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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 121, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/121\hfil Minimal wave speeds of predator-prey systems]
{Minimal wave speeds of delayed dispersal predator-prey systems with stage structure}

\author[S. Pan \hfil EJDE-2016/121\hfilneg]
{Shuxia Pan}

\address{Shuxia Pan \newline
Department of Applied Mathematics,
Lanzhou University of Technology,
Lanzhou, Gansu 730050, China}
\email{shxpan@yeah.net}


\thanks{Submitted January 27, 2016. Published May 13, 2016.}
\subjclass[2010]{47G20, 35J50, 35B65}
\keywords{Contracting rectangle; upper and lower solutions; 
\hfill\break\indent asymptotic spreading}

\begin{abstract}
 This article  concerns the minimal wave speed of delayed predator-prey
 systems with nonlocal dispersal and stage structure.
 By the method of upper and lower solutions, we prove the existence of positive
 traveling wave  solutions. With the help of a contracting rectangle,
 we establish the limit behavior of traveling wave solutions.
 The nonexistence of traveling wave solutions is obtained using the theory
 of asymptotic spreading, and therefore, the minimal wave speed is obtained.
\end{abstract}

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


\section{Introduction}

In this article, we sutdy the  delayed predator-prey systems with nonlocal
dispersal and stage structure,
\begin{equation}
\begin{gathered}
\frac {\partial u(x,t)}{\partial t}=(D_1u)(x,t)+\alpha e^{-\gamma
\tau _1}u(x,t-\tau _1)-mu^2(x,t)-a_1u(x,t)v(x,t), \\
\frac {\partial v(x,t)}{\partial
t}=(D_2v)(x,t)+r_1v(x,t)+a_2u(x,t-\tau _2)v(x,t-\tau _2)-bv^2(x,t),
\end{gathered}  \label{1}
\end{equation}
in which all the parameters are positive and
\begin{gather*}
( D_1u) (x,t) =\int_{\mathbb{R}}J_1(x-y)[u(y,t)-u(x,t)]dy,\\
( D_2v) (x,t) =\int_{\mathbb{R}}J_2(x-y)[v(y,t)-v(x,t)]dy,
\end{gather*}
herein $J_1,J_2: \mathbb{R}\to \mathbb{R}^+$ are integrable functions
satisfying some conditions specified later.

Zhang et al \cite{zll} gave this model with state structure and nonlocal dispersal.
Moreover, they also established the existence of traveling wave solutions
connecting the trivial steady state with the positive equilibrium
if the wave speed is larger than a threshold. Such a traveling wave
solution could formulate the existence of
a transition zone moving from the steady state with no species to the steady state
with the coexistence of both species in mathematical biology \cite{zll}.

Although the existence of traveling wave solutions could reflect some phenomena
of population dynamics, the minimal wave speed depending on the existence and
nonexistence of traveling wave solutions is one of the most important thresholds
in mathematical biology. However, the estimation of minimal wave speed is not
an easy job. Before presenting our methods and results of minimal wave speeds,
we first recall some important results on the topic.
After the pioneer works of Fisher \cite{fisher} and Kolmogorov et al \cite{kpp}
on traveling wave solutions of reaction-diffusion equations, Aronson and Weinberger
 \cite{aron1} studied the asymptotic spreading of some population models
with reaction and diffusion, which describes some dynamical results different
from those in \cite{fisher,kpp}. Besides some results for reaction-diffusion systems,
integral equations and integrodifference equations,  there are some results
appealing to abstract monotone semiflows, see some results by Chen \cite{chen},
Fang and Zhao \cite{fz}, Liang and Zhao \cite{liangzhao},
Weinberger \cite{wein}, Weinberger et al \cite{weinberger}, Yi et al \cite{yi}
and a survey paper by Zhao \cite{zhao}.

However, for non-cooperation systems, it is difficult to obtain the minimal wave
speed due to the deficiency of comparison principle appealing to cooperative
systems. On the traveling wave solutions of predator-prey systems,
some classical conclusions were established about three decades ago by
Dunbar \cite{dun,dunbar,du1},  Gardner and Smoller \cite{g1}, Gardner
 and Jones \cite{g2}. After 2000, several investigators further studied
the problem by phase analysis, perturbation theory and fixed point theory,
we refer to some results by Huang et al \cite{hlr}, Huang \cite{huangw}, Hsu et al
 \cite{hsuyang}, Liang et al \cite{lww}, Lin \cite{lin}, Lin et al \cite{linweng}
 and Wang et al \cite{www}. In particular, Zhang et al \cite{zll} proved
the existence of traveling wave solutions by constructing upper and lower
solutions if the wave speed is larger than the threshold, and we shall
investigate the existence or nonexistence of traveling wave solutions when
the wave speed is the threshold and smaller than the threshold.

To further study the existence of traveling wave solutions, we shall
first present a result via generalized upper and lower solutions
motivated by Lin and Ruan \cite{linruan}. Then the asymptotic behavior will
be established by the idea of contracting rectangles \cite{linruan}
(see the definition of contracting rectangle for functional differential
equations by Smith \cite{smith}) as well as the theory of asymptotic spreading
given by Fang and Zhao \cite{fz}, Jin and Zhao \cite{jz}. Finally, the
nonexistence of traveling wave solutions is confirmed by combining the
asymptotic behavior of traveling wave solutions with the theory of asymptotic
 spreading.

\section{Main Results}

In this section, we shall present our main results. We first give some notation
 and definitions. In what follows, we use the standard partial ordering and
order intervals in $\mathbb{R}$ or $\mathbb{R}^2$, and apply
 $\| \cdot \|$ to denote the norm in $\mathbb{R}^2$. 
That is, for $u=(u_1,u_2)$ and $v=(v_1,v_2)$, we denote
$u\leq v$ if $u_i\leq v_i$ for $i=1,2$, and 
$u<v$ if $u\leq v$ but $u\neq v$.
In particular, we denote $u \ll v$ if $u\leq v$ but $u_i\neq v_i$ for $i=1,2$.
If $u\leq v$, we denote $(u,v]=\{w\in \mathbb{R}^2,u<w\leq v\}$, 
$[u,v)=\{w\in \mathbb{R}^2,u\leq w<v\}$, and 
$[u,v]=\{w\in \mathbb{R}^2,u\leq w\leq v\}$.

Define
\[
X=\{U:U\text{ is a bounded and uniformly
continuous function from }\mathbb{R}\text{ to }\mathbb{R}^2\},
\]
then $X$ is a Banach space equipped with the standard supremum norm.
If $\mathbf{a}, \mathbf{b}\in \mathbb{R}^2$ with $\mathbf{a}\le
\mathbf{b}$, then
\[
X_{[\mathbf{a},\mathbf{b}]}=\{U\in X: \mathbf{a}\le
U(\xi) \le \mathbf{b}, \xi \in \mathbb{R}\}.
\]
$C^1(\mathbb{R},\mathbb{R}^2)$ is defined by
\[
C^1(\mathbb{R},\mathbb{R}^2)=\{(u,v): (u,v),(u',v')\in X\}.
\]

By scaling, it suffices to investigate
\begin{equation}
\begin{gathered}
\frac {\partial u(x,t)}{\partial t}=(D_1u)(x,t)+\alpha e^{-\gamma
\tau
_1}[u(x,t-\tau _1)-u^2(x,t)-au(x,t)v(x,t)], \\
\frac {\partial v(x,t)}{\partial
t}=(D_2v)(x,t)+r_1[v(x,t)+bu(x,t-\tau _2)v(x,t-\tau _2)-v^2(x,t)],
\end{gathered}  \label{1-1}
\end{equation}

