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


\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 197, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/197\hfil Traveling waves in a diffusive model ]
{Traveling waves in a diffusive predator-prey model with
general functional response}

\author[Z. Xu, P. Weng \hfil EJDE-2012/??\hfilneg]
{Zhaoquan Xu, Peixuan Weng}  % in alphabetical order

\address{Zhaoquan Xu \newline
School of Mathematics, South China Normal University\\
Guangzhou 510631, China}
\email{xiaozhao20042008@163.com}

\address{Peixuan Weng \newline
School of Mathematics, South China Normal University \\
Guangzhou 510631, China}
\email{wengpx@scnu.edu.cn, Tel: 0086-20-85213533}

\thanks{Submitted February 20, 2012. Published November 10, 2012.}
\subjclass[2000]{34C12, 34C37, 35K57, 92D25}
\keywords{Traveling waves; general functional response; \hfill\break\indent
  Schauder's fixed theorem}

\begin{abstract}
 This article concerns the existence of traveling waves in a diffusive
 predator-prey model with general functional response.
 By applying the Schauder fixed theorem, we establish existence results
 of traveling wave solutions. The results are then applied to the
 predator-prey model with Holling type-II response.
 Our results indicate that there is a transition zone moving from the
 state with no species to the coexistence state of both species.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

Dynamical relations among species can be very
complicated.  Due to their presence in
natural environments, various types of predator-prey models have been
widely studied, for example, see
\cite{2a}--\cite{15a}. Nonlinear
reaction-diffusion equations describe the dynamical
relationship between predator and prey. In many situations,  
traveling waves determine the
long term behavior of predator and prey.


Fundamental and important predator-prey models with diffusion are given by:
\begin{equation}\label{eq1.1}
\begin{gathered}
u_{t}= D_1u_{xx}+Au(1-\frac{u}{K})-UW,\\
w_{t}= D_2w_{xx}-Cw +Duw ;\\
\end{gathered}
\end{equation}
\begin{equation}\label{eq1.2}
\begin{gathered}
u_{t}= D_1u_{xx}+Au(1-\frac{u}{K})-B \frac{uw}{1+Eu},\\
w_{t}= D_2w_{xx}-Cw +D \frac{uw}{1+Eu};\\
\end{gathered}
\end{equation}
and
\begin{equation}\label{eq1.3}
\begin{gathered}
u_{t}= D_1u_{xx}+Au(1-\frac{u}{K})-B\frac{u^2w}{1+Eu^2},\\
w_{t}= D_2w_{xx}-Cw +D\frac{u^2w}{1+Eu^2};\\
\end{gathered}
\end{equation}
where $u(t,x)$,  $w(t,x)$ are the density functions of prey and
predator, respectively; $D_1>0$ and $D_2>0$ represent the
diffusive rates; $A$ is the
growth factor for the prey species, $K>0$ is the carrying capacity
of prey species, $C>0$ is the death rate for the predator in the absence of
prey. For more details about the biological meaning of the
parameters, we refer the readers to
\cite{4a,7a,9a,13a}.

System \eqref{eq1.1} is the familiar Lotka-Volterra model (with Holling type-I functional response) and
the systems \eqref{eq1.2}, \eqref{eq1.3} have the Holling type-II
and  Holling type-III functional response, respectively. In  \cite{2a,1a,3a},
Dunbar obtained the existence of several kinds of
traveling wave solutions for diffusive predator-prey systems with type I and type II functional responses ($D_1=0$ \cite{2a} and
$D_1\not=0$ \cite{1a} for  \eqref{eq1.1}, $D_1=0$ \cite{3a} for \eqref{eq1.2}). He
considered the existence of small
amplitude periodic traveling waves, and ``heteroclinic traveling waves'' that correspond to heteroclinic orbits
connecting two equilibria (point-to-point) or an equilibrium and a periodic orbit (point-to-periodic).  The methods used by Dunbar include the invariant manifold theory, the shooting method, Hopf bifurcation analysis, and LaSalle's invariance principle.  Huang, Lu \& Ruan
\cite{9a} extended the work  in \cite{1a} to $\mathbb{R}^4$ ($D_1\not=0$ for  \eqref{eq1.2})   using Dunbar's method in \cite{1a}. An interesting question is whether those results can be extended to a system with type III functional
response. Recently, Li \& Wu \cite{12a} proved the existence of
traveling waves  in a diffusive predator-prey system \eqref{eq1.3} with $D_1=0$ by employing a method similar
to that used in \cite{2a, 1a}. We emphasize that in \cite{9a} and \cite{12a}  only heteroclinic orbits
connecting equilibrium-to-equilibrium (point-to-point) are considered.

The shooting method used by Dunbar is based on a variant of Wazewski's theorem
\cite{2a,1a,3a}.  In Dunbar and  Wazewski set $\mathbb{W}$,
 there is an orbit starting at  the unstable manifold of an equilibrium  
that stays in $\mathbb{W}$ in the future. However,
the Wazewski set $\mathbb{W}$ constructed in \cite{2a,1a,3a} is unbounded. 
To ensure the boundedness of the orbit, several
additional  lemmas were proved to rule out
the possibility that the constructed orbit may escape to infinity.
The use of unbounded sets
$\mathbb{W}$ in $\mathbb{R}^3$ or $\mathbb{R}^4$ makes the argument long and hard to read. In a recent work, Lin, Weng \& Wu\cite{LWW2011}
 constructed a simple bounded Wazewski set
$\mathbb{W}$ and use the original Wazewski's theorem to simplify the proof of the existence of
heteroclinic traveling waves  connecting two equilibria related to the following  predator-prey system with Sigmoidal response function:
\begin{equation}\label{equ2}
\begin{gathered}
\frac{\partial u}{\partial t}=ru(1-\frac{u}{K})-\frac{u^2}{a_1+b_1u+u^2}v\\
\frac{\partial v}{\partial t}=D_2\frac{\partial^2 v}{\partial
x^2}+v\Big(\frac{\alpha u^2}{a_1+b_1u+u^2}-e\Big).
\end{gathered}
\end{equation}

Liang, Weng and Wu  \cite{LWW2008} considered the delayed  diffusive 
predator-prey system
\begin{equation}\label{*}
\begin{gathered}
 \frac{\partial u(t,x)}{\partial t}=Au(t,x)\Big
(1-\frac{u(t,x)}{K}\Big )
-B\frac{u(t-\tau,x) w(t,x)}{1+Eu(t-\tau,x)},\\
 \frac{\partial w(t,x)}{\partial t}=D_2
\frac{\partial^2w(t,x)}{\partial
x^2}-Cw(t,x)+D\frac{u(t-\tau,x)w(t,x)}{1+Eu(t-\tau,x)},
\end{gathered}
\end{equation}
where $\tau\geq 0$ measures the retarded response of growth
for the prey species or the time for the prey species taken from
birth to maturity.  They proved
the existence of small amplitude periodic traveling wave solutions
of \eqref{*} for small $\tau> 0$. Furthermore, they developed a new
method for combining the singular
limit argument and the singular perturbation technique to establish
the existence of the point-to-periodic traveling wave solutions
 for \eqref{*} with small
delay $\tau >0$, and also proved the existence of point-to-point 
traveling wave solutions
 for the any given $\tau>0$.


