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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 192, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/192\hfil A density-dependent predator-prey model]
{A density-dependent predator-prey model of Beddington-DeAngelis type}

\author[H. Y. Li, Z. K. She \hfil EJDE-2014/192\hfilneg]
{Haiyin Li, Zhikun She}  % in alphabetical order

\address{Haiyin Li \newline
LMIB and School of Mathematics and Systems Science,
Beihang University, Beijing, China.\newline
Department of Mathematics and Information, Henan University of
Economics and Law, Zhengzhou, China}
\email{lihaiyin2013@163.com}

\address{Zhikun She \newline
SKLSDE, LMIB and School of Mathematics and Systems Science,
Beihang University, Beijing, China}
\email{zhikun.she@buaa.edu.cn}

\thanks{Submitted November 16, 2013. Published Septgember 16, 2014.}
\subjclass[2000]{34D23, 92D25}
\keywords{Density dependence; global attractiveness;
 $\omega$-limit set; permanence}

\begin{abstract}
 In this article, we study the dynamics of a density-dependent
 predator-prey system of Beddington-DeAngelis type. We obtain
 sufficient and necessary conditions for the existence
 of a unique positive equilibrium, the global attractiveness of the 
 boundary equilibrium, and the permanence of the system, respectively.
 Moreover, we derive a sufficient condition for the locally asymptotic
 stability of the positive equilibrium by the Lyapunov function theory
 and a sufficient condition for the global attractiveness of the positive
 equilibrium by the comparison theory.
\end{abstract}

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

\section{Introduction}

The study of dynamics of predator-prey systems is one of the
importan subjects in mathematical ecology and mathematical biology.
The basic predator-prey model for a prey population density $x(t)$
and a predator population density $y(t)$ is
\begin{equation}
\begin{gathered}
x'(t)=x(t)(a-bx(t))-f(x,y)y(t)\\
y'(t)=-dy(t)+hf(x,y)y(t)
\end{gathered} \label{1.2}
\end{equation}
where $a$ is the intrinsic growth rate of the prey, $b$ measures the
intensity of intraspecific action of the prey, $h$ denotes the
conversion coefficient, $d$ denotes the predator's death rate, and
the function $f(x,y)$ is the predator's functional response.

The above basic model has been extensively studied in the
literature \cite{du, du1, hu, ku, so, su, te, ws, xi, z}. Since one of the
central goals of ecologists is to understand the relationship
between predator and prey, the predator's functional response, as
one significant component of the predator-prey relationship, has
also been considered \cite{b, cui, d, f, sg}. Beddington \cite{b}
and DeAngelis \cite{d} originally proposed the predator-prey system
with the Beddington-DeAngelis functional response, described by the
 model
\begin{equation}
\begin{gathered}
x'(t)=x(t)(a-bx(t)-\frac{cy(t)}{m_1+m_2x(t)+m_3y(t)})\\
y'(t)=y(t)(-d+\frac{fx(t)}{m_1+m_2x(t)+m_3y(t)}).
\end{gathered}\label{1.1}
\end{equation}
Skalski and Gilliam \cite{sg} further presented the statistical
evidence for predator-prey systems 
that three predator-dependent functional response: Beddington-DeAngelis, 
Crowley-Martin and Hassell-Varley can provide better description 
of predator feeding over a range of predator-prey abundances.

Moreover, certain environments confine the predator to be density
dependent and there are also considerable evidences that some
predator species may be density dependence because of the
environmental factors \cite{bs, bss}. Further, Kratina \cite{hhw}
showed that predator dependence is important not only at very high
predator densities on per capita predation rate but also at low
predator densities. So, it is not enough to only require the prey to
be density dependent and we also need to take into account realistic
levels of predator dependence.

In \cite{lt}, the following model is used to describe the growth of
a prey $x(t)$ and a predator $y(t)$ with density dependence:
\begin{equation}
\begin{gathered}
x'(t)=x(t)\Big(a-bx(t)-\frac{cy(t)}{m_1+m_2x(t)+m_3y(t)}\Big)\\
y'(t)=y(t)\Big(-d-ry(t)+\frac{fx(t)}{m_1+m_2x(t)+m_3y(t)}\Big)
\end{gathered} \label{1.3}
\end{equation}
where $x(t)$ is the prey population density, $y(t)$ is the predator
population density, $r$ stands for predator density dependence rate,
and the predator consumes prey with functional response of the
Beddington-DeAngelis type
$\frac{cx(t)y(t)}{m_1+m_2x(t)+m_3y(t)}$ and contributes to its
growth with the rate $\frac{fx(t)y(t)}{m_1+m_2x(t)+m_3y(t)}$. Note
that compared with the system \eqref{1.1}, the system \eqref{1.3}
contains not only  $bx^{2}(t)$ (which stands for intraspecific
action of prey species) but also $ry^{2}(t)$ (which stands for
intraspecific action of predator species).

In this article, we  investigate the dynamics of
the model described by the differential equations \eqref{1.3}.
We start with a sufficient and necessary condition for the existence
of a unique positive equilibrium by analyzing the corresponding
locations of hyperbolic curves while  the
same condition  was provided in \cite{lt} only as a sufficient condition.

Then, by using the corresponding characteristic equations of the
origin and the boundary equilibrium, we analyze their locally
asymptotic stability, respectively. Additionally, we analyze the
locally asymptotic stability of the positive equilibrium by
constructing a Lyapunov function.

Afterwards, based on a sufficient and necessary condition for the
global attractiveness of the boundary equilibrium, we further obtain
a sufficient and necessary condition for the permanence of the
system \eqref{1.3} by investigating types of the limit set
\cite{hahn67} instead of making use of the the persistence
theory \cite{hw, lb, hrt}. Note that \cite{lt} does not consider the
necessary condition for the global attractiveness of the boundary
equilibrium and thus can only provide a stronger, sufficient
condition for the permanence of the system. Here, the following
definition of permanence is used.

\begin{definition}  \rm
The system \eqref{1.3} is said to be permanent if there exist
positive constants $\delta$ and $\Delta$ with $0<\delta\leq\Delta$
such that
$$
\min\{\liminf_{t\to+\infty}x(t), \liminf_{t\to+\infty}y(t)\}\geq\delta,\quad
\max\{\limsup_{t\to+\infty}x(t), \limsup_{t\to+\infty}y(t)\}\leq\Delta
$$
for all solutions $(x(t),y(t))$ of  \eqref{1.3} with
positive initial conditions. \label{def:1.1}
\end{definition}

Since the permanence of the system shows that the time evolution of
the two species eventually either forms a cyclic loop or attracts to
the positive equilibrium, we finally derive a sufficient condition
for assuring the global attractiveness of the positive equilibrium
by the comparison theorem.

The rest of this article is organized as follows. In
Section~\ref{sec:stability}, we obtain a sufficient and necessary condition for the
existence of a unique positive equilibrium and analyze the local
stability of the non-negative equilibria of the system \eqref{1.3}.
In Section~\ref{sec:attractive}, we present a sufficient and
necessary condition for the global attractiveness of the boundary
equilibrium. In Section~\ref{sec:Permanent}, we derive a sufficient
and necessary condition for the permanence of the system
\eqref{1.3}. In Section~\ref{sec:Coexistence}, we consider the
global attractiveness of the positive equilibrium by using the
comparison theorem. We conclude our discussions in
Section~\ref{sec:Conclusion}.