A \emph{traveling wave solution} of \eqref{1-1} is a special
translation invariant solution of the form
\[
(u(x,t),v(x,t))=(\phi(\xi),\psi(\xi)), \quad \xi= x+ct,
\]
in which $(\phi,\psi)\in C^1$ is the
profiles of the wave that propagate through the one-dimensional
spatial domain at a constant velocity $c>0$. If we substitute
$(\phi,\psi)$ into \eqref{1-1}, then
\begin{equation}\label{2}
\begin{gathered}
\begin{aligned}
c\phi'(\xi)&=\int_{\mathbb{R}}J_1(\xi -y)[\phi(y)-\phi(\xi)]dy\\
&\quad +\alpha e^{-\gamma \tau_1}[\phi(\xi-c\tau _1)-\phi^2(\xi)-a\phi(\xi)\psi(\xi)],\\
c\psi'(\xi)&=\int_{\mathbb{R}}J_2(\xi -y)[\psi(y) -\psi(\xi)]dy \\
&\quad +r_1[\psi(\xi)+b\phi(\xi-c\tau _2)\psi(\xi-c\tau _2)-\psi ^2(\xi)],
\end{aligned}
\end{gathered}
\end{equation}
where $\xi\in \mathbb{R}$.
Same as that in \cite{zll}, we also require that $(\phi,\psi)$ satisfy the
asymptotic boundary conditions
\begin{equation}\label{3}
\lim_{\xi\to -\infty}(\phi(\xi),\psi(\xi))=(0,0)\quad
\text{and}\quad
\lim_{\xi\to \infty} (\phi(\xi),\psi(\xi))=(k_1,k_2),
\end{equation}
where
\[
k_1=\frac{1-a}{1+ab}, \quad k_2=\frac{1+b}{1+ab}
\]
provided that
$a<1$ which will be imposed throughout this paper.

For $J_1,J_2$, we assume that
\begin{itemize}
\item[(J1)] $J_i:\mathbb{R}\to \mathbb{R}^+$ is  symmetric and
 Lebesgue measurable for each $i=1,2$;
\item[(J2)] for any $\lambda \in \mathbb{R}$,
$0< \int_{\mathbb{R}}J_i(y)e^{\lambda y} dy < \infty$, $i=1,2$.
\end{itemize}
Define
\begin{gather*}
\Delta_1(\lambda,c)=\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy-c\lambda +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1},\\
\Delta_2(\lambda,c)=\int^{+\infty}_{-\infty}J_2(y)(e^{\lambda
y}-1)dy-c\lambda +r_1.
\end{gather*}
Using (J1) and (J2), we have the following results.

\begin{lemma}\label{lem1}
There exists $c^*>0$ such that the following four items hold.
\begin{itemize}
\item[(i)] For any given $c>c^*$, $\Delta_1(\lambda,c)$  has two
distinct positive roots $\lambda_1(c)$ and $\lambda_3(c)$.
Moreover, assume that $0<\lambda_1(c) < \lambda_3(c) $ holds.  Then
\[
\Delta_1 (\lambda ,c)
\begin{cases}
>0 & \text{for }0<\lambda <\lambda _1(c) \text{  or  } \lambda >\lambda _3(c), \\
<0 & \text{for }\lambda _1(c)<\lambda <\lambda _3(c).
\end{cases}
\]

\item[(ii)] For any given $c>c^*$, $\Delta_2(\lambda,c)$  has two
distinct positive roots $\lambda_2(c)$ and $\lambda_4(c)$. Moreover,
assume that $0<\lambda_2 (c)< \lambda_4 (c)$ holds.  Then
\[
\Delta_2 (\lambda ,c)
\begin{cases}
>0 &\text{for }0<\lambda <\lambda _2(c) \text{  or  } \lambda >\lambda _{4}(c) \\
<0 &\text{for }\lambda _2(c)<\lambda <\lambda _{4}(c).
\end{cases}
\]

\item[(iii)] If $c=c^*$, then at least one of
$\Delta_1(\lambda,c)=0,\Delta_2(\lambda,c)=0$ has a double root.

\item[(iv)] If $c<c^*$, then at least one of $\Delta_1(\lambda,c)$
and  $\Delta_2(\lambda,c)$ has no real root.
\end{itemize}
\end{lemma}

\begin{remark} \rm
By Fang and Zhao \cite{fz}, Liang and Zhao \cite{liangzhao}, Jin
and Zhao \cite{jz}, $c^*$ can also be defined as
\begin{align*}
c^*&= \max\Big\{ \inf_{\lambda >0 }
\Big[ \frac{\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1}}{\lambda}\Big],\\
&\quad \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_2(y)(e^{\lambda
y}-1)dy +r_1}{\lambda}\Big]\Big\}.
\end{align*}
\end{remark}

Our main results reads as follows.

\begin{theorem}\label{thm1}
Assume that {\rm (J1)--(J2)} and
\begin{equation}
a(1+b)<1. \label{01}
\end{equation}
\begin{itemize}
\item[(1)] If $c>c^*$, then \eqref{2} has a positive solution satisfying \eqref{3}.

\item[(2)] If
\[
\inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1}}{\lambda}\Big]< c^*
\]
and $\overline{\lambda}_2\le \lambda_1(c^*)$, where $\overline{\lambda}_2$ is the positive root of $\Delta_2(\lambda,c^*)=0$,
then \eqref{2} has a positive solution satisfying \eqref{3}.
\item[(3)] If $c<c^*$, then \eqref{2} does not have a positive solution satisfying \eqref{3}.
\end{itemize}
\end{theorem}

\begin{remark} \rm
Zhang et al \cite[Condition (3.2)]{zll} proved  the existence of traveling wave
 solutions when
\begin{equation}
1-a >a(1+b).  \label{p}
\end{equation}
Clearly,  \eqref{01} is weaker than \eqref{p}.
\end{remark}

\begin{remark} \rm
Theorem \ref{thm1} implies that $c^*$ is the minimal wave speed. However,
 when $c=c^*$, the result needs further investigation.
\end{remark}


\section{Existence of traveling wave solutions: $c\ge c^*.$}

In this section, we shall prove the existence of positive solutions of \eqref{2}
by several lemmas throughout which (J1)-(J2) hold without further illustration.

\begin{lemma}\label{lem3}
Assume that there exist $\overline{\Phi}=(\overline{\phi},\overline{\psi})\in
C_{[0,M]}$ and $\underline{\Phi}=(\underline{\phi},\underline{\psi})\in
C_{[0,M]}$ with $M=(1,1+b)$ satisfy
\begin{itemize}
\item[(1)] for $\mathbb{T}=\{T_i\in \mathbb{R},i=1,\dots,m\}$,
$\overline{\Phi}'$ and $\underline{\Phi}'$ exist and are bounded for
$t\in \mathbb{R}\backslash \mathbb{T}$;

