\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 81, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/81\hfil Permanence of predator-prey system] {Permanence of predator-prey system with infinite delay} \author[J. Cui \& Y. Sun\hfil EJDE-2004/81\hfilneg] {Jingan Cui, Yonghong Sun} % in alphabetical order \address{Jingan Cui \hfill\break School of Mathematics and Computer Science\\ Nanjing Normal University, Nanjing 210097, China} \email{cuija@njnu.edu.cn} \address{Yonghong Sun \hfill\break School of Mathematics and Computer Science\\ Nanjing Normal University, Nanjing 210097, China} \email{loop1979@sohu.com} \date{} \thanks{Submitted May 25, 2004. Published June 8, 2004.} \thanks{This work is supported by Jiangsu Provincial Department of Education (02KJB110004)} \subjclass[2000]{92D25, 34C25} \keywords{Predator-prey system, stage structure, permanence, periodic solution, \hfill\break\indent infinite delay} \begin{abstract} In this paper we consider a predator-prey system with periodic coefficients and infinite delay, in which the prey has a history that takes them through two stages, immature and mature. We provide a sufficient and necessary condition to guarantee the permanence of the system. The system has a positive periodic solution under this condition. Some known results are extended to the delay case. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this paper, we consider the following periodic predator-prey system with infinite delay and stage structure $$\begin{gathered} \dot{x_1}=a(t)x_2-b(t)x_1-d(t)x_1^2-p(t)x_1\int_{-\infty}^0k_{12}(s)y(t+s)\,ds,\\ \dot{x_2}=c(t)x_1-f(t)x_2^2,\\ \dot{y}=y\big[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1(t+s)\,ds-q(t) \int_{-\infty}^0k_{22}(s)y(t+s)\,ds\big], \end{gathered} \label{e1}$$ where $x_1$ and $x_2$ denote the density of immature and mature population A(prey) respectively, and $y$ is the density of the predator B that prey on $x_1$. The coefficients in \eqref{e1} are all $\omega$-periodic and continuous for $t\geq0$, $a(t),b(t),c(t),d(t)$ and $f(t)$ are all positive, $p(t),h(t)$ and $q(t)$ are nonnegative, and $\int_0^\omega q(t)dt>0$, $\int_0^\omega g(t)dt\geq0$. The functions $k_{ij}(s)(i,j=1,2)$ defined on $\mathbb{R}_- =(-\infty,0]$ are nonnegative and integrable, $\int_{-\infty}^0 k_{ij}(s)=1$. The biological background for \eqref{e1} can be found in \cite{c1, z1}. Predator-prey systems have been studied in many articles; see for example \cite{f1,l1,m1,s1, t1}. However, for the predator-prey systems with stage structure and infinite time delay, we have not obtained necessary and sufficient conditions for its permanence. In the natural world, however, there are many species whose individual members have a life history that take them through two stages, immature and mature. In