\section{Equilibria and their local stability}\label{sec:stability}

It is clear that for all parameter values,
the system \eqref{1.3} has the equilibria $E_0(0,0)$ and
$E_1(\frac{a}{b},0)$, denoted as the origin and the boundary
equilibrium, respectively. For studying the existence of positive
equilibria, we analyze the following two equations:
\begin{equation}
\begin{gathered}
(a-bx)(m_1+m_2x+m_3y)-cy=0\\
(-d-ry)(m_1+m_2x+m_3y)+fx=0.
\end{gathered}
\label{2.1}
\end{equation}

For the equation $(a-bx)(m_1+m_2x+m_3y)-cy=0$, it is clear
that $(a/b,0)$ and $(-m_1/m_2,0)$ are on its
corresponding curves and if $c-am_3\neq 0$,
$(0,\frac{am_1}{c-am_3})$ is also on its corresponding curves. In
addition, when $\frac{c-am_3}{bm_3}\neq \frac{m_1}{m_2}$, this
equation is a hyperbolic equation and its two asymptotic lines are
$x+\frac{c-am_3}{bm_3}=0$ and
\[
y+\frac{m_2}{m_3}x+\frac{bm_1m_3-cm_2}{bm_3^2}=0.
\]
 Thus, the locations of its corresponding curves can be roughly shown from
Figure~\ref{fig:6}. When $\frac{c-am_3}{bm_3}= \frac{m_1}{m_2}$, the
equation is equivalent to $(m_1+m_2x)(am_2-bm_2x-bm_3y)=0$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.32\textwidth]{fig1a}    % x1.eps
\includegraphics[width=0.32\textwidth]{fig1b}    % x2.eps
\includegraphics[width=0.32\textwidth]{fig1c} \\ % x3.eps
(a) $\frac{m_1}{m_2}>\frac{c-am_3}{bm_3}\geq 0$ \hfil
(b) $\frac{c-am_3}{bm_3}>\frac{m_1}{m_2}>0$ \hfil
(c) $\frac{c-am_3}{bm_3}\leq 0$
\end{center}
\caption{Curves of the hyperbolic equation
$(a-bx)(m_1+m_2x+m_3y)-cy=0$.} \label{fig:6}
\end{figure}

For the equation $(-d-ry)(m_1+m_2x+m_3y)+fx=0$, it is clear
that $(0,-d/r)$ and $(0,-m_1/m_3)$ are on its
corresponding curves and if $f-dm_2\neq 0$,
$(\frac{dm_1}{f-dm_2},0)$ is also on its corresponding curves. In
addition, when $\frac{m_1}{m_3}\neq \frac{dm_2-f}{rm_2}$, this
equation is a hyperbolic equation and its two asymptotic lines are
$y+\frac{dm_2-f}{rm_2}=0$ and
$y+\frac{m_2}{m_3}x+\frac{rm_1m_2+fm_3}{rm_2}=0$. Thus, the
locations of its corresponding curves can be roughly seen from
Figure~\ref{fig:4}. When $\frac{m_1}{m_3}= \frac{dm_2-f}{rm_2}$, the
equation is equivalent to $(m_1+m_3y)(dm_3+rm_2x+rm_3y)=0$.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.32\textwidth]{fig2a}  % y1.eps
\includegraphics[width=0.32\textwidth]{fig2b}  % y2.eps
\includegraphics[width=0.32\textwidth]{fig2c} \\ %y3.eps
(a) $\frac{dm_2-f}{rm_2}\leq 0$ \hfil
(b) $\frac{m_1}{m_3}>\frac{dm_2-f}{rm_2}\geq 0$ \hfil
(c) $\frac{dm_2-f}{rm_2}>\frac{m_1}{m_3}>0$
\end{center}
\caption{Curves of the hyperbolic equation
$(a-bx)(m_1+m_2x+m_3y)-cy=0$.}\label{fig:4}
\end{figure}

Thus, by combining Figures~\ref{fig:6} and~\ref{fig:4} with the
above discussions, we  have the following theorem.

\begin{theorem}\label{exist}
System \eqref{1.3} has a unique positive
equilibrium $E^{*}(x^{*},y^{*})$ if and only if
\begin{equation}
(f-dm_2)a/b>dm_1\,.\label{2.2}
\end{equation}
\end{theorem}

\begin{remark}\label{re:2.3} \rm
In \cite{lt} it is used
$(f-dm_2)a/b>dm_1$ as the sufficient condition of the
existence of a unique positive equilibrium.
\end{remark}


\begin{remark} \rm
From  \eqref{2.2}, we can easily see that the predator
density dependent rate $r$ does not affect the existence of the
positive equilibrium.
\end{remark}

In the rest of this section, we  study the stability of the
non-negative equilibria $E_0(0,0), E_1(\frac{a}{b},0)$ and
$E^{*}(x^{*},y^{*})$, respectively. For this, we first write the
system \eqref{1.3} as
$\overline{X}'(t)=\overline{F}(\overline{X}(t))$, where
$\overline{X}(t)=(x(t), y(t))$. Then, for an arbitrary but the fixed
point $\overline{X}^{*}=(x,y)$, we consider its corresponding
characteristic equation as follows.

Let $G=(\frac{\partial \overline{F}}{\partial
\overline{X}(t)})_{\overline{X}^{*}}$, then
$$
G=\begin{bmatrix}a-2bx-cq_{x}'&-cq_{y}'\\fq_{x}'&-d-2ry+fq_{y}'
\end{bmatrix}_{\overline{X}^{*}}\,,
$$
where
\begin{gather*}
q(x,y)=\frac{xy}{m_1+m_2x+m_3y},\quad
q_{x}'=\frac{y(m_1+m_3y)}{(m_1+m_2x+m_3y)^{2}}, \\
q_{y}'=\frac{x(m_1+m_2x)}{(m_1+m_2x+m_3y)^{2}}.
\end{gather*}
Thus, the characteristic equation of  \eqref{1.3} at the
point $\overline{X}^{*}$ is
$$
|G-\lambda I|
=\begin{vmatrix}
    a-2bx-cq_{x}'-\lambda & -cq_{y}' \\
    fq_{x}' & -d-2ry+fq_{y}'-\lambda \\
  \end{vmatrix}=P(\lambda,\tau)  =0,
$$
where
\begin{gather*}
P(\lambda)=\lambda^{2}+P_1\lambda+P_0,\quad
P_1=-a+2bx+cq_{x}'-R,\\
P_0=(a-2bx)R+cq_{x}'(d+2ry),\quad
R=fq_{y}'-d-2ry.
\end{gather*}

Based on the characteristic equation of the point $E_0$, we have:

\begin{theorem}
The equilibrium $E_0(0,0)$ is unstable. \label{th:2.1}
\end{theorem}

\begin{proof}
The characteristic equation of \eqref{1.3} at the point
$E_0$ is
\[
|G-\lambda I |_{(0,0)}=(\lambda-a)(\lambda+d)=0.
\]
Clearly, $\lambda=-d$ is a negative eigenvalue and $\lambda=a$ is a
positive eigenvalue, implying that $E_0$ is an unstable saddle.
\end{proof}

Additionally, based on the characteristic equation of the point
$E_1$, we have:

\begin{theorem}
The equilibrium $E_1(\frac{a}{b},0)$ is
\begin{itemize}
\item[(i)]  unstable if $(f-dm_2)a/b>dm_1$;

\item[(ii)]  locally asymptotically stable if
$(f-dm_2)a/b<dm_1$. \label{th:2.2}
\end{itemize}
\end{theorem}

\begin{proof}
Since the characteristic equation of  \eqref{1.3} at the
point $E_1$ is
$(\lambda+a)\big[\lambda-(\frac{af}{m_1b+m_2a}-d)\big]=0$,
it follows that
$\lambda=-a$ and $\lambda=\frac{af}{m_1b+m_2a}-d$ are two
eigenvalues.

(i)  If $(f-dm_2)a/b>dm_1$,
$\lambda=\frac{af}{m_1b+m_2a}-d$ is positive and $E_1$ is a unstable saddle.

(ii) If $(f-dm_2)a/b<dm_1$, then
$\lambda=\frac{af}{m_1b+m_2a}-d$ is negative, implying that
$E_1$ is a locally asymptotically stable node.
\end{proof}

\begin{remark}\label{re:2.2} \rm
If $(f-dm_2)a/b=dm_1$, we can easily
prove that $E_1(\frac{a}{b},0)$ is linearly neutrally stable. But,
whether $E_1(\frac{a}{b},0)$ is stable when
$(f-dm_2)a/b=dm_1$ is unknown. However, we can prove
that when $(f-dm_2)a/b=dm_1$, $E_1(\frac{a}{b},0)$ is
globally attractive, which will be discussed in
Section \ref{sec:attractive}.
\end{remark}

Further, instead of considering the negativeness of the real parts
of the eigenvalues \cite{lt}, we analyze the locally asymptotically
stable analysis of $E^{*}(x^{*},y^{*})$ by constructing a Lyapunov
function for its linearization as follows.