It is very interesting to develop simpler methods to treat the problem of
traveling waves for diffusive predator-prey systems.
Recently, Lin et al  \cite{La} studied the existence of point-to-point traveling wave
solutions of the following  Lotka-Volterra
system:
\begin{equation}\label{eq1.6}
\begin{gathered}
\frac{\partial u_1(t,x)}{\partial t}= d_1\Delta
u_1(x,t)+r_1u_1[1-a_{11}u_1(t-\tau_1,x)-a_{12}u_2(t-\tau_2,x)],
\\
\frac{\partial u_2(t,x)}{\partial t}= d_2\Delta
u_2(x,t)+r_2u_2[1+a_{21}u_1(t-\tau_3,x)-a_{22}u_2(t-\tau_4,x)]
\end{gathered}
\end{equation}
by introducing the mixed quasi-monotone condition(MQM) and the
exponentially mixed quasi-monotone condition(EMQM).

Motivated by the work in \cite{La},
in the present article, we  consider the existence of traveling
wave solutions of the following predator-prey system with general
functional response:
\begin{equation}\label{eq1.6b}
\begin{gathered}
\frac{\partial u}{\partial t}= D_1\frac{\partial^2
u}{\partial x^2}+h_1(u)
-f(u)w,\\
\frac{\partial w}{\partial t}= D_2\frac{\partial^2
w}{\partial x^2}+h_2(w)
+\mu f(u)w,\\
\end{gathered}
\end{equation}
where $D_1>0$, $D_2>0$ are the diffusive rates of the prey and
predator, respectively. Also $h_1(u)$ denotes the growth function of
prey which is a positive function within  the maximal carrying
capacity of the prey, and  $h_2(v)$ denotes the growth function of
predator. If the predator only depends on the prey given in
\eqref{eq1.6}, then $h_2(v)$ is a negative function.  The function $f$ denotes the predator response
function.

For the functions $h_1$, $h_2$ and $f$, we make assumptions as follows.
\begin{itemize}
\item[(H1)] There exist two positive numbers $u_0$,$w_0$
such that $h_1(u_0)-f(u_0)w_0=0$, $h_2(w_0)+\mu
f(u_0)w_0=0$, and $f(0)=h_1(0)=h_2(0)=0$;

\item[(H2)] $f$, $h_1$ and $h_2$ are Lipschitz continuous functions
on any compact interval;

\item[(H3)] $f$ is nondecreasing on $[0,+\infty)$.
\end{itemize}


\begin{remark}\label{R1.1} \rm
The hypothesis (H1) guarantees that $(0,0)$ is a steady state for  the
system \eqref{eq1.6} and  it has another  positive
steady state $(u_0,w_0)$. Moreover
all the response functions in
\eqref{eq1.1}-\eqref{eq1.3} satisfy the conditions (H2) and (H3).
 On the other hand, (H1) and (H3) imply that $f(u)\geq 0$ for $u\in \mathbb{R}$.
\end{remark}


The rest of the paper is organized as follows. In section 2, some
preliminaries  are given. In section 3,  we show the main results
on the existence of traveling wave solutions for
\eqref{eq1.6}. In the last section,  as an application of our main results,
we shall establish the existence
results of traveling wave solutions for system
\begin{equation}\label{eq1.7}
\begin{gathered}
u_{t}= D_1u_{xx}+\alpha u(\beta-u)-wf(u),\\
w_{t}= D_2w_{xx}+\gamma w(\delta-w)+\mu wf(u),
\end{gathered}
\end{equation}
with $f(u)=\frac{u}{1+u}$.


\section{Preliminaries}

In this article, we adopt the usual notation for the standard partial
ordering in $\mathbb{R}^2$; i.e.,
if $a_1\leq a_2$ and $b_1\leq b_2$, we say that 
$(a_1,b_1)\leq (a_2,b_2)$.
 Let $|\cdot|$ denote the Euclidean norm
in $\mathbb{R}^2$ and $\|\cdot\|$ denote the supermum norm in space
$C(\mathbb{R},\mathbb{R}^2)$.

A traveling wave solution of \eqref{eq1.6} is a solution
with the form $(u(t,x),w(t,x))=(\varphi(x+ct),\psi(x+ct))$, where
$(\phi,\psi)\in C^2(\mathbb{R},\mathbb{R}^2)$ is the wave profile 
which propagates at a constant velocity $c>0$.

We study traveling wave solutions of \eqref{eq1.6} that connect $(0,0)$ and
$(u_0,v_0)$. By substituting such $(\varphi,\psi)$ into
\eqref{eq1.6} and replacing $x+ct$ by $t$, we know that
$(\varphi,\psi)$ satisfy the wave profile system
\begin{equation}\label{eq2.1}
\begin{gathered}
c\varphi'(t)=D_1\varphi''(t)+h_1(\varphi(t))
-f(\varphi(t))\psi(t),\\
c\psi'(t)=D_2\psi''(t)+h_2(\psi(t))
+\mu f(\varphi(t))\psi(t)
\end{gathered}
\end{equation}
accompanied with  asymptotic boundary  conditions
\begin{equation} \label{eq2.2}
\lim_{t\to-\infty}(\varphi(t),\psi(t))=(0,0),\quad
\lim_{t\to+\infty}(\varphi(t),\psi(t))=(u_0,w_0).
\end{equation}
If, for some $c>0$, system \eqref{eq2.1} has a solution
$(\varphi(t),\psi(t))$ satisfying the asymptotic boundary
conditions \eqref{eq2.2}, then $(u(t,x),v(t,x))=(\varphi(x+ct),\psi(x+ct))$
is the traveling wave solution of system \eqref{eq1.6}.

Let
\begin{equation*}
C_{[0,K]}(\mathbb{R},\mathbb{R}^2)=\{(\varphi,\psi)\in
C(\mathbb{R},\mathbb{R}^2): 0\leq(\varphi,\psi)(t)\leq K\text{ for } t\in
\mathbb{R}\},
\end{equation*}
where $K=(k_1,k_2)$ is some constant vector such that
$(u_0,w_0)\leq (k_1,k_2)$.