particularly, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages. Recently, autonomous systems with stage structure of species have been considered in \cite{a1,a2,w1,z1}, in particularly, the effect of dispersal on the permanence of a single species with stage structure was discussed in [11]. And two species predator-prey Lotka-Volterra type dispersal systems with periodic coefficients and infinite delays have been studied in \cite{t2}. In this paper, we consider system \eqref{e1} with periodic coefficients. Our purpose is to establish sufficient and necessary conditions of integrable form for the permanence of system \eqref{e1} . \section{Main Results} When $f(t)$ is a continuous $\omega$-periodic function defined on $[0,+\infty)$, we set $$A_{\omega}(f)=\omega^{-1}\int_0^\omega f(t)dt,\quad f^M=\max_{t\geq0}f(t),\quad f^L=\min_{t\geq0}f(t).$$ Let set $C_+=$\{$\phi=(\phi_1,\phi_2,\phi_3):\phi_{i}(t)$ is continuous and nonnegative on $\mathbb{R}_-$ and $\phi_i(0)>0$, $i=1,2,3\}$ \noindent {\bf Definition} The system $\dot{x}=F(t,x)$, $x\in \mathbb{R}^n$ is said to be permanent if there are constants $M\geq m>0$ such that every positive solution $x(t)=(x_1(t),\dots ,x_n(t))\in R_{+}^n=\{(x_1,\dots ,x_n): x_i>0,\;i=1, \dots ,n\}$ of this system, satisfies $$m\leq\liminf_{t\to\infty}x_i(t) \leq\limsup_{t\to\infty}x_i(t)\leq M\,.$$ \begin{lemma}[\cite{c1}] \label{lm1} The system $$\begin{gathered} \dot{x_1}=a(t)x_2-b(t)x_1-d(t)x_1^2, \\\dot{x_2}=c(t)x_1-f(t)x_2^2 \end{gathered} \label{e2}$$ has a positive $\omega$-periodic solution $(x_1^*(t),x_2^*(t))$ which is globally asymptotically stable with respect to $R_{+0}^2=\{(x_1,x_2):x_1>0,x_2>0\}$. \end{lemma} \begin{lemma}[\cite{t3}] \label{lm2} For the well-known periodic logistic equation $$\dot{u}=u[a(t)-b(t)u] \label{e3}$$ where $a(t)$ and $b(t)$ are $\omega-$periodic continuous functions, $b^l\geq 0$ and $A_{\omega}(b)>0$, there is a constant $M>0$ such that every positive solution $u(t)$ of \eqref{e3} satisfies $\limsup_{t\to\infty}u(t)\leq M$. \end{lemma} \begin{theorem} \label{main} System \eqref{e1} is permanent if and only if $$\int_0^\omega[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s)\,ds]dt>0, \label{e4}$$ where $(x_1^*(t),x_2^*(t))$ is the positive $\omega$-periodic solution of \eqref{e2}. \end{theorem} To prove this theorem, we need several Lemmas. In this paper we always assume that solutions of system \eqref{e1} satisfy the initial conditions: $$x_i(s)=\varphi_i(s),y(s)=\psi(s),\quad (i=1,2),\; (\varphi_1,\varphi_2,\psi)\in C_{+},\; s\in(-\infty,0].$$ \begin{lemma} \label{lm3} There exist positive constants $M_{x}$ and $M_{y}$ such that $$\limsup_{t\to\infty}x_i(t)\leq M_x,\quad \limsup_{t\to\infty}y(t)\leq M_y.