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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 131, pp. 1--11.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/131\hfil Asymptotic behavior]
{Asymptotic behavior of a predator-prey diffusion system
with time delays}
\author[Y. Meng, Y. Wang\hfil EJDE-2005/131\hfilneg]
{Yijie Meng, Yifu Wang}  % in alphabetical order

\address{Yijie Meng \hfill\break
Department of Mathematics,  Xiang Fan University\\
  Xiangfan, 441053,  China}
\email{yijie\_meng@sina.com}

\address{Yifu Wang \hfill\break
 Department of Applied Mathematics, Beijing Institute of
Technology\\
Beijing, 100081, China}
\email{yifu\_wang@163.com}

\date{}
\thanks{Submitted September 12, 2005. Published November 24, 2005.}
\thanks{Supported by grants 10226013 and 19971004 from
 the  National Nature Science Foundation\hfill\break\indent
  of China.}
\subjclass[2000]{35B40}
\keywords{Predator-prey diffusion system; asymptotic behavior;
\hfill\break\indent time delays; upper-lower solutions}

\begin{abstract}
 In this paper, we study a class of reaction-diffusion systems
 with time delays, which models the dynamics  of predator-prey species.
 The global asymptotic convergence is established by the upper-lower
 solutions and iteration method in terms of the rate constants of
 the reaction  function, independent of the time delays and the
 effect of diffusion
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction}

The purpose of this paper is to study the asymptotic behavior of
solutions to the predator-prey diffusion system with time delays:
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}= \Delta
u+u[a_1-b_1u-\int_{0}^{\infty}f_1(\tau)u(t-\tau,x)d\tau - d_1v(t-r_2,x)],\\
 t>0, \; x\in\Omega, \\
\frac{\partial v}{\partial t}= \Delta
v+v[a_2-b_2v-\int_{0}^{\infty}f_2(\tau)v(t-\tau,x)d\tau + d_2u(t-r_1,x)],
\end{gathered}
\label{e1.1}
\end{equation}
subjected to the boundary conditions
\begin{equation}
\frac{\partial u}{\partial n}=\frac{\partial v}{\partial
n}=0,\quad t>0,\; x\in\partial\Omega,
\label{e1.2}
\end{equation}
and to the nonnegative initial conditions
\begin{equation}
u(t,x)=\phi_1(t,x),~~v(t,x)=\phi_2(t,x),\quad i=1,2,\; t\leq
0,\; x\in\overline{\Omega},\label{e1.3}
\end{equation}
where $\Omega\subseteq \mathbb{R}^N$ ($N\geq 1$) is a bounded domain with
smooth boundary $\partial\Omega$, and $\partial/\partial n$ denotes
differentiation in the direction of the outward normal. $a_i$,
$b_i$, $d_i$ and $r_i(i=1,2)$ are positive constants.
$f_i\in C(\mathbb{R}^+)\cap L^1(\mathbb{R}^+)$, and the integral part
means the hereditary term concerning the effect of the past history
on the present growth rate.
$\phi_i\in C^1((-\infty,0]\times\overline{\Omega})$ is bounded nonnegative.

We write $f_i=f_i^+-f_i^-(i=1,2)$, where $f_i^+(s)=\max(0,f(s))$, and
$f_i^-(s)=\max(0,-f_i(s))$ for $s\geq 0$. We  set
$$
c_i^+=\int_0^{\infty}f_i^+(s)ds,\quad
c_i^-=\int_0^{\infty}f_i^-(s)ds\quad i=1,2.
$$
Throughout the paper, we assume that
\begin{equation}
b_1>\int_{0}^{\infty}|f_1(s)|ds,\quad
b_2>\int_{0}^{\infty}|f_2(s)|ds,
\label{e1.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\frac{d_2c_2^+}{(b_1-c_1^+-c_1^-)(b_2-c_2^+-c_2^-)-d_1d_2}\\
&<\frac{a_2}{a_1}<\frac{(b_1-c_1^+-c_1^-)(b_2-c_2^+-c_2^-)-d_1d_2}{d_1(b_1-c_1^-)},
\end{aligned}\label{e1.5}
\end{equation}
 Our result can be stated as follows.

\begin{theorem} \label{thm1}
Assume that $f_1$ and $f_2$ belong to $C(\mathbb{R}^+)\cap L^1(\mathbb{R}^+)$
and \eqref{e1.4}--\eqref{e1.5} hold. Then for every
$\phi_i\in C^1((\infty,0])\times\overline{\Omega}$ with
$\phi_i(0,x)\not\equiv 0$, the solution of \eqref{e1.1}--\eqref{e1.3}
 satisfies
\begin{equation}
\lim_{t\to\infty}u(t,x)=\frac{a_1(b_2+c_2^+-c_2^-)-a_2d_1}
{(b_1+c_1^+-c_1^-)(b_2+c_2^+-c_2^-)+d_1d_2}, \label{e1.6}
\end{equation}
uniformly for $x\in\overline{\Omega}$.
Also \begin{equation}
\lim_{t\to\infty}v(t,x)=\frac{a_2(b_1+c_1^+-c_1^-)+a_1d_2}
{(b_1+c_1^+-c_1^-)(b_2+c_2^+-c_2^-)+d_1d_2}, \label{e1.7}
\end{equation}
uniformly for $x\in\overline{\Omega}$.
\end{theorem}

\noindent\textbf{Remark.}
If $f_1\equiv 0$ and $f_1\equiv 0$, then
\[
\lim_{t\to\infty}u(t,x)=\frac{a_1b_2+a_2d_1}
{b_1b_2+d_1d_2}, \quad\mbox{and}\quad
\lim_{t\to\infty}v(t,x)=\frac{a_2b_1+a_1d_2}
{b_1b_2+d_1d_2},
\]
 uniformly for $x\in\overline{\Omega}$, which
coincides with the result of \cite{p1}. \smallskip