For  $(\varphi,\psi)\in
C_{[0,K]}(\mathbb{R},\mathbb{R}^2)$, define the operator
$Q=(Q_1,Q_2):C_{[0,K]}(\mathbb{R},\mathbb{R}^2)\to
C(\mathbb{R},\mathbb{R}^2)$ by
\begin{equation}\label{eq2.4}
\begin{gathered}
Q_1(\varphi,\psi)(t)=d_1\varphi(t)
+h_1(\varphi(t)) -f(\varphi(t))\psi(t),\\
Q_2(\varphi,\psi)(t)=d_2\psi(t)
+h_2(\psi(t))+\mu f(\varphi(t))\psi(t),
\end{gathered}
\end{equation}
where $d_1=L_{h_1}+k_2L_{f}$, $d_2=L_{h_2}$, $L_{h_2}$
is the Lipschitz constant of $h_2$ on $[0,k_2]$ and $L_{f},
L_{h_1}$ are the Lipschitz constants of $f, h_1$ on $[0,k_1]$, respectively.
Hence,  \eqref{eq2.1} is equivalent to
\begin{equation}\label{eq2.4b}
\begin{gathered}
c\varphi'(t)=D_1\varphi''(t)-d_1\varphi(t)+Q_1(\varphi,\psi)(t),\\
c\psi'(t)=D_2\psi''(t)-d_2\psi(t)+Q_2(\varphi,\psi)(t)\,.
\end{gathered}
\end{equation}
Let
$$
r_{i1}=\frac{c-\sqrt{c^2+4D_id_i}}{2D_i},\quad
r_{i2}=\frac{c+\sqrt{c^2+4D_id_i}}{2D_i},\quad i=1,2.
$$
Clearly, we have  $r_{i1}<0<r_{i2}$ and
$$
D_ir^2_{ij}-cr_{ij}-d_i=0,\quad i,j=1,2.
$$
For $(\varphi,\psi)\in
C_{[0,K]}(\mathbb{R},\mathbb{R}^2)$, define an operator
$P=(P_1,P_2):C_{[0,K]}(\mathbb{R},\mathbb{R}^2)\to
C(\mathbb{R},\mathbb{R}^2)$ by
\begin{equation}\label{eq2.5}
\begin{gathered}
P_1(\varphi,\psi)(t)=\frac{1}{D_1(r_{12}-r_{11})}
\Big[\int^{t}_{-\infty}e^{r_{11}(t-s)}
+\int^{+\infty}_{t}e^{r_{12}(t-s)}\Big]
Q_1(\varphi,\psi)(s)ds,\\
P_2(\varphi,\psi)(t)=\frac{1}{D_2(r_{22}-r_{21})}
\Big[\int^{t}_{-\infty}e^{r_{21}(t-s)}
+\int^{+\infty}_{t}e^{r_{22}(t-s)}\Big]
Q_2(\varphi,\psi)(s)ds.
\end{gathered}
\end{equation}
Note that fixed points of $P$ are solutions to  \eqref{eq2.1}.
 Therefore, to prove the existence of traveling wave solutions of
\eqref{eq1.3} connecting $(0,0)$ and $(u_0,w_0 )$, it is
sufficient to consider  fixed points  of $P$  that  satisfy
the asymptotic boundary conditions \eqref{eq2.2}.


\section{Main results}

 We first give the definition of upper-lower solutions of \eqref{eq2.1}
which is crucial in proving our main results.


\begin{definition} \label{def3.1}\rm
  A pair of continuous functions
$\overline{\Phi}=(\overline{\varphi},\overline{\psi})$ and
$\underline{\Phi}=(\underline{\varphi},\underline{\psi})\in
C_{[0,K]}(\mathbb{R},\mathbb{R}^2)$ is called an upper solution
and a lower solution of \eqref{eq2.1}, respectively, if
$(\overline{\varphi}'(t),{\overline{\psi}}'(t))$,
$(\underline{\varphi}''(t),\underline{\psi}''(t))$
exist and bounded on $R\setminus\Upsilon$ and satisfy
\begin{gather*}
D_1\overline{\varphi}''(t)-c\overline{\varphi}'(t)+h_1(\overline{\varphi}(t))
-f(\overline{\varphi}(t))\underline{\psi}(t)\leq0,\\
D_2{\overline{\psi}}''(t)-c\overline{\psi}'(t)+h_2(\overline{\psi}(t))
+\mu f(\overline{\varphi}(t))\overline{\psi}(t)\leq0,
\end{gather*}
and
\begin{gather*}
D_1\underline{\varphi}''(t)-c\underline{\varphi}'(t)+h_1(\underline{\varphi}(t))
-f(\underline{\varphi}(t))\overline{\psi}(t)\geq0,\\
D_2\underline{\psi}''(t)-c\underline{\psi}'(t)+h_2(\underline{\psi}(t))
+\mu f(\underline{\varphi}(t))\underline{\psi}(t)\geq0
\end{gather*}
for $ R\setminus\Upsilon$, where
$\Upsilon=\{t_1,t_2,\dots,t_{n}\}$, with $t_1<t_2<\dots<t_{n}$, is
a finite set of points.
\end{definition}

In what follows, we assume that \eqref{eq2.1} admits an upper
solution
$\overline{\Phi}=(\overline{\varphi},\overline{\psi})$
and a lower solution
$\underline{\Phi}=(\underline{\varphi},\underline{\psi})$
such that
\begin{itemize}
\item[(G1)] $(0,0)\leq(\underline{\varphi},\underline{\psi})(t)
\leq(\overline{\varphi},\overline{\psi})(t)\leq(k_1,k_2),\
t\in\mathbb{R}$;

\item[(G2)] $\lim_{t\to -\infty}(\overline{\varphi},\overline{\psi})(t)=(0,0)$,\\
 $\lim_{t\to+\infty}(\underline{\varphi},\underline{\psi})(t)
=\lim_{t\to+\infty}(\overline{\varphi},\overline{\psi})(t)
=(u_0,w_0)$;

\item[(G3)] $(\overline{\varphi}',{\overline{\psi}}')(t_i^{+})
\leq (\overline{\varphi}',{\overline{\psi}}')(t_i^{-})$,\;
$(\underline{\varphi}',\underline{\psi}')(t_i^{+})
\geq (\underline{\varphi}',\underline{\psi}')(t_i^{-})$.
\end{itemize}


\begin{lemma}\label{L3.1}
If $\Phi_1=(\varphi_1,\psi_1)$,
$\Phi_2=(\varphi_2,\psi_2)\in
C(\mathbb{R},\mathbb{R}^2)$ with
$0\leq\Phi_2(t)\leq\Phi_1(t)\leq K$, $t\in \mathbb{R}$, then
\begin{itemize}
\item[(1)]  $Q_1(\varphi_2,\psi_1)(t)\leq Q_1
(\varphi_1,\psi_2)(t)$, $P_1(\varphi_2,\psi_1)(t)\leq
P_1 (\varphi_1,\psi_2)(t)$,
\item[(2)] $Q_2(\varphi_2,\psi_2)(t)\leq Q_2
(\varphi_1,\psi_1)(t)$, $P_2(\varphi_2,\psi_2)(t)\leq
P_2 (\varphi_1,\psi_1)(t)$\,.
\end{itemize}
\end{lemma}