$$ \end{lemma} \begin{proof} Obviously, $R_{+}^3$ is a positively invariant set of system \eqref{e1}. Given any positive solution $(x_1(t),x_2(t),y(t))$, we have \begin{gather*} \dot{x_1}\leq a(t)x_2-b(t)x_1-d(t)x_1^2,\\ \dot{x_2}=c(t)x_1-f(t)x_2^2. \end{gather*} By the vector comparison theorem \cite{l2}, we obtain $$x_i(t)\leq\bar{x}_i(t)\quad (i=1,2)$$ for all $t\geq0$, where $\bar{x}(t)=(\bar{x}_1(t),\bar{x}_2(t))$ is the solution of \eqref{e2} with $\bar{x}(0)=x(0)$. By the global asymptotic stability of $x^*(t)$, there is a $T_0>0$ such that for all $t\geq T_0$, $$\bar{x}_i(t)\leq M_x\quad (i=1,2);$$ hence, for all $t\geq T_0$, $$x_i(t)\leq M_x \quad (i=1,2). \label{e7}$$ Consequently, $\limsup_{t\to\infty}x_i(t)\leq M_x$, $i=1,2$. Let $\bar{e}(t)=-g(t)+2M_xh(t)$ and $H_0=\sup\{x_1(t+s)+x_2(t+s)\;|\;t\geq0,s\leq0\}$ and let the constant $\tau>0$ be such that \begin{gather} H_0\int_{-\infty}^{-\tau}k_{21}(s)\,ds0. \label{e9} \end{gather} From \eqref{e7}, for any $t\geq T_0+\tau$ we have \begin{align*} \dot{y}&\leq y[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1(t+s)\,ds]\\ &\leq y\big[-g(t)+h(t)\int_{-\infty}^{-\tau}k_{21}(s)H_0\,ds+h(t) \int_{-\tau}^0k_{21}(s)M_x\,ds\big]\\ &\leq y[-g(t)+2M_xh(t)]=y\bar{e}(t). \end{align*} Hence, for any $t\geq t+s\geq T_0+\tau$ we obtain $$y(t+s)\geq y(t)\exp\int_t^{t+s}\bar{e}(u)du\geq y(t)\exp(\bar{e}^ms).$$ From this, for any $t\geq T_0+2\tau$, we have \begin{align*} \dot{y}&\leq y[\bar{e}(t)-q(t)\int_{-\infty}^0k_{22}(s)y(t+s)\,ds]\\ &\leq y\big[\bar{e}(t)-q(t)\int_{-\tau}^0k_{22}(s)y(t+s)\,ds\big]\\ &\leq y\big[\bar{e}(t)-q(t)\int_{-\tau}^0k_{22}(s)\exp(\bar{e}^ms)\,ds\;y(t)\big]. \end{align*} Let $u(t)$ be the solution of the auxiliary equation $$\dot{u}=u[\bar{e}(t)-q(t)\int_{-\tau}^0k_{22}(s)\exp(\bar{e}^ms)\,ds u]$$ with the initial condition $u(T_0+2\tau)=y(T_0+2\tau)$. Then we obtain $$y(t)\leq u(t) \label{e10}$$ for all $t\geq T_0+2\tau$. From Lemma \ref{lm2} , \eqref{e9}, $q(t)\geq0$ and $A_{\omega}(q)>0$, we know that there is a constant $M_y>0$ such that $$\limsup_{t\to\infty}u(t)\leq M_y.$$ Consequently, by \eqref{e10} we have $$\limsup_{t\to\infty}y(t)\leq M_y. \label{e11}$$ \end{proof} \begin{lemma} \label{lm4} There is a positive constant $\delta_x$ ($\delta_x0$ such that H_1\int_{-\infty}^{-\sigma}k_{12}(s)\,ds0 such that for all t\geq T_1 y(t)\leq M_y. For every t\geq T_1+\sigma, we have \begin{align*} \dot{x_1}&=a(t)x_2-b(t)x_1-d(t)x_1^2 -p(t)x_1\int_{-\infty}^{-\sigma}k_{12}(s)y(t+s) \,ds\\ &\quad -p(t)x_1\int_{-\sigma}^0k_{12}(s)y(t+s)\,ds \\ &\geq a(t)x_2-b(t)x_1-d(t)x_1^2-2M_yp(t)x_1,\\ \dot{x_2}&=c(t)x_1-f(t)x_2^2\,. \end{align*} Consider the auxiliary system $$\begin{gathered} \dot{u_1}=a(t)u_2-[b(t)+2M_yp(t)]u_1-d(t)u_1^2,\\ \dot{u_2}=c(t)u_1-f(t)u_2^2. \end{gathered} \label{e14}$$ Let u(t)=(u_1(t),u_2(t)) is the solution of system \eqref{e14} with the initial condition u(T_1+\sigma)=x(T_1+\sigma), then for all t\geq T_1+\sigma, x_i(t)\geq u_i(t).