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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 86, pp. 1--9.\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-2013/86\hfil Singular perturbation method]
{Singular perturbation method for global stability
of ratio-dependent predator-prey models with stage structure for the prey}

\author[L. Nie, Z. Teng \hfil EJDE-2013/86\hfilneg]
{Linfei Nie, Zhidong Teng}  % in alphabetical order

\address{
College of Mathematics and Systems Science,
Xinjiang University, Urumqi,  830046,  China}
\email[Linfei Nie]{lfnie@163.com}
\email[Zhidong Teng]{zhidong@xju.edu.cn}

\thanks{Submitted October 23, 2012. Published April 5, 2013.}
\subjclass[2000]{39A30, 34D23, 34D15}
\keywords{Singular perturbation; stage structure;ratio-dependence; 
\hfill\break\indent 
predator-prey; global asymptotically stability}

\begin{abstract}
 In this article, a singular perturbation is introduced
 to analyze the global asymptotic stability of positive equilibria
 of ratio-dependent predator-prey models with stage structure for the prey.
 We prove theoretical results and  show numerically that the proposed approach
 is feasible and efficient.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

One of the most important and interesting topics in both ecology and
mathematical ecology is the analysis between predators and their
preys. This has long been and will continue to be one of the
dominant themes  due to its universal importance.
There are many mathematical models for predator-prey behavior. The
ratio-dependent type systems are very basic and important in the
models of multi-species population dynamics. This can be roughly
stated as that the per capita predator growth rate should be a
function of the ratio of prey to predator abundance, and so should
be the so-called predator functional responses. This is strongly
supported by numerous field and laboratory experiments and
observations; see for example Arditi and Ginzburg \cite{Arditi-1},
Arditi et al. \cite{Arditi-2}, Hanski \cite{Hanski}. Generally, a
ratio-dependent predator-prey model takes the form
\begin{equation}
\begin{gathered}
\frac{\mathrm{d}x}{\mathrm{d}t}=xf(x)-yp(\frac{x}{y}),\\
\frac{\mathrm{d}y}{\mathrm{d}t}=cyq(\frac{x}{y})-dy.
\end{gathered}\label{system-7}
\end{equation}

In previous decades, the dynamics of the ratio-dependent
predator-prey system \eqref{system-7} has been systematically
studied by Kuang and Beretta \cite{Kuangyang-1}, Hsu el at.
\cite{Hsu-1}, Berezovskaya el at. \cite{Berezovskaya}, Xiao and Ruan
\cite{Xiaodongmei-1}, Li and Kuang \cite{Libingtuan} and Ginzburg el
at. \cite{Ginzburg-1}. These authors have shown that system
\eqref{system-7} has very rich dynamics.

In the natural world, there are many species whose
individual members have a life history that take them through two
stages: immature and mature. Stage-structured models have been
received much attention in recent years; see for example
\cite{Brauer,Walter,Freedman-1,Stephen}. Recently, Wang and Chen
\cite{Wangwendi-1}, Magnusson \cite{Magnusson}, Zhang el at.
\cite{Zhangxingan} proposed and investigated predator-prey models
with stage structure for prey or predator to analyze the influence
of a stage structure for the prey or the predator on the dynamics of
predator-prey models. In particular, Xu el at. \cite{Xurui-1}
studied a ratio-dependent predator-prey model with stage structure
for the prey. Their model appears as
\begin{equation}
\begin{gathered}
\frac{\mathrm{d}x_1}{\mathrm{d}t}=ax_2-r_1x_1-bx_1,\\
\frac{\mathrm{d}x_2}{\mathrm{d}t}=bx_1-b_1x_2^2-\frac{a_1x_2x_3}{mx_3+x_2},\\
\frac{\mathrm{d}x_3}{\mathrm{d}t}=x_3\big(-r+\frac{a_2x_2}{mx_3+x_2}\big),\\
\end{gathered} \label{system-1}
\end{equation}
where $x_1$ represents the density of immature individual preys at
time $t$, and $X_2$ denotes the density of mature individual preys
at time $t$, $y$ represents the density of the predator at time $t$.
By constructing Lyapunov functions, sufficient conditions are
derived for the global asymptotic stability of nonnegative
equilibria of the model.