\begin{proof}
From the definition of $Q$, we have
\begin{align*}
&Q_1(\varphi_1,\psi_2)(t)-Q_1(\varphi_2,\psi_1)(t)\\
&= d_1(\varphi_1(t)-\varphi_2(t))+
[h_1(\varphi_1(t))-h_1(\varphi_2(t))]
-f(\varphi_1(t))\psi_2(t)+f(\varphi_2(t))\psi_1(t)
\\
&= d_1(\varphi_1(t)-\varphi_2(t))+
[h_1(\varphi_1(t))-h_1(\varphi_2(t))]
-f(\varphi_1(t))[\psi_2(t)-\psi_1(t)]
\\
&\quad -\psi_1(t)[f(\varphi_1(t))-
f(\varphi_2(t))]\\
&\geq (d_1-L_{h_1}-k_2L_{f})(\varphi_1(t)-\varphi_2(t))
-f(\varphi_1(t))[\psi_2(t)-\psi_1(t)]
\geq  0,
\end{align*}
\begin{align*}
&Q_2(\varphi_1,\psi_1)(t)-Q_2(\varphi_2,\psi_2)(t)\\
&= d_2(\psi_1(t)-\psi_2(t))+
[h_2(\psi_1(t))-h_2(\psi_2(t))]
+\mu f(\varphi_1(t))\psi_1(t)-\mu f(\varphi_2(t))\psi_2(t)\\
&= d_2(\psi_1(t)-\psi_2(t))+
[h_2(\psi_1(t))-h_2(\psi_2(t))]
+\mu f(\varphi_1(t))[\psi_1(t)-\psi_2(t)]\\
&\quad +\mu \psi_2(t)[ f(\varphi_1(t))-f(\varphi_2(t))]
\\
&\geq (d_2-L_{h_2})(\psi_1(t)-\psi_2(t))
+\mu \psi_2(t)[f(\varphi_1(t))-f(\varphi_2(t))]
\geq 0.
\end{align*}
A similar argument leads to the inequalities about $P$. We omit the details.
\end{proof}

Define a set
\begin{equation*}
\Omega=\{\Phi=(\varphi,\psi)\in
C_{[0,K]}(\mathbb{R},\mathbb{R}^2):
(\underline{\varphi},\underline{\psi})\leq(\varphi,\psi)
\leq(\overline{\varphi},\overline{\psi})\}.
\end{equation*}
Clearly, $\Omega$ is nonempty, bounded, closed and convex subset of
$C(\mathbb{R},\mathbb{R}^2)$ with respect to the norm $\|\cdot\|$.

\begin{lemma}\label{L3.2}
The operator
$P=(P_1,P_2):C_{[0,K]}(\mathbb{R},\mathbb{R}^2)\to
C(\mathbb{R},\mathbb{R}^2)$ is continuous with respect to the norm
$\|\cdot\|$.
\end{lemma}

\begin{proof}
 For any $\Phi_1=(\varphi_1,\psi_1)$,
$\Phi_2=(\varphi_2,\psi_2)\in C_{[0,K]}(\mathbb{R},\mathbb{R}^2)$, we have
\begin{align*}
&|Q_1(\varphi_1,\psi_1)(t)-Q_1(\varphi_2,\psi_2)(t)|\\
&=  |d_1(\varphi_1(t)-\varphi_2(t))+
[h_1(\varphi_1(t))-h_1(\varphi_2(t))]
-f(\varphi_1(t))\psi_1(t)+f(\varphi_2(t))\psi_2(t)|\\
&=  |d_1(\varphi_1(t)-\varphi_2(t))+
[h_1(\varphi_1(t))-h_1(\varphi_2(t))]
-f(\varphi_1(t))[\psi_1(t)-\psi_2(t)]\\
&\quad -\psi_2(t)[f(\varphi_1(t))-f(\varphi_2(t))]|\\
&\leq (d_1+L_{h_1}+k_2L_{f})|\varphi_1(t)-\varphi_2(t)|
+f(k_1)|\psi_1(t)-\psi_2(t)|
\end{align*}
which implies
$$
\sup_{t\in\mathbb{R}}|Q_1(\varphi_1,\psi_1)(t)-Q_1(\varphi_2,\psi_2)(t)|\ \to0
 \quad\text{as } \|\Phi_1-\Phi_2\|\to0.
$$
By the definition of $P$, we have
\begin{align*}
&|P_1(\varphi_1,\psi_1)(t)-P_1(\varphi_2,\psi_2)(t)|\\
&= \frac{1}{D_1(r_{12}-r_{11})}
\Big[\int^{t}_{-\infty}e^{r_{11}(t-s)}
+\int^{+\infty}_{t}e^{r_{12}(t-s)}\Big]\\
&\quad\times \big|Q_1(\varphi_1,\psi_1)(s)-Q_1(\varphi_2,\psi_2)(s)\big|ds\\
&\leq  \frac{1}{D_1(r_{12}-r_{11})} \sup_{s\in\mathbb{R}}
\big|Q_1(\varphi_1,\psi_1)(s)-Q_1(\varphi_2,\psi_2)(s)\big|\\
&\quad\times \Big[\int^{t}_{-\infty}e^{r_{11}(t-s)}ds
+\int^{+\infty}_{t}e^{r_{12}(t-s)}ds\Big]
\\
&=\frac{-1}{D_1r_{11}r_{12}}\sup_{s\in\mathbb{R}}
 |Q_1(\varphi_1,\psi_1)(s)-Q_1(\varphi_2,\psi_2)(s)|\\
&= \frac{1}{d_1}\|Q_1(\varphi_1,\psi_1)-Q_1(\varphi_2,\psi_2)\|.
\end{align*}
Therefore,
$$
\sup_{t\in\mathbb{R}}|P_1(\varphi_1,\psi_1)(t)-P_1(\varphi_2,\psi_2)(t)|\ \to0
 \text{\ if\ } \|\Phi_1-\Phi_2\|\to0.
$$
which implies $P_1$ is continuous. In a similar way, we can get
that $P_2$ is also continuous.
\end{proof}


\begin{lemma}\label{L3.3} For $P$ and $\Omega$ as above,
$P(\Omega)\subset\Omega$.
\end{lemma}