$$By Lemma \ref{lm1}, \eqref{e14} has a positive \omega-periodic solution u^*(t)=(u_1^*(t),u_2^*(t)), which is globally asymptotically stable. By the global asymptotic stability of u^*(t), there exist constants \delta_x>0 and T_2>T_1+\sigma such that for all t\geq T_2,$$u_i(t)\geq\delta_x.$$Hence, for all t\geq T_2, x_i(t)\geq\delta_x. So we have$$ \liminf_{t\to\infty}x_i(t)\geq\delta_x. $$\end{proof} \begin{lemma} \label{lm5} There is a positive constant \beta (\beta\beta. \label{e16} \end{lemma} \begin{proof} By \eqref{e4}, we can choose constant \varepsilon_0, 0<\varepsilon_0<1, such that $$\int_0^\omega[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s)\,ds-2h(t)\varepsilon_0-2q(t) \varepsilon_0]dt>\varepsilon_0. \label{e17}$$ Consider the following equations with a positive parameter \alpha, $$\begin{gathered} \dot{x_1}=a(t)x_2-[b(t)+2\alpha p(t)]x_1-d(t)x_1^2,\\ \dot{x_2}=c(t)x_1-f(t)x_2^2. \end{gathered} \label{e18}$$ By Lemma \ref{lm1}, \eqref{e18} has a positive \omega-periodic solution x_{\alpha}^*(t)= (x_{1\alpha}^*(t),x_{2\alpha}^*(t)), which is globally asymptotically stable. Let x_{\alpha}(t)=(x_{1\alpha}(t),x_{2\alpha}(t)) be the solution of \eqref{e18} with initial condition x_{\alpha}(0)=x^*(0), where x^*(t)=(x_1^*(t),x_2^*(t))is the positive periodic solution of \eqref{e2}. Then for the above \varepsilon_0, there exists T_2>0 such that for all t\geq T_2,$$|x_{i\alpha}^*(t)-x_{i\alpha}(t)|<\varepsilon_0/4,\quad i=1,2. $$By the continuity of the solution in the parameter, we have x_{\alpha}(t)\to x^*(t) uniformly in [T_2,T_2+\omega] as \alpha\to 0. Hence for \varepsilon_0>0, there exists \alpha_0=\alpha_0(\varepsilon_0) (0<\alpha_0<\varepsilon_0) such that$$|x_{i\alpha}(t)-x_i^*(t)|<\varepsilon_0/4,\quad i=1,2,\; 0\leq\alpha\leq\alpha_0,\; t\in[T_2,T_2+\omega]. $$So we have$$ |x_{i\alpha}^*(t)-x_i^*(t)|<\varepsilon_0/2,\quad i=1,2,\; 0\leq\alpha\leq\alpha_0,\; t\in[T_2,T_2+\omega]. $$Since x_{\alpha}^*(t) and x^*(t) are all \omega-periodic, we have $$|x_{i\alpha}^*(t)-x_i^*(t)|<\varepsilon_0/2,\quad i=1,2,\;0\leq\alpha\leq\alpha_0,\;t\geq0. \label{e22}$$ Suppose that the condition \eqref{e16} is not true, then for any positive constant \alpha<\alpha_0 we have$$\limsup_{t\to\infty}y(t)<\alpha.$$So there exists T_3>0 such that for all t\geq T_3 y(t)<\alpha. We can choose constant \tau_0>0 such that$$H_2\int_{-\infty}^{-\tau_0}k(s)\,ds<\alpha, where k(s)=k_{12}(s)+k_{21}(s)+k_{22}(s), H_2=H_1+\max\{x_1^*(t)\}. For any t\geq T_3+\tau_0, we have \begin{align*} \dot{x_1}&=a(t)x_2-b(t)x_1-d(t)x_1^2 -p(t)x_1\int_{-\tau_0}^0k_{12}(s)y(t+s)\,ds \\ &\quad -p(t)x_1\int_{-\infty}^{-\tau_0}k_{12}(s) y(t+s)\,ds\\ &\geq a(t)x_2-b(t)x_1-d(t)x_1^2-2p(t)x_1\alpha\\ &=a(t)x_2-[b(t)+2\alpha p(t)]x_1-d(t)x_1^2,\\ \dot{x_2}&=c(t)x_1-f(t)x_2^2. \end{align*} Then by the vector comparison theorem we obtain $$x_i(t)\geq x_{i\alpha}(t),\quad i=1,2,\; t\geq T_3+\tau_0, \label{e25}$$ where x_{\alpha}(t)=(x_{1\alpha}(t),x_{2\alpha}(t)) is the solution of \eqref{e18} with initial condition x_{\alpha}(T_3+\tau_0)=x(T_3+\tau_0). By the global asymptotic stability of x_{\alpha}^*(t), there exists T_4>T_3+\tau_0 such that $$|x_{i\alpha}(t)-x_{i\alpha}^*(t)|<\varepsilon_0/2,\quad i=1,2,\; t\geq T_4. \label{e26}$$ Hence, by \eqref{e22}, \eqref{e25} and \eqref{e26}x_i(t)\geq x_i^*(t)-\varepsilon_0,\quad i=1,2,\;t\geq T_4. Then for any t\geq T_4+\tau_0, we have \begin{align*} \dot{y}&\geq y\big[-g(t)+h(t)\int_{-\tau_0}^0k_{21}(s)x_1(t+s)\,ds -q(t)\int_{-\tau_0}^0k_{22}(s)y(t+s)\,ds\\ &\quad -q(t)\int_{-\infty}^{-\tau_0}k_{22}(s)y(t+s)\,ds\big]\\ &\geq y\big[-g(t)+h(t)\int_{-\tau_0}^0k_{21}(s)(x_1^*(t+s) -\varepsilon_0)\,ds-q(t)\alpha\int_{-\tau_0}^0k_{22}(s)\,ds-q(t)\alpha\big]\\ &\geq y\big[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s)\,ds-2h(t)\varepsilon_0 -2q(t)\varepsilon_0\big](\alpha<\varepsilon_0) \end{align*} Integrating the above inequality from T_4+\tau_0 to t (t\geq T_4+\tau_0) yieldsy(t)\geq y(T_4+\tau_0)\exp\int_{T_4+\tau_0}^t[-g(t) +h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s)-2h(t)\varepsilon_0-2q(t) \varepsilon_0]\,ds. $$By \eqref{e17} we know that y\to\infty as t\to\infty, which is a contradiction. This completes the proof. \end{proof} \begin{lemma} \label{lm6} There exists a positive constant \delta_y (\delta_y\beta, k=1,2,\dots . Hence, for each k there are time sequences \{s_q^{(k)}\} and \{t_q^{(k)}\}, satisfying 00, there is a T^{(k)}>0 such that x_i(t,\phi_k)\leq M_x (i=1,2) and y(t,\phi_k)\leq M_y for all t\geq T^{(k)}. Further, there is a constant \sigma^{(k)}>0 such that$$ H_1^{(k)}\int_{-\infty}^{-\sigma^{(k)}}k(s)\,dsT^{(k)}+\sigma^{(k)}$as$q\geq K_1^{(k)}$. For any$t\geq T^{(k)}+\sigma^{(k)}, we have \begin{align*} \frac {dy(t,\phi_k)}{dt} &\geq y(t,\phi_k)\big[-g(t)-q(t)\int_{-\sigma^{(k)}}^0 k_{22}(s)y(t+s,\phi_k)\,ds\\ &\quad -q(t)\int_{-\infty}^{-\sigma^{(k)}}k_{22}(s)y(t+s,\phi_k)\,ds\big]\\ &\geq y(t,\phi_k)\big[-g(t)-2q(t)M_y\big]. \end{align*} Integrating the above inequality froms_q^{(k)}$to$t_q^{(k)}$, for any$q\geq K_1^{(k)}$we get $$y(t_q^{(k)},\phi_k)\geq y(s_q^{(k)},\phi_k)\exp\int_{s_q^{(k)}}^{t_q^{(k)}}[-g(t)-2q(t)M_y]dt.$$ Consequently $$t_q^{(k)}-s_q^{(k)}\geq\frac{\ln(k+1)}{r_1},\;q\geq K_1^{(k)}, \label{e31}$$ where$r_1=\max_{t\geq0}\{|g(t)|+2M_yq(t)\}$. By \eqref{e17}, there are constants$P>0$and$r$such that for any$a\geq P$we have $$\int_0^a[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s)\,ds-2h(t)\varepsilon_0-2q(t) \varepsilon_0]dt>r.