\item[(2)]  for $\xi\in \mathbb{R}\backslash \mathbb{T}$,
$\overline{\Phi}'$ and $\underline{\Phi}'$ satisfy
\begin{equation}\label{4}
\begin{aligned}
c\overline{\phi}'(\xi)
&\ge \int_{\mathbb{R}}J_1(\xi -y)
[\overline{\phi}(y)-\overline{\phi}(\xi)]dy\\
&\quad +\alpha e^{-\gamma
\tau_1}[\overline{\phi}(\xi-c\tau _1)-\overline{\phi}^2(\xi)
-a\overline{\phi}(\xi)\underline{\psi}(\xi)],\\
c\overline{\psi}'(\xi)
&\ge \int_{\mathbb{R}}J_2(\xi -y)
[\overline{\psi}(y)-\overline{\psi}(\xi)]dy \\
&\quad +r_1[\overline{\psi}(\xi)+b\overline{\phi}(\xi-c\tau _2)
\overline{\psi}(\xi-c\tau _2)-\overline{\psi} ^2(\xi)]
\end{aligned}
\end{equation}
and
\begin{equation}\label{5}
\begin{aligned}
c\underline{\phi}'(\xi)
&\le \int_{\mathbb{R}}J_1(\xi -y)
[\underline{\phi}(y)-\underline{\phi}(\xi)]dy\\
&\quad +\alpha e^{-\gamma \tau _1}[\underline{\phi}(\xi-c\tau _1)
-\underline{\phi}^2(\xi)-a\underline{\phi}(\xi)\overline{\psi}(\xi)],
\\
c\underline{\psi}'(\xi)
&\le \int_{\mathbb{R}}J_2(\xi -y)
[\underline{\psi}(y)-\underline{\psi}(\xi)]dy\\
&\quad+r[\underline{\psi}(\xi)+b\underline{\phi}(\xi-c\tau _2)
\underline{\psi}(\xi-c\tau _2)-\underline{\psi} ^2(\xi)].
\end{aligned}
\end{equation}
\end{itemize}
Then \eqref{2} has a positive solution $(\phi(\xi),\psi(\xi))$ satisfying
\[
(\underline{\phi}(\xi),\underline{\psi}(\xi))\le (\phi(\xi),\psi(\xi)) \le (\overline{\phi}(\xi),\overline{\psi}(\xi)), \xi \in \mathbb{R}.
\]
\end{lemma}

The proof of the above lemma is similar to that in Pan \cite[Theorem 3.2]{pan-k},
so we omit it here. Different from that in  Zhang et al \cite{zll}, we do not
require the asymptotic behavior when $\xi\to \infty$. Of course, this leads
to a weaker result than that in  \cite{zll}.

\begin{lemma}\label{lem40}
If  $c>c^*$, then \eqref{2} has a positive solution
$(\phi(\xi),\psi(\xi))\in C_{[0,M]}$.
\end{lemma}

\begin{proof}
Define continuous functions as follows
\begin{gather*}
\overline{\phi }(\xi )
=\min \{e^{\lambda _1(c)\xi },1\},\overline{\psi}(\xi )
=\min \{e^{\lambda _2(c)\xi }+p_1e^{\eta \lambda _2(c)\xi},1+b\}, \\
\underline{\phi }(\xi )=\max \{e^{\lambda _1(c)\xi }-p_2e^{\eta \lambda
_1(c)\xi },0\},\underline{\psi}(\xi )=\max \{e^{\lambda _2(c)\xi
}-p_3e^{\eta \lambda _3(c)\xi },0\},
\end{gather*}
where $p_1,p_2,p_3$ are constants which will be defined later, and
$\eta $ is a constant satisfying
\[
1<\eta <\min \big\{ \frac{\lambda _3(c)}{\lambda _1(c)},\frac{\lambda
_{4}(c)}{\lambda _2(c)}, 2\big\}.
\]
We shall prove that these functions satisfy \eqref{4} and \eqref{5}
by eight steps.
\smallskip

\noindent\textbf{Step 1.}
 If $\overline{\phi }(\xi )=e^{\lambda _1(c)\xi }<1$, then
\begin{align*}
&\int_{\mathbb{R}}J_1(\xi -y)[\overline{\phi }(y)-\overline{\phi }(\xi
)]dy+\alpha e^{-\gamma \tau _1}\big[ \overline{\phi }(\xi -c\tau _1)-%
\overline{\phi }^2(\xi )-a\overline{\phi }(\xi )\underline{\psi}(\xi )%
\big]  \\
&\leq \int_{\mathbb{R}}J_1(\xi -y)[\overline{\phi }(y)-\overline{\phi }%
(\xi )]dy+\alpha e^{-\gamma \tau _1}\overline{\phi }(\xi -c\tau _1) \\
&\leq \int_{\mathbb{R}}J_1(\xi -y)[e^{\lambda _1(c)y}-e^{\lambda
_1(c)\xi }]dy+\alpha e^{-\gamma \tau _1}e^{\lambda _1(c)(\xi -c\tau
_1)} \\
&=e^{\lambda _1(c)\xi }\big[ \int_{\mathbb{R}}J_1(y)[e^{\lambda
_1(c)y}-1]dy+\alpha e^{-\gamma \tau _1}e^{-\lambda _1(c)c\tau _1}%
\big]  \\
&=c\lambda _1(c)e^{\lambda _1(c)\xi } \\
&=c\overline{\phi }'(\xi ).
\end{align*}
\smallskip

\noindent\textbf{Step 2.}
  If $\overline{\phi }(\xi )=1<e^{\lambda _1(c)\xi }$,
then
\begin{align*}
&\int_{\mathbb{R}}J_1(\xi -y)[\overline{\phi }(y)-\overline{\phi }(\xi
)]dy+\alpha e^{-\gamma \tau _1}\big[ \overline{\phi }(\xi -c\tau _1)-
\overline{\phi }^2(\xi )-a\overline{\phi }(\xi )\underline{\psi}(\xi )
\big]  \\
&\leq \int_{\mathbb{R}}J_1(\xi -y)[\overline{\phi }(y)-\overline{\phi }
(\xi )]dy+\alpha e^{-\gamma \tau _1}\big[ \overline{\phi }(\xi -c\tau
_1)-\overline{\phi }^2(\xi )\big]  \\
&\leq \alpha e^{-\gamma \tau _1}\big[ \overline{\phi }(\xi -c\tau _1)-%
\overline{\phi }^2(\xi )\big]  \\
&\leq 0
=c\overline{\phi }'(\xi ).
\end{align*}
\smallskip

\noindent\textbf{Step 3.}
  If $\overline{\psi}(\xi )=1+b<e^{\lambda _2(c)\xi }+p_1e^{\eta \lambda
_2(c)\xi }$, then
\begin{align*}
&\int_{\mathbb{R}}J_2(\xi -y)[\overline{\psi}(y)-\overline{\psi}(\xi
)]dy+r_1\big[ \overline{\psi}(\xi )+b\overline{\phi }(\xi -c\tau _2)%
\overline{\psi}(\xi -c\tau _2)-\overline{\psi}^2(\xi )\big]  \\
&\leq r_1\big[ \overline{\psi}(\xi )+b\overline{\phi }(\xi -c\tau _2)%
\overline{\psi}(\xi -c\tau _2)-\overline{\psi}^2(\xi )\big]  \\
&\leq r_1\big[ \overline{\psi}(\xi )+b\overline{\psi}(\xi -c\tau _2)-%
\overline{\psi}^2(\xi )\big]  \\
&\leq 0=c\overline{\psi}'(\xi ).
\end{align*}
\smallskip