On the other hand, in a wide class of large-scale interconnected
systems such as in power systems, large economies or even in
networks one encounters dynamics with different speeds or multiple
time scales. Singular perturbation technique is an adequate tool to
describe such systems. Singular perturbation problems are of common
occurrence in many branches of applied mathematics such as fluid
dynamics, elasticity, chemical reactor theory, neural networks,
etc.. In particular, by singular perturbation methods,
\cite{Meyer-Baese,Meyer-Baese-1,Luhongtao,Meyer-Baese-2} analyzed
the exponential stability of the competitive neural networks,
\cite{Songbaojun,Zhangzhonghua} discussed the dynamic behavior of
the epidemic models, \cite{Arino} considered a general linear
population model with both a continuous age structure and a finite
spatial structure.

 Motivated by the literature
survey, in this paper, we use singular perturbation theory to
simplify the study of system \eqref{system-1} and analysis the
global asymptotic stability of positive equilibria of system
\eqref{system-1}.

The paper is organized as follows. In the next section, a singular
perturbed system is introduced. We state and prove a general
criterion for the global asymptotically stability of positive
equilibrium of system \eqref{system-1} in Section 3. In Section 4,
specific examples are given to illustrate our results.

\section{Model description}

Obviously, system \eqref{system-1} always has equilibria
$E_0(0,0,0)$, $E_1(\widetilde{x}_1,\widetilde{x}_2,0)$, where
$$
\widetilde{x}_1=\frac{a^2b}{b_1(r_1+b)^2},\quad
\widetilde{x}_2=\frac{ab}{b_1(r_1+b)}
$$
and has a positive equilibrium $E_2(x_1^*,x_2^*,x_3^*)$ if and only
if $ab/(r_1+b)>a_1(a_2-r)/(ma_2)>0$, where
\begin{equation}
x_1^*=\frac{ax_2^*}{r_1+b},\quad
x_2^*=\frac{ab}{b_1(r_1+b)}-\frac{a_1(a_2-r)}{ma_2b_1},\quad
x_3^*=\frac{x_2^*(a_2-r)}{mr}. \label{equa-1}
\end{equation}

On the global asymptotic stability of equilibria $E_0$, $E_1$ and
$E_2$ of system \eqref{system-1}, we have the following result.

\begin{theorem}[\cite{Xurui-1}]\label{theorem-3}
If $a_2<r$, $E_1$ is locally asymptotically stable, if $a_2>r$,
which is locally unstable; the positive equilibrium $E_2$ is global
asymptotically stable if
\begin{equation}\label{equa-22}
ab/(r_1+b)>a_1(a_2-r)/(ma_2)>0,\quad ab/(r_1+b)>2a_1/m.
\end{equation}
\end{theorem}
According to above discussion, to obtain the global asymptotically
stability of positive equilibrium $E_2^*$ of system
\eqref{system-1}, it is necessary to construct the Lyapunov
functions. However, it is usually difficult for nonlinear systems.
In the present paper, by choosing a reasonable transformation, we
transform system \eqref{system-1} into the standard singular
perturbation system.

It is well know that the survival of individual eggs may be very
low, so millions of eggs must be produced in order for the species
to successfully survive the larval stage and then to persist for a
long time. The fish species provides an exact example of this
phenomenon. So, we suppose $r/a$ is small enough and re-scale time
by $rt=\tau$. Further, let $x=x_1-x_1^*$, $y=x_2-x_2^*$,
$z=x_3-x_3^*$, then the equilibria $E_2(x_1^*,x_2^*,x_3^*)$ of
system \eqref{system-1} has been shift to the origin $O(0,0,0)$.
Thus, we note that system \eqref{system-1} can be rewritten as the
following singular perturbation form
\begin{equation}
\begin{gathered}
\frac{\mathrm{d}\theta}{\mathrm{d}\tau}=f(x,\theta),\\
\varepsilon\frac{\mathrm{d}x}{\mathrm{d}\tau}= g(x,\theta),
\end{gathered} \label{system-2}
\end{equation}
where $\varepsilon=r/a$, $\theta=(y,z)\in D_{\theta}=\{(y,z):y>-
x_2^*,z>-x_3^*\}$ and $x\in D_x=\{x:x>-x_1^*\}$ and
\begin{gather*}
 g(x,\theta)=y-\frac{r_1}{a}x-\frac{b}{a}x,
\\
 f(x,\theta)= \begin{pmatrix}\frac{1}{r}
\big[b(x+x_1^*)-b_1(y+x_2^*)^2-\frac{a_1(y+x_2^*)(z+x_3^*)}{m(z+x_3^*)
+(y+x_2^*)}\big]\\
(z+x_3^*)\big[-1+\frac{a_2y(y+x_2^*)}{r(m(z+x_3^*)+(y+x_2^*))}\big]
\end{pmatrix}.
\end{gather*}