Let us introduce the following result (see \cite{r1}) on the asymptotic
behavior of the diffusion logistic equation with time delays,
which plays an important role in the proof of Theorem.
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}= \Delta
u+u[a-bu-\int_{0}^{\infty}f(\tau)u(t-\tau,x)d\tau], \quad
t>0,\; x\in\Omega,\\
\frac{\partial u}{\partial n}=0,\quad
t>0,\; x\in\partial\Omega,\\
u(t,x)=\phi(t,x), \quad t\leq 0,\; x\in\overline{\Omega},
\end{gathered}
\label{e1.8}
\end{equation}
where $a$ and $b$ are positive constants,  $\phi\in
C^1((-\infty,0]\times\overline{\Omega})$ is a bounded nonnegative
function.

\begin{lemma} \label{lem1.1}
Assume that $f\in C(\mathbb{R}^+)\cap L^1(\mathbb{R}^+)$ and
$b>\int_0^{\infty}|f(s)|ds$. Then \eqref{e1.8} has a unique bounded
nonnegative solution. Moreover, if $\phi(0,x)\not\equiv 0$, then
$u(t,x)>0$ for all $(t,x)\in (0,\infty)\times\overline{\Omega}$
and
\[
\lim_{t\to\infty}u(t,x)=\frac{a}{b+\int_{0}^{\infty}f(s)ds},
\]
uniformly for $x\in\overline{\Omega}$.
\end{lemma}

Reaction-diffusion systems with delays have been studied by many
authors. However, most of the systems are mixed quasimonotone, and
most of the discussions are in the framework of semi-group theory
of dynamical systems \cite{m1,m2,r2,r3}. The method of upper and lower
solutions and its associated monotone iterations have been used to
investigate the dynamic property of the system, which is mixed
quasimonotone with discrete delays \cite{l1,p2,p3}. In this paper, the
method of proof is via successive improvement of upper-lower
solutions of some suitable systems, and the fact that we are
dealing with system (1.1) without mixed quasimonotone forces us to
develop some significance in the process of proof.

\section{Proof of Main Results}

In this section, we first introduce the following
existence-comparison result  for the predator-prey system
\eqref{e1.1}--\eqref{e1.3}. \smallskip

\noindent\textbf{Definition} \cite{p2} % def. 2.1
A pair of smooth functions
$(\tilde{u},\tilde{v})$ and $(\hat{u},\hat{v})$ are called
upper-lower solutions of \eqref{e1.1}--\eqref{e1.3},
if $\tilde{u}\geq \hat{u}$,
$\tilde{v}\geq \hat{v}$ in $\mathbb{R}^1\times\overline{\Omega}$, and if
for all $\hat{u}\leq\psi_1\leq\tilde{u}$,
$\hat{v}\leq\psi_2\leq\tilde{v}$, the following differential
inequalities hold.
\begin{equation}
\begin{gathered}
\frac{\partial \tilde{u}}{\partial t}-\Delta \tilde{u}\geq
\tilde{u}[a_1-b_1\tilde{u}-
\int_{0}^{\infty}f_{1}(\tau)\psi_1(t-\tau,x)d\tau-d_1\hat{v}(t-r_2,x)],\\
 t>0,\; x\in\Omega,
\\
\frac{\partial \tilde{v}}{\partial t}-\Delta \tilde{v}\geq
\tilde{v}[a_2-b_2\tilde{v}-
\int_{0}^{\infty}f_{2}(\tau)\psi_2(t-\tau,x)d\tau+d_2\tilde{u}(t-r_1,x)],
\\
\frac{\partial \hat{u}}{\partial t}-\Delta \hat{u}\leq
\hat{u}[a_1-b_1\hat{u}-
\int_{0}^{\infty}f_{1}(\tau)\psi_1(t-\tau,x)d\tau-d_1\tilde{v}(t-r_2,x)],
\\
\frac{\partial \hat{v}}{\partial t}-\Delta \hat{v}\leq
\hat{v}[a_1-b_1\hat{v}-
\int_{0}^{\infty}f_{2}(\tau)\psi_2(t-\tau,x)d\tau+d_2\hat{u}(t-r_1,x)],
\\
\frac{\partial \hat{u}}{\partial n} \leq 0\leq \frac{\partial
\tilde{u}}{\partial n},~~ \frac{\partial \hat{v}}{\partial n} \leq
0\leq
\frac{\partial \tilde{v}}{\partial n},\quad t>0,x\in\partial\Omega,
\\
\hat{u}(t,x)\leq \phi_1(t,x)\leq \tilde{u}(t,x),\quad
\hat{v}(t,x)\leq
\phi_2(t,x)\leq \tilde{v}(t,x),\quad t\leq 0,\; x\in\overline{\Omega}.
\end{gathered} \label{e2.1}
\end{equation}

With these definitions of upper-lower solutions, we can state the
following lemma.

\begin{lemma}[\cite{p2}] \label{lem2.1}
 If there exists a pair of upper-lower solutions
$(\tilde{u},\tilde{v})$, $(\hat{u},\hat{v})$ of \eqref{e1.1}--\eqref{e1.3}.
Then  the problem \eqref{e1.1}--\eqref{e1.3} has a unique solution
$(u^*,v^*)$ satisfying $\hat{u}\leq u^*\leq\tilde{u}$,
$\hat{v}\leq v^*\leq\tilde{v}$.
\end{lemma}