\noindent\textbf{Step 4.}
 If $\overline{\psi}(\xi )=e^{\lambda _2(c)\xi }+p_1e^{\eta \lambda
_2(c)\xi }<1+b$, then
\[
\xi <\frac{\ln \frac{1+b}{p_1}}{\eta \lambda _2(c)}
\]
and
\begin{align*}
&\int_{\mathbb{R}}J_2(\xi -y)[\overline{\psi}(y)-\overline{\psi}(\xi
)]dy+r_1\big[ \overline{\psi}(\xi )+b\overline{\phi }(\xi -c\tau _2)%
\overline{\psi}(\xi -c\tau _2)-\overline{\psi}^2(\xi )\big]  \\
&\leq \int_{\mathbb{R}}J_2(\xi -y)[e^{\lambda _2(c)y}+p_1e^{\eta
\lambda _2(c)y}-e^{\lambda _2(c)\xi }-p_1e^{\eta \lambda _2(c)\xi
}]dy \\
&\quad +r_1\Big[ \big[ e^{\lambda _2(c)\xi }+p_1e^{\eta \lambda
_2(c)\xi }\big] +be^{\lambda _1(c)\xi }\big[ e^{\lambda _2(c)\xi
}+p_1e^{\eta \lambda _2(c)\xi }\big] \\
&\quad -\Big( e^{\lambda _2(c)\xi
}+p_1e^{\eta \lambda _2(c)\xi }\Big) ^2\Big]  \\
&\leq \int_{\mathbb{R}}J_2(\xi -y)[e^{\lambda _2(c)y}+p_1e^{\eta
\lambda _2(c)y}-e^{\lambda _2(c)\xi }-p_1e^{\eta \lambda _2(c)\xi
}]dy \\
&\quad +r_1\big[ e^{\lambda _2(c)\xi }+p_1e^{\eta \lambda _2(c)\xi }%
\big] +be^{\lambda _1(c)\xi }\big[ e^{\lambda _2(c)\xi }+p_1e^{\eta
\lambda _2(c)\xi }\big]  \\
&=\big[ \Delta _2(\lambda _2(c),c)+c\lambda _2(c)\big] e^{\lambda
_2(c)\xi }+p_1\big[ \Delta _2(\eta \lambda _2(c),c)+c\eta \lambda
_2(c)\big] e^{\eta \lambda _2(c)\xi } \\
&\quad +be^{( \lambda _1(c)+\lambda _2(c)) \xi }+bp_1e^{(\eta
+1)\lambda _2(c)\xi } \\
&= c\lambda _2(c)e^{\lambda _2(c)\xi }+p_1c\eta \lambda _2(c)e^{\eta
\lambda _2(c)\xi } \\
&\quad +p_1\Delta _2(\eta \lambda _2(c),c)e^{\eta \lambda _2(c)\xi
}+be^{( \lambda _1(c)+\lambda _2(c)) \xi }+bp_1e^{(\eta
+1)\lambda _2(c)\xi } \\
&\leq c\lambda _2(c)e^{\lambda _2(c)\xi }+p_1c\eta \lambda
_2(c)e^{\eta \lambda _2(c)\xi } \\
&= c\overline{\psi}'(\xi )
\end{align*}
if
\begin{equation}
p_1\Delta _2(\eta \lambda _2(c),c)e^{\eta \lambda _2(c)\xi
}+be^{( \lambda _1(c)+\lambda _2(c)) \xi }+bp_1e^{(\eta
+1)\lambda _2(c)\xi }\leq 0.  \label{p1}
\end{equation}

Clearly, \eqref{p1} holds provided that
\begin{gather}
p_1\Delta _2(\eta \lambda _2(c),c)e^{\eta \lambda _2(c)\xi
}+2be^{( \lambda _1(c)+\lambda _2(c)) \xi }\leq 0,  \label{p2} \\
p_1\Delta _2(\eta \lambda _2(c),c)e^{\eta \lambda _2(c)\xi
}+2bp_1e^{(\eta +1)\lambda _2(c)\xi }\leq 0.  \label{p3}
\end{gather}
Note that $\eta \lambda _2(c)<\lambda _1(c)+\lambda _2(c)$, then
\eqref{p2} is true if
\[
p_1>1-\frac{2b}{\Delta _2(\eta \lambda _2(c),c)}>1.
\]%
At the same time, \eqref{p3} is true if $\xi <0$ and
\[
\lambda _2(c)\xi \leq \ln \frac{2b}{-\Delta _2(\eta \lambda _2(c),c)},
\]
which holds provided that
\[
\ln \frac{1+b}{p_1}\leq 0\leq \eta \ln \frac{2b}{-\Delta _2(\eta \lambda
_2(c),c)};
\]
that is,
\[
p_1\geq ( 1+b) \Big[ \Big( \frac{2b}{-\Delta _2(\eta
\lambda _2(c),c)}\Big) ^{\eta }+1\Big]
+1-\frac{2b}{\Delta _2(\eta \lambda _2(c),c)}:=\overline{p}_1.
\]
What we have done implies that if $p_1=\overline{p}_1$, then \eqref{4}
is true.
\smallskip

\noindent\textbf{Step 5.}
  If $\underline{\phi }(\xi )=e^{\lambda _1(c)\xi }-p_2e^{\eta \lambda_1(c)\xi }>0$,
 then
\begin{align*}
&\int_{\mathbb{R}}J_1(\xi -y)[\underline{\phi }(y)-\underline{\phi }(\xi
)]dy+\alpha e^{-\gamma \tau _1}\left[ \underline{\phi }(\xi -c\tau _1)-%
\underline{\phi }^2(\xi )-a\underline{\phi }(\xi )\overline{\psi}(\xi )%
\right]  \\
&\geq \int_{\mathbb{R}}J_1(\xi -y)[e^{\lambda _1(c)y}-p_2e^{\eta
\lambda _1(c)y}-e^{\lambda _1(c)\xi }+p_2e^{\eta \lambda _1(c)\xi
}]dy \\
&\quad +\alpha e^{-\gamma \tau _1}\left[ e^{\lambda _1(c)(\xi -c\tau
_1)}-( e^{\lambda _1(c)\xi }-p_2e^{\eta \lambda _1(c)\xi
}) ^2\right]  \\
&\quad -a\alpha e^{-\gamma \tau _1}( e^{\lambda _1(c)\xi }-p_2e^{\eta
\lambda _1(c)\xi }) ( e^{\lambda _2(c)\xi }+\overline{p}%
_1e^{\eta \lambda _2(c)\xi })  \\
&\geq c\lambda _1(c)e^{\lambda _1(c)\xi }-c\eta p_2\lambda
_1(c)e^{\eta \lambda _1(c)\xi }-p_2\Delta _1(\eta \lambda
_1(c),c)e^{\eta \lambda _1(c)\xi } \\
&\quad -\alpha e^{-\gamma \tau _1}e^{2\lambda _1(c)\xi }-a\alpha e^{-\gamma
\tau _1}e^{( \lambda _1(c)+\lambda _2(c)) \xi }-a\alpha
e^{-\gamma \tau _1}\overline{p}_1e^{(\lambda _1(c)+\eta \lambda
_2(c))\xi } \\
&\geq c\lambda _1(c)e^{\lambda _1(c)\xi }-c\eta p_2\lambda
_1(c)e^{\eta \lambda _1(c)\xi }=c\underline{\phi }'(\xi )
\end{align*}
provided that
\begin{align*}
&-p_2\Delta _1(\eta \lambda _1(c),c)e^{\eta \lambda _1(c)\xi }\\
&\geq e^{2\lambda _1(c)\xi }+a\alpha e^{-\gamma \tau _1}e^{( \lambda
_1(c)+\lambda _2(c)) \xi }+a\alpha e^{-\gamma \tau _1}\overline{p%
}_1e^{(\lambda _1(c)+\eta \lambda _2(c))\xi },
\end{align*}
which holds when
\[
p_2=\frac{1+a\alpha e^{-\gamma \tau _1}+a\alpha e^{-\gamma \tau _1}%
\overline{p}_1}{-\Delta _1(\eta \lambda _1(c),c)}+1>1.
\]
\smallskip

\noindent\textbf{Step 6.}
  If $\underline{\phi }(\xi )=0>e^{\lambda _1(c)\xi }-p_2e^{\eta \lambda
_1(c)\xi }$, then the result is clear.
\smallskip