\section{Main results}

In this section, we are concerned with the global asymptotically
stable of nonnegative equilibria of system \eqref{system-1} by using
the singular perturbation.

Now, we proceed to the discussion on the stability of the origin
$O(0,0,0)$ by examining the reduced and boundary-layer models. Let
$\varepsilon$ tend to zero in system \eqref{system-2}, we can get
the first equation of system \eqref{system-2} has a unique real
function root
\begin{equation}
x=h(\theta)=\frac{ay}{r_1+b}.\label{equa-13}
\end{equation}
It is more convenient to work in the $(\vartheta,y,z)$ coordinates,
where
$$
\vartheta=x-h(\theta)
$$
because this change of variables shifts the equilibrium of the
boundary layer model to the origin. In the new coordinates, the
singularly perturbed system \eqref{system-2} can be rewritten as
\begin{equation}
\begin{aligned}
\frac{\mathrm{d}\theta}{\mathrm{d}\tau}
&=  f(\vartheta+h(\theta),\theta),\\
\varepsilon\frac{\mathrm{d}\vartheta}{\mathrm{d}\tau}
&= g(\vartheta+h(\theta),\theta)-\varepsilon\frac{\partial h}{\partial
\theta}f(\vartheta+h(\theta),\theta).
\end{aligned} \label{system-3}
\end{equation}
Then, the reduced system
\begin{equation}
\frac{\mathrm{d}\theta}{\mathrm{d}\tau}=f(h(\theta),\theta)
\label{system-4}
\end{equation}
has equilibrium at $(0,0)$ and boundary-layer system
\begin{equation}
\frac{\mathrm{d}\vartheta}{\mathrm{d}s}
=g(\vartheta+h(\theta),\theta)\\
=-\frac{r_1+b}{a}\vartheta, \label{system-5}
\end{equation}
where $s=\tau/\varepsilon$, has equilibrium at $\vartheta=0$.

To discuss the globally asymptotically stable of
equilibrium $O(0,0,0)$ of system \eqref{system-2}, we first derive
certain upper bound and lower bound estimates for solution of
reduced system \eqref{system-4}.

\begin{theorem}  \label{theorem-1}
Let $(y(\tau),z(\tau))$ denote any solutions of system
\eqref{system-4} corresponding to initial conditions $y(0)>0$ and
$z(0)>0$. If $a_2>r$ and $mab>a_1(r_1+b)$, then there is a constant
$T>0$ such that if $t\geq T$,
$$
m_1-x_2^*\leq y(\tau)\leq M_1+x_2^*,\quad
m_2<z(\tau)\leq M_2,
$$
where
\begin{equation}
m_1=\frac{mab-a_1(r_1+b)}{mb_1(r_1+b)},\quad
M_1=\frac{ab}{b_1(r_1+b)},\quad m_2=0, \quad M_2=\frac{a_2M_1}{mr}.
\label{equa-5}
\end{equation}
\end{theorem}

The proof of the above theorem  is similar to that of 
\cite[Theorem 2.1]{Xurui-1}; therefore we omit it here.
Now, we state and prove our result on the globally asymptotically
stable of system \eqref{system-1}.