\begin{proof}
For any $\Phi=(\varphi,\psi)\in\Omega$, we have from
Lemma \ref{L3.1} that
\begin{equation}
\begin{gathered}
Q_1(\underline{\varphi},\overline{\psi})(t)\leq Q_1
(\varphi,\psi)(t)\leq
Q_1(\overline{\varphi},\underline{\psi})(t),\\
Q_2(\underline{\varphi},\underline{\psi})(t)\leq Q_2
(\varphi,\psi)(t)\leq
Q_2(\overline{\varphi},\overline{\psi})(t),\\
P_1(\underline{\varphi},\overline{\psi})(t)\leq P_1
(\varphi,\psi)(t)\leq
P_1(\overline{\varphi},\underline{\psi})(t),\\
P_2(\underline{\varphi},\underline{\psi})(t)\leq P_2
(\varphi,\psi)(t)\leq P_2(\overline{\varphi},\overline{\psi})(t).
\end{gathered}
\end{equation}
Now, it is sufficient to show that
\begin{equation}\label{eq3.2}
\begin{gathered}
\underline{\varphi}(t)\leq
P_1(\underline{\varphi},\overline{\psi})(t)\leq
P_1(\overline{\varphi},\underline{\psi})(t)\leq\overline{\varphi}(t)\,,\\
\underline{\psi}(t)\leq
P_2(\underline{\varphi},\underline{\psi})(t)\leq
P_2(\overline{\varphi},\overline{\psi})(t)\leq\overline{\psi}(t)\,.
\end{gathered}
\end{equation}
According to the definitions of upper-lower solutions and the operator
$P$, we have that
$$
Q_1(\underline{\varphi},\overline{\psi})(t)
\geq d_1\underline{\varphi}(t)+c\underline{\varphi}'(t)-D_1\underline{\varphi}''(t),
\quad t\in\mathbb{R}\setminus \Upsilon.
$$
 Let $t_0=-\infty$ and $t_{n+1}=+\infty$, then for
$t_{k-1}<t<t_{k}$ with $k=1, 2, \dots, n+1,$ we have from (G3) that
\begin{align*}
&P_1(\underline{\varphi},\overline{\psi})(t)\\
&= \frac{1}{D_1(r_{12}-r_{11})}
\Big[\int^{t}_{-\infty}e^{r_{11}(t-s)}
+\int^{+\infty}_{t}e^{r_{12}(t-s)}\Big]
Q_1(\underline{\varphi},\overline{\psi})(s)ds\\
&\geq  \frac{1}{D_1(r_{12}-r_{11})}
\Big[\int^{t}_{-\infty}e^{r_{11}(t-s)}
+\int^{+\infty}_{t}e^{r_{12}(t-s)}\Big]
(d_1\underline{\varphi}(s)+c\underline{\varphi}'(s)-D_1\underline{\varphi}''(s))ds\\
&= \underline{\varphi}(t)+\frac{1}{r_{12}-r_{11}}
\Big[\sum_{i=1}^{k}e^{r_{11}(t-t_i)}
(\underline{\varphi}'(t^{+}_i)-\underline{\varphi}'(t^{-}_i)) \\
&\quad +\sum_{i=k+1}^{n}e^{r_{12}(t-t_i)}(\underline{\varphi}'(t^{+}_i)-
\underline{\varphi}'(t^{-}_i))\Big]\\
 &\geq \underline{\varphi}(t)\quad  \text{for } t\in \mathbb{R}\setminus \Upsilon.
\end{align*}
By the continuity of
$P_1(\underline{\varphi},\overline{\psi})(t)$ and
$\underline{\varphi}(t)$, we obtain
$$
\underline{\varphi}(t)\leq P_1(\underline{\varphi},\overline{\psi})(t)
\quad \text{for }t\in\mathbb{R}.
$$
In a similar way, we can show that  \eqref{eq3.2}  holds for $t\in\mathbb{R}$.
\end{proof}

\begin{lemma}\label{L3.4}
The operator $P:\Omega\to\Omega$ is compact with respect to the norm
$\|\cdot\|$.
\end{lemma}

 The proof of of the above lemma is similar
to that of \cite[Lemma 3.5]{La}; since it is independent of the monotone
condition, so we omit it here.
Now, we are in a position to state and prove  our main results.

\begin{theorem}\label{T3.1}
Assume {\rm (H1)--(H3)} hold. If \eqref{eq2.1} has a pair of
upper-lower solutions
$\overline{\Psi}=(\overline{\varphi},\overline{\psi})$
and
$\underline{\Psi}=(\underline{\varphi},\underline{\psi})$
satisfying (G1)-(G3). Then \eqref{eq1.6} admits a
traveling wave solution connecting $(0,0)$ and $(u_0,w_0)$.
\end{theorem}

\begin{proof}
By Lemma \ref{L3.2}-\ref{L3.4} and  the Schauder's fixed point
theorem,  we know that the operator $P$ admits a fixed point
$(\varphi{*},\psi^{*})\in\Omega$ which is a solution of
\eqref{eq2.1}. Noting the fact that
 $$(
\underline{\varphi},\underline{\psi})\leq(\varphi^{*},\psi^{*})
\leq(\overline{\varphi},\overline{\psi}),
$$
 then we have from (G2) that
$$
\lim_{t\to-\infty}(\varphi^{*},\psi^{*})=(0,0)\quad \text{and}
\quad \lim_{t\to+\infty}(\varphi^{*},\psi^{*})=
(u_0,w_0).
$$
Therefore, the fixed point
$(\varphi^{*},\psi^{*})$ satisfies the asymptotic boundary condition
\eqref{eq2.2}, and thus it is a traveling wave solution of \eqref{eq1.6}
connecting $(0,0)$ and
 $(u_0,w_0)$.
\end{proof}

\section{Applications}

In this section we apply our results in section 3 to establish the
existence of traveling wave solution for  \eqref{eq1.7} with 
$f(u)=\frac{u}{1+u}$. In view of Theorem \ref{T3.1}, 
the key point is to construct a pair of upper-lower solutions
satisfying (G1)--(G3).

 \begin{example}\label{E4.1} \rm
Consider  the existence of traveling wave solution for the
system
\begin{equation}\label{eq4.1}
\begin{gathered}
u_{t}= D_1u_{xx}+\alpha u(\beta-u)-\frac{uw}{1+u}
,\\
w_{t}= D_2w_{xx}+\gamma w(\delta-w)+\frac{\mu uw}{1+u}\,.
\end{gathered}
\end{equation}
\end{example}

 We are interested in the co-existence of species, so
we assume that  \eqref{eq4.1} has a unique
positive  equilibrium $(u_0,w_0)$ satisfying
\begin{equation}\label{eq4.2}
\alpha\beta-\alpha u_0-\frac{w_0}{1+u_0}=0,\quad
\gamma\delta-\gamma w_0+\frac{\mu u_0}{1+u_0}=0.
\end{equation}
It is clear that $\gamma w_0>\frac{\mu u_0}{1+u_0}$.
Moreover, for the technique reason, we assume that
\begin{equation}\label{a}
\alpha u_0>2w_0.
\end{equation}
Clearly, the wave system corresponding to \eqref{eq4.1} is
\begin{equation}\label{eq4.3}
\begin{gathered}
c\varphi'=
D_1\varphi''+\alpha\varphi(\beta-\varphi)-\frac{\varphi\psi}{1+\varphi}
,\\
c\psi'= D_2\psi'+\gamma\psi(\delta-\psi)+\frac{\mu
\varphi\psi}{1+\varphi}.
\end{gathered}
\end{equation}
 As mentioned above, we are interested in the solution of
 \eqref{eq4.3} with asymptotic boundary  conditions
\begin{equation} \label{eq4.4}
\lim_{t\to-\infty}(\varphi(t),\psi(t))=(0,0),\quad
\lim_{t\to+\infty}(\varphi(t),\psi(t))=(u_0,w_0).
\end{equation}