\noindent\textbf{Step 7.}
 If $\underline{\psi}(\xi )=e^{\lambda _2(c)\xi }-p_3e^{\eta \lambda
_3(c)\xi }>0$, then
\begin{align*}
&\int_{\mathbb{R}}J_2(\xi -y)[\underline{\psi}(y)-\underline{\psi}(\xi)]dy
+r_1[ \underline{\psi}(\xi )+b\underline{\phi }(\xi -c\tau _2)
\underline{\psi}(\xi -c\tau _2)-\underline{\psi}^2(\xi )]  \\
&\geq \int_{\mathbb{R}}J_2(\xi -y)[\underline{\psi}(y)
 -\underline{\psi}(\xi )]dy+r_1[ \underline{\psi}(\xi )-\underline{\psi}^2(\xi )
]  \\
&\geq \int_{\mathbb{R}}J_2(\xi -y)[e^{\lambda _2(c)y}-p_3e^{\eta
\lambda _3(c)y}-e^{\lambda _2(c)\xi }+p_3e^{\eta \lambda _3(c)\xi
}]dy \\
&\quad +r_1[ e^{\lambda _2(c)\xi }-p_3e^{\eta \lambda _3(c)\xi }]
-r_1( e^{\lambda _2(c)\xi }-p_3e^{\eta \lambda _3(c)\xi }) ^2 \\
&\geq c\lambda _2(c)e^{\lambda _2(c)\xi }-p_3c\eta \lambda
_2(c)e^{\eta \lambda _2(c)\xi }-p_3\Delta _2(\eta \lambda
_2(c),c)e^{\eta \lambda _2(c)\xi }-r_1e^{2\lambda _2(c)\xi } \\
&\geq c\lambda _2(c)e^{\lambda _2(c)\xi }-p_3c\eta \lambda
_2(c)e^{\eta \lambda _2(c)\xi } \\
&=c\underline{\psi}'(\xi )
\end{align*}
provided that $p_3=\frac{r_1}{-\Delta _2(\eta \lambda _2(c),c)}+1$,
and so \eqref{5} holds.
\smallskip

\noindent\textbf{Step 8.}
  If $\underline{\psi}(\xi )=0>e^{\lambda _2(c)\xi }-p_3e^{\eta \lambda
_3(c)\xi }$, then the result is clear.

By Lemma \ref{lem3}, the proof is complete.
\end{proof}


By Carr and Chmaj \cite{cc}, Li et al. \cite{lisw} and Wu and Ruan \cite{wuruan},
we have the following result of scalar equations.

PAGE 7

\begin{lemma}\label{lem2}
Assume that
\[
\inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1}}{\lambda}\Big]
< \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_2(y)(e^{\lambda
y}-1)dy +r_1}{\lambda}\Big].
\]
Then when $c=c^*$,
the scalar equation
\begin{equation}\label{au}
c\psi'(\xi)=\int_{\mathbb{R}}J_2(\xi -y)[\psi(y)-\psi(\xi)]dy+r_1\psi(\xi)-\psi ^2(\xi),\xi\in \mathbb{R}
\end{equation}
has a strictly positive solution satisfying
\[
\lim_{\xi\to -\infty } \psi(\xi)=0,\quad
\lim_{\xi\to \infty } \psi(\xi)=r_1,\quad
\lim_{\xi\to -\infty } \frac{\psi(\xi)}{\xi e^{-\overline{\lambda}_2 \xi}}=-1.
\]
\end{lemma}


\begin{lemma}
Assume that
\begin{align*}
&\inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1}}{\lambda}\Big] \\
&< \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_2(y)(e^{\lambda
y}-1)dy +r_1}{\lambda}\Big]=c^*.
\end{align*}
If $\overline{\lambda}_2\le \lambda_1(c^*)$, then \eqref{2} with $c=c^*$ 
has a positive solution $(\phi(\xi),\psi(\xi))\in C_{[0,M]}$.
\end{lemma}

\begin{proof}
We shall construct continuous functions satisfying  \eqref{4} and \eqref{5}.
Let
\[
\overline{\phi}(\xi)=\min\{ e^{\lambda_1 (c) \xi}, 1\},\quad
\underline{\phi}(\xi)=\max\{ e^{\lambda_1 (c) \xi} 
-p_2 e^{\eta\lambda_1 (c) \xi}, 0\},
\]
where $p_2>1$ is a positive constants specified later and $\eta$ satisfies
\[
1< \eta \lambda_1 (c) < \min\{ \lambda_3 (c), \lambda_1 (c)
+ \overline{\lambda}_2 /4 , 3\lambda_1 (c) /2 \}.
\]
Define
\[
\underline{\psi}(\xi)=\widetilde{\psi}(\xi),
\]
where $\widetilde{\psi}(\xi)$ is the positive solution of \eqref{au} 
and satisfies Lemma \ref{lem2}. Further define
\[
\overline{\psi}(\xi)= \begin{cases}
1+b, & \xi \ge \xi_1,\\
(M- 2\xi)e^{\overline{\lambda}_2 \xi }, & \xi <\xi_1,
\end{cases}
\]
where $M>0$ is a constant clarified later. Clearly, if $M>1+1/b$ is large,
 then  $(M- 2\xi)e^{\overline{\lambda}_2 \xi }=1+b$ has two real roots, 
and here $\xi_1$ is the smaller root.

We now verify that these functions satisfy \eqref{4} and \eqref{5}. 
In particular, the inequalities about $\overline{\phi}(\xi), \underline{\psi}(\xi)$
 are clear. Moreover, if $\xi <\xi_1$, then
\[
b\phi(\xi-c\tau _2)\psi(\xi-c\tau _2)-\psi ^2(\xi)<0
\]
and the inequalities on $\underline{\psi}(\xi)$ is true.

Moreover $p_2>1$ such that
\[
e^{\lambda_1 (c) \xi} -p_2 e^{\eta\lambda_1 (c) \xi} >0
\]
implies that $\xi <\xi_1$ and
\[
0< (M- 2\xi)e^{\overline{\lambda}_2 \xi } < e^{\overline{\lambda}_2 \xi /2 }.
\]
Similar to that in Lemma \ref{lem40}, we see that
\[
\int_{\mathbb{R}}J_1(\xi -y)[\underline{\phi }(y)-\underline{\phi }(\xi
)]dy+\alpha e^{-\gamma \tau _1}\left[ \underline{\phi }(\xi -c\tau _1)-%
\underline{\phi }^2(\xi )-a\underline{\phi }(\xi )\overline{\psi}(\xi )%
\right] \ge c^* \underline{\phi }'(\xi )
\]
if $p_2>1$ is large enough. By Lemma \ref{lem3}, the result follows.
\end{proof}

\section{Nonexistence of traveling wave solutions: $c<  c^*$}

In this section, we shall prove that if $c< c^*$, then \eqref{2} does not 
have a positive solution satisfying \eqref{3}.
We first consider the following initial value problem by Fang and
 Zhao \cite{fz}, Jin and Zhao \cite{jz}
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac {\partial w(x,t)}{\partial t}
&=\int_{\mathbb{R}}J(x-y)[w(y,t)-w(x,t)] \\
&\quad +dw(x,t-\tau )+fw(x,t)-g w^2(x,t),\quad t>0,
\end{aligned} \\
w(x,s)=\varphi (x,s),\quad s\in [-\tau, 0],
\end{gathered}  \label{i}
\end{equation}
where $x\in \mathbb{R}, \tau \ge 0, d\ge 0,  d+f >0, g>0$ and 
$\varphi (x,s)$ is bounded and uniformly continuous in 
$(x,s)\in \mathbb{R} \times [-\tau ,0]$.
\begin{lemma}\label{lem4}
Assume that $J$ satisfies (J1) and (J2). Define
\[
c_0= \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J(y)(e^{\lambda
y}-1)dy +d e^{-\lambda c\tau} +f}{\lambda}\Big] .
\]
If $\varphi (x,s)$ has nonempty support for each $s\in [-\tau, 0]$, then
\[
\lim_{t\to\infty}\sup_{|x|\le ct}w(x,t)
=\lim_{t\to\infty}\inf_{|x|\le ct}w(x,t)=\frac{g}{d+f}
\]
for each $c<c_0$.
\end{lemma}