\begin{theorem} \label{theorem-2}
Let $\varepsilon^*$ be defined by \eqref{equa-17}.  If
\begin{equation}
\frac{ab}{r_1+b}+\frac{ra_1}{ma_2}-\frac{2a_1}{ma_2}>0,\quad
\frac{ab}{r_1+b}-\frac{a_1}{m}>0 \label{equa-6}
\end{equation}
hold, then the equilibrium $O(0,0,0)$ of system \eqref{system-2} is
globally asymptotically stable for all
$\varepsilon\in(0,\varepsilon^*)$, that is, the equilibrium
$E_2(x_1^*,x_2^*,x_3^*)$ of system \eqref{system-1} is globally
asymptotically stable for all
$\varepsilon\in(0,\varepsilon^*)$.
\end{theorem}

\begin{proof}
Let $(y(t),z(t))$ be any positive solution of system
\eqref{system-4} with initial conditions $y(0)>0$ and $z(0)>0$. In
view of the $E_2(x_1^*,x_2^*,x_3^*)$ is positive equilibrium of
system \eqref{system-1}, we note that system \eqref{system-4} can be
rewritten as
\begin{equation}
\begin{gathered}
\frac{\mathrm{d}y}{\mathrm{d}\tau}
=\frac{y+x_2^*}{r}
\big[-b_1y+\frac{a_1x_3^*y-a_1x_2^*z}{(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big],\\
\frac{\mathrm{d}z}{\mathrm{d}\tau}
= \frac{ma_2(z+x_3^*)}{r}\big[\frac{x_3^*y-x_2^*z}{(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big].\\
\end{gathered} \label{system-6}
\end{equation}
Define a Lyapunov function candidate
\begin{equation}
V_1(\theta)=c_1\big[y-x_2^*\ln(y+x_2^*)-x_2^*\ln x_2^*\big]
+c_2\big[z-x_3^*\ln(z+x_3^*)-x_3^*\ln x_3^*\big] \label{equa-2}
\end{equation}
and calculating the derivative of $V_1(\theta)$ along solutions of
system \eqref{system-6}, it follows that
\begin{equation}
\begin{aligned}
\frac{\mathrm{d}V_1}{\mathrm{d}\tau}
&= \frac{c_1y}{y+x_2^*}\frac{\mathrm{d}y}{\mathrm{d}\tau}
+\frac{c_2z}{z+x_3^*}\frac{\mathrm{d}z}{\mathrm{d}\tau}\\
&= -c_1\big[-\frac{b_1}{r}y^2+\frac{a_1x_3^*y^2-a_1x_2^*yz}
{r(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big]\\
&\quad +c_2\big[\frac{-ma_2x_2^*z^2-ma_2x_2^*yz}
{r(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big].
\end{aligned}\label{equa-3}
\end{equation}
Let $c_2=1$ and $c_1=ma_2x_3^*/a_1x_2^*$. We derive from
\eqref{equa-3} that
\begin{equation}
\begin{aligned}
\frac{\mathrm{d}V_1}{\mathrm{d}\tau}
&= -\frac{c_1}{r}\big[b_1-\frac{a_1x_3^*}{(mx_3^*+x_2^*)
[m(z+x_3^*)+y+x_2^*]}\big]y^2\\
&\quad -\frac{ma_2c_2x_2^*}{r(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}z^2.
\end{aligned}\label{equa-4}
\end{equation}
From \eqref{equa-6}, we can choose a positive constant $\epsilon$
such that
$$
\frac{ab}{r_1+b}+\frac{ra_1}{ma_2}-\frac{2a_1}{ma_2}-\epsilon>0.
$$
Further, from Theorem \eqref{theorem-1}, there is a $\tau_1\geq0$
such that $y(\tau)>m_1-\epsilon$ for all $\tau\geq\tau_1$.
Therefore, from Theorem \eqref{theorem-1} and \eqref{equa-4}, we
obtain
\begin{equation}
\begin{aligned}
\frac{\partial V_1}{\partial \theta}f(h(\theta),\theta)
&= -\big[b_1-\frac{a_1(a_2-r)}{ma_2(m_1-\epsilon)}\big]y^2
-\frac{m}{mM_2+M_1}z^2\\
&\leq -\alpha_1\phi_1^2(\theta)
\end{aligned}\label{equa-7}
\end{equation}
for all $\tau\geq\tau_1$, where $\phi_1(\theta)=\sqrt{y^2+z^2}$ and
\begin{equation}
\alpha_1=\min\big\{b_1-\frac{a_1(a_2-r)}{ma_2(m_1-\epsilon)},
\frac{m}{mM_2+M_1}\big \}.\label{equa-21}
\end{equation}

