq0$ and $g(u)$ has
exactly one positive zero 
$u_0=\frac{-(1+\gamma\delta d-2M/K)}{1+\gamma\delta d-\gamma\delta-2m/K}$ 
if $\gamma\delta>\frac{1-2m/K}{1-d}$.
In last case, we have $g(u)(u-u_0)<0$ for $u\neq u_0$.

From system \eqref{e7}, we have that the prey isocline, $z=h(u)$ is
implicitly defined by $f_1(u,z)=0$. Since
$\lim_{z\to\infty}f_1(u,z)=-\infty$ and 
$\frac{\partial f_1}{\partial z}(u,z)<0$, it follows from the implicit function
theorem that $z=h(u)$ is $C^1$ function defined on $(0,\infty)$ if
$\gamma\delta\leq\frac{1-2m/K}{1-d}$ or on $(0,u_0]$ if
$\gamma\delta>\frac{1-2m/K}{1-d}$. Moreover,
\begin{equation} \label{e8}
\begin{aligned}
h'(u)
&= -\frac{{\partial f_1(u,h(u))}/{\partial u}}{{\partial f_1(u,h(u))}
 /{\partial z}},\\
&= {\frac{\big(\frac{g(u)}{\varphi_2(u)}\big)'
 -\big(\frac{\varphi_1(u)}{\varphi_2(u)}\big)'
 h^{\sigma_1}(u)-\frac{1-m/K}{\varphi^2_2(u)} \p'_2(u) h^{-1}(u)}
 {\big(\frac{\varphi_1(u)}{\varphi_2(u)}\big)\sigma_1 h^{\sigma_1 -1}(u)
 +\sigma_2 h^{\sigma_2 -1}(u)+\frac{1-m/K}{\varphi_2(u)}h^{-2}(u)}}.
\end{aligned}
\end{equation}
The qualitative behavior of $z=h(u)$ is given in the following
lemma.

\begin{lemma} \label{lem1}
\begin{itemize}
\item[(a)] If $\gamma\delta\in(\frac{1-2m/K}{1-d},\infty)$, 
then $h(u)>0>h'(u)$ for all $u\in[0,u_0]$.

\item[(b)] If $\gamma\delta\in(0,\frac{1-2m/K}{1-d}]$ and
$h(u)>{[\frac{K(1+\gamma\delta
d-\gamma\delta-2m/k)}{m}]}^{1/{\sigma_1}}$,
 then $h(u)>0>h'(u)$ for all $u\in (0,\infty)$.
\end{itemize}
\end{lemma}

\begin{proof} From \eqref{e8}, we have $h'(u)<0$ as long as
 $\gamma\delta\in(\frac{1-2m/K}{1-d},\infty)$, This proves the assertion (a).

Now let $1+\gamma\delta d-\gamma\delta-2m/K\geq0$ , we have
\begin{align*}
h'(u)\\
&={\frac{\frac1{s}(1+\gamma\delta d-\gamma\delta-2\frac{m}{K})
 -\frac{m}{sK}(1+2m)h^\sigma_1(u)-
su^{-2}(1-\frac{m}{K})h^{-1}(u)}{(\frac{\varphi_1(u)}{\varphi_2(u)})
\sigma_1 h^{\sigma_1 -1}(u)+\sigma_2 h^{\sigma_2 -1}(u)
 +\frac{1-m/K}{\varphi_2(u)}h^{-2}(u)}}
<0
\end{align*}
as long as
 $$
\frac{m}{sK}h^{\sigma_1}(u)>\frac{1}{s(1+\gamma\delta
d-\gamma\delta-2m/K)}.
$$
 This proves assertion (b).
\end{proof}

System \eqref{e7} has one positive equilibrium $e_*=(u_*,z_*)$ 
where $z_*=h(u_*)$. The variational matrix of system \eqref{e7} 
is given by
\begin{align*}
&J(u,z)\\
&= \begin{pmatrix}
 g'(u)-\varphi_1'(u)z^{\sigma_1}-\varphi_2'(u)z^{\sigma_2 }
 &-\sigma_1\varphi_1(u)z^{\sigma_1-1}-\sigma_2\varphi_2(u)z^{\sigma_2-1}
 -(1-\frac{m}{K})/z^{2} \\
 \sigma\delta z/(1+u)^2  & \sigma\delta(u/(1+u)-d) \\
 \end{pmatrix}.
\end{align*}
The stability of equilibrim $e_*$ is determined by the eigenvalues
of the matrix $J(e_*)$ and is given in the following lemma.

