\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 $$\begin{gathered} \frac{\mathrm{d}x}{\mathrm{d}t}=xf(x)-yp(\frac{x}{y}),\\ \frac{\mathrm{d}y}{\mathrm{d}t}=cyq(\frac{x}{y})-dy. \end{gathered}\label{system-7}$$ 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 $$\begin{gathered} \frac{\mathrm{d}x_1}{\mathrm{d}t}=ax_2-r_1x_1-bx_1,\\ \frac{\mathrm{d}x_2}{\mathrm{d}t}=bx_1-b_1x_2^2-\frac{a_1x_2x_3}{mx_3+x_2},\\ \frac{\mathrm{d}x_3}{\mathrm{d}t}=x_3\big(-r+\frac{a_2x_2}{mx_3+x_2}\big),\\ \end{gathered} \label{system-1}$$ 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 $$x_1^*=\frac{ax_2^*}{r_1+b},\quad x_2^*=\frac{ab}{b_1(r_1+b)}-\frac{a_1(a_2-r)}{ma_2b_1},\quad x_3^*=\frac{x_2^*(a_2-r)}{mr}. \label{equa-1}$$ 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 $a_2r$, which is locally unstable; the positive equilibrium $E_2$ is global asymptotically stable if $$\label{equa-22} ab/(r_1+b)>a_1(a_2-r)/(ma_2)>0,\quad ab/(r_1+b)>2a_1/m.$$ \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 $$\begin{gathered} \frac{\mathrm{d}\theta}{\mathrm{d}\tau}=f(x,\theta),\\ \varepsilon\frac{\mathrm{d}x}{\mathrm{d}\tau}= g(x,\theta), \end{gathered} \label{system-2}$$ 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 $$x=h(\theta)=\frac{ay}{r_1+b}.\label{equa-13}$$ 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{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} Then, the reduced system $$\frac{\mathrm{d}\theta}{\mathrm{d}\tau}=f(h(\theta),\theta) \label{system-4}$$ has equilibrium at $(0,0)$ and boundary-layer system $$\frac{\mathrm{d}\vartheta}{\mathrm{d}s} =g(\vartheta+h(\theta),\theta)\\ =-\frac{r_1+b}{a}\vartheta, \label{system-5}$$ 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_20,\quad \frac{ab}{r_1+b}-\frac{a_1}{m}>0 \label{equa-6} 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 $$\begin{gathered} \frac{\mathrm{d}y}{\mathrm{d}\tau} =\frac{y+x_2^*}{r} \big[-b_1y+\frac{a_1x_3^*y-a_1x_2^*z}{(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big],\\ \frac{\mathrm{d}z}{\mathrm{d}\tau} = \frac{ma_2(z+x_3^*)}{r}\big[\frac{x_3^*y-x_2^*z}{(mx_3^*+x_2^*)[m(z+x_3^*)+y+x_2^*]}\big].\\ \end{gathered} \label{system-6}$$ Define a Lyapunov function candidate $$V_1(\theta)=c_1\big[y-x_2^*\ln(y+x_2^*)-x_2^*\ln x_2^*\big] +c_2\big[z-x_3^*\ln(z+x_3^*)-x_3^*\ln x_3^*\big] \label{equa-2}$$ and calculating the derivative of V_1(\theta) along solutions of system \eqref{system-6}, it follows that \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} Let c_2=1 and c_1=ma_2x_3^*/a_1x_2^*. We derive from \eqref{equa-3} that \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} 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{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} for all \tau\geq\tau_1, where \phi_1(\theta)=\sqrt{y^2+z^2} and $$\alpha_1=\min\big\{b_1-\frac{a_1(a_2-r)}{ma_2(m_1-\epsilon)}, \frac{m}{mM_2+M_1}\big \}.\label{equa-21}$$ 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 $$V_2(\vartheta)=\frac{1}{2}\vartheta^2\label{equa-14}$$ and calculating the derivative of V_2(\vartheta) along solutions of system \eqref{system-5}, it follows that $$\frac{\partial V_2}{\partial\vartheta}g(\vartheta+h(\theta),\theta)=-\alpha_2\phi_2^2(\vartheta), \label{equa-8}$$ where \phi_2(\vartheta)=|\vartheta| and $$\alpha_2=\frac{r_1+b}{a}.\label{equa-20}$$ Now, for the singularly perturbed system \eqref{system-3}, we consider the composite Lyapunov function candidate $$V(\theta,\vartheta)=(1-\delta)V_1(\theta)+\delta V_2(\vartheta),\label{equa-9}$$ 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{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} Further, from \eqref{equa-2}, systems \eqref{system-3} and \eqref{system-4} we have $$\frac{\partial V_1}{\partial \theta}[f(\vartheta+h(\theta),\theta)-f(h(\theta),\theta)] = (\frac{y}{y+x_2^*},\frac{z}{z+x_3^*}) \begin{pmatrix} \frac{b\vartheta}{r}\\ 0 \end{pmatrix} \leq \beta_1\phi_1(\theta)\phi_2(\vartheta) \label{equa-11}$$ for all \tau\geq\tau_1, where $$\beta_1=\frac{b_1}{m_1r}.