\begin{lemma}\label{lem5}
Assume that $J$ satisfies {\rm (J1)} and {\rm (J2)}. If $\overline{w}(x,t)$ satisfies
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac {\partial \overline{w}(x,t)}{\partial t}
&\ge \int_{\mathbb{R}}J(x-y)[\overline{w}(y,t)-\overline{w}(x,t)]\\
&\quad +d\overline{w}(x,t-\tau )+f\overline{w}(x,t)-g \overline{w}^2(x,t),\quad t>0,
\end{aligned} \\
\overline{w}(x,s)\ge \varphi (x,s),\quad s\in [-\tau, 0]
\end{gathered}  \label{u-i}
\end{equation}
for $x\in \mathbb{R}$, then
\[
\overline{w}(x,t) \ge w(x,t),\quad  x\in \mathbb{R},\; t>0.
\]
\end{lemma}


By analysis, we have the following result.

\begin{lemma}\label{lem6}
Assume that $(\phi (\xi), \psi (\xi))$ is a bounded positive solution of \eqref{2}. 
If  $\phi (\xi_1)>0$ for some $\xi_1 \in \mathbb{R}$, then $\phi (\xi) >0$ 
for all $\xi \in \mathbb{R}$, if  $\psi (\xi_2)>0 $  for some
 $\xi_2 \in \mathbb{R}$, then $\psi (\xi) >0$ for all $\xi \in \mathbb{R}$. 
Moreover, $\phi (\xi), \psi (\xi)$  satisfy
\[
0\le \phi (\xi)\le 1, 0\le \psi (\xi)\le 1+b, \xi\in \mathbb{R}.
\]
\end{lemma}

\begin{theorem}
If $c<c^*$, then \eqref{2} does not have a positive solution satisfying \eqref{3}.
\end{theorem}

\begin{proof}
Were the statement false, then for some $c_1 <c^*$, \eqref{2} has a positive 
solution satisfying \eqref{3}. That is, there exist $(\phi(\xi), \psi(\xi))$ 
satisfying
\begin{equation}\label{2-0}
\begin{aligned}
c_1\phi'(\xi)
&=\int_{\mathbb{R}}J_1(\xi -y)[\phi(y)-\phi(\xi)]dy\\
&\quad +\alpha e^{-\gamma \tau_1}[\phi(\xi-c\tau _1)
 -\phi^2(\xi)-a\phi(\xi)\psi(\xi)],
\\
c_1\psi'(\xi)
&=\int_{\mathbb{R}}J_2(\xi -y)[\psi(y)-\psi(\xi)]dy \\
&\quad +r_1[\psi(\xi)+b\phi(\xi-c\tau _2)\psi(\xi-c\tau _2)-\psi ^2(\xi)],
\end{aligned}
\end{equation}
and
\begin{equation}\label{3-0}
\lim_{\xi\to -\infty}(\phi(\xi),\psi(\xi))=(0,0),\quad 
\lim_{\xi\to \infty} (\phi(\xi),\psi(\xi))=(k_1,k_2).
\end{equation}
If
\[
c^*= \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1}}{\lambda}\Big],
\]
then there exists $\epsilon \in (0,\alpha e^{-\gamma \tau_1}) $ such that
\[
c_1 < \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_1(y)(e^{\lambda
y}-1)dy +\alpha e^{-\gamma\tau_1}e^{-\lambda c\tau_1} -2\epsilon}{\lambda}\Big]
=: c_2.
\]
By \eqref{3-0}, there exists $T\in \mathbb{R}$ such that
\[
a\alpha e^{-\gamma
\tau_1}\psi(\xi)< \epsilon, \quad \xi \le T,
\]
and so
\begin{align*}
&\alpha e^{-\gamma \tau _1}[\phi (\xi -c\tau _1)-\phi ^2(\xi )-a\phi
(\xi )\psi (\xi )] \\
&\geq \alpha e^{-\gamma \tau _1}\phi (\xi -c\tau _1)-\epsilon \phi (\xi
)-\alpha e^{-\gamma \tau _1}\phi ^2(\xi ),\quad \xi \leq T.
\end{align*}
If $\xi >T$, then \eqref{3-0} and Lemma \ref{lem6} imply that there exists 
$M>0$ such that
\[
a \alpha e^{-\gamma \tau_1} \phi(\xi)\psi(\xi) < M \phi^2(\xi).
\]
Therefore, $\psi(\xi)$ satisfies
\[
c_1\phi'(\xi)\ge \int_{\mathbb{R}}J_1(\xi -y)[\phi(y)-\phi(\xi)]dy+\alpha e^{-\gamma
\tau_1}\phi(\xi-c\tau _1)-\epsilon \phi(\xi)- (M+ \alpha e^{-\gamma
\tau_1})\phi^2(\xi)
\]
for all $\xi\in \mathbb{R}$. Since $\xi=x+c_1 t$, we have
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac {\partial u(x,t)}{\partial t}
&\ge (D_1u)(x,t)+\alpha e^{-\gamma \tau _1}u(x,t-\tau _1)-\epsilon u(x,t)\\
&\quad -(M+ \alpha e^{-\gamma \tau_1}) u^2(x,t)], \quad t>0,
\end{aligned} \\
u(x,-s)= \phi (x+c_1s), \quad s\in [-\tau_1, 0].
\end{gathered}  \label{u-1}
\end{equation}
By Lemmas \ref{lem4} and \ref{lem5}, we have
\[
\lim_{t\to \infty} \inf_{|x|\le c_2 t} u(x,t) 
\ge  \frac{e^{-\gamma\tau_1}-\epsilon}{M+ \alpha e^{-\gamma\tau_1}} >0.
\]
On the other hand, letting $-x=c_2t$, we have
\[
x+c_1t=(c_1-c_2)t \to -\infty , \quad t\to \infty
\]
and so
$u(-c_2t,t)=\phi (-c_2t+c_1t)\to 0$, as $t\to \infty$,
which is a contradiction.

If
\[
c^*= \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_2(y)
(e^{\lambda y}-1)dy +r_1}{\lambda}\Big],
\]
then there exists $\iota \in (0,1)$ such that
\[
c_1 < \inf_{\lambda >0 }\Big[ \frac{\int^{+\infty}_{-\infty}J_2(y)(e^{\lambda
y}-1)dy +r_1(1-\iota)}{\lambda}\Big] := c_3.
\]
At the same time, $\psi (x+c_1 t)= v(x,t)$ satisfies
\begin{equation}
\begin{gathered}
\frac {\partial v(x,t)}{\partial
t}\ge (D_2v)(x,t)+r_1[v(x,t)-v^2(x,t)],\quad t>0,\\
v(x,0)= \psi (x),
\end{gathered}  \label{u-00}
\end{equation}
where $x\in \mathbb{R}$.
By Lemmas \ref{lem4} and \ref{lem5}, we see that
\[
\lim_{t\to \infty} \inf_{|x|\le c_3 t} v(x,t) \ge 1  >0.
\]
On the other hand, letting $-x=c_3t$, we have
\[
x+c_1t=(c_1-c_3)t \to -\infty , t\to \infty
\]
and so
$v(-c_3 t,t)=\phi (-c_3 t+c_1t)\to 0$, as $t\to \infty$,
which is also a contradiction. The proof is complete.
\end{proof}

By the same process as above, we can obtain the following result.