In this example, we choose $k_1=\beta$,  $k_2=\delta+\frac{\mu
\beta}{\gamma(1+\beta)}$, then we have $k_1>u_0$ and $k_2>w_0$.
Let $c>c^{*}:=\max\{2\sqrt{D_1\alpha
k_1},2\sqrt{D_2\gamma k_2}\}$, then there exist
$$
0<\lambda_{11}<\lambda_{12},\quad 0<\lambda_{21}<\lambda_{22}
$$
such that
$$
D_1\lambda_{1i}-c\lambda_{1i}+\alpha k_1=0,\quad
D_2\lambda_{2i}-c\lambda_{2i}+\gamma k_2=0,\quad i=1,2.
$$
Since $\gamma w_0>\frac{\mu u_0}{1+u_0}$,
$\alpha u_0>2w_0$, there exist $\varepsilon_1\in(0,u_0)$,
$\varepsilon_2\in(0,w_0)$ such that
\begin{equation} \label{eq4.5}
\gamma\varepsilon_2>\frac{\mu u_0}{1+u_0}, \quad
\alpha\varepsilon_1>2w_0.
\end{equation}

For a small $\lambda>0$, let $f(t):=\min\{
e^{\lambda_{11}t},u_0+u_0e^{-\lambda t}\}$,
 $g(t):=\min\{ e^{\lambda_{21}t},w_0+w_0e^{-\lambda t}\}$ and denote
$$
m_1=\max_{t\in\mathbb{R}}\{f(t)\},\quad \quad m_2=\max_{t\in\mathbb{R}}\{g(t)\}.
$$
 If $m_1>k_1$, $m_2>k_2$, define the
following  continuous functions:
\begin{gather*}
\overline{\varphi}(t)=
\begin{cases}
 e^{\lambda_{11}t}, & t\leq t_1,\\
k_1, & t_1< t< t_2,\\
 u_0+u_0e^{-\lambda t},& t\geq t_2,
\end{cases} \qquad
\underline{\varphi}(t)=\begin{cases}
 0,&t\leq t_3,\\
 u_0-\varepsilon_1e^{-\lambda t}, & t>t_3,
\end{cases}
\\
\overline{\psi}(t)=\begin{cases}
 e^{\lambda_{21}t}, & t\leq t_4,\\
k_2, & t_4< t< t_{5},\\
w_0+w_0e^{-\lambda t},& t\geq t_{5},
\end{cases} \qquad
\underline{\psi}(t)=\begin{cases}
 0, &t\leq t_6,\\
w_0-\varepsilon_2e^{-\lambda t},&t> t_6.
\end{cases}
\end{gather*}
If $m_1\leq k_1$, $m_2\leq k_2$, then redefine the above
$\overline{\varphi}(t),\overline{\psi}(t)$ as
\begin{equation*}
\overline{\varphi}(t)=\begin{cases}
 e^{\lambda_{11}t}, & t\leq t_1,\\
 u_0+u_0e^{-\lambda t},& t\geq t_1,
\end{cases} \qquad
\overline{\psi}(t)=\begin{cases}
 e^{\lambda_{21}t}, & t\leq t_4,\\
w_0+w_0e^{-\lambda t},& t\geq t_4.
\end{cases}
\end{equation*}
The other two cases: either $m_1> k_1$, $m_2\leq k_2$, or
 $m_1\leq k_1$, $m_2> k_2$ can be considered similarly.

In what follows, we consider only the situation: $m_1>k_1$, $m_2>k_2$.
The discussions of other cases will be omitted. It is easily seen that
$(\overline{\varphi}(t),\overline{\psi}(t))$,
$(\underline{\varphi}(t),\underline{\psi}(t))$
satisfy (G1)-(G3). Furthermore,  we have from \eqref{eq4.2} and \eqref{a} that
$$
k_1-u_0=\beta- u_0=\frac{\frac{w_0}{\alpha}}{1+u_0}<\frac{ u_0}{2+2u_0}<u_0,
$$
which leads to
$\frac{u_0}{k_1-u_0}>1$. Note that $\frac{\varepsilon_2}{w_0}<1$, and thus we have
$$
t_2=\frac{1}{\lambda}\ln\frac{u_0}{k_1-u_0}>0>t_6=\frac{1}{\lambda}
\ln\frac{\varepsilon_2}{w_0}.
$$

\begin{lemma}\label{L4.4}
If $\lambda>0$ is small enough, then
$\overline{\Phi}(t)=(\overline{\varphi},\overline{\psi})(t)$
and
$\underline{\Phi}(t)=(\overline{\varphi},\underline{\psi})(t)$
is a pair of  upper-lower solutions of \eqref{eq4.3}.
\end{lemma}