For a given $\phi=(\phi_1,\phi_2)$, let $M_1,M_2$ be constants such
that
$$
M_1\geq
\max\Big\{\|\phi_1\|,\frac{a_1}{b_1-\int_{0}^{\infty}|f_1(s)|ds}\Big\},\quad
M_2\geq\max\Big\{\|\phi_2\|,\frac{a_2+d_2M_1}{b_2-\int_{0}^{\infty}
|f_2(s)|ds}\Big\}
$$
where $\|\phi_i\|=\sup_{(t,x)\in
(-\infty,0]\times\overline{\Omega}}|\phi_i(t,x)|$, $i=1,2$. Then
$(0,0)$ and $(M_1,M_2)$ are clearly a pair of lower-upper
solutions of \eqref{e1.1}--\eqref{e1.3}.
By Lemma \ref{lem2.1}, a unique global
nonnegative solution $(u,v)$ to \eqref{e1.1}--\eqref{e1.3} exists
and satisfies
$0\leq u\leq M_1,0\leq v\leq M_2$, moreover $(u,v)$ is positive in
$(0,+\infty)\times\overline{\Omega}$ if $\phi_i(0,x)\not\equiv
0(i=1,2)$ by maximal principle.

Define $\overline{u}_1(t,x)$ by
\begin{equation}
\begin{gathered}
\frac{\partial \overline{u}_1}{\partial t}= \Delta
\overline{u}_1+\overline{u}_1[a_1-b_1\overline{u}_1+
\int_{0}^{\infty}f_1^-(\tau)\overline{u}_1(t-\tau,x)d\tau],\quad
 t>0,\; x\in\Omega,\\
\frac{\partial \overline{u}_1}{\partial
n}=0, \quad  t>0,\; x\in\partial\Omega,\\
\overline{u}_1(t,x)=M_1,\quad  t\leq 0,\; x\in\overline{\Omega}.
\end{gathered}
\label{e2.2}
\end{equation}
By Lemma \ref{lem1.1}, we have
$$
\lim_{t\to\infty}\overline{u}_1(t,x)=
\frac{a_1}{b_1-c_1^-}=\overline{\alpha}_1, \quad\hbox{uniformly
for } x\in\overline{\Omega}.
$$
So, for all sufficiently small $\varepsilon>0$, there exists a
$t_1>0$, such that
\begin{equation}
\max_{x\in\overline{\Omega}}\overline{u}_1(t,x)<
\overline{\alpha}_1+\varepsilon,\quad  \hbox{for } t>t_1. \label{e2.3}
\end{equation}
Define $\overline{v}_1(t,x)$ by
\begin{equation}
\begin{gathered}
\frac{\partial \overline{v}_1}{\partial t}= \Delta
\overline{v}_1+\overline{v}_1[a_2-b_2\overline{v}_1+
\int_{0}^{\infty}f_2^-(\tau)\overline{v}_1(t-\tau,x)d\tau+d_2\overline{u}_1],
\quad  t>t_1,\; x\in\Omega,\\
\frac{\partial \overline{v}_1}{\partial
n}=0, \quad t>t_1,\; x\in\partial\Omega,\\
\overline{v}_1(t,x)=M_2,\quad t\leq t_1,\; x\in\overline{\Omega}.
\end{gathered} \label{e2.4}
\end{equation}
It is easy to check that $(0,0)$ and
$(\overline{u}_1,\overline{v}_1)$ are the lower and upper
solutions of \eqref{e1.1}--\eqref{e1.3}. Therefore, Lemma \ref{lem2.1} implies
$$
0\leq u\leq  \overline{u}_1,\quad 0\leq v\leq \overline{v}_1.
$$
 From \eqref{e2.3} and \eqref{e2.4}, it follows that
$$
\frac{\partial \overline{v}_1}{\partial t}\leq \Delta
\overline{v}_1+\overline{v}_1[a_2-b_2\overline{v}_1+
\int_{0}^{\infty}f_2^-(\tau)\overline{v}_1(t-\tau,x)d\tau+d_2(\overline{\alpha}_1+\varepsilon)].
$$
By the comparison principle,
$$
\overline{v}_1\leq \overline{V}_1,
$$
where $\overline{V}_1$ is the solution of the problem
\begin{gather*}
\frac{\partial \overline{V}_1}{\partial t}=\Delta
\overline{V}_1+\overline{V}_1[a_2-b_2\overline{V}_1+
\int_{0}^{\infty}f_2^-(\tau)\overline{V}_1(t-\tau,x)d\tau
+d_2(\overline{\alpha}_1+\varepsilon)],\\
 t>t_1,\; x\in\Omega,\\
\frac{\partial \overline{V}_1}{\partial
n}=0, \quad t>t_1,\; x\in\partial\Omega,\\
\overline{V}_1(t,x)=M_2,\quad t\leq t_1,\; x\in\overline{\Omega}.
\end{gather*}
 From Lemma \ref{lem1.1},
$$
\lim_{t\to\infty}\overline{V}_1(t,x)=
\frac{a_2+d_2\overline{\alpha}_1}{b_2-c_2^-}+
\varepsilon\frac{d_2}{b_2-c_2^-}, \quad \hbox{uniformly for }
x\in\overline{\Omega}.
$$
So, for all sufficiently small $\varepsilon$, there exists a
$t_2>t_1$ such that
\begin{equation}
\max_{x\in\overline{\Omega}}\overline{v}_1(t,x)<
\overline{\beta}_1+\varepsilon,\quad \hbox{for } t>t_2, \label{e2.5}
\end{equation}
where
$\overline{\beta}_1=(a_2+d_2\overline{\alpha}_1)/(b_2-c_2^-)$.
Define $\underline{u}_1$ by
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial \underline{u}_1}{\partial t}
&= \Delta \underline{u}_1+\underline{u}_1\Big[a_1-b_1\underline{u}_1
+\int_{0}^{\infty}f_1^-(\tau)\underline{u}_1(t-\tau,x)d\tau\\
&\quad -\int_{0}^{\infty}f_1^+(\tau)\overline{u}_1(t-\tau,x)d\tau
-d_1\overline{v}_1(t-r_2,x)\Big],\quad t>t_2,\; x\in\Omega,
\end{aligned}
\\