\begin{lemma} \label{lem2}
For system \eqref{e7}, the following statements are true.
\begin{itemize}
\item[(a)] If $tr(J(e_*))<0$, then $e_*$ is locally asymptotically stable;
\item[(b)] If $tr(J(e_*))>0$, then $e_*$ is an unstable focus or node.
\end{itemize}
\end{lemma}

\begin{remark} \label{rmk1} \rm
Since
$(x_*,y_*)=(u_*z_*^{\frac{\gamma}{\sigma}+1},z_*^{\frac{1}{\sigma}})$,
we have
\begin{align*}
\operatorname{tr} A(E_*) 
&= 1-2m/Kx_*-\frac{sy_*}{y_*^\gamma+x_*1}+\frac{(x_*-1)y_*(s-\gamma\delta y_*^{\gamma-1})}{(y_*^\gamma+x_*-1)^2} \\
&= 1-2m/K(u_*z_*^{\gamma/\sigma}+1)-\frac{sz_*^{(1-\gamma)/\sigma}}{(u_*+1)}+\frac{u_*z_*^{(1-\gamma)/\sigma}(s-\delta\gamma z_*^{(\gamma-1)/\sigma})}{(u_*+1)^2} \\
&= \operatorname{tr}J(e_*) .
\end{align*}
So, the locally stability of $E_*$ and $e_*$ are the same.
\end{remark}

\begin{lemma} \label{lem3}
System \eqref{e7} is uniformly persistent in $\mathbb{R}_+^2$.
\end{lemma}

\begin{proof}
 Let $(u(t),z(t))$ be the solution starting at $A=(u_*,M_*+1)$
where $M_*=[\frac{(1+d \delta)^2K}{4sdm}]^\sigma$ and $\Gamma$ be
its orbit. Then since
$(x(t),y(t))=(u(t)z^{\frac{\gamma}{\sigma}+1}(t),z^\frac{1}{\sigma}(t))$
is a solution of system \eqref{e6} and by Proposition \ref{prop1}, we have
 $\lim\sup_{t\to\infty}z(t)\leq M_*$.
 Hence $\Gamma\subseteq\mathbb{R}_+\times(0,M_*+1)$. 
The flow analysis gives that $\Gamma$ must intersect
 the prey isocline$\{(u,h(u))|0<u<u_*\}$. Let B be the first
 point that they intersect. There are two possibilities for $\Gamma$.

\noindent\textbf{Case 1: $\Gamma\cap\{(u_*,z)|z\in(0,h(u_*))\}\neq\emptyset$.}
Let $C=(u_*,z_1)$ be the first point of
$\Gamma\cap\{(u_*,z)|z\in(0,h(u_*))\}$, $D=(\bar{u},z_1)$ be the
intersection of $\{(u,z_1)|u>u_*\}$ and $z=h(u)$. Consider the bounded
region $\overline{\Omega}_1$, enclosed by
$\Gamma,\overline{CD},\overline{DE}$ and $\overline{EA}$ where
$E=(\bar{u},M_*+1)$. Clearly, every trajectory will enter and stay
in $\overline{\Omega}_1$ for all $t$ sufficiently large.

\noindent\textbf{Case 2: $\Gamma\cap\{(u_*,z)|z\in(0,h(u_*))\}=\emptyset$.}
This implies $\lim_{t\to\infty}(u(t),z(t))=e_*$. Let
$\overline{\Omega}_2$ be the bounded region enclosed by $\Gamma$ and
$\overline{e_*A}$. Thus every trajectory will either enter
$\overline{\Omega}_2$ or tend to $e_*$ as $t$ goes to $\infty$.