Let
\begin{gather*}
x(t)=x^{*}+u(t)\\
y(t)=y^{*}+v(t),
\end{gather*}
then the linearization of \eqref{1.3} is
\begin{equation}
\begin{gathered}
u'(t)=Au(t)-Cv(t)\\
v'(t)=-Dv(t)+Fu(t),
\end{gathered}\label{2.7}
\end{equation}
where
\begin{equation}
\begin{gathered}
A=a-2bx^{*}-\frac{cy^{*}(m_1+m_3y^{*})}{(m_1+m_2x^{*}+m_3y^{*})^{2}},\quad
C=\frac{cx^{*}(m_1+m_2x^{*})}{(m_1+m_2x^{*}+m_3y^{*})^{2}}\\
D=d+2ry^{*}-\frac{fx^{*}(m_1+m_2x^{*})}{(m_1+m_2x^{*}+m_3y^{*})^{2}},\quad
F=\frac{fy^{*}(m_1+m_3y^{*})}{(m_1+m_2x^{*}+m_3y^{*})^{2}}.
\end{gathered}
\label{2.8}
\end{equation}
Clearly, $C$ and $F$ are positive.
Therefore, by the construction of a Lyapunov function, we have the
following result for the positive equilibrium $E^{*}(x^{*},y^{*})$.

\begin{theorem} If \eqref{2.2} holds and
\begin{equation}
|F-C|<\min\{2D,-2A\}, \label{2.9}
\end{equation}
then the equilibrium $(0, 0)$ of the system \eqref{2.7} is
locally asymptotically stable, implying that the positive
equilibrium $E^{*}(x^{*}, y^{*})$ of
\eqref{1.3} is locally asymptotically stable.
\label{th:2.3}
\end{theorem}

\begin{proof}
For proving the locally asymptotic stability of the equilibrium
$(0,0)$ of the system \eqref{2.7}, it it sufficient to consider the
existence of a strict Lyapunov function.

Letting $W(t)=u^{2}(t)+v^{2}(t)$, the time derivative of $W(t)$ is
$$
W'(t)=2Au^{2}(t)-2Dv^{2}(t)+2(F-C)u(t)v(t).$$
Clearly, $W(t)\geq 0$ and $W(t)=0$ if and only if $u(t)=v(t)=0$. In
addition, if $u(t)=v(t)=0$, then $W'(t)=0$. Moreover,
\begin{align*}
W'(t)
&\leq 2Au^{2}(t)-2Dv^{2}(t)+2|F-C||u(t)||v(t)|\\
&\leq [2A+|F-C|]u^{2}(t)+(-2D+|F-C|)v^{2}(t).
\end{align*}

From \eqref{2.2} and \eqref{2.9}, we have that: if
$u^2(t)+v^2(t)>0$, then $W'(t)<0$. Thus, $W(t)$ is a strict Lyapunov
function. Due to the Lyapunov stability theorem \cite{h}, the
equilibrium $(0, 0)$ of the system \eqref{2.7} is locally
asymptotically stable, implying that the positive equilibrium
$E^{*}(x^{*}, y^{*})$ of the system \eqref{1.3} is locally
asymptotically stable.
\end{proof}


\section{Global attractiveness of the boundary equilibrium}\label{sec:attractive}

 From Theorem \ref{th:2.2}, if $(f-dm_2)a/b<dm_1$,
$E_1$ is locally attractive. However, for the qualitative
analysis, it is far from enough. So, in this section, we try to
derive a sufficient and necessary condition for assuring the global
attractiveness of $E_1$. For this, the following lemma is first
introduced.

\begin{lemma}
Let $S=\{(x,y):x>0,y>0\}$ and $\overline{S}=\{(x,y):x\geq 0,y\geq
0\}$. Then, the sets $S$ and $\overline{S}$ are both invariant sets.
\label{th:4.2}
\end{lemma}

\begin{proof}
Since $x=0$ and $y=0$ are both solutions to the system \eqref{1.3},
due to the uniqueness of the solution to the system \eqref{1.3}, the
lemma directly holds.
\end{proof}


Then, based on Lemma \ref{th:4.2}, we have the following result on
the global attractiveness of $E_1$.

\begin{theorem}\label{th:3.1}
For any solution $(x(t),y(t))$ of
\eqref{1.3} with $x(0)>0$ and $y(0)>0$,
$\lim_{t\to+\infty}(x(t),y(t))=(\frac{a}{b}, 0)$ if and only if
\begin{equation}
(f-dm_2)a/b\leq dm_1. \label{3.1}
\end{equation}
\end{theorem}

\begin{proof}
For proving the necessity, we consider the following two cases:

Case 1: $(f-dm_2)a/b< dm_1$.
First, we want to prove that $\lim_{t\to+\infty}y(t)=0$.
Due to Lemma~\ref{th:4.2}, $x'(t)\leq ax(t)-bx^{2}(t)$. Then, by
considering the  comparison equation
$$
p'(t)=ap(t)-bp^{2}(t),~p(0)=x(0)>0,
$$
we have that $x(t)\leq p(t)$ for all $t\geq 0$, and
$\lim_{t\to+\infty}p(t)=\frac{a}{b}$. Thus, there exists a
sufficiently small positive constant $\varepsilon$ with
$(f-dm_2)(\frac{a}{b}+\varepsilon)<dm_1$ such that for this
$\varepsilon$, there exists a $T_{\varepsilon}>0$ such that
$x(t)<\frac{a}{b}+\varepsilon$ for all $t>T_{\varepsilon}$.
Substituting it into the second equation of the system \eqref{1.3},
we get that for all $t>T_{\varepsilon}$,
$$
y'(t)\leq \big[\frac{(f-dm_2)(\frac{a}{b}+\varepsilon)
-dm_1}{m_1+m_2(\frac{a}{b}+\varepsilon)}\big]y(t)-ry^{2}(t).
$$
So, let us consider the  comparison equation
$$
q'(t)= \big[\frac{(f-dm_2)(\frac{a}{b}+\varepsilon)
-dm_1}{m_1+m_2(\frac{a}{b}+\varepsilon)}\big]q(t)-rq^{2}(t),\quad
q(T_{\varepsilon})=y(T_{\varepsilon})>0,
$$
whose solution is
$$
q(t)=\frac{Fq'(T_{\varepsilon})e^{F(t-T_{\varepsilon})}}{1+rq'(T_{\varepsilon})
e^{F(t-T_{\varepsilon})}},
$$
where
$$
F=\frac{(f-dm_2)(\frac{a}{b}+\varepsilon)-dm_1}{m_1+m_2(\frac{a}{b}+\varepsilon)},\quad
q'(T_{\varepsilon})=\frac{q(T_{\varepsilon})}{F-rq(T_{\varepsilon})}.
$$
Clearly, by the comparison theorem, we have that $y(t)\leq q(t)$ for
all $t\geq T$. In addition, since
$(f-dm_2)(\frac{a}{b}+\varepsilon)<dm_1$, then $F<0$, implying
that $\lim_{t\to+\infty}q(t)=0$ and thus
$\lim_{t\to+\infty}y(t)=0$.

Second, we want to prove that $x(t)\to \frac{a}{b}$ as
$t\to +\infty$, that is, to prove that for any
$\varepsilon_1\in (0,\frac{a}{b})$, there exists a $T_0>0$ such that
for all $t> T_0$, $-\varepsilon_1<x(t)-\frac{a}{b}<\varepsilon_1$.