\frac{\partial \underline{u}_1}{\partial n}=0,\quad  t>t_2,\;
x\in\partial\Omega,
\\
\underline{u}_1(t,x)=\frac{1}{2}u(t,x), \quad
(t,x)\in(-\infty,t_2]\times\overline{\Omega}.
\end{gathered}\label{e2.6}
\end{equation}
 From \eqref{e2.5} and \eqref{e2.6}, for $t>t_2$, $x\in\Omega$ we have
$$
\frac{\partial \underline{u}_1}{\partial t}\geq \Delta
\underline{u}_1+\underline{u}_1[a_1-b_1\underline{u}_1
+\int_{0}^{\infty}f_1^-(\tau)\underline{u}_1(t-\tau,x)d\tau
-c_1^+(\overline{\alpha}_1+\varepsilon)
-d_1(\overline{\beta}_1+\varepsilon)].
$$
By the comparison principle,
$$
\underline{u}_1\geq \underline{U}_1, \quad t>t_2,\; x\in\Omega,
$$
where $\underline{U}_1$ is defined by
\begin{gather*}
\begin{aligned}
\frac{\partial \underline{U}_1}{\partial t}
&= \Delta \underline{U}_1+\underline{U}_1[a_1-b_1\underline{U}_1
+\int_{0}^{\infty}f_1^-(\tau)\underline{U}_1(t-\tau,x)d\tau \\
&\quad -c_1^+(\overline{\alpha}_1+\varepsilon)
-d_1(\overline{\beta}_1+\varepsilon)], \quad t>t_2,\; x\in\Omega,
\end{aligned} \\
\frac{\partial \underline{U}_1}{\partial n}=0,\quad
t>t_2, \; x\in\partial\Omega,\\
\underline{U}_1(t,x)=\frac{1}{2}u(t,x), \quad
(t,x)\in(-\infty,t_2]\times\overline{\Omega}.
\end{gather*}
By \eqref{e1.5} with $\varepsilon$ sufficiently small,
$$
a_1-c_1^+(\overline{\alpha}_1+\varepsilon)-d_1(\overline{\beta}_1
+\varepsilon)>0.
$$
Thus from Lemma \ref{lem1.1}, we have
$$
\lim_{t\to\infty}\underline{U}_1(t,x)=
\frac{a_1-c_1^+\overline{\alpha}_1-d_1\overline{\beta}_1}{b_1-c_1^-}-
\varepsilon\frac{c_1^++d_1}{b_1-c_1^-} , \quad\hbox{uniformly for }
x\in\overline{\Omega}.
$$
Hence for any sufficiently small $\varepsilon>0$, there exists a
$t_3>t_2$ such that
\begin{equation}
\min_{x\in\overline{\Omega}}\underline{u}_1(t,x)>\underline{\alpha}_1
-\varepsilon,\quad t>t_3,
\label{e2.7}
\end{equation}
where
$\underline{\alpha}_1=
(a_1-c_1^+\overline{\alpha}_1-d_1\overline{\beta}_1)/(b_1-c_1^-)$.
Define $\underline{v}_1$ by
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial \underline{v}_1}{\partial t}
&= \Delta \underline{v}_1+\underline{v}_1[a_2-b_2\underline{v}_1
+\int_{0}^{\infty}f_2^-(\tau)\underline{v}_1(t-\tau,x)d\tau\\
&\quad -\int_{0}^{\infty}f_2^+(\tau)\overline{v}_1(t-\tau,x)d\tau
+d_2\underline{u}_1(t-r_1,x)],\quad t>t_3, x\in\Omega,
\end{aligned}\\
\frac{\partial \underline{v}_1}{\partial
n}=0,\quad  t>t_3,\; x\in\partial\Omega,\\
\underline{v}_1(t,x)=\frac{1}{2}v(t,x), \quad
(t,x)\in(-\infty,t_3]\times\overline{\Omega}.
\end{gathered}
\label{e2.8}
\end{equation}
It is easy to check that $(\overline{u}_1,\overline{v}_1)$ and
$(\underline{u}_1,\underline{v}_1)$ are the upper and lower
solutions of \eqref{e1.1}--\eqref{e1.3}, and from Lemma \ref{lem2.1} we get
$$
\underline{u}_1\leq u\leq\overline{u}_1,\quad
\underline{v}_1\leq v\leq\overline{v}_1.
$$
 From \eqref{e2.5}, \eqref{e2.7} and \eqref{e2.8}, we have
$$
\frac{\partial \underline{v}_1}{\partial t}\geq \Delta
\underline{v}_1+\underline{v}_1[a_2-b_2\underline{v}_1
+\int_{0}^{\infty}f_2^-(\tau)\underline{v}_1(t-\tau,x)d\tau
-c_2^+(\overline{\beta}_1+\varepsilon)
+d_2(\underline{\alpha}_1-\varepsilon)].
$$
By the comparison principle,
$$
\underline{v}_1\geq \underline{V}_1, \quad t>t_3,\; x\in\Omega,
$$
where $\underline{V}_1$ is defined by
\begin{gather*}
\begin{aligned}
\frac{\partial \underline{V}_1}{\partial t}
&= \Delta \underline{V}_1+\underline{V}_1[a_2-b_2\underline{V}_1
+\int_{0}^{\infty}f_2^-(\tau)\underline{V}_1(t-\tau,x)d\tau\\
&\quad -c_2^+(\overline{\beta}_1+\varepsilon)
+d_2(\underline{\alpha}_1-\varepsilon)], \quad t>t_3,\; x\in\Omega,
\end{aligned}\\
\frac{\partial \underline{V}_1}{\partial n}=0,\quad
 t>t_3,\; x\in\partial\Omega,\\
\underline{V}_1(t,x)=\frac{1}{2}v(t,x), \quad
(t,x)\in(-\infty,t_3]\times\overline{\Omega}.
\end{gather*}
Note that from \eqref{e1.5},
$$
a_2-c_2^+(\overline{\beta}_1+\varepsilon)+d_2(\underline{\alpha}_1
-\varepsilon)>0.
$$
for sufficiently small $\varepsilon$. From Lemma \ref{lem1.1}, we get
$$
\lim_{t\to\infty}\underline{V}_1(t,x)=