Hence, from the above discussion, we show that system \eqref{e7} is
permanent.
\end{proof} 

Since every solution of \eqref{e6} takes the form of
$(x(t),y(t))=(u(t)z^{\frac{\gamma}{\sigma}+1}(t),
z^\frac{1}{\sigma}(t))$, where $(u(t),z(t))$ is some solution of
system \eqref{e7}. Thus, as a consequence of Lemma \ref{lem3}, 
we have the following theorem for system \eqref{e6}.

\begin{theorem} \label{thm1}
System \eqref{e6} is uniformly persistent in $\Omega$.
\end{theorem}

\section{Global stability results}

To study the global behavior of \eqref{e6}, we need following lemma.

\begin{lemma} \label{lem4}
Let $1+\gamma\delta d-\gamma\delta-2m/K\leq0$, Then the equilibrium 
$e_*$ is globally asymptotically stable for system \eqref{e7}
 in $\mathbb{R}_+^2$.
\end{lemma}

\begin{proof}
 To show that $e_*$ is globally asymptotically stable in
$\mathbb{R}_+^2$. Consider the following Lyapunov function
\[
V(u,z)=z^{-\frac{g(u_*)}{\varphi_2(u_*)}}\exp
\Big(\frac{\varphi_1(u_*)}{\varphi_2(u_*)}
\frac{z^\sigma_1}{\sigma_1}+\frac{z^{\sigma_2}}{\sigma_2}+
\frac{(1-m/K)z^{-1}}{\varphi_2(u_*)}
+\int_{u_*}^u\frac{\psi(\xi)}{\varphi_2(\xi)}d\xi\Big)
\]
for $(u,z)\in\mathbb{R}_+^2$. The derivative of $V$ along the solution
 of  \eqref{e7} is
\begin{align*}
\frac{\dot{V}(u,z)}{V(u,z)}
&= \psi(u)\Big[\frac{g(u)}{\varphi_2(u)}-\frac{g(u_*)}{\varphi_2(u_*)}\Big]
 -\psi(u)z^{\sigma_1}
 \Big[\frac{\varphi_1(u)}{\varphi_2(u)}-\frac{\varphi_1(u_*)}{\varphi_2(u_*)}
 \Big]\\
&\quad +\psi(u)z^{-1}(1-m/K)\Big[\frac{1}{\varphi_2(u)}
 -\frac{1}{\varphi_2(u_*)}\Big]\\
&= \frac{1}{s}\psi(u)(u-u_*)\Big[1+\gamma\delta d-\gamma\delta-2m/K-m/K(1+u+u_*)z^{\sigma_1}\\
&\quad -(1-m/K)\frac{1-d}{ud}z^{-1}\Big].
\end{align*}
Clearly, $1+\gamma\delta d-\gamma\delta-2m/K\leq0$ implies 
$\dot{V}(u,z)\leq0$ for $(u,z)\in\mathbb{R}_+^2$. 
Hence, the lemma follows from Lyapunov-LaSalle's
 invariance principle \cite{Hale}.
\end{proof}

\begin{theorem} \label{thm2}
Let $1+\gamma\delta d-\gamma\delta-2m/K\leq0$, then the equilibrium
 $E_*$ is globally asymptotically stable for system \eqref{e6} in $\Omega$.
\end{theorem}

\subsection{Discussion} 
To facilitate the discussion section, we summarize our findings.
Recall that 
\[
s=\frac{b}{a}(cm)^{\frac{1}{\gamma}-1},\quad
\delta=\frac{be}{ac}, \quad
d=\frac{Dc}{be}.
\]
 Since $0<d<1$ is
equivalent to $Dc<be$. Predator death rate $D$ multiplied by a
constants $c$ is smaller than its growth rate $e$ multiplied by
capturing rate $b$. From Theorem \ref{thm1}, system \eqref{e6} or 
equivalently \eqref{e5}, is uniformly persistent. 
This means that neither predator nor prey
can die out. Moreover, there is only one positive equilibrium. From
\eqref{e8}(8), Lemma \ref{lem1}, Lemma \ref{lem2} and Remark 
\ref{rmk1}, we have, if $\operatorname{tr} A(E_*)<0$ $(>0)$, then
$E_*$ is locally asymptotically stable (unstable). Moreover, if
$1+\gamma\delta d-\gamma\delta-2m/K\leq0$ , $E_*$ is globally
asymptotically stable.