Since $\lim_{t\to+\infty}y(t)=0$, from Lemma~\ref{th:4.2}, for any
given $\varepsilon_1\in (0,\frac{a}{b})$, there exists a $T_1>0$
such that for all $t\geq T_1$,
$0<y(t)<\frac{bm_1}{2c}\varepsilon_1$. Thus, for all $t\geq T_1$, we
have
\begin{equation}
\big(a-\frac{b\varepsilon_1}{2}\big) x(t)- bx^{2}(t) \leq
x'(t)\leq ax(t)-bx^{2}(t). \label{3.2}
\end{equation}
Let us consider the  comparison equation
$$
\widetilde{p}'(t)=\big(a-\frac{b\varepsilon_1}{2}\big)\widetilde{p}(t)
-b\widetilde{p}^{2}(t),\quad \widetilde{p}(T_1)=x(T_1)>0.
$$
Since $a>b\varepsilon_1$, we have
$\lim_{t\to+\infty}\widetilde{p}(t)=\frac{a}{b}-\frac{\varepsilon_1}{2}$.
In addition, we have that for all $t\geq T_1$, $\widetilde{p}(t)
\leq x(t)\leq p(t)$.

 Since $\lim_{t\to+\infty}p(t)=\frac{a}{b}$,
for the above $\varepsilon_1$, there exists a $T_2>0$ such that for
all $t>T_2$, $p(t)\leq \frac{a}{b}+\varepsilon_1$. Similarly, since
$\lim_{t\to+\infty}\widetilde{p}(t)=\frac{a}{b}-\frac{\varepsilon_1}{2}$,
for the above $\varepsilon_1$, there exists a $T_3>0$ such that for
all $t>T_3$,
$\widetilde{p}(t)-\frac{a}{b}+\frac{\varepsilon_1}{2}>-\frac{\varepsilon_1}{2}$.
Thus, letting $T_0=\max\{T_1,T_2,T_3\}$, for all $t>T_0$,
$-\varepsilon_1<x(t)-\frac{a}{b}<\varepsilon_1$, implying that
$\lim_{t\to+\infty}x(t)=\frac{a}{b}$.

(2) $(f-dm_2)a/b= dm_1$.
 First, we want to prove that
$\lim_{t\to+\infty}y(t)=0$.
Similarly, for an arbitrary $\varepsilon_2>0$, there exists
$T_{\varepsilon_2}>0$ such that $x(t)<\frac{a}{b}+\varepsilon_2$ for
all $t>T_{\varepsilon_2}$. Thus, due to Lemma~\ref{th:4.2}, for all
$t>T_{\varepsilon_2}$,
\begin{align*}
y'(t)
&<-dy(t)-ry^{2}(t)+\frac{fx(t)}{m_1+m_2x(t)}y(t)\\
&<y(t)\big(\frac{f(\frac{a}{b}+\varepsilon_2)}{m_1+m_2(\frac{a}{b}+\varepsilon_2)}
 -d\big)-ry^{2}(t)\\
&=\frac{\varepsilon_2(f-dm_2)}{m_1+m_2(\frac{a}{b}+\varepsilon_2)}y(t)
 -ry^{2}(t).
\end{align*}
So, let us consider the  comparison equation
$$
\widetilde{q}'(t)=\frac{\varepsilon_2(f-dm_2)}{m_1+m_2
(\frac{a}{b}+\varepsilon_2)}\widetilde{q}(t)-r\widetilde{q}^{2}(t), \quad
\widetilde{q}(T_{\varepsilon_2})=y(T_{\varepsilon_2})>0,
$$
whose solution is
$\widetilde{q}(t)=\frac{F\widetilde{q}'(T_{\varepsilon_2})e^{F(t-T_{\varepsilon_2})}}
{1+r\widetilde{q}'(T_{\varepsilon_2})e^{F(t-T_{\varepsilon_2})}}$,
where
$F=\frac{\varepsilon_2(f-dm_2)}{m_1+m_2(\frac{a}{b}+\varepsilon_2)}$
and
$\widetilde{q}'(T)=\frac{\widetilde{q}
(T_{\varepsilon_2})}{F-r\widetilde{q}(T_{\varepsilon_2})}$.
Clearly, by the comparison theorem, we have that $y(t)\leq
\widetilde{q}(t)$ for all $t\geq T_{\varepsilon_2}$. In addition,
since $(f-dm_2)a/b= dm_1$, then $F>0$, implying that
$\lim_{t\to+\infty}\widetilde{q}(t)=\frac{\varepsilon_2(f-dm_2)}{r(m_1+m_2(\frac{a}{b}+\varepsilon_2))}$.\\
Thus, for the above $\varepsilon_2$, there exists a $T'>0$ such that
for all $t\geq T'$,
$$
\widetilde{q}(t)-\frac{\varepsilon_2(f-dm_2)}{r(m_1+m_2
(\frac{a}{b}+\varepsilon_2))}<\varepsilon_2.
$$
Letting $T_0'=\max\{T_{\varepsilon_2},T_1'\}$, then for all
$t>T_0'$,
$y(t)<\frac{\varepsilon_2(f-dm_2)}{r(m_1+m_2(\frac{a}{b}+\varepsilon_2))}
+\varepsilon_2
<\frac{f-dm_2+r(m_1+m_2\frac{a}{b})}{r(m_1+m_2\frac{a}{b})}\varepsilon_2$,
implying that $\lim_{t\to+\infty}y(t)=0$.
The proof of $\lim_{t\to +\infty}x(t)=\frac{a}{b}$ is
similar to the case $(f-dm_2)a/b< dm_1$.
\smallskip

For proving the sufficiency, we assume that
$(f-dm_2)a/b> dm_1$ and try to derive a contradiction.
Due to the assumption that $(f-dm_2)a/b> dm_1$,
system \eqref{1.3} has a unique positive equilibrium
$(x^{*},y^{*})$, which is also a solution to  \eqref{1.3},
contradicting with $\lim_{t\to
+\infty}(x^{*},y^{*})=(\frac{a}{b},0)$. Thus, condition
\eqref{3.1} must hold.
\end{proof}

\begin{remark} \rm
In \cite{lt} it is only provided $f<dm_2$ as a sufficient condition for the
globally asymptotic stability of $E_1(\frac{a}{b},0)$.
\end{remark}

\begin{remark} \rm
From Theorems~\ref{th:2.2} and \ref{th:3.1}, we can directly derive
that $E_1(\frac{a}{b},0)$ is a saddle if and only if
$(f-dm_2)a/b>dm_1$.
\end{remark}

\section{Permanence analysis}\label{sec:Permanent}

 From Theorem \ref{th:3.1}, $(f-dm_2)a/b\leq dm_1$ is a sufficient
condition for the predator to be extinctive. In this section,
we will like to derive a sufficient and necessary condition for the permanence
(or equivalently, the extinction).

Firstly, we introduce the following boundedness result for \eqref{1.3}.

\begin{lemma} \label{th:4.1}
All solutions of \eqref{1.3} with positive initial
conditions are bounded for $t\geq 0$.
\end{lemma}

\begin{proof}
Due to Lemma~\ref{th:4.2}, for all $t>0$, $x'(t)\leq
ax(t)-bx^{2}(t)$. Similar to the proof of Theorem~\ref{th:3.1},
there exists a $T>0$ such that for all $t>T$, $x(t)\leq
\frac{a}{b}+1$, implying that $x(t)$ is bounded for all $t\geq 0$.