\begin{corollary} \label{coro4.5}
If $c<c^*$, then \eqref{2} does not have a positive solution satisfying
\[
\lim_{\xi\to -\infty}(\phi(\xi),\psi(\xi))=(0,0), \quad
\liminf_{\xi\to \infty} (\phi(\xi),\psi(\xi))\gg (0,0).
\]
\end{corollary}

\section{Asymptotic behavior of traveling wave solutions}

In this section, we study the asymptotic behavior of the traveling wave 
solutions obtained in Section 3. The method is based on the idea 
of contracting rectangles, which was earlier used by Lin and Ruan \cite{linruan} 
in studying the asymptotic behavior of traveling wave solutions of delayed 
reaction-diffusion systems.
For $s\in [0,1]$, define
\begin{gather*}
\underline{a}(s) =sk_1+(1-s)(1-ab)(1-a)(1-\varepsilon _1), \\
\overline{a}(s) =sk_1+(1-s)(1-a)(1+\varepsilon _2), \\
\underline{b}(s) =sk_2+(1-s)(1-\varepsilon _3), \\
\overline{b}(s) = sk_2+(1-s)(1+b(1-a))(1+\varepsilon _{4}),
\end{gather*}
where $\varepsilon_1 , \varepsilon_2, \varepsilon_3, \varepsilon_4 \in (0,1)$ with
\begin{gather}
(1-ab)(1-a)\varepsilon _1 = 2a(1+b(1-a))\varepsilon _{4},  \label{e1} \\
(1+b(1-a))\varepsilon _{4} = 2b(1-a)\varepsilon _2,  \label{e2} \\
(1-a)\varepsilon _2  = 2a\varepsilon _3.  \label{e3}
\end{gather}

We now illustrate that 
$\varepsilon_1 , \varepsilon_2, \varepsilon_3, \varepsilon_4 \in (0,1)$ 
are admissible. Let $\overline{\varepsilon}_1=1$ and
\[
\overline{\varepsilon }_{4}=\frac{(1-ab)(1-a)}{2a(1+b(1-a))},\quad
\overline{\varepsilon }_2=\frac{(1+b(1-a))}{2b(1-a)}\overline{\varepsilon }_{4},\quad
\overline{\varepsilon }_3=\frac{1-a}{2a}\overline{\varepsilon }_2.
\]
For any $c>0$, then
\[
(\varepsilon_1 , \varepsilon_2, \varepsilon_3, \varepsilon_4)
= (c, c \overline{\varepsilon }_2, c \overline{\varepsilon }_3, 
 c \overline{\varepsilon }_{4} )
\]
satisfy \eqref{e1}-\eqref{e3}. Clearly, $\varepsilon_1 , \varepsilon_2, 
\varepsilon_3, \varepsilon_4 \in (0,1)$  if $c>0$ is small enough.

\begin{lemma} 
For each $s\in (0,1)$, we have
\begin{gather}
1-\underline{a}(s)-a\overline{b}(s) > 0,  \label{cc1} \\
1-\overline{a}(s)-a\underline{b}(s) < 0,  \label{cc2} \\
1+b\underline{a}(s)-\underline{b}(s) > 0,  \label{cc3} \\
1+b\overline{a}(s)-\overline{b}(s) < 0.  \label{cc4}
\end{gather}
\end{lemma}

\begin{proof}
If $s\in (0,1)$, then
\begin{align*}
1-\underline{a}(s)-a\overline{b}(s) 
&= 1-sk_1-(1-s)(1-ab)(1-a)(1-\varepsilon_1) \\
&\quad -ask_2-a(1-s)(1+b(1-a))(1+\varepsilon _{4}) \\
&= (1-s)[ 1-(1-ab)(1-a)(1-\varepsilon _1)-a(1+b(1-a))
 (1+\varepsilon_{4})]  \\
&> (1-s)[ (1-ab)(1-a)\varepsilon _1-a(1+b(1-a))\varepsilon _{4}]\\
&=(1-s)a(1+b(1-a))\varepsilon _{4} 
>0,
\end{align*}
by $(1-ab)(1-a)\varepsilon _1 =2a(1+b(1-a))\varepsilon _{4}$.
The above inequality implies \eqref{cc1}.

Since $2a\varepsilon _3 =(1-a)\varepsilon _2$, we have
\begin{align*}
1-\overline{a}(s)-a\underline{b}(s) 
&=1-sk_1-(1-s)(1-a)(1+\varepsilon _2)-ask_2-a(1-s)(1-\varepsilon _3)
\\
&=(1-s)[1-(1-a)(1+\varepsilon _2)-a(1-\varepsilon _3)] \\
&=(1-s)\left[ a\varepsilon _3-(1-a)\varepsilon _2\right]  \\
&=- (1-s)a\varepsilon _3 < 0,
\end{align*}
which implies \eqref{cc2}.

Moreover, \eqref{cc3} holds since
\begin{align*}
1+b\underline{a}(s)-\underline{b}(s) 
&=1-sk_2-(1-s)(1-\varepsilon _3)  +bsk_1 \\
&\quad +b(1-s)(1-ab)(1-a)(1-\varepsilon _1) \\
&=(1-s)(\varepsilon _3+b(1-ab)(1-a)(1-\varepsilon _1))
>0.
\end{align*}

Note that $2b(1-a)\varepsilon _2 =(1+b(1-a))\varepsilon _{4}$. Then
\begin{align*}
1+b\overline{a}(s)-\overline{b}(s) 
&= 1-sk_2-(1-s)(1+b(1-a))(1+\varepsilon _{4})+sbk_1\\
&\quad +b(1-s)(1-a)(1+\varepsilon _2) \\
&= (1-s)(1-(1+b(1-a))(1+\varepsilon _{4})+b(1-a)(1+\varepsilon _2)) \\
&= (1-s)(b(1-a)\varepsilon _2-(1+b(1-a))\varepsilon _{4}) \\
&= -(1-s)b(1-a)\varepsilon _2 
< 0,
\end{align*}
which implies \eqref{cc4}. The proof is complete.
\end{proof}

\begin{lemma}\label{lem7}
If $(\phi (\xi), \psi (\xi))$ is a positive solution of \eqref{2}, then
\begin{gather*}
(1-ab)(1-a) \leq \liminf_{\xi \to \infty }\phi (\xi )\leq
\limsup_{\xi \to \infty }\phi (\xi )\leq 1-a, \\
1 \leq \liminf_{\xi \to \infty }\psi (\xi )\leq \limsup_{\xi
\to \infty }\psi (\xi )\leq 1+b(1-a).
\end{gather*}
\end{lemma}

\begin{proof}
By the definition, $\psi (x+c t)= v(x,t)$ satisfies
\begin{equation}
\begin{gathered}
\frac {\partial v(x,t)}{\partial
t}\ge (D_2v)(x,t)+r_1[v(x,t)-v^2(x,t)],\quad t>0,\\
v(x,0)= \psi (x),
\end{gathered}  \label{u-00b}
\end{equation}
where $x\in \mathbb{R}$.
By Lemmas \ref{lem4} and \ref{lem5}, we have
\[
\liminf_{t\to \infty}  v(0,t) \ge 1  >0.
\]
which implies 
\[
\liminf_{\xi \to \infty }\psi (\xi )\ge 1.
\]