\subsection*{Acknowledgments}
 The authors want to thank the anonymous referees for their 
valuable comments and suggestions.

This work was supported by grants 12JCYBJC10600 from
the Natural Science Foundation of Tianjin of China, and
20081003 from the Technology Development Foundation of Higher
Education of Tianjin of China.


\begin{thebibliography}{23}
\bibitem{Arditi} R. Arditi, L.R. Ginzburg;
 \emph{Coupling in predator-prey dynamics: ratio-dependence}, 
J. Theor. Biol., 139 (1989), 311-326.

\bibitem{Cosner} C. Cosner, D. L. De Angellis, J. S. Ault,  D. B. Olson;
 \emph{Effects of spatial grouping on the functional response of predators}, 
Theor. Pop. Biol., 565 (1999), 65-75.

\bibitem{Freedman} H.I. Freedman;
 \emph{Deterministic Mathematical Models in Population Ecology},
 Marcel Dekker, 1980 New York.

\bibitem{Gonzalez} E. Gonzalez-Olivares, R. Ramos-Jiliberto;
 \emph{Dynamic consequences of prey rufuges in a simple model system:
More prey, fewer predators and enhanced stability}, 
Ecol. Model. 166 (2003), 135-146.

\bibitem{Hairston} N. G. Hairston, F. E. Simth, L. B. Slobodkin;
\emph{Community structure, population control and competition}, 
American Naturalist, 94 (1960), 42-425.

\bibitem{Hale} J. Hale;
 \emph{Ordinary Differential Equtions}, Krieger Publ. Co., Malabar., 1980.

\bibitem{Hassel} M. P. Hassel;
\emph{The Dynamics of Arthropod Pradator-Prey System},
Princetion University Press, 1978.
\bibitem{Hassell} M. P. Hassell, G. C. Varley;
 \emph{New inductive population model for insect parasites and its bearing 
on biological control}, Nature, 223 (1969), 1133-1136.

\bibitem{Harrision} G. W. Harrision;
\emph{Comparing pradator-prey models to Luckinbill's experiment with Didinium 
and Paramecium}, Ecology, 76(1995), 357-374.

\bibitem{Hsu} S.-B. Hsu, T.-W. Hwang, Y. Kuang;
\emph{Global analysis of a predator-prey model with Hassell-Varley type function 
response}, Discrete Contin. Dyn. Syst. Ser. B., 10 (2008), 857-871.

\bibitem{Hwang1} T.-W. Hwang;
\emph{Uniqueness of limit cycle for Gause-type predator-prey systems}, 
J. Math. Anal. Appl., 238 (1990), 179-195.

\bibitem{Hwang2} T.-W. Hwang, Y. Kuang;
\emph{Host extincton dynamics in a simple parasite-host interaction model}, 
Math. Biosc. and Eng., 2 (2005), 743-751.

\bibitem{Kuang1} Y. Kuang;
\emph{Rich dynamics of Gause-tyoe ratio-dependent pradator-prey system}, 
The Fields Institute Communications, 21 (1999), 325-337.

\bibitem{Kuang2} Y. Kuang, E. Beretta;
 \emph{Global qualitative analysis of a ratio-dependent predator-prey system}, 
J. Math. Biol., 36 (1998), 389-406.

\bibitem{Rosenzweig} M. L. Rosenzweig;
\emph{Paradox of enrichment, destabilization of exploitation systems 
in ecologicaltime}, Science, 171 (1961), 385-387.

\bibitem{Sih} A. Sih;
\emph{Prey refuges and predator-prey stability},
 Theoret. Popul. Biol. 31 (1987), 1-12.

\bibitem{Xiao} D. Xiao, S. Ruan;
\emph{Global dynamics of a ratio-dependent predator-prey system}, 
J. Math. Biol., 43 (2001), 268-290.

\end{thebibliography}

\end{document}