Letting $\omega(t)=\frac{f}{c}x(t)+y(t)$, we have
$$
\frac{d\omega(t)}{dt}\leq-dy(t)+\frac{af}{c}x(t)
=-d\omega(t)+\frac{(a+d)f}{c}x(t).
$$
Clearly, there exist $M>0$ and $T_1>0$ such that for all
$t\geq T_1$,
$$
\frac{d\omega(t)}{dt}\leq M-d\omega(t).
$$
Let $\frac{dp(t)}{dt}= M-dp(t)$ with $p(T_1)=\omega(T_1)$, then
$\omega(t)\leq p(t)$ for all $t\geq T_1$ and
$\lim_{t\to+\infty}p(t)\leq\frac{M}{d}$. Thus, there exists a
$T_2>\max\{T,T_1\}$ such that for all $t>T_2$, $\omega(t)\leq
p(t)\leq \frac{M}{d}+1$, implying that $y(t)$ is bounded for all
$t\geq 0$.
\end{proof}

Secondly, based on Lemma \ref{th:4.1} and \cite[Lemma 4.1]{h},
 we  have the following property about the $\omega$-limit set.

\begin{lemma} \label{limit}
For any point in $S=\{(x,y): x>0,y>0\}$, its
$\omega$-limit set is nonempty, compact, connected, and invariant.
\end{lemma}

Thirdly, by Lemma~\ref{th:4.1}, Lemma~\ref{limit} and
Poincar\'{e}-Bendixson theorem \cite{hahn67}, we have the following
theorem describing the possible types of the $\omega$-limit set of
any initial point in $S=\{(x(0),y(0)): x(0)>0,y(0)>0\}$.

\begin{theorem}\label{type}
If $(f-dm_2)a/b> dm_1$, then for any
initial point in $S$, its $\omega$-limit set consists of either only
the positive equilibrium $E^*$ or a closed orbit.
\end{theorem}

\begin{proof}
For any point $(x_0,y_0)$ in $S$, let $(x(t),y(t))$ be the orbit of
the system \eqref{1.3} with $(x(0),y(0))=(x_0,y_0)$. By
Lemma~\ref{limit} and Poincar\'{e}-Bendixson theorem \cite{hahn67},
\begin{itemize}
\item[(a)] the $\omega$-limit set of $(x_0,y_0)$ consists of a single point
$p$ which is a equilibrium point such that
$\lim_{t\to+\infty}(x(t),y(t))=p$, or

\item[(b)] the $\omega$-limit set of $(x_0,y_0)$ is a closed orbit, or

\item[(c)] the $\omega$-limit set of $(x_0,y_0)$ consists of equilibrium
points together with their connecting orbits. Each such orbit approaches
an equilibrium point as $t\to+\infty$ and $t\to-\infty$.
\end{itemize}

In addition, it is clear that if $(f-dm_2)a/b> dm_1$,
 system \eqref{1.3} has only three equilibria $E_0$, $E_1$ and
$E^*$ in the first quadrant. Moreover, by the proof of
Theorem~\ref{th:2.1}, $E_0$ is a saddle; by the proof of
Theorem~\ref{th:2.2}, if $(f-dm_2)a/b> dm_1$, $E_1$ is
also a saddle.

So, for the above case (a), the $\omega$-limit set consists of only
the equilibrium $E^*$. Moreover, we can prove that the above case
(c) cannot occur as follows.

First, we can prove that the $\omega$-limit set cannot contain $E_0$ and
$E^*$ together.
Otherwise, there exists an orbit $\gamma_0(t)$ connecting $E_0$ and
$E^*$. Since $E_0$ is a saddle, $\lim_{t\to+\infty}
\gamma_0(t)=E_0$, contradicting with the fact that $(0,y(t))$ is the
unique orbit of the system \eqref{1.3} with
$\lim_{t\to+\infty} (0,y(t))=E_0$.

 Second, we assume that the $\omega$-limit set consists of $E_0$
and $E_1$ together with their connecting orbit $(x(t),0)$ with
$0<x(t)<a/b$, $\lim_{t\to-\infty}(x(t),0)=E_0$ and
$\lim_{t\to+\infty}(x(t),0)=E_1$, and try to derive a
contradiction as follows.

Since $E_0$ is a saddle, there exists a constant $\delta>0$ such
that the orbit $(x(t),y(t))$ infinitely enters and then leaves the
region $\{(x,y): x^2+y^2\leq \delta\}$. Let $t_n$ be the $n$-th time
instant for the orbit to enter the region. Due to
Lemma~\ref{th:4.1}, $\{(x(t_n),y(t_n))\}$ is a bounded sequence.
Thus, there exist a subsequence $\{(x(t_{n_k}),y(t_{n_k})\}$ and a
$(\bar{x},\bar{y})$ such that $\lim_{k\to
+\infty}(x(t_{n_k}),y(t_{n_k}))=(\bar{x},\bar{y})$ and $\bar{y}\neq
0$, contradicting with the assumption that the $\omega$-limit set
consists of $E_0$ and $E_1$ together with their connecting orbit
$(x(t),0)$.

 Third, we can similarly prove that the $\omega$-limit set cannot consist of $E_1$
and $E^*$ together with their connecting orbit.

 Fourth, the $\omega$-limit set cannot consist of $E_0$ and a homoclinic orbit
since $(0,y(t))$ is the unique orbit of the system \eqref{1.3} with
$\lim_{t\to+\infty} (0,y(t))=E_0$.

 Fifth, the $\omega$-limit set cannot consist of $E_1$ and a homoclinic orbit
since $(x(t),0)$ with $0<x(t)<a/b$ is the unique orbit of
the system \eqref{1.3} with
$\lim_{t\to+\infty}(x(t),0)=E_1$.

 Sixth, assume that the $\omega$-limit set contains $E^*$ and a homoclinic orbit. Then,
there exists at least one positive equilibrium inside the region
enclosed by the homoclinic orbit, contradicting with the result that
$E^*$ is the unique positive equilibrium.

Thus, we have proved that if $(f-dm_2)a/b> dm_1$, then
for any point in $S$, its $\omega$-limit set consists of either only
the positive equilibrium $E^*$ or a closed orbit.
\end{proof}


Finally, based on Theorems~\ref{type} and \ref{th:3.1} and
Definition~\ref{def:1.1}, we have the following result for the
permanence of the system \eqref{1.3}.

\begin{theorem}
System \eqref{1.3} is permanent if and only if
$(f-dm_2)a/b> dm_1$ (i.e., positive equilibria exist).
\label{th:4.3}
\end{theorem}

\begin{proof}
Due to Theorem~\ref{type} and Definition~\ref{def:1.1}, if
$(f-dm_2)a/b> dm_1$, then the system \eqref{1.3} is
permanent.
In addition, due to Theorem \ref{th:3.1} and
Definition~\ref{def:1.1}, if $(f-dm_2)a/b\leq dm_1$, the
system \eqref{1.3} is not permanent.
Thus, the sufficiency and necessity are both proved.
\end{proof}


\begin{remark} \rm
By Theorem~\ref{th:4.3}, the predator density dependent rate $r$
does not affect the permanence of the system \eqref{1.3}.
\end{remark}

\section{Permanent coexistence to the positive equilibrium}\label{sec:Coexistence}

From Theorems~\ref{type} and~\ref{th:4.3},
the permanence of the system shows that the time evolution of the
two species eventually either forms a cyclic loop or attracts to the
positive equilibrium. In this section, we try to use the comparison
theorem to provide a sufficient condition for the global asymptotic
stability of $E^{*}(x^{*},y^{*})$.

Let the initial point be in the set $S=\{(x,y):x> 0,y> 0\}$. We need
the following preparations by iteratively making use of the
comparison theorem.

Similar to the proof in Theorem~\ref{th:3.1}, for an arbitrary
sufficiently small $\varepsilon_1'>0$, there exists a $T_1$ such
that for all $t\geq T_1$,
\begin{equation}
x(t)<\frac{a}{b}+\varepsilon_1'. \label{3.3}
\end{equation}
Let $A_1=\frac{a}{b}+\varepsilon_1'$. In addition, from the first
equation of the system \eqref{1.3}, we can also obtain that: for all
$t>0$,
$$
x'(t)>ax(t)-bx^{2}(t)-\frac{c}{m_3}x(t).
$$
When $a>\frac{c}{m_3}$, for any given $\varepsilon_{1,B}'>0$ with
$\varepsilon_{1,B}'< \min \{\varepsilon_1',
\frac{1}{b}(a-\frac{c}{m_3})\}$, there exists a $T_2>T_1$ such
that for all $t>T_2$,
\begin{equation}
x(t)>\frac{1}{b}(a-\frac{c}{m_3})-\varepsilon_{1,B}'>0.
\label{3.5}\end{equation} Let
$B_1=\frac{1}{b}(a-\frac{c}{m_3})-\varepsilon_{1,B}'$.