\begin{proof}
For $t<t_1$, we have $\overline{\varphi}(t)=e^{\lambda_{11}t}$ and
\begin{align*}
&D_1\overline{\varphi}''(t)-c\overline{\varphi}'(t)+\alpha\overline{\varphi}(t)
(\beta-\overline{\varphi}(t))-
\frac{\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\underline{\psi}(t)\\
&\leq  D_1 \lambda_{11}^2e^{\lambda_{11}t}-c\lambda_{11}e^{\lambda_{11}t}
 +\alpha\beta e^{\lambda_{11}t}\\
&=  e^{\lambda_{11}t}[D_1\lambda_{11}^2-c\lambda_{11}+\alpha k_1] =0.
\end{align*}
For $t_1<t<t_2$, then we have $\overline{\varphi}(t)=k_1=\beta$ and
\begin{align*}
& D_1\overline{\varphi}''(t)-c\overline{\varphi}'(t)
 +a\overline{\varphi}(t)(\beta-\overline{\varphi}(t))-
\frac{\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\underline{\psi}(t)\\
&\leq D_1\overline{\varphi}''(t)-c\overline{\varphi}'(t)
 -\alpha\overline{\varphi}(t)(\beta-\overline{\varphi}(t)) =0.
\end{align*}
For $t> t_2$, by \eqref{eq4.2}, we have
$\overline{\varphi}(t)=u_0+u_0e^{-\lambda t}$,
$\underline{\psi}(t)=w_0-\varepsilon_2e^{-\lambda t}$ and
\begin{align*}
&D_1\overline{\varphi}''(t)-c\overline{\varphi}'(t)
 +\alpha\overline{\varphi}(t)(\beta-\overline{\varphi}(t))-
\frac{\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\underline{\psi}(t)
\\
&=  D_1u_0\lambda^2e^{-\lambda t}+cu_0\lambda e^{-\lambda
t}+(u_0+u_0e^{-\lambda t})\\
&\quad\times \big[\alpha\beta-\alpha(u_0+u_0e^{-\lambda
t})-\frac{w_0-\varepsilon_2e^{-\lambda
t}}{1+(u_0+u_0e^{-\lambda t})}\big]
\\
&=   D_1u_0\lambda^2e^{-\lambda t}+cu_0\lambda e^{-\lambda
t}+(u_0+u_0e^{-\lambda t})\\
&\quad\times \big[-\alpha u_0e^{-\lambda t}+\frac{w_0}{1+u_0}
-\frac{w_0-\varepsilon_2e^{-\lambda t}}{1+(u_0+u_0e^{-\lambda t})}\big]
\\
&= :I_1(\lambda,t)=p_1(\lambda,t)+q_1(\lambda,t),
\end{align*}
where
\begin{gather*}
p_1(\lambda,t)= D_1u_0\lambda^2e^{-\lambda t}+cu_0\lambda e^{-\lambda t}, \quad
q_1(\lambda,t)=\overline{q}_1(\lambda,t)\cdot \underline{q}_1(\lambda,t),\\
\overline{q}_1(\lambda,t)=u_0+u_0e^{-\lambda t},\quad
\underline{q}_1(\lambda,t)=-\alpha u_0e^{-\lambda
t}+\frac{w_0}{1+u_0}-\frac{w_0-\varepsilon_2e^{-\lambda
t}}{1+(u_0+u_0e^{-\lambda t})}.
\end{gather*}
Since $\alpha u_0>2w_0$, then for $t> t_2$ uniformly, we have
$$
I_1(0,t)=2u_0(-\alpha u_0+\frac{w_0}{1+u_0}
-\frac{w_0-\varepsilon_2}{1+2u_0})<2u_0(-\alpha u_0+w_0)<0.
$$
Furthermore, $\underline{q}_1(\lambda,0)=-\alpha u_0+\frac{w_0}{1+u_0}
-\frac{w_0-\varepsilon_2}{1+2u_0}<0$, and for any fixed
$\lambda>0$,  $I(\lambda, \infty)=0$.
 Note that for any fixed $\lambda>0$,
$\overline{q}_1(\lambda,t)=u_0+u_0e^{-\lambda t}>0$ and  is decreasing on $t>0$,
and $\underline{q}_1(\lambda,t)<0$ and is increasing on $t>0$.
We know that $q_1(\lambda,t)<0$ and is increasing on  $t>0$.
On the other hand, $p_1(\lambda,t)>0$ and is  decreasing on $t>0$.
 For all $\lambda_1>0$, $t>t_2$, we have
$$
p_1(\lambda_1,t)=D_1u_0\lambda^2e^{-\lambda t}+cu_0\lambda e^{-\lambda
t}<D_1u_0\lambda_1^2+cu_0\lambda_1\ \mbox {\ for \ } \lambda\in(0,\lambda_1).
$$
From the monotone property of $p_1(\lambda,t)$ and $q_1(\lambda,t)$,
one can choose $\lambda_1>0$ small such that $D_1u_0\lambda_1^2+cu_0\lambda_1$
is small and $I_1(\lambda,t)=p_1(\lambda,t)+q_1(\lambda,t)<0$ for $t> t_2$
and $\lambda\in(0,\lambda_1)$. That is,
\[
D_1\overline{\varphi}''(t)-c\overline{\varphi}'(t)
 +\alpha\overline{\varphi}(t)(\beta-\overline{\varphi}(t))
 -\frac{\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\underline{\psi}(t)<0
\]
 for $\lambda\in(0,\lambda_1),\;t> t_2$.

 Note  that $f(u)=\frac{u}{1+u}$ is nondecreasing on
$[0,+\infty)$, then for $t< t_4$, we have
$\overline{\psi}(t)=e^{\lambda_{21}t}$ and
\begin{align*}
& D_2\overline{\psi}''(t)-c\overline{\psi}'(t)+\gamma\overline{\psi}(t)
(\delta-\overline{\psi}(t))+\frac{\mu\overline{\varphi}(t)}{1+\overline{\varphi}(t)}
\overline{\psi}(t)
\\
&\leq  D_2 \lambda_{21}^2e^{\lambda_{21}t}-c\lambda_{21}e^{\lambda_{21}t}
+\gamma\delta e^{\lambda_{21}t}+\frac{\mu k_1}{1+k_1}e^{\lambda_{21}t}
\\
&= e^{\lambda_{21}t}[D_2\lambda_{21}^2-c\lambda_{21}+\gamma\delta+\frac{\mu
k_1}{1+k_1}]\\
&= e^{\lambda_{21}t}[D_2\lambda_{21}^2-c\lambda_{21}+\gamma k_2] =0.
\end{align*}
For $t_4<t< t_{5}$, we have $\overline{\psi}(t)=k_2$ and
\begin{align*}
& D_2\overline{\psi}''(t)-c\overline{\psi}'(t)
 +\gamma\overline{\psi}(t)(\delta-\overline{\psi}(t))+
\frac{\mu\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\overline{\psi}(t)\\
&\leq D_2\overline{\psi}''(t)-c\overline{\psi}'(t)
 +\overline{\psi}(t)(\gamma\delta-\gamma\overline{\psi}(t)+\frac{\mu k_1}{1+k_1})
=0.
\end{align*}

For $t>t_{5}$, by \eqref{eq4.2}, we have
$\overline{\psi}(t)=w_0+w_0e^{-\lambda t}$,
$\overline{\varphi}(t)\leq u_0+u_0e^{-\lambda t}$, and
\begin{align*}&
D_2\overline{\psi}''(t)-c\overline{\psi}'(t)+\gamma\overline{\psi}(t)
(\delta-\overline{\psi}(t))+
\frac{\mu\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\overline{\psi}(t)\\
&\leq  D_2w_0\lambda^2e^{-\lambda t}+cw_0\lambda
e^{-\lambda t}+(w_0+w_0e^{-\lambda t})\\
&\quad\times  \big[\gamma\delta-\gamma(w_0+w_0e^{-\lambda
t})+\frac{\mu(u_0+u_0e^{-\lambda
t})}{1+(u_0+u_0e^{-\lambda t})}\big]\\
&= D_2w_0\lambda^2e^{-\lambda t}+cw_0\lambda e^{-\lambda
t}+(w_0+w_0e^{-\lambda t})\\
&\quad \times \big[-\gamma
w_0e^{-\lambda t}-\frac{\mu
u_0}{1+u_0}+\frac{\mu(u_0+u_0e^{-\lambda t})}{1+(u_0+u_0e^{-\lambda t})}\big]
= :I_2(\lambda,t).
\end{align*}
Since $\gamma w_0>\frac{\mu u_0}{1+u_0}$, then for $t>t_5$ uniformly, we have
$$
I_2(0,t)=2w_0(-\gamma w_0-\frac{\mu u_0}{1+u_0}
+\frac{2\mu u_0}{1+2u_0})<2w_0(-\gamma w_0+
\frac{\mu u_0}{1+u_0})<0.
$$
 Similar to the discussion of $I_1(\lambda,t)$,  there exists
$\lambda_2>0$ such that
\[
D_2\overline{\psi}''(t)-c\overline{\psi}'(t)
 +\gamma\overline{\psi}(t)(\delta-\overline{\psi}(t))
 + \frac{\mu\overline{\varphi}(t)}{1+\overline{\varphi}(t)}\overline{\psi}(t)<0
\]
for $\lambda\in(0,\lambda_2)$ and $t>t_5$.