On the other hand, Let $\vartheta(s)$ be any positive solution of
the boundary-layer system \eqref{system-5} with initial condition
$\vartheta(0)>0$. We define a Lyapunov function candidate
\begin{equation}
V_2(\vartheta)=\frac{1}{2}\vartheta^2\label{equa-14}
\end{equation}
and calculating the derivative of $V_2(\vartheta)$ along solutions
of system \eqref{system-5}, it follows that
\begin{equation}
\frac{\partial
V_2}{\partial\vartheta}g(\vartheta+h(\theta),\theta)=-\alpha_2\phi_2^2(\vartheta),
\label{equa-8}
\end{equation}
where $\phi_2(\vartheta)=|\vartheta|$ and
\begin{equation}
\alpha_2=\frac{r_1+b}{a}.\label{equa-20}
\end{equation}

Now, for the singularly perturbed system \eqref{system-3}, we
consider the composite Lyapunov function candidate
\begin{equation}
V(\theta,\vartheta)=(1-\delta)V_1(\theta)+\delta
V_2(\vartheta),\label{equa-9}
\end{equation}
where $0<\theta<1$ is to be chosen. Calculating the derivative of
$V(\theta,\vartheta)$ along the solutions of the full system
\eqref{system-3}, we obtain
\begin{equation}
\begin{aligned}
\frac{\mathrm{d}V}{\mathrm{d}\tau}
&= (1-\delta)\frac{\partial V_1}{\partial
 \theta}f(\vartheta+h(\theta),\theta)
 +\frac{\delta}{\varepsilon}\frac{\partial
 V_2}{\partial\vartheta}g(\vartheta+h(\theta),\theta)
-\delta\frac{\partial V_2}{\partial \vartheta}
 \frac{\partial h}{\partial \theta}f(\vartheta+h(\theta),\theta)\\
&= (1-\delta)\frac{\partial V_1}{\partial
\theta}f(h(\theta),\theta)+\frac{\delta}{\varepsilon}\frac{\partial
V_2}{\partial\vartheta}g(\vartheta+h(\theta),\theta)\\
&+(1-\delta)\frac{\partial V_1}{\partial
 \theta}[f(\vartheta+h(\theta),\theta)-f(h(\theta),\theta)]
+\delta\big[\frac{\partial V_2}{\partial
 \theta}-\frac{\partial V_2}{\partial \vartheta} 
 \frac{\partial h}{\partial \theta}\big]f(\vartheta+h(\theta),\theta).
\end{aligned}\label{equa-10}
\end{equation}
Further, from \eqref{equa-2}, systems \eqref{system-3} and
\eqref{system-4} we have
\begin{equation}
\frac{\partial V_1}{\partial
\theta}[f(\vartheta+h(\theta),\theta)-f(h(\theta),\theta)]
= (\frac{y}{y+x_2^*},\frac{z}{z+x_3^*})
 \begin{pmatrix}
\frac{b\vartheta}{r}\\
0
\end{pmatrix}
\leq \beta_1\phi_1(\theta)\phi_2(\vartheta) \label{equa-11}
\end{equation}
for all $\tau\geq\tau_1$, where
\begin{equation}
\beta_1=\frac{b_1}{m_1r}.\label{equa-19}
\end{equation}
By \eqref{equa-13}, \eqref{equa-14} and system \eqref{system-3}, we
obtain
\begin{equation}
\begin{aligned}
&\big[\frac{\partial V_2}{\partial
\theta}-\frac{\partial V_2}{\partial \vartheta} \frac{\partial
h}{\partial \theta}\big]f(\vartheta+h(\theta),\theta)\\
&= \frac{-a\vartheta}{r(r_1+b)}