\frac{a_2-c_2^+\overline{\beta}_1+d_2\underline{\alpha}_1}{b_2-c_2^-}-
\varepsilon\frac{c_2^++d_2}{b_2-c_2^-} ,
$$
uniformly for $x\in\overline{\Omega}$.
So for any sufficiently small $\varepsilon$, there exists a
$t_4>t_3$ such that
\begin{equation}
\min_{x\in\overline{\Omega}}\underline{v}_1(t,x)>\underline{\beta}_1-\varepsilon,
\quad t>t_4,
\label{e2.9}
\end{equation}
where
$\underline{\beta}_1=\frac{a_2-c_2^+\overline{\beta}_1+d_2\underline{\alpha}_1}
{b_2-c_2^-}$.
Hence for all sufficiently small $\varepsilon$, we can conclude
\begin{equation}
0<\underline{\alpha}_1\leq
\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}u(t,x)\leq
\limsup_{t\to\infty}\max_{x\in\overline{\Omega}}u(t,x)\leq\overline{\alpha}_1,
\label{e2.10}
\end{equation}
and
\begin{equation}
0<\underline{\beta}_1\leq
\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}v(t,x)\leq
\limsup_{t\to\infty}\max_{x\in\overline{\Omega}}v(t,x)\leq\overline{\beta}_1.
\label{e2.11}
\end{equation}
Define $\overline{u}_2$ by
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial \overline{u}_2}{\partial t}
&= \Delta \overline{u}_2+\overline{u}_2[a_1-b_1\overline{u}_2
+\int_{0}^{\infty}f_1^-(\tau)\overline{u}_2(t-\tau,x)d\tau\\
&\quad -\int_{0}^{\infty}f_1^+(\tau)\underline{u}_1(t-\tau,x)d\tau
-d_1\underline{v}_1(t-r_2,x)],& t>t_4, x\in\Omega,
\end{aligned}\\
\frac{\partial \overline{u}_2}{\partial
n}=0,\quad  t>t_4,\; x\in\partial\Omega,\\
\overline{u}_2(t,x)=M_1, \quad (t,x)\in(-\infty,t_4]\times\overline{\Omega}.
\end{gathered}
\label{e2.12}
\end{equation}
 From \eqref{e2.7}, \eqref{e2.9} and \eqref{e2.12}, for $t>t_4$, we have
$$
\frac{\partial \overline{u}_2}{\partial t}\leq \Delta
\overline{u}_2+\overline{u}_2[a_1-b_1\overline{u}_2
+\int_{0}^{\infty}f_1^-(\tau)\overline{u}_2(t-\tau,x)d\tau
-c_1^+(\underline{\alpha}_1-\varepsilon)
-d_1(\underline{\beta}_1-\varepsilon)].
$$
By the comparison principle, we get
$\overline{u}_2\leq\overline{U}_1$, $t>t_4$, where
$\overline{U}_1$ is defined by
\begin{gather*}
\begin{aligned}
\frac{\partial \overline{U}_1}{\partial t}
&\leq \Delta \overline{U}_1+\overline{U}_1[a_1-b_1\overline{U}_1
 +\int_{0}^{\infty}f_1^-(\tau)\overline{U}_1(t-\tau,x)d\tau\\
&\quad -c_1^+(\underline{\alpha}_1-\varepsilon)
-d_1(\underline{\beta}_1-\varepsilon)],\quad
 t>t_4,\; x\in\Omega,
 \end{aligned}\\
\frac{\partial \overline{U}_1}{\partial n}=0,\quad
 t>t_4,\; x\in\partial\Omega,\\
\overline{U}_1(t,x)=K_1, \quad
(t,x)\in(-\infty,t_4]\times\overline{\Omega}.
\end{gather*}
 For sufficiently small $\varepsilon$, It is easy to show that
$$
a_1-c_1^+(\underline{\alpha}_1-\varepsilon)-d_1(\underline{\beta}_1
-\varepsilon)>0.
$$
Thus, from lemma \ref{lem1.1}, we have
$$
\lim_{t\to\infty}\overline{U}_1(t,x)=
\frac{a_1-c_1^+\underline{\alpha}_1-d_1\underline{\beta}_1}{b_1-c_1^-}+
\varepsilon\frac{c_1^++d_1}{b_1-c_1^-} , \quad \hbox{uniformly for }
x\in\overline{\Omega}.
$$
Hence, for any sufficiently small $\varepsilon>0$, there exists a
$t_5>t_4$ such that
\begin{equation}
\max_{x\in\overline{\Omega}}\overline{u}_2(t,x)
<\overline{\alpha}_2+\varepsilon,\quad t>t_5,
\label{e2.13}
\end{equation}
where
$\overline{\alpha}_2=\frac{a_1-c_1^+\underline{\alpha}_1
-d_1\underline{\beta}_1} {b_1-c_1^-}$.
\par
Define $\overline{v}_2$ by
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial \overline{v}_2}{\partial t}
&= \Delta \overline{v}_2+\overline{v}_2[a_2-b_2\overline{v}_2
+\int_{0}^{\infty}f_2^-(\tau)\overline{v}_2(t-\tau,x)d\tau\\
&\quad -\int_{0}^{\infty}f_2^+(\tau)\underline{v}_1(t-\tau,x)d\tau
+d_2\overline{u}_2(t-r_1,x)],& t>t_5, x\in\Omega,
\end{aligned}\\
\frac{\partial \overline{v}_2}{\partial n}=0,\quad t>t_5,\;
x\in\partial\Omega,\\
\overline{v}_2(t,x)=M_2, \quad
(t,x)\in(-\infty,t_5]\times\overline{\Omega}.
\end{gathered}
\label{e2.14}
\end{equation}
It is easy to check that $(\underline{u}_1,\underline{v}_1)$ and
$(\overline{u}_2,\overline{v}_2)$ are the lower and upper
solutions of \eqref{e1.1}--\eqref{e1.3}, and thus from Lemma \ref{lem2.1}, we get