\label{equa-19}$$ By \eqref{equa-13}, \eqref{equa-14} and system \eqref{system-3}, we obtain \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} for all \tau\geq\tau_1, where $$\alpha_3=\frac{ab}{r(r_1+b)},\quad \beta_2=\frac{a}{r(r_1+b)}\big[\frac{a_1(a_2-r+mr)}{ma_2} +\frac{ab}{r_1+b}\big]\label{equa-18}$$ Using \eqref{equa-8}, \eqref{equa-10}, inequalities \eqref{equa-11} and \eqref{equa-12}, we obtain \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} 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}. $$The right-hand side of inequality \eqref{equa-15} is a quadratic form in \phi. The quadratic form is negative definite when $$\alpha_1[\frac{\alpha_2}{\varepsilon}+\alpha_3] >\frac{[(1-\delta)\beta_1+\delta\beta_2]^2}{4\delta(1-\delta)}.\label{equa-16}$$ It can be easily seen that the minimum value of inequality \eqref{equa-16} at \delta^*=\beta_1/(\beta_1+\beta_2) and is given by \beta_1\beta_2. So, the inequality \eqref{equa-16} is equivalent to$$ \alpha_1[\frac{\alpha_2}{\varepsilon}+\alpha_3] >\beta_1\beta_1. $$Therefore, The quadratic form is negative definite for all \varepsilon<\varepsilon^*, where $$\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}$$ and \alpha_1, \alpha_2, \alpha_3, \beta_1 and \beta_2 be defined by \eqref{equa-21}, \eqref{equa-20}, \eqref{equa-19} and \eqref{equa-18}, respectively. It follows that the origin of system \eqref{system-3} is global asymptotically stable for all \varepsilon<\varepsilon^*. That is the equilibrium E_2(x_1^*,x_2^*,x_3^*) of system \eqref{system-1} is globally asymptotically stable for all \varepsilon<\varepsilon^*. This completes the proof of this theorem. \end{proof} \begin{remark} \rm Xu et al \cite{Xurui-1} studied the globally asymptotically stable of the positive equilibrium of system \eqref{system-1} by using the technique of directly constructing Lyapunov function. Obviously, their method is different from our method, and our result improve theirs, in Theorem \ref{theorem-3} for \varepsilon=a/r small enough. So our results are more general. \end{remark} From the proof of Theorem \eqref{theorem-2}, we have the following corollary. \begin{corollary} Suppose that$$ \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{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}$$ It is easy to compute that$$ \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). \begin{thebibliography}{00} \bibitem{Arditi-1} R. Arditi, L. R. Ginzburg; \emph{Coupling in predator-prey dynamics: ratio-dependence}, J. Theoretical Biology 139(1989), 311-326. \bibitem{Arditi-2} R. Arditi, L. R. Ginzburg, H. R. Akcakaya; \emph{Variation in plankton densities among lakes: a case for ratio-dependent models}, American Naturalist 138(1991), 1287-1296. \bibitem{Arino} O. Arino, E. Sanchen, R. B. D. L. Parra, P Auger; \emph{A singular perturbation in an age-Structured population model}, SIAM J. Appl. Mah. 60(1999), 408-436. \bibitem{Barao} M. Barao, J. M. Lemos; \emph{Nonlinear control of HIV-1 infection with a singular perturbation model}, Biomedical Signal Processing and Control 2(2007), 248-257. \bibitem{Berezovskaya} F. Berezovskaya, G. Karev, R. Arditi; \emph{Parametric analysis of the ratio-dependent predator-prey model}, J. Math. Biol. 43(2001), 221-246. \bibitem{Brauer} F. Brauer, Z. E. Ma; \emph{Stability of stage-structured population models}, J. Math. Anal. Appl. 126(1987), 