\big\{b\vartheta-(y+x_2^*)
\big[-b_1y+\frac{a_1x_3^*y-a_1x_2^*z}{(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big]\big\}\\
&\leq -\alpha_3\phi_2^2(\vartheta)
+\beta_2\phi_1(\theta)\phi_2(\vartheta)
\end{aligned}\label{equa-12}
\end{equation}
for all $\tau\geq\tau_1$, where
\begin{equation}
\alpha_3=\frac{ab}{r(r_1+b)},\quad
\beta_2=\frac{a}{r(r_1+b)}\big[\frac{a_1(a_2-r+mr)}{ma_2}
+\frac{ab}{r_1+b}\big]\label{equa-18}
\end{equation}
Using \eqref{equa-8}, \eqref{equa-10}, inequalities \eqref{equa-11}
and \eqref{equa-12}, we obtain
\begin{equation}
\begin{aligned}
\frac{\mathrm{d}V}{\mathrm{d}\tau}
&\leq-(1-\delta)\alpha_1\phi_1^2(\theta)
-\delta[\frac{\alpha_2}{\varepsilon}+\alpha_3]\phi_2^2(\vartheta)
+(1-\delta)\beta_1\phi_1(\theta)\phi_2(\vartheta)
+\delta\beta_2\phi_1(\theta)\phi_2(\vartheta)\\
&=-\phi^T(\theta,\vartheta)\Lambda\phi(\theta,\vartheta)
\end{aligned}\label{equa-15}
\end{equation}
for all $\tau\geq\tau_1$, where
$$
\phi^T(\theta,\vartheta)=(\phi_1(\theta),\phi_2(\theta))
$$
and
$$
\Lambda=\begin{bmatrix}
 (1-\delta)\alpha_1 &
-\frac{(1-\delta)\beta_1+\delta\beta_2}{2}\\
-\frac{(1-\delta)\beta_1+\delta\beta_2}{2}
& \delta[\frac{\alpha_2}{\varepsilon}+\alpha_3]
\end{bmatrix}.
$$
The right-hand side of inequality \eqref{equa-15} is a quadratic
form in $\phi$. The quadratic form is negative definite when
\begin{equation}
\alpha_1[\frac{\alpha_2}{\varepsilon}+\alpha_3]
>\frac{[(1-\delta)\beta_1+\delta\beta_2]^2}{4\delta(1-\delta)}.\label{equa-16}
\end{equation}
It can be easily seen that the minimum value of inequality
\eqref{equa-16} at $\delta^*=\beta_1/(\beta_1+\beta_2)$ and is given
by $\beta_1\beta_2$. So, the inequality \eqref{equa-16} is
equivalent to
$$
\alpha_1[\frac{\alpha_2}{\varepsilon}+\alpha_3]
>\beta_1\beta_1.
$$
Therefore, The quadratic form is negative definite for all
$\varepsilon<\varepsilon^*$, where
\begin{equation}
\varepsilon^*=
\begin{cases}
+\infty,&\text{if }\alpha_1\alpha_3\geq\beta_1\beta_2;\\
\frac{\alpha_1\alpha_2}{\beta_1\beta_2-\alpha_1\alpha_3},
&\text{if } \alpha_1\alpha_3<\beta_1\beta_2,
\end{cases}\label{equa-17}
\end{equation}
and $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$ and $\beta_2$ be
defined by \eqref{equa-21}, \eqref{equa-20}, \eqref{equa-19} and
\eqref{equa-18}, respectively. It follows that the origin of system
\eqref{system-3} is global asymptotically stable for all
$\varepsilon<\varepsilon^*$. That is the equilibrium
$E_2(x_1^*,x_2^*,x_3^*)$ of system \eqref{system-1} is globally
asymptotically stable for all $\varepsilon<\varepsilon^*$. This
completes the proof of this theorem.
\end{proof}