$$
\underline{u}_1\leq u\leq\overline{u}_2,\quad
\underline{v}_1\leq v\leq\overline{v}_2
$$
 From \eqref{e2.9}, \eqref{e2.13} and \eqref{e2.14}, for $t>t_5$, we have
$$
\frac{\partial \overline{v}_2}{\partial t}\leq \Delta
\overline{v}_2+\overline{v}_2[a_2-b_2\overline{v}_2
+\int_{0}^{\infty}f_2^-(\tau)\overline{v}_2(t-\tau,x)d\tau
-c_2^+(\underline{\beta}_1-\varepsilon)
+d_2(\underline{\alpha}_2+\varepsilon)].
$$
By the comparison principle, we get
$\overline{v}_2\leq\overline{V}_2$, $t>t_5$, where
$\overline{V}_2$ is defined by
\begin{gather*}
\begin{aligned}
\frac{\partial \overline{V}_2}{\partial t}
&=\Delta \overline{V}_2+\overline{V}_2[a_2-b_2\overline{V}_2
+\int_{0}^{\infty}f_2^-(\tau)\overline{V}_2(t-\tau,x)d\tau\\
&\quad -c_2^+(\underline{\beta}_1-\varepsilon)
+d_2(\overline{\alpha}_2+\varepsilon)],\quad
 t>t_5,\; x\in\Omega,
\end{aligned}\\
\frac{\partial \overline{V}_2}{\partial
n}=0,\quad t>t_5,\; x\in\partial\Omega,\\
\overline{V}_2(t,x)=M_2, \quad
(t,x)\in(-\infty,t_5]\times\overline{\Omega}.
\end{gather*}
 For sufficiently small $\varepsilon$, it is easy to show that
$$
a_2-c_2^+(\underline{\beta}_1-\varepsilon)-d_2(\overline{\alpha}_2
+\varepsilon)>0.
$$
Thus from lemma \ref{lem1.1}, we have
$$
\lim_{t\to\infty}\overline{V}_2(t,x)=
\frac{a_2-c_2^+\underline{\beta}_1+d_2\overline{\alpha}_2}{b_2-c_2^-}+
\varepsilon\frac{c_2^++d_2}{b_2-c_2^-} , \quad \hbox{uniformly for }
x\in\overline{\Omega}.
$$
Hence for any sufficiently small $\varepsilon>0$, there exists
$t_6>t_5$ such that
\begin{equation}
\max_{x\in\overline{\Omega}}\overline{v}_2(t,x)<\overline{\beta}_2
+\varepsilon,\quad t>t_6,
\label{e2.15}
\end{equation}
where
$\overline{\beta}_2=\frac{a_2-c_2^+\underline{\beta}_1+d_2\overline{\alpha}_2}
{b_2-c_2^-}$.
Define $\underline{u}_2$ by
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial \underline{u}_2}{\partial t}
&= \Delta
\underline{u}_2+\underline{u}_2[a_1-b_1\underline{u}_2
+\int_{0}^{\infty}f_1^-(\tau)\underline{u}_2(t-\tau,x)d\tau\\
&\quad -\int_{0}^{\infty}f_1^+(\tau)\overline{u}_2(t-\tau,x)d\tau
-d_1\overline{v}_2(t-r_2,x)],\quad  t>t_6, x\in\Omega,
\end{aligned}\\
\frac{\partial \underline{u}_2}{\partial
n}=0,\quad  t>t_6,\; x\in\partial\Omega,\\
\underline{u}_2(t,x)=\frac{1}{2}u(t,x), \quad
(t,x)\in(-\infty,t_6]\times\overline{\Omega}.
\end{gathered}
\label{e2.16}
\end{equation}
 From \eqref{e2.13}, \eqref{e2.15} and \eqref{e2.16},
 for $t>t_6, x\in\Omega$ we get
$$
\frac{\partial \underline{u}_2}{\partial t}\geq \Delta
\underline{u}_2+\underline{u}_2[a_1-b_1\underline{u}_2
+\int_{0}^{\infty}f_1^-(\tau)\underline{u}_2(t-\tau,x)d\tau
-c_1^+(\overline{\alpha}_2+\varepsilon)
-d_1(\overline{\beta}_2+\varepsilon)].
$$
By the comparison principle,
$$
\underline{u}_2\geq \underline{U}_2, \quad t>t_6,\; x\in\Omega,
$$
where $\underline{U}_2$ is defined by
\begin{gather*}
\begin{aligned}
\frac{\partial \underline{U}_2}{\partial t}
&= \Delta \underline{U}_2+\underline{U}_2[a_1-b_1\underline{U}_2
+\int_{0}^{\infty}f_1^-(\tau)\underline{U}_2(t-\tau,x)d\tau\\
&\quad -c_1^+(\overline{\alpha}_2+\varepsilon)
-d_1(\overline{\beta}_2+\varepsilon)], \quad
 t>t_6,\; x\in\Omega,
\end{aligned}\\
\frac{\partial \underline{U}_2}{\partial
n}=0,\quad  t>t_6,\; x\in\partial\Omega,\\
\underline{U}_2(t,x)=\frac{1}{2}u(t,x), \quad
(t,x)\in(-\infty,t_6]\times\overline{\Omega}.
\end{gather*}
For sufficiently small $\varepsilon$, we can get
$$
a_1-c_1^+(\overline{\alpha}_2+\varepsilon)-d_1(\overline{\beta}_2
+\varepsilon)>0.
$$
Thus from Lemma \ref{lem1.1}, we have
$$
\lim_{t\to\infty}\underline{U}_2(t,x)=
\frac{a_1-c_1^+\overline{\alpha}_2-d_1\overline{\beta}_2}{b_1-c_1^-}-
\varepsilon\frac{c_1^++d_1}{b_1-c_1^-} , ~~\hbox{uniformly for}~
x\in\overline{\Omega}.
$$
Hence, for any sufficiently small $\varepsilon>0$, there exists a
$t_7>t_6$ such that
\begin{equation}
\min_{x\in\overline{\Omega}}\underline{u}_2(t,x)
>\underline{\alpha}_2-\varepsilon,\quad t>t_3,
\label{e2.17}
\end{equation}
where
$\underline{\alpha}_2=\frac{a_1-c_1^+\overline{\alpha}_2