 For $t< t_3$, we have $\underline{\varphi}(t)=0$,
 and
$$
D_1\underline{\varphi}''(t)-c\underline{\varphi}'(t)
+\alpha\underline{\varphi}(t)(\beta-\underline{\varphi}(t))-
\frac{\underline{\varphi}(t)}{1+\underline{\varphi}(t)}\overline{\psi}(t)=0.
$$
For $t> t_3$, by \eqref{eq4.2}, we have
$\underline{\varphi}(t)=u_0-\varepsilon_1 e^{-\lambda t}$,
$\overline{\psi}(t)\leq w_0+w_0e^{-\lambda t}$ and
\begin{align*}
& D_1\underline{\varphi}''(t)-c\underline{\varphi}'(t)+\alpha\underline{\varphi}(t)
(\beta-\underline{\varphi}(t))-
\frac{\underline{\varphi}(t)}{1+\underline{\varphi}(t)}\overline{\psi}(t)\\
&\geq
 -D_1\varepsilon_1\lambda^2e^{-\lambda t}-
c\varepsilon_1\lambda e^{-\lambda t}+
(u_0-\varepsilon_1e^{-\lambda t})\\
&\quad\times \big[\alpha\beta-\alpha(u_0-\varepsilon_1e^{-\lambda t})
-\frac{w_0+w_0e^{-\lambda t}}{1+(u_0-\varepsilon_1e^{-\lambda t})}big]\\
&=  -D_1\varepsilon_1\lambda^2e^{-\lambda t}-
c\varepsilon_1\lambda e^{-\lambda t}+
(u_0-\varepsilon_1e^{-\lambda t})\\
&\quad\times\big[\alpha\varepsilon_1e^{-\lambda t}+\frac{w_0}{1+u_0}
-\frac{w_0+w_0e^{-\lambda t}}{1+(u_0-\varepsilon_1e^{-\lambda t})}\big]
= :I_3(\lambda,t).
\end{align*}
It follows from \eqref{eq4.5} that for $t>t_3$ uniformly, we have
$$
I_3(0,t)=(u_0-\varepsilon_1)
\Big[\alpha\varepsilon_1+\frac{w_0}{1+u_0}-\frac{2w_0}{1+(u_0-\varepsilon_1)}\Big]
>(u_0-\varepsilon_1)(\alpha\varepsilon_1-2w_0)>0.
$$
Therefore, there exists $\lambda_3>0$ such that
\[
D_1\underline{\varphi}''(t)-c\underline{\varphi}'(t)+\alpha\underline{\varphi}(t)
(\beta-\underline{\varphi}(t))-
\frac{\underline{\varphi}(t)}{1+\underline{\varphi}(t)}\overline{\psi}(t)
>0
\]
 for $\lambda\in(0,\lambda_3)$ and $t>t_3$.


 For $t< t_6$,  we have $\underline{\psi}(t)= 0$ and
$$
D_2\underline{\psi}''(t)-c\underline{\psi}'(t)
+\gamma\underline{\psi}(t)(\delta-\underline{\psi}(t))+
\frac{\mu\underline{\varphi}(t)}{1+\underline{\varphi}(t)}\underline{\psi}(t)=0
$$
For $t> t_6$, by \eqref{eq4.2}, we have
$\underline{\psi}(t)= w_0-\varepsilon_2e^{-\lambda t}$, and
\begin{align*}
& D_2\underline{\psi}''(t)-c\underline{\psi}'(t)
+\gamma\underline{\psi}(t)(\delta-\underline{\psi}(t))+
\frac{\mu\underline{\varphi}(t)}{1+\underline{\varphi}(t)}\underline{\psi}(t)\\
&\geq
 -D_2\varepsilon_2\lambda^2e^{-\lambda t}-
c\varepsilon_2\lambda e^{-\lambda t}+
(w_0-\varepsilon_2e^{-\lambda t})
[\gamma\delta-\gamma(w_0-\varepsilon_2e^{-\lambda
t})] \\
&=  -D_2\varepsilon_2\lambda^2e^{-\lambda t}-
c\varepsilon_2\lambda e^{-\lambda t}+
(w_0-\varepsilon_2e^{-\lambda t})
(\gamma\varepsilon_2e^{-\lambda t}-\frac{\mu u_0}{1+u_0})
 = :I_4(\lambda,t).
\end{align*}
It follows from \eqref{eq4.5} that for $t>t_6$ uniformly, we have
$$
I_4(0,t)=(w_0-\varepsilon_2)(\gamma\varepsilon_2-\frac{\mu u_0}{1+u_0})
>0.$$
Therefore, there exists $\lambda_4>0$ such that
\[
 D_2\underline{\psi}''(t)-c\underline{\psi}'(t)
+\gamma\underline{\psi}(t)(\delta-\underline{\psi}(t))+
\frac{\mu\underline{\varphi}(t)}{1+\underline{\varphi}(t)}\underline{\psi}(t)
<0
\]
 for $\lambda\in(0,\lambda_4)$ and $t>t_6$.
By the above argument, we know that
$\overline{\Phi}=(\overline{\varphi},\overline{\psi})(t)$
and $\underline{\Phi}=(\overline{\varphi},\underline{\psi})$
is a pair of  upper-lower solutions of \eqref{eq4.3} for  $\lambda>0$ small
enough.
\end{proof}

As a direct consequence of Theorem \ref{T3.1},  we have the
following result.

\begin{theorem}\label{T4.1}
Assume that \eqref{a} holds and $c>c^{*}:=\max\{2\sqrt{D_1\alpha
k_1},2\sqrt{D_2\gamma k_2}\}$, where $k_1=\beta$,
 $k_2=\delta+\frac{\mu \beta}{\gamma(1+\beta)}$.
For $c>c^{*}$, system \eqref{eq4.1} has a traveling wave
$\Psi(t)=(\varphi(t),\psi(t))$ satisfying
$\Psi(-\infty)=(0,0)$, $\Psi(\infty)=(u_0,w_0).$
 \end{theorem}


\section{Concluding discussion}%5

In this article we have dealt with the existence of traveling wave
solutions for a reaction-diffusion system based on a predator-prey
model with a general functional response. By constructing an
admissible pair of upper and lower solutions and using Schauder's
fixed point theorem, we show that there is a traveling wave solution
connecting the trivial equilibrium $(0,0)$ and the positive
equilibrium $(u_0,w_0)$. That is, there is a zone of transition
from the steady state with no species to the steady state with the
coexistence of both species.  In comparison, the technique used here
is simpler than those of the works mentioned in the introduction.

Predator-prey systems admit
 multiple equilibria. Our work here considered  only one
case. Traveling waves connecting other
pairs of equilibrium are also possible. It would be interesting to
use the techniques in the present paper to investigate the existence
of traveling waves connecting $(\beta,0)$ and $(u_0,w_0)$ which
would explain the situation where the habitat is first saturated with
prey to its carrying capacity, then the invasion of predator may
result in co-existence of both species in the long term.


\subsection*{Acknowledgments}
 Research is supported by grants 11171120 from  the Natural Science
Foundation of China, 20094407110001 from the Doctoral Program of Higher
Education of China,  and 10151063101000003 from the Natural Science Foundation of
Guangdong Province. We  want to thank the anonymous referee
for his/her helpful suggestions which led to improvements of this article.

\begin{thebibliography}{16}

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