\begin{remark} \rm
Xu et al \cite{Xurui-1} studied the globally asymptotically
stable of the positive equilibrium of system \eqref{system-1} by
using the technique of directly constructing Lyapunov function.
Obviously, their method is  different from our method,
and our result improve theirs, in Theorem \ref{theorem-3}
for $\varepsilon=a/r$ small enough. So our results are 
more general.
\end{remark}

From the proof of Theorem \eqref{theorem-2}, we have the following
corollary.

\begin{corollary}
Suppose that
$$
\frac{ab}{r_1+b}+\frac{ra_1}{ma_2}-\frac{2a_1}{ma_2}>0,\quad
\frac{ab}{r_1+b}-\frac{a_1}{m}>0.
$$
If $\alpha_1\alpha_3\geq\beta_1\beta_2$ holds, then the positive
equilibrium $E_2(x_1^*,x_2^*,x_3^*)$ of system \eqref{system-1} is
globally asymptotically stable, where $\alpha_1$, $\alpha_2$,
$\alpha_3$, $\beta_1$ and $\beta_2$ be defined by \eqref{equa-21},
\eqref{equa-20}, \eqref{equa-19} and \eqref{equa-18}, respectively.
\end{corollary}


\section{Example and numerical simulation}

To check the validity of our results we consider the
 ratio-dependent predator-prey model with stage structure
for the prey,
\begin{equation}
\begin{gathered}
\frac{\mathrm{d}x_1}{\mathrm{d}t}=25x_2-22x_1-1.9x_1,\\
\frac{\mathrm{d}x_2}{\mathrm{d}t}
=1.9x_1-2.8x_2^2-\frac{2.2x_2x_3}{2.2x_3+x_2},\\
\frac{\mathrm{d}x_3}{\mathrm{d}t}=x_3\big(-1+\frac{1.5x_2}{2.2x_3+x_2}\big).\\
\end{gathered} \label{system-8}
\end{equation}
It is easy to compute that
$$
\frac{ab}{r_1+b}-\frac{2a_1}{m}=\frac{25\times1.9}{22+1.9}
-\frac{2\times2.2}{2.2}\approx-0.0126<0.
$$
So,  conditions \eqref{equa-22} of Theorem \ref{theorem-3} do not
hold. Thus, we cannot guarantee the global asymptotically stability
of positive equilibrium of system \eqref{system-8} from Theorem
\ref{theorem-3}. However, it is also easy to verify that
$$
\frac{ab}{r_1+b}+\frac{ra_1}{ma_2}-\frac{2a_1}{ma_2}
=\frac{25\times1.9}{22+1.9}+\frac{1\times2.2}{2.2\times1.5}
-\frac{2\times2.2}{2.2}\approx0.6541>0
$$
and
$$
\varepsilon=\frac{r}{a}=0.0400<\varepsilon^*
=\frac{\alpha_1\alpha_2}{\beta_1\beta_2-\alpha_1\alpha_3}\approx0.0409.
$$
Therefore, from Theorem \ref{theorem-2}, the positive equilibrium
$E_2$ of system \eqref{system-1} is globally asymptotically stable.
Which is shown in Figure \ref{fig1}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig1a}
\includegraphics[width=0.48\textwidth]{fig1b}
\end{center}
\caption{Trajectory of system
\eqref{system-8} with $a=25$, $r_1=22$, $b=1.9$, $b_1=2.8$,
$a_1=m=2.2$, $a_2=1.5$, $r=1$ and $\varepsilon=0.04$} \label{fig1}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig2a}
\includegraphics[width=0.48\textwidth]{fig2b}
\includegraphics[width=0.48\textwidth]{fig2c}
\includegraphics[width=0.48\textwidth]{fig2d}
\end{center}
\caption{Trajectory of system
\eqref{system-8} and its reduced system with $a=25$, $r_1=22$,
$b=1.9$, $b_1=2.8$, $a_1=m=2.2$, $a_2=1.5$, $r=1$ and
$\varepsilon=0.04$} \label{fig2}
\end{figure}

Further,  to show how the reduced system \eqref{system-4}
approximates to the full system \eqref{system-1} and how the small
parameter $\varepsilon$ affects the stability of zero solution of
system \eqref{system-1}. By the equivalence of systems
\eqref{system-1} and \eqref{system-2}, we only focus on the
numerical analysis of system \eqref{system-2} and its reduced system
\eqref{system-4}. Let
$(x(t,\varepsilon),y(t,\varepsilon),z(t,\varepsilon))$ be solution
of system \eqref{system-2}, $(y(t),z(t))$ be the solution of system
\eqref{system-4}. If $\varepsilon$ is small enough, the solutions of
the reduced system \eqref{system-4} closely approximate to the
solutions of the full system \eqref{system-2} and the errors (i.e.
$x(t,\varepsilon)-h(y)$, $y(t,\varepsilon)-y(t)$,
$z(t,\varepsilon)-z(t)$) quickly converge to zero after oscillation,
and all solutions of system \eqref{system-2} approach to zero
solution. Which are shown in Figure \ref{fig2}(a)-(d).

\subsection*{Acknowledgments}
The authors are deeply indebted to the anonymous referee for the 
valuable suggestions and comments which improved this manuscript.
This work is supported by the
Natural Science Foundation of Xinjiang (Grant Nos. 2011211B08).


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\end{document}