Let $\beta >0$.
Note that $\phi (\xi )$ and $\psi (\xi )$ are bounded and positive, then
there exists $\beta >0$ such that
\[
\beta \phi (s)-\phi (s)\int_{\mathbb{R}}J_1(y)dy+\alpha e^{-\gamma \tau _1}[\phi
(s-c\tau _1)-\phi ^2(s)-a\phi (s)\psi (s)]
\]
is monotone increasing in $\phi (s)$ and
\[
\beta \psi (s)-\psi (s)\int_{\mathbb{R}} J_2(y)dy+r_1[\psi (s)-\psi ^2(s)+b\phi
(s-c\tau _2)\psi (s-c\tau _2)]
\]
is monotone increasing in $\psi (s)$. 
Moreover, $\phi (\xi )$ and $\psi (\xi)$ also satisfy
\begin{align*}
\phi (\xi ) 
&=\frac{1}{c}\int_{-\infty }^{\xi }e^{-\frac{\beta (\xi -s)}{c}%
}\int_{\mathbb{R}}J_1(s-y)[\phi (y)-\phi (s)]dyds \\
&\quad +\int_{-\infty }^{\xi }\big\{ \beta \phi (s)+\alpha e^{-\gamma \tau
_1}[\phi (s-c\tau _1)-\phi ^2(s)-a\phi (s)\psi (s)]\big\} ds, 
\\
\psi (\xi ) 
&=\frac{1}{c}\int_{-\infty }^{\xi }e^{-\frac{\beta (\xi -s)}{c}%
}\int_{\mathbb{R}}J_2(s-y)[\psi (y)-\psi (s)]dyds \\
&\quad +\int_{-\infty }^{\xi }\big\{ \beta \psi (s)+r_1[\psi (s)-\psi
^2(s)+b\phi (s-c\tau _2)\psi (s-c\tau _2)]\big\} ds.
\end{align*}
Since $\liminf_{\xi \to \infty }\psi (\xi )\geq 1$. Applying 
Fatou's lemma in the integral equation of $\phi (\xi )$, we see that
\[
\alpha e^{-\gamma \tau _1}\Big[\limsup_{\xi \to \infty }\phi (\xi
)-( \limsup_{\xi \to \infty }\phi (\xi )) ^2-a\limsup_{\xi \to \infty }
\phi (\xi )\Big]\geq 0;
\]
then the boundedness of $\limsup_{\xi \to \infty }\phi (\xi )$
indicates that
\[
\limsup_{\xi \to \infty }\phi (\xi )\leq 1-a.
\]
Further applying Fatou's lemma in the integral equation of $\psi (\xi )$,
we see that
$\limsup_{\xi \to \infty }\psi (\xi )\geq 1$, and
\[
\limsup_{\xi \to \infty }\psi (\xi )-\Big( \limsup_{\xi
\to \infty }\psi (\xi )\Big) ^2+b(1-a)\limsup_{\xi \to
\infty }\psi (\xi )\geq 0,
\]
which leads to
\[
\limsup_{\xi \to \infty }\psi (\xi )\leq 1+b(1-a).
\]
Returning to the integral equation of $\phi (\xi)$, we see that
\[
\liminf_{\xi \to \infty }\phi (\xi )\ge (1-ab)(1-a)
\]
if $\liminf_{\xi \to \infty }\phi (\xi )>0$. 
In fact, by Lemma \ref{lem6}, we see that $\phi (\xi)$ satisfies
\[
c\phi'(\xi)\ge \int_{\mathbb{R}}J_1(\xi -y)[\phi(y)-\phi(\xi)]dy
 +\alpha e^{-\gamma \tau_1}[\phi(\xi-c\tau _1)-a(1+b)\phi(\xi)-\phi^2(\xi)].
\]
That is, $u(x,t)=\phi(x+ct)$ satisfies
\begin{gather*}
\frac {\partial u(x,t)}{\partial t}=(D_1u)(x,t)+\alpha e^{-\gamma
\tau _1}[u(x,t-\tau _1)-a(1+b)u(x,t)-u^2(x,t)],\quad t>0, \\
u(x,s)=\phi (x+cs), \quad s\in [-\tau_1, 0],
\end{gather*}
where $x\in \mathbb{R}$. By Lemmas \ref{lem4} and \ref{lem5}, we see that
\[
\liminf_{t\to \infty} u(0, t)\ge 1-a(1+b)>0,
\]
which implies that
\[
\liminf_{\xi \to \infty }\phi (\xi )> 1-a(1+b)>0
\]
by the invariant form of traveling wave solutions. The proof is complete.
\end{proof}

\begin{lemma}
If $(\phi (\xi), \psi (\xi))$ is a positive solution of \eqref{2}, then
\begin{align*}
\lim_{\xi\to \infty} (\phi(\xi),\psi(\xi))=(k_1,k_2).
\end{align*}
\end{lemma}

\begin{proof}
By Lemma \ref{lem7}, we see that there exists $s_1 \in (0,1)$ such that
\begin{equation}
\begin{gathered}
\underline{a}(s)\leq \liminf_{\xi \to \infty }\phi (\xi )\leq
\limsup_{\xi \to \infty }\phi (\xi )\leq \overline{a}(s),\\
\underline{b}(s)\leq \liminf_{\xi \to \infty }\psi (\xi )\leq \limsup_{\xi
\to \infty }\psi (\xi )\leq \overline{b}(s)
\end{gathered}\label{ss4}
\end{equation}
for all $s\le s_1$ since
$\underline{a}(s),\overline{a}(s), \underline{b}(s), \overline{b}(s)$
are continuous and monotone, and
\begin{gather*}
\underline{a}(0) <  (1-ab)(1-a)\leq 1-a <\overline{a}(0), \\
\underline{b}(0) <  1 <  1+b(1-a)< \overline{b}(0).
\end{gather*}
Define
\[
s_0=\sup_{s\in (0,1]}\{ \eqref{ss4} \text{ hold}\}.
\]
Then $s_0$ is well defined.

If $s_0=1$, then the result is true. We now assume that $s_0<1$. 
Without loss of generality, we suppose that
\begin{equation}
\begin{gathered}
\underline{a}(s_0)= \liminf_{\xi \to \infty }\phi (\xi ), \\
\underline{a}(s_0)= \liminf_{\xi \to \infty }\phi (\xi )
\leq \limsup_{\xi \to \infty }\phi (\xi )\leq \overline{a}(s_0),\\
\underline{b}(s_0)\leq \liminf_{\xi \to \infty }\psi (\xi )
\leq \limsup_{\xi \to \infty }\psi (\xi )\leq \overline{b}(s_0).
\end{gathered}\label{ss5}
\end{equation}

By the definition of $\liminf$, there exist a sequence $\{\xi_m\}$ such that
\begin{gather*}
\lim_{m\to \infty} \xi_m =\infty,\quad
\lim_{m\to\infty}\phi (\xi_m)=\underline{a}(s_0), \quad
\lim_{m\to\infty}\phi '(\xi_m)=0,\\
\liminf_{m\to \infty}\left[\int_{\mathbb{R}}J_1(\xi_m -y)
[\phi(y)-\phi(\xi_m)]dy\right]\ge 0.
\end{gather*}
At the same time, \eqref{cc1} implies that
\begin{align*}
&\liminf_{m\to \infty }\alpha e^{-\gamma \tau _1}[\phi (\xi
_{m}-c\tau _1)-\phi ^2(\xi _{m})-a\phi (\xi _{m})\psi (\xi _{m})] \\
&\geq \alpha e^{-\gamma \tau _1}[\underline{a}(s_{0})
-\underline{a}^2(s_{0})-a\underline{a}(s_{0})\overline{b}(s_{0})] \\
&= \alpha e^{-\gamma \tau _1}\underline{a}(s_{0})[1
-\underline{a}(s_{0})-a\overline{b}(s_{0})]
> 0\,.
\end{align*}
This is a contradiction, so $s_0=1$. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The author would like to express her sincere
gratitude to the anonymous referee for his/her careful reading.
 This work is supported by NSF of China (11461040, 11471149).

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\end{document}