-d_1\overline{\beta}_2}{b_1-c_1^-}$.
Define $\underline{v}_2$ by
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial \underline{v}_2}{\partial t}
&= \Delta \underline{v}_2+\underline{v}_2[a_2-b_2\underline{v}_2
+\int_{0}^{\infty}f_2^-(\tau)\underline{v}_2(t-\tau,x)d\tau\\
&\quad -\int_{0}^{\infty}f_2^+(\tau)\overline{v}_2(t-\tau,x)d\tau
+d_2\underline{u}_2(t-r_1,x)],\quad t>t_7, x\in\Omega,
\end{aligned}\\
\frac{\partial \underline{v}_2}{\partial
n}=0,\quad  t>t_7,\; x\in\partial\Omega,\\
\underline{v}_2(t,x)=\frac{1}{2}v(t,x), \quad
(t,x)\in(-\infty,t_7]\times\overline{\Omega}.
\end{gathered} \label{e2.18}
\end{equation}
It is easy to check that $(\overline{u}_2,\overline{v}_2)$ and
$(\underline{u}_2,\underline{v}_2)$ are the upper and lower
solutions of \eqref{e1.1}--\eqref{e1.3}, and thus from
Lemma \ref{lem2.1}, we get
$$
\underline{u}_2\leq u\leq\overline{u}_2,~~\underline{v}_2\leq
v\leq\overline{v}_2.
$$
 From \eqref{e2.15}, \eqref{e2.17} and \eqref{e2.18}, we have
$$
\frac{\partial \underline{v}_2}{\partial t}\geq \Delta
\underline{v}_2+\underline{v}_2[a_2-b_2\underline{v}_2
+\int_{0}^{\infty}f_2^-(\tau)\underline{v}_2(t-\tau,x)d\tau
-c_2^+(\overline{\beta}_2+\varepsilon)
+d_2(\underline{\alpha}_2-\varepsilon)].
$$
By the comparison principle,
$$
\underline{v}_2\geq \underline{V}_2, \quad t>t_7,\; x\in\Omega,
$$
where $\underline{V}_2$ is defined by
\begin{gather*}
\begin{aligned}
\frac{\partial \underline{V}_2}{\partial t}
&= \Delta \underline{V}_2+\underline{V}_2[a_2-b_2\underline{V}_2
+\int_{0}^{\infty}f_2^-(\tau)\underline{V}_2(t-\tau,x)d\tau\\
&\quad -c_2^+(\overline{\beta}_2+\varepsilon)
+d_2(\underline{\alpha}_2-\varepsilon)],\quad
 t>t_7,\; x\in\Omega,
\end{aligned}\\
\frac{\partial \underline{V}_2}{\partial
n}=0,\quad  t>t_7,\; x\in\partial\Omega,\\
\underline{V}_2(t,x)=\frac{1}{2}v(t,x), \quad
(t,x)\in(-\infty,t_7]\times\overline{\Omega}.
\end{gather*}
For sufficiently small $\varepsilon$, we can show that
$$
a_2-c_2^+(\overline{\beta}_2+\varepsilon)+d_2(\underline{\alpha}_2
-\varepsilon)>0.
$$
 From lemma \ref{lem1.1}, we get
$$
\lim_{t\to\infty}\underline{V}_2(t,x)=
\frac{a_2-c_2^+\overline{\beta}_2+d_2\underline{\alpha}_2}{b_2-c_2^-}-
\varepsilon\frac{c_2^++d_2}{b_2-c_2^-} , \quad \hbox{uniformly for }
x\in\overline{\Omega}.
$$
So for any sufficiently small $\varepsilon$, there exists a
$t_8>t_7$ such that
\begin{equation}
\min_{x\in\overline{\Omega}}\underline{v}_2(t,x)>\underline{\beta}_2
-\varepsilon,\quad t>t_8,
\label{e2.19}
\end{equation}
where
$\underline{\beta}_2=\frac{a_2-c_2^+\overline{\beta}_2+d_2\underline{\alpha}_2}
{b_2-c_2^-}$.
Therefore, for all sufficiently small $\varepsilon$, we can
conclude
\begin{gather}
\underline{\alpha}_2\leq
\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}u(t,x)\leq
\limsup_{t\to\infty}\max_{x\in\overline{\Omega}}u(t,x)\leq
\overline{\alpha}_2, \label{e2.20}
\\
\underline{\beta}_2\leq
\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}v(t,x)\leq
\limsup_{t\to\infty}\max_{x\in\overline{\Omega}}v(t,x)\leq\overline{\beta}_2.
\label{e2.21}
\end{gather}
It is obvious that
\begin{equation}
\underline{\alpha}_1\leq\underline{\alpha}_2\leq\overline{\alpha}_2
\leq\overline{\alpha}_1,
\quad
\underline{\beta}_1\leq\underline{\beta}_2\leq\overline{\beta}_2\leq\overline{\beta}_1
\end{equation}
Define the sequences $\underline{\alpha}_k$,
$\overline{\alpha}_k$, $\underline{\beta}_k$,
$\overline{\beta}_k(k\geq 1)$ as follows
\begin{equation}
\begin{gathered}
\overline{\alpha}_k=
\frac{a_1-c_1^+\underline{\alpha}_{k-1}-d_1\underline{\beta}_{k-1}}{b_1-c_1^-},
\overline{\beta}_k=\frac{a_2-c_2^+\underline{\beta}_{k-1}
+d_2\overline{\alpha}_k}{b_2-c_2^-},\\
\underline{\alpha}_k= \frac{a_1-c_1^+\overline{\alpha}_k
-d_1\overline{\beta}_k}{b_1-c_1^-},\quad
\underline{\beta}_k= \frac{a_2-c_2^+\overline{\beta}_k+d_2\underline
{\alpha}_k}{b_2-c_2^-}.
\end{gathered}
\label{e2.23}
\end{equation}
where $\underline{\alpha}_0=\underline{\beta}_0=0$.