From the second equation of the system \eqref{1.3}, we can obtain
that: for all $t>0$,
$y'(t)<y(t)[\frac{f}{m_2}-d-ry(t)]$. Due to the
condition \eqref{2.2}, $f>dm_2$ directly holds. Thus, similar to
the proof of the second case in Theorem~\ref{th:3.1}, for the above
$\varepsilon_1'$, there exists a $T_3>T_2$ such that for all
$t>T_3$,
\begin{equation}
y(t)<\frac{1}{r}(\frac{f}{m_2}-d)+\varepsilon_1'. \label{3.4}
\end{equation}
Let $C_1=\frac{1}{r}(\frac{f}{m_2}-d)+\varepsilon_1'$. In
addition, by using the inequalities \eqref{3.5} and \eqref{3.4} for
the second equation of \eqref{1.3}, we also obtain that: for all
$t>T_3$
$$
y'(t)>y(t)[-d-ry(t)+\frac{fB_1}{m_1+m_2B_1+m_3C_1}].
$$
If $\frac{fB_1}{m_1+m_2B_1+m_3C_1}>d$, similar to the
proof of the second case in Theorem~\ref{th:3.1}, for any given
$\varepsilon_{1,D}'>0$ with $\varepsilon_{1,D}'< \min
\{\varepsilon_1',
\frac{1}{r}\big(\frac{fB_1}{m_1+m_2B_1+m_3C_1}-d\big)$,
there exists a $T_{4}>T_3$ such that for all $t>T_4$,
\begin{equation}
y(t)>\frac{1}{r}\big(\frac{fB_1}{m_1+m_2B_1+m_3C_1}-d\big)-\varepsilon_{1,D}'>0.
\label{3.6}
\end{equation}
Let
$D_1=\frac{1}{r}\big(\frac{fB_1}{m_1+m_2B_1+m_3C_1}-d\big)-\varepsilon_{1,D}'$.
Therefore, for  system \eqref{1.3}, we have
$$
B_1<x(t)<A_1,\quad D_1<y(t)<C_1,\quad t\geq T_{4}.
$$

Provided that $a>\frac{c}{m_3}$ and
$\frac{fB_1}{m_1+m_2B_1+m_3C_1}>d$, by using
 \eqref{3.3} and \eqref{3.6} in the first equation of
 \eqref{1.3}, we obtain
$$
x'(t)<ax(t)-bx^{2}(t)-\frac{cD_1x(t)}{m_1+m_2A_1+m_3D_1},\quad t>T_{4}.
$$
If $a>\frac{c}{m_3}$ holds, then
$a>\frac{cD_1}{m_1+m_2A_1+m_3D_1}$. Similarly, for the
above $\varepsilon_1'$, there exists a $T_5>T_{4}$ such that for
all $t>T_5$,
\begin{equation}
x(t)<\frac{1}{b}\big(a-\frac{cD_1}{m_1+m_2A_1+m_3D_1}\big)+\varepsilon_1'.
\label{3.7}
\end{equation}
Let
$A_2=\frac{1}{b}\big(a-\frac{cD_1}{m_1+m_2A_1+m_3D_1}\big)+\varepsilon_1'$.
Clearly, $A_2<A_1$. In addition, by using
\eqref{3.5} and \eqref{3.4} in  the first equation of
\eqref{1.3}, we have
$$
x'(t)>ax(t)-bx^{2}(t)-\frac{cx(t)C_1}{m_1+m_2B_1+m_3C_1},\quad t>T_{4}.
$$
When $a>\frac{c}{m_3}$ holds, then
$a>\frac{cC_1}{m_1+m_2B_1+m_3C_1}$. Similarly, for any
given $\varepsilon_{2,B}'>0$ with
$\varepsilon_{2,B}'< \min \{\varepsilon_1', \varepsilon_{1,B}',
\frac{1}{b}\big(a-\frac{cC_1}{m_1+m_2B_1+m_3C_1}\big)\}$,
there exists a $T_6>T_5$ such that for all $t>T_6$,
\begin{equation}
x(t)>\frac{1}{b}\Big(a-\frac{cC_1}{m_1+m_2B_1+m_3C_1}\Big)-\varepsilon_{2,B}'>0.
\label{3.9}
\end{equation}
Let
$B_2=\frac{1}{b}\big(a-\frac{cC_1}{m_1+m_2B_1+m_3C_1}\big)-\varepsilon_{2,B}'$.
Clearly, $B_2>B_1$.

Moreover, provided that $a>\frac{c}{m_3}$ and
$\frac{fB_1}{m_1+m_2B_1+m_3C_1}>d$, by using the
inequalities \eqref{3.3} and \eqref{3.6} in the second equation of
the system \eqref{1.3}, we obtain
$$
y'(t)<y(t)[\frac{fA_1}{m_1+m_2A_1+m_3D_1}-d-ry(t)],\quad
t>T_{4}.
$$
If $\frac{fB_1}{m_1+m_2B_1+m_3C_1}>d$ holds, then
$\frac{fA_1}{m_1+m_2A_1+m_3D_1}>d$. Similarly, for the
above $\varepsilon_1'$, there exists a $T_{7}>T_6$ such that for
all $t>T_7$,
\begin{equation}
y(t)<\frac{1}{r}\Big(\frac{fA_1}{m_1+m_2A_1+m_3D_1}-d\Big)+\varepsilon_1',
\label{3.8}
\end{equation}
 Let
$C_2=\frac{1}{r}\big(\frac{fA_1}{m_1+m_2A_1+m_3D_1}-d\big)+\varepsilon_1'$.
So $C_2<C_1$. In addition, by using
\eqref{3.5} and \eqref{3.4} in the second equation of \eqref{1.3},
we have
$$
y'(t)>y(t)[-d-ry(t)+\frac{fB_1}{m_1+m_2B_1+m_3C_1}],\quad
t>T_{4}.
$$
Similarly, for any given $\varepsilon_{2,D}'>0$,
$\varepsilon_{2,D}'< \min \{\varepsilon_1',\varepsilon_{1,D}',
\frac{1}{r}\big(\frac{fB_1}{m_1+m_2B_1+m_3C_1}-d\big)\}$,
there exists a $T_{8}>T_{7}$ such that for all $t>T_8$,
\begin{equation}
y(t)>\frac{1}{r}\big(\frac{fB_1}{m_1+m_2B_1+m_3C_1}-d\big)-\varepsilon_{2,D}',
\label{3.10}
\end{equation}
Let
$D_2=\frac{1}{r}\big(\frac{fB_1}{m_1+m_2B_1+m_3C_1}-d\big)-\varepsilon_{2,D}'$.
So it has $D_2>D_1$.


Thus, combining the above discussions, we  have
$$
B_1<B_2<x(t)<A_2<A_1,\quad D_1<D_2<y(t)<C_2<C_1,\quad t\geq T_{8}.
$$

By repeating the above procedure, we can get five sequences
$\{T_n\}_{n=1}^{+\infty}$, $\{A_n\}^{\infty}_{n=1}$,
$\{C_n\}^{\infty}_{n=1}$, $\{B_n\}^{\infty}_{n=1}$ and
$\{D_n\}^{\infty}_{n=1}$. Here, by defining $\Delta(x, y)$ to be
$m_1+m_2x+m_3y$, then for all $n\geq 2$, $A_n$, $C_n$,
$B_n$ and $D_n$ have the following expressions£º
\begin{gather*}
A_n=\frac{1}{b}(a-\frac{cD_{n-1}}{\Delta(A_{n-1}, D_{n-1})})+\varepsilon_1', \quad
B_n=\frac{1}{b}(a-\frac{cC_{n-1}}{\Delta(B_{n-1},  C_{n-1})})-\varepsilon_{n,B}',\\
C_n=\frac{1}{r}(\frac{fA_{n-1}}{\Delta(A_{n-1}, D_{n-1})}-d)+\varepsilon_1', \quad
D_n=\frac{1}{r}(\frac{fB_{n-1}}{\Delta(B_{n-1},
C_{n-1})}-d)-\varepsilon_{n,D}',
\end{gather*}
respectively, satisfying
\begin{equation}
\begin{gathered}
0<\varepsilon_{n,B}'< \min \{\varepsilon_1',\varepsilon_{n-1,B}',
\frac{1}{b}(a-\frac{cC_{n-1}}{\Delta(B_{n-1}, C_{n-1})})\}\\
0<\varepsilon_{n,D}'< \min \{\varepsilon_1',\varepsilon_{n-1,D}',
\frac{1}{r}(\frac{fB_{n-1}}{\Delta(B_{n-1}, C_{n-1})}-d)\}\\
0<B_1<B_2<\dots <B_n<x(t)<A_n<\dots <A_2<A_1,\quad t\geq T_{4n}\\
0<D_1<D_2<\dots <D_n<y(t)<C_n<\dots <C_2<C_1,\quad t\geq T_{4n}.
\end{gathered}
\label{3.12}
\end{equation}