301-315. \bibitem{Freedman-1} H. I. Freedman, J. H. Wu; \emph{Persistence and global asymptotic stability of single species dispersal models with stage structure}, Quart. Appl. Math. 49(1991), 351-371. \bibitem{Ginzburg-1} L. R. Ginzburg, H. R. Akcakaya; \emph{Consequences of ratio-dependent predation for steady state properties of ecosystems}, Ecology 73(1992), 1536-1543. \bibitem{Gutierrez} A. P. Gutierrez; \emph{The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson's blowflies as an example}, Ecology 73(1992), 1552-1563. \bibitem{Hanski} I. Hanski; \emph{The functional response of predator: worries about scale}, TREE 6(1991), 141-142. \bibitem{Hsu-1} S. B. Hsu, T. W. Hwang, Y. Kuang; \emph{Global analysis of the Michaelis-Menten type ratiodependent predator-prey system}, J. Math. Biol. 42(2001), 489-506. \bibitem{Kuangyang-1} Y. Kuang, E. Beretta; \emph{Global qualitative analysis of a ratio-dependent predator-prey system}, J. Math. Biol. 36(1998), 389-406. \bibitem{Libingtuan} B. T. Li, Y. Kuang; \emph{LiHeteroclinic bifurcation in the Michaelis-Menten-Type ratio-dependent predator-prey system}, SIAM J. Appl. Math. 67(2007), 1453-1464. \bibitem{Liushengqiang} S. Q. Liu, L. S. Chen, Z. J. Liu; \emph{Extinction and permanence in nonautonomous competitive system with stage structure}, J. Math. Anal. Appl. 274(2002), 667-684. \bibitem{Luhongtao} H. T. Lu, Z .Y. He; \emph{Global exponential stability of delayed competitive neural networks with different time scales}, Neural Networks 18(2005), 243-250. \bibitem{Magnusson} K. G. Magnusson; \emph{Destabilizing effect of cannibalism on a structured predator-prey system}, Math. Biosci. 155(1999), 61-75. \bibitem{Meyer-Baese} A. Meyer-Base, F. Ohl, Scheich H.; \emph{Singular perturbation analysis of competitive neural networks with different time-scales}, Neural Computation 8(1996), 1731-1742. \bibitem{Meyer-Baese-1} A. Meyer-Baese, S. Pilyugin, Y. Chen; \emph{Global exponential stability of competitive neural networks with different time scales}, IEEE Transactions on Neural Networks 14(2003), 716-719. \bibitem{Meyer-Baese-2} A. Meyer-Baese, S. Pilyugin, A. Wismuller, S. Foo; \emph{Local exponential stability of competitive neural networks with different time scales}, Engineering Applications of Artificial Intelligence 17(2004), 227-232. \bibitem{Songbaojun} B. J. Song, C. Castillo-Chavez, J. P. Aparicio; \emph{Tuberculosis models with fast and slow dynamics: the role of close and casual contacts}, Math. Biosci. 180(2002), 187-205. \bibitem{Stephen} A. G. Stephen, Y. Kuang; \emph{A stage structure predator-prey model and its dependence on maturation delay and death rate}, J. Math. Biol. 49(2004), 188-200. \bibitem{Walter} A. Walter, H. J. Freedman; \emph{A time-delay model of single species growth with stage-structure}, Math. Biosci. 101(1991), 139-153. \bibitem{Wangwendi-1} W. D. Wang, L. S. Chen; \emph{A predator-prey system with stage-structure for predator}, Comput. Math. Appl. 33(1997), 83-91. \bibitem{Xiaodongmei-1} D. M. Xiao, S. G. Ruan; \emph{Global dynamics of a ratio-dependent predator-prey system}, J. Math. Biol. 43(2001), 268-290. \bibitem{Xurui-1} R. Xu, M. A.J. Chaplain, F. A. Davidson; \emph{Persistence and global stability of a ratio-dependent predator-prey model with stage structure}, Appl. Math. Comput. 158(2004), 729-744. \bibitem{Zhangxingan} X. Zhang, L. S. Chen, U. A. Numan; \emph{The stage-structured predator-prey model and optimal harvesting policy}, Math. Biosci. 168(2000), 201-210. \bibitem{Zhangzhonghua} Z. H. Zhang, Y. H. Suo, J. G. Peng, W. H. Lin; \emph{Singular perturbation approach to stability of a SIRS epidemic system}, Nonlinear Anal.: RWA 10(2009), 2688-2699. \end{thebibliography} \end{document}