$$ Obviously, for any$k$there is$K_2^{(k)}>K_1^{(k)}$such that for all$q\geq K_2^{(k)}$, $$H_1^{(k)}\int_{-\infty}^{T^{(k)}-s_q^{(k)}}k(s)\,ds<\frac{1}{2}\beta. \label{e33}$$ Further, we can choose a constant$\sigma_0$such that $$H_2\int_{-\infty}^{-\sigma_0}k(s)\,ds<\frac{1}{2}\beta, \label{e34}$$ where$H_2=M_y+\max_{t\geq0}\{x_1^*(t)+x_2^*(t)\}$. by \eqref{e31}, there is a integer$N_1>0$such that$t_q^{(k)}-s_q^{(k)}>\sigma_0$for all$k\geq N_1,q\geq K_2^{(k)}$. For any$k\geq N_1,q\geq K_2^{(k)}$and$t\in[s_q^{(k)}+\sigma_0,t_q^{(k)}], by \eqref{e29}, \eqref{e33} and \eqref{e34} we have \begin{align*} \frac{dx_1(t,\phi_k)}{dt} &=a(t)x_2(t,\phi_k)-b(t)x_1(t,\phi_k)-d(t)x_1^2(t,\phi_k)\\ &\quad -p(t)x_1(t,\phi_k)\int_{-\infty}^{T^{(k)}} k_{12}(u-t)y(u,\phi_k)\,du\\ &\quad -p(t)x_1(t,\phi_k)\int_{T^{(k)}}^{s_q^{(k)}}k_{12} (u-t)y(u,\phi_k)\,du\\ &\quad -p(t)x_1(t,\phi_k)\int_{s_q^{(k)}}^tk_{12}(u-t)y(u,\phi_k)du\\ &\geq a(t)x_2(t,\phi_k)-b(t)x_1(t,\phi_k)-d(t)x_1^2 (t,\phi_k)\\ &\quad -p(t)x_1(t,\phi_k)H_1^{(k)}\int_{-\infty}^{T^{(k)}-t}k_{12}(s)\,ds\\ &\quad -p(t)x_1(t,\phi_k)M_y\int_{-\infty}^{s_q^{(k)}-t}k_{12} (s)\,ds\\ &\quad -p(t)x_1(t,\phi_k)\beta\int_{-\infty}^0k_{12}(s)\,ds\\ &\geq a(t)x_2(t,\phi_k)-b(t)x_1(t,\phi_k)-d(t) x_1^2(t,\phi_k)-2p(t)x_1(t,\phi_k)\beta\\ &=a(t)x_2(t,\phi_k)-[b(t) +2\beta p(t)]x_1(t,\phi_k)-d(t)x_1^2(t,\phi_k) , \end{align*} $$\frac{dx_2(t,\phi_k)}{dt} =c(t)x_1(t,\phi_k)-f(t)x_2^2(t,\phi_k).$$ Letx_{\beta}(t)=(x_{1\beta}(t),x_{2\beta}(t))$be the solution of \eqref{e18} for$\alpha=\beta$with the initial condition$x_{\beta}(s_q^{(k)}+\sigma_0)=x(s_q^{(k)}+\sigma_0,\phi_k)$. Then by the vector comparison theorem, it follows that $$x_i(t,\phi_k)\geq x_{i\beta}(t),\quad i=1,2,\;t\in[s_q^{(k)}+\sigma_0,t_q^{(k)}]. \label{e37}$$ From$\lim_{q\to\infty}s_q^{(k)}=\infty$and Lemmas \ref{lm3} and \ref{lm4}, we obtain that for any$k$there is a$K_3^{(k)}>K_2^{(k)}$such that for any$q\geq K_3^{(k)}$, $$\delta_x\leq x_i(s_q^{(k)}+\sigma_0,\phi_k)\leq M_x,\quad i=1,2.$$ For the parameter$\alpha=\beta$, Equation \eqref{e18} has a globally asymptotically stable positive$\omega$-periodic solution$x_{\beta}^*(t)= (x_{1\beta}^*(t),x_{2\beta}^*(t))$. >From the periodicity of \eqref{e18} we know that the periodic solution$x_{\beta}^*(t)$also is globally uniformly asymptotically stable. Hence, there is a$T_5>P$, and$T_5$is independent of any$k$and$q$, such that $$x_{i\beta}(t)>x_{i\beta}^*(t)-\frac{1}{2}\varepsilon_0$$ for all$t\geq T_5+s_q^{(k)}+\sigma_0$and$q\geq K_3^{(k)}$. Consequently, by \eqref{e22}, $$x_{i\beta}(t)>x_i^*(t)-\varepsilon_0 \label{e39}$$ for all$t\geq T_5+s_q^{(k)}+\sigma_0$and$q\geq K_3^{(k)}$. By\eqref{e31}, there is a$N_2\geq N_1$such that$t_q^{(k)}-s_q^{(k)}\geq 2W$for all$k\geq N_2$and$q\geq K_3^{(k)}$, where$W\geq T_5+\sigma_0$. Hence, from \eqref{e37} and \eqref{e39} we finally obtain $$x_i(t,\phi_k)\geq x_i^*(t)-\varepsilon_0. \label{e40}$$ for all$t\in[W+s_q^{(k)},t_q^{(k)}]$,$k\geq N_2$and$q\geq K_3^{(k)}$. Since, for any$t\in[W+s_q^{(k)}+\sigma_0,t_q^{(k)}]$,$k\geq N_2$and$q\geq K_3^{(k)}, by \eqref{e40}, \eqref{e33} and \eqref{e34}, we have \begin{align*} \frac{dy(t,\phi_k)}{dt} &\geq y(t,\phi_k)[-g(t)+h(t)\int_{-\sigma_0}^0k_{21}(s)x_1 (t+s,\phi_k)\,ds\\ &\quad -q(t)\int_{-\infty}^{T^{(k)}}k_{22}(u-t)y(u,\phi_k)du -q(t)\int_{T^{(k)}}^{s_q^{(k)}}k_{22}(u-t)y(u,\phi_k)du\\ &\quad -q(t)\int_{s_q^{(k)}}^tk_{22}(u-t)y(u,\phi_k)du]\\ &\geq y(t,\phi_k)[-g(t)+h(t)\int_{-\sigma_0}^0k_{21}(s)(x_1^*(t+s,\phi_k) -\varepsilon_0)\,ds\\ &\quad -q(t)H_1^{(k)}\int_{-\infty}^{T^{(k)}-t}k_{22}(s)\,ds -q(t)M_y\int_{-\infty}^{s_q^{(k)}-t}k_{22}(s)\,ds\\ &\quad -q(t)\beta\int_{-\infty}^0k_{22}(s)\,ds]\\ &\geq y(t,\phi_k)[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s,\phi_k)\,ds\\ &\quad -2h(t)\varepsilon_0-q(t)\frac{1}{2}\beta-q(t)\frac{1}{2}\beta-q(t)\beta]\\ &\geq y(t,\phi_k)[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s,\phi_k)\,ds-2h(t) \varepsilon_0-2q(t)\varepsilon_0]\,. \end{align*} Integrating fromW+s_q^{(k)}+\sigma_0$to$t_q^{(k)}$for any$k\geq N_2$and$q\geq K_3^{(k)}we obtain \begin{align*} y(t_q^{(k)},\phi_k)&\geq y(W+s_q^{(k)}+\sigma_0,\phi_k) \exp\int_{W+s_q^{(k)}+\sigma_0}^{t_q^{(k)}}\Big[-g(t)\\ &\quad +h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s,\phi_k)\,ds-2h(t)\varepsilon_0-2q(t) \varepsilon_0\Big]dt. \end{align*} Hence, by \eqref{e28} and \eqref{e29} we finally have \begin{align*} \frac{\beta}{k+1} &\geq \frac{\beta}{k+1}\exp\int_{W+s_q^{(k)}+\sigma_0}^{t_q^{(k)}} \big[-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s,\phi_k)\,ds\\ &\quad -2h(t)\varepsilon_0-2q(t) \varepsilon_0\big]dt\\ &>\frac{\beta}{k+1}\,, \end{align*}t_q^{(k)}-(W+s_q^{(k)}+\sigma_0)\geq T_5>P$, which leads to a contradiction. This completes the proof. \end{proof} \subsection*{Proof of main Theorem} The sufficiency of this theorem now follows from Lemmas \ref{lm3} \ref{lm4} \ref{lm5} \ref{lm6}. We thus only need to prove the necessity of theorem. Suppose that $$\int_0^{\omega}[-g(t)+h(t)\int_{-\infty}^0k_{12}(s)x_1^*(t+s)\,ds]dt\leq0.$$ We will show that$\lim_{t\to\infty}y(t)=0$. In fact, we know that for any given$0<\varepsilon<1$, there exist$\varepsilon_1<\varepsilon$and$\varepsilon_0>0such that \begin{aligned} &\int_0^{\omega}[-g(t)+h(t)\int_{-\infty}^0k_{12}(s)(x_1^*(t+s)+\varepsilon_1)\,ds- \frac{1}{2}q(t)l\varepsilon]dt\\ &\leq\varepsilon_1\int_0^{\omega}h(t)dt-\frac{1}{2}l \varepsilon\int_0^{\omega}q(t)dt<-\varepsilon_0, \end{aligned} \label{e43} wherel=\int_{-\infty}^0k_{22}(s)\exp(e^ms)\,ds$,$e(t)=-g(t)+h(t)\int_{-\infty}^0k_{21}(s)x_1^*(t+s)\,ds+h(t)\varepsilon_1. Since \begin{align*} \dot{x}_1\leq a(t)x_2-b(t)x_1-d(t)x_1^2,\\ \dot{x}_2=c(t)x_1-f(t)x_2^2, \end{align*} for allt\geq0$. Let$\overline{x}(t)=(\overline{x}_1(t),\overline{x}_2(t))$be the solution of \eqref{e2} with initial condition$\overline{x}(0)=x(0)$. By the vector comparison theorem we obtain$x_i(t)\leq\overline{x}_i(t)$($i=1,2$),$t\geq0$. Obviously, by the global asymptotic stability of$x^*(t)$, there is a$T_6>0$such that$\overline{x}_i(t)\leq x_i^*(t)+\frac{1}{2}\varepsilon_1$($i=1,2$) for all$t\geq T_6$. Hence, we have $$x_i(t)\leq x_i^*(t)+\frac{1}{2}\varepsilon_1\quad (i=1,2) \label{e44}$$ for all$t\geq T_6$. Choose a constant$\tau_1>0$such that \begin{gather} H_0\int_{-\infty}^{-\tau_1}k(s)\,ds<\frac{1}{2}\varepsilon_1 \label{e45}\\ \int_{-\tau_1}^0k_{22}(s)\exp(e^ms)\,ds>\frac{1}{2}l. \label{e46} \end{gather} For any$t\geq T_6+\tau_1, by \eqref{e44} and \eqref{e45} we have \begin{align*} \dot{y} &\leq y\big[-g(t)+h(t)\int_{-\tau_1}^0k_{21}(s)x_1(t+s)\,ds +h(t)\int_{-\infty}^{-\tau_1}k_{21}(s)x_1(t+s)\,ds\big] \\ &\leq y\big[-g(t)+h(t)\int_{-\tau_1}^0k_{21}(s)(x_1^*(t+s) +\frac{1}{2}\varepsilon_1)\,ds+\frac{1}{2}h(t)\varepsilon_1\big] \\ &\leq y e(t) \end{align*} Hence, by \eqref{e46}, for anyt\geq t+s\geq T_6+\tau_1we obtain \begin{align*} \dot{y}&\leq y\big[e(t)-q(t)\int_{-\tau_1}^0k_{22}(s)y(t+s)\,ds\big]\\ &\leq y\big[e(t)-q(t)\int_{-\tau_1}^0k_{22}(s)\exp(e^ms)\,ds\,y\big]\\ &\leq y\big[e(t)-\frac{1}{2}lq(t)y\big]. \end{align*} Ify(t)\geq \varepsilon$for all$t\geq T_6+2\tau_1$, then we have $$\dot{y}\leq y[e(t)-\frac{1}{2}lq(t)\varepsilon]. \label{e48}$$ Consequently, by \eqref{e43} we obtain $$y(t)\leq y(T_6+2\tau_1)\exp\int_{T_6+2\tau_1}^t[e(u)-\frac{1}{2}lq(u) \varepsilon]du\to 0$$ as$t\to\infty$, which leads to a contradiction. Hence, there is a$t_1\geq T_6+2\tau_1$such that$y(t_1)<\varepsilon$. Let$M(\varepsilon)=\max_{t\geq0}\{|e(t)|+\frac{1}{2}lq(t) \varepsilon\}$. We know that$M(\varepsilon)$is bounded for$\varepsilon\in[0,1]$. We then show that $$y(t)\leq\varepsilon\exp(M(\varepsilon)\omega),\quad t\geq t_1. \label{e49}$$ Otherwise, there are$t_3>t_2>t_1$such that$y(t_3)>\varepsilon\exp(M(\varepsilon)\omega)$,$y(t_2)=\varepsilon$and$y(t)>\varepsilon$for all$t\in(t_2,t_3]$. Let$p\geq 0$be an integer such that$t_3\in(t_2+p\omega,t_2+(p+1)\omega]. Then from \eqref{e48} we have \begin{align*} \varepsilon\exp(M(\varepsilon)\omega)&< y(t_3)\\ &\leq y(t_2)\exp\int_{t_2}^{t_3}[e(t)-\frac{1}{2}lq(t)\varepsilon]dt\\ &=\varepsilon\exp(\int_{t_2}^{t_2+p\omega}+\int_{t_2+p\omega}^{t_3}) [e(t)-\frac{1}{2}lq(t)\varepsilon]dt\\ &<\varepsilon\exp(\int_{t_2+p\omega}^{t_3}[e(t)-\frac{1}{2}lq(t)\varepsilon]dt)\\ &\leq \varepsilon\exp(M(\varepsilon)\omega). \end{align*} This leads to a contradiction. 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