\begin{lemma} \label{lem2.2}
For the above defined sequences, we have
\begin{equation}
[\underline{\alpha}_{k+1},\overline{\alpha}_{k+1}]\subseteq
[\underline{\alpha}_k,\overline{\alpha}_k], \quad
[\underline{\beta}_{k+1},\overline{\beta}_{k+1}]\subseteq
[\underline{\beta}_k,\overline{\beta}_k],\quad k\geq 1. \label{e2.24}
\end{equation}
\end{lemma}

For $k=1$, it has been shown that
$[\underline{\alpha}_{2},\overline{\alpha}_{2}]\subseteq
[\underline{\alpha}_1,\overline{\alpha}_1]$,
$[\underline{\beta}_{2},\overline{\beta}_{2}]\subseteq
[\underline{\beta}_1,\overline{\beta}_1]$. Using  induction, we
can easily complete the proof, and omit the detail.

Note that Lemma \ref{lem2.2} implies that the following limits exist:
$\lim_{k\to\infty}\underline{\alpha}_k=\underline{\alpha}$,
$\lim_{k\to\infty}\overline{\alpha}_k=\underline{\alpha}$,
$\lim_{k\to\infty}\underline{\beta}_k=\underline{\beta}$ and
$\lim_{k\to\infty}\overline{\beta}_k=\overline{\beta}$.
 By straightforward computation,
we can obtain
\begin{equation}
\begin{gathered}
\underline{\alpha}=\overline{\alpha}=\frac{a_1(b_2+c_2^+-c_2^-)-a_2d_1}
{(b_1+c_1^+-c_1^-)(b_2+c_2^+-c_2^-)+d_1d_2},\\
\underline{\beta}=\overline{\beta}=\frac{a_2(b_1+c_1^+-c_1^-)+a_1d_2}
{(b_1+c_1^+-c_1^-)(b_2+c_2^+-c_2^-)+d_1d_2}.
\end{gathered}
\label{e2.25}
\end{equation}

\begin{lemma} \label{lem2.3}
For the solutions of \eqref{e1.1}--\eqref{e1.3}, we have
\begin{gather}
\underline{\alpha}_k\leq
\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}u(t,x)\leq
\limsup_{t\to\infty}\max_{x\in\overline{\Omega}}v(t,x)\leq
\overline{\alpha}_k,\quad \hbox{for } k\geq 1, \label{e2.26}
\\
\underline{\beta}_k\leq
\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}u_2(t,x)\leq
\limsup_{t\to\infty}\max_{x\in\overline{\Omega}}u_2(t,x)
\leq\overline{\beta}_k,\quad \hbox{for } k\geq 1. \label{e2.27}
\end{gather}
\end{lemma}

We have shown that \eqref{e2.26} and \eqref{e2.27} are valid for
$k=1,2$. Using induction and repeating the above process, we can
complete the proof of Lemma \ref{lem2.3}.

Combining the above lemmas, we can complete the proof of the main
theorem.

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\end{document}