Clearly,  $\{A_n\}$ and $\{C_n\}$ are bounded decreasing
sequences and $\{B_n\}$ and $\{D_n\}$ are bounded increasing
sequences. Thus, there exist $\overline{A}$, $\overline{C}$,
$\overline{B}$ and $\overline{D}$ such that
$\lim_{t\to+\infty}A_n=\overline{A}$,
$\lim_{t\to+\infty}C_n=\overline{C}$,
$\lim_{t\to+\infty}B_n=\overline{B}$ and
$\lim_{t\to+\infty}D_n=\overline{D}$. In addition, from the
formula \eqref{3.12}, $\overline{A}\geq \overline{B}$ and
$\overline{C}\geq \overline{D}$.

Further, from the expressions  of $A_n$, $C_n$, $B_n$ and
$D_n$, we  obtain
\begin{align*}
A_n-B_n
&=\varepsilon_1'+\varepsilon_{n,B}'
+\Big(cm_1(C_{n-1}-D_{n-1})+cm_2\big[A_{n-1}(C_{n-1} -D_{n-1})\\
&\quad +D_{n-1}(A_{n-1}-B_{n-1})\big]\Big)/\big(b\Delta(B_{n-1},
C_{n-1})\Delta(A_{n-1}, D_{n-1})\big).
\end{align*}
Thus, when $n\to+\infty$, we have
\begin{equation}
\overline{A}-\overline{B}
=\frac{cm_1(\overline{C}-\overline{D})+cm_2[\overline{A}(\overline{C}-\overline{D})
+\overline{D}(\overline{A}-\overline{B})]}{b\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})}+\varepsilon_1'+\varepsilon_{n,B}'. \label{3.13}
\end{equation}
Similarly, we can obtain
\begin{align*}
C_n-D_n
&=\varepsilon_1'+\varepsilon_{n,D}'
+\Big(fm_1(A_{n-1}-B_{n-1})+fm_3\big[A_{n-1}(C_{n-1}-D_{n-1})\\
&\quad +D_{n-1}(A_{n-1}-B_{n-1})\big]\Big)/\big(r\Delta(B_{n-1},
C_{n-1})\Delta(A_{n-1}, D_{n-1})\big).
\end{align*}
Thus, when $n\to+\infty$, we have
\begin{equation}
\overline{C}-\overline{D}=\frac{f(m_1+m_3\overline{D})(\overline{A}-\overline{B})+(\varepsilon_1'+\varepsilon_{n,D}')r\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})}{r\Delta(\overline{B},
\overline{C})\Delta(\overline{A}, \overline{D})-fm_3\overline{A}}.
\label{3.14}
\end{equation}
Putting  \eqref{3.14} in  \eqref{3.13}, we have
\[
\overline{A}-\overline{B}
\leq\bigg|\frac{2(\frac{cr(m_1+m_2\overline{A})}{b[r\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})-fm_3\overline{A}]}+1)}{1-\frac{c}{b\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})}[\frac{f(m_1+m_2\overline{A})(m_1+m_3\overline{D})}{r\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})-fm_3\overline{A}}+m_2\overline{D}]}\bigg|\varepsilon_1'.
\]
Then, by the arbitrariness of $\varepsilon_1'$, we have
$\overline{A}=\overline{B}$.

Similarly, by equation \eqref{3.14} and the relation
$\overline{A}=\overline{B}$, we have
\[
\overline{C}-\overline{D}
\leq \Big|\frac{2r\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})}{r\Delta(\overline{B},
\overline{C})\Delta(\overline{A},
\overline{D})-fm_3\overline{A}}\Big|\varepsilon_1'.
\]
Then, by the arbitrariness of $\varepsilon_1'$, we have
$\overline{C}=\overline{D}$.

Combining the above preparations, we can prove the
following theorem.

\begin{theorem}
If \eqref{2.2} and the following condition
\begin{equation}
am_3>c,\quad
\frac{\frac{f}{bm_3}(am_3-c)}{m_1+\frac{m_2}{bm_3}(am_3-c)
+\frac{m_3}{rm_2}(f-dm_2)}>d
\label{3.11}
\end{equation}
hold, then for any solution
$(x(t),y(t))$ of  \eqref{1.3} with the positive initial
condition in $S$, $\lim_{t\to+\infty}(x(t),y(t))=E^*$; this implies that
the positive equilibrium $E^{*}$ of  \eqref{1.3} is
globally attractive. \label{th:3.2}
\end{theorem}

\begin{proof}
The condition \eqref{3.11} can assure that $a>\frac{c}{m_3}$ and
$\frac{fB_1}{m_1+m_2B_1+m_3C_1}>d$. Thus, provided that
the condition \eqref{2.2} holds, from the above preparations and
the formula \eqref{3.12}, for any solution $(x(t),y(t))$ of the
system \eqref{1.3} with the positive initial condition in $S$, there
exist $\overline{A}$ and $\overline{C}$ such that
$\lim_{t\to+\infty}(x(t),y(t))=(\overline{A},\overline{C})$.

Since $(\overline{A},\overline{C})$ is the unique $\omega$-limit
point of $(x(0),y(0))$, due to the property of the $\omega$-limit
set, $(\overline{A},\overline{C})$ must be an equilibrium in the set
$\overline{S}=\{(x,y):x\geq 0,y\geq 0\}$. Further, due to
Theorem~\ref{th:2.1}, the condition \eqref{2.2} and
Theorem~\ref{th:3.1}, this equilibrium must be the positive
equilibrium $E^*$.
\end{proof}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig3} % yprim41xy.eps
\end{center}
\caption{Four phase diagrams of system \eqref{3.30}.}\label{fig:5.1}
\end{figure}

\begin{example} \label{ex:5.1} \rm
Let $a=2$, $b=16$, $c=1$, $d=0.01$, $r=3$, $f=2$, $m_1=1$, $m_2=2$ and
$m_3=3$, then system \eqref{1.3} becomes
\begin{equation}
\begin{gathered}
x'=x[2-16x-\frac{y}{1+2x+3y}],\\
y'=y[-\frac{1}{100}-3y+\frac{2x}{1+2x+3y}].
\end{gathered}
\label{3.30}
\end{equation}
Clearly, $(f-dm_2)a/b-dm_1\approx0.238$,
$am_3-c=5$, $\frac{\frac{f}{bm_3}(am_3-c)}{m_1+\frac{m_2}{bm_3}(am_3-c)
+\frac{m_3}{rm_2}(f-dm_2)}-d\approx0.102$.
Thus, the conditions \eqref{2.2} and \eqref{3.11} hold. By
Theorem \ref{th:3.2}, the positive equilibrium point
$E^{*}=(0.123,0.055)$ of  \eqref{3.30} is globally attractive, which
can also be seen from Figure \ref{fig:5.1}. Note that in Figure
\ref{fig:5.1}, the four phase diagrams start from initial points
$(0.2, 0.1)$, $(0.05, 0.01)$, $(0.1, 0.1)$ and $(0.18, 0.04)$, respectively,
and all approach $E^{*}=(0.123, 0.055)$ as $t\to +\infty$.
\end{example}


\begin{remark} \rm
The conditions for global
attractiveness of the positive equilibrium
provided in Theorem~\ref{th:3.2}  depend only on
parameters, while the conditions in \cite{lt}  depend  on
parameters and on the positive equilibrium $(x^*,y^*)$.
That additionally need requires solving numerically for $(x^*,y^*)$ in equations
\eqref{2.1}.
\end{remark}


\section{Conclusion} \label{sec:Conclusion}

In this paper, we further investigated the dynamics of a
density-dependent predator-prey system developed by Li and
Takeuchi \cite{lt} and obtained the following results:
\begin{enumerate}
\item The system has a unique positive equilibrium if and only
if $(f-dm_2)a/b>dm_1$;

\item The boundary equilibrium $E_1(\frac{a}{b}, 0)$ is a saddle if and only if $(f-dm_2)a/b>
dm_1$. Moreover, $E_1(\frac{a}{b}, 0)$ is global attractive if and
only if $(f-dm_2)a/b\leq dm_1$;

\item The system is permanent if and only if $(f-dm_2)a/b>dm_1$.
\end{enumerate}
In addition, we have provided a sufficient condition for locally
asymptotic stability of $E^{*}(x^{*}, y^{*})$ by constructing a
Lyapunov function and a sufficient condition for global
attractiveness of $E^{*}(x^{*}, y^{*})$ by making use of the
comparison theorem.

Further, we derived that the predator density dependent rate $r$
does not affect the existence of a positive equilibrium and the
permanence (or equivalently, the extinction) of the system
\eqref{1.3}. However, whether $r$ will affect locally asymptotic
stability of $E^{*}(x^{*}, y^{*})$ and global attractiveness of
$E^{*}(x^{*}, y^{*})$ is still an unsolved problem, which will be
our future work. It is also interesting to:
\begin{enumerate}
\item provide weaker conditions for global attractiveness of the
positive equilibrium;

\item derive conditions to assure the (unique) existence of periodic
 orbits \cite{CC, H, hh};

\item analyze bifurcations \cite{hh, lb, ly, z} about the stability of
the positive equilibrium.
\end{enumerate}

\subsection*{Acknowledgments}
This work was supported by grants NSFC-11422111,
NSFC-11290141, NSFC-11371047, SKLSDE-2013ZX-10, and by the Innovation
Foundation of BUAA for PhD Granduates.


\begin{thebibliography}{00}

\bibitem{bs}
D. D. Bainov, P. S. Simeonov; \emph{Systems with impulse effect: stability
theory and applications}, Ellis Horwood Limited, Chichester, 1989.

\bibitem{bss}
D. D. Bainov, P. S. Simeonov; \emph{Impulsive differential equations:
periodic solutions and applications}, Longman Scientific and
Technical, New York, 1993.

\bibitem{b}
J. R. Beddington; \emph{Mutual interference between parasites or predators
and its effect on searching efficiency}, J. Animal Ecol.
44(1975), 331--340.

\bibitem{CC}
R. S. Cantrell, C. Cosner; \emph{On the dynamics of predator-prey models
with the Beddington-DeAngelis functional response}, J. Math. Anal.
Appl. 257(2001), 206--222.

\bibitem{cui}
J. Cui, Y. Takeuchi; \emph{Permanence, extinction and periodic solution of
predator-prey system with Beddington-DeAngelis functional response},
J. Math. Anal. Appl. 317(2006), 464--474.

\bibitem{d}
D. L. DeAngelis, R. A. Goldstein, R. V.O'Neil; \emph{A model of trophic
interaction}, Ecology 56(1975), 881--892.

\bibitem{du}
Z. Du and Z. Feng, \emph{Periodic solutions of a neutral impulsive predator-prey model
with Beddington-DeAngelis functional response with delays}, J. Comput. Appl. Math. 258(2014), 87-98.

\bibitem{du1}Z. Du, X. Chen and Z. Feng, \emph{Multiple positive periodic solutions to
a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms},
Discrete Contin. Dyn. Syst. Ser. S 7(2014), 1203-1214.

\bibitem{f}
M. Fan, Y. Kuang; \emph{Dynamics of a nonautonomous predator-prey system
with the Beddington-DeAngelis functional response}, J. Math. Anal.
Appl. 295(2004), 15--39.

\bibitem{hahn67}
W. Hahn; \emph{Stability of motion}, Springer, 1967.

\bibitem{hw}
J. K. Hale, P. Waltman; \emph{Persistence in infinite-dimensional systems},
SIAM J. Math. Anal. 20(1989), 388--395.

%\bibitem{he}
%Z. He, J. Xu, Overtaking optimal harvesting of age-structured
%predator-prey systems, Int. J. Biomath. 3(2010)277--298.

\bibitem{hhw}
S. B. Hsu, S. P. Hubbell, P. Waltman; \emph{Competing predators}, SIAM J.
Appl. Math. 35(1978), 617--625.

\bibitem{hh}
S. B. Hsu, S. P. Hubbell, P. Waltman; \emph{A contribution to the theory
of competing predators}, Ecological Monogr. 35(1978), 617--625.

\bibitem{hu}
G. Huang, W. Ma, Y. Takeuchi; \emph{Global analysis for delay virus
dynamics model with Beddington-DeAngelis functional response}, Appl.
Math. Lett. 24(2011), 1199--1203.

\bibitem{H}
T. Hwang; \emph{Uniqueness of limit cycles of the predator-prey system
with Beddington-DeAngelis functional response}, J. Math. Anal. Appl.
290(2004), 113--122.

\bibitem{h}
H. K. Khalil; \emph{Nonlinear systems (third edition)}, Prentice Hall,
2002.

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

\bibitem{lt}
H. Li, Y. Takeuchi; \emph{Dynamics of the density dependent predator-prey
system with Beddington-DeAngelis functional response}, J. Math. Anal.
Appl. 374(2011), 644--654.

\bibitem{lb}
S. Liu, E. Beretta; \emph{A stage-structured predator-prey model of
Beddington-DeAngelis type}, SIAM J. Appl. Math. 66(2006), 1101--1129.

\bibitem{ly}
Z. Liu, R. Yuan; \emph{Stability and bifurcation in a delayed
predator-prey system with Beddington-DeAngelis functional response},
J. Math. Anal. Appl. 296(2004), 521--537.

\bibitem{sg}
G. T. Skalski, J. F. Gilliam; \emph{Functional responses with predator
interference: Viable alternatives to the Holling type II model},
Ecology, 82(2001), 3083--3092.

\bibitem{so}
X. Song, L. Chen; \emph{Optimal harvesting and stability for a
predator-prey system with stage structure}, Acta Math. Appl. Sini.
Eng ser. 18(3)(2002), 423--430.

\bibitem{su}
G. Sun, Z. Jin, Q. Liu, B. Li; \emph{Rich dynamics in a predator-prey with
both noise and periodic force}, Biosystems 100(2010), 14--22.

\bibitem{te}
Z. Teng; \emph{Uniform persistence of the periodic predator-prey
Lotka-Volterra systems}, Appl. Anal. 72(3-4)(1998), 339--352.

\bibitem{hrt}
H. R. Thieme; \emph{Persistence under relaxed point-dissipativity (with
application to an endemic model)}, SIAM J. Math. Anal.
24(1993), 407--435.

\bibitem{ws}
W. Wang, J. Sun; \emph{On the predator-prey system with Holling-(n+1)
functional response}, Acta Math. Sini. Eng ser. 23(2007), 1--6.

\bibitem{xi}
D. Xiao, H. Zhu; \emph{Multiple focus and hopf bifurcations in a predator
prey system with nonmonotonic functional response}, SIAM J. Appl.
Math. 66(3)(2006), 802--819.

\bibitem{z}
 H. Zhu, S. Campbell, G. Wolkowicz; \emph{Bifurcation
analysis of a predator-prey system with nonmonotonic functional
response}, SIAM J. Appl. Math. 63(2002), 636--682.

\end{thebibliography}

\end{document}