\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 161, pp. 1--15.\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-2012/161\hfil Traveling waves and spreading speed] {Traveling waves and spreading speed on a lattice model with age structure} \author[Z. Wang \hfil EJDE-2012/161\hfilneg] {Zongyi Wang} \address{Zongyi Wang \newline Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, China} \email{wzy@hzu.edu.cn} \thanks{Submitted June 9, 2012. Published September 20, 2012.} \subjclass[2000]{45J05, 34A33, 34K31, 92D25} \keywords{Lattice differential system; spreading speed; traveling wave; \hfill\break\indent minimal wave speed} \begin{abstract} In this article, we study a lattice differential model for a single species with distributed age-structure in an infinite patchy environment. Using method of approaches by Diekmann and Thieme, we develop a comparison principle and construct a suitable sub-solution to the given model, and show that there exists a spreading speed of the system which in fact coincides with the minimal wave speed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Assume $u(t,a,x)$ is the population density at time $t$, age $a$ and spatial location $x$, and $x$ denotes the point coordinate which may be an integer, in $\mathbb{Z}$, or real number in $\mathbb{R}$. We study the species in a patchy environment with infinite number of patches connected by diffusion of population within the neighboring islands, where we can describe the patches as integer nodes of a one-dimensional lattice. In this case we change $x$ to $j$, and let $u(t,a,j)=u_j(t,a)$ denote the population density of the species at $j$-th patch. Let $f(r)$ be a probability density function which specifies the probability of maturing of an individual with age $a\geq r$. This function satisfies $f(0)=0$, $f(\infty)=0$ and $\int_0^{\infty}f(r)dr=1$. Let $w_j(t)$ denotes the total of mature population at time $t$ and location $j$: $$w_j(t)=\int_0^\infty f(r)\Big(\int_r^{\infty}u_j(t,a)da\Big)dr.$$ Ling \cite{LW} derived the lattice model $$\label{1.1} \begin{split} \frac{dw_j(t)}{dt} &= D[w_{j+1}(t)+w_{j-1}(t)-2w_j(t)] - dw_j(t)\\ &\quad +\frac{1}{2\pi}\int_0^{\infty}e^{-da}f(a)\sum_{l=-\infty}^{\infty} \beta(a,l)b(w_{l+j}(t-a))da, \quad t>0, \end{split}$$ where $\beta(a,l)=2\int_{0}^{\pi}\cos(l\omega)e^{-4Da\sin^2(\frac{\omega}{2})} d\omega.$ Note that this equation has a nonlocal term $\sum_{l=-\infty}^{\infty} \beta(a,l)b(w_{l+j}(t-a))$ and a delay that is continuously distributed and infinite. Ling studied the existence and uniqueness of solutions to \eqref{1.1} with an initial value, also discussed the global attractivity of the zero solution, and the existence of wavefronts with speed greater than the spreading speed $c_*$ of traveling wave. Motivated by the method in Diekmann and Thieme \cite{Th2}, in this article, we give a study on the traveling wave and spreading speed for \eqref{1.1}. More information on the traveling waves for lattice differential systems can be found in \cite{CMS,HLR,HLZ,LW,MZ,Mallet2,WHW} and the references therein. Let $\mathbb{R}_+:=[0,+\infty)$ and $\tilde f(d):=\int_{0}^{\infty}f(a)e^{-da}da<1$. We will use the following assumptions: \begin{itemize} \item[(H0)] $b(0)=0$, $b(w)\leq b'(0)w$ for $w\geq 0$; $b(w)\tilde f(d)0$, and $b'(0)\tilde f(d)d$, $b(w)\tilde f(d)=dw$ admits a positive solution $w^+$ on $(0,K]$. $b(w)\tilde f(d)>dw$ for $0w^+$. \end{itemize} This article is organized as follows. In Section 2, we introduce some definitions and properties of the characteristic equations. In Section 3, we establish the well-posedness and the comparison principle for \eqref{1.1}, and obtain our main result on the existence of the spreading speed $c_*$ of traveling wave of \eqref{1.1}. We also give an estimate for $c_*$ and study the relation between the spreading speed with the minimal wave speed. \section{Preliminaries} A solution $\{w_j(t)\}_{j\in \mathbb{Z}}$ is called a traveling wave of \eqref{1.1} provided that it has the form $w_j(t)=\phi(j+ct)=\phi(s)$. A sequence of functions $W(t)=\{w_j(t)\}_{j\in \mathbb{Z}}$ is called isotropic on an interval $I$ if $w_j(t)=w_{-j}(t)$ for $j \in \mathbb{Z}$ and $t\in I$. Define $$C_{K}^+(-\infty,T]= \{\phi: \phi \text{ is continuous function defined from (-\infty,T] to [0,K]}\}.$$ We need also the following notation. \begin{gather*} B_N=\{j\in \mathbb{N}: |j|\leq N, N\in \mathbb{N}\},\\ w_j(t)=w(t,j)\text{ for } j\in \mathbb{Z},\quad W(t)=W(t,\cdot)=\{w_j(t)\}_{j\in \mathbb{Z}},\\ \operatorname{supp}W(t,\cdot)=\{j: w(t,j)\neq 0\}\text{ is the support of } W(t,\cdot),\\ W(t)\geq V(t) \text{ if } w_j(t)\geq v_j(t) \text{ for } j\in \mathbb{Z},\\ W(t)\succ V(t) \text{ if } W(t)\geq V(t) \text{ and } w_j(t)>v_j(t) \text{ for } j\in \operatorname{supp}V(t,\cdot). \end{gather*} A constant $c_*>0$ is called the spreading speed of \eqref{1.1} provided that \begin{gather}\label{2.1} \lim_{t\to\infty}\sup\{w_j(t): |j|\geq ct\}=0\quad \text{for }c> c_*,\\ \label{2.2} \lim_{t\to\infty}\inf\{w_j(t): |j|\leq ct\}\geq w^+>0 \quad\text{for }c\in(0, c_*). \end{gather} where $\{w_j(t)\}_{j\in \mathbb{Z}}$ is a solution of \eqref{1.1}. Substituting $w_j(t)=\phi(j+ct)=\phi(s)$ into \eqref{1.1}, we obtain the wave equation $$\label{2.3} \begin{split} c\phi'(s) &= D[\phi(s+1) +\phi(s-1) -2\phi(s)] - d\phi(s)\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(\phi(s+l-ca))da. \end{split}$$ The following assumption is needed for considering characteristic equation. \begin{itemize} \item[(H4)] Assume that for a given $c>0$, one of the following two conditions is satisfied, \begin{itemize} \item[(i)] For any $\lambda>0$, $\int_{0}^{\infty}f(a)e^{-da}e^{2D(\cosh\lambda-1)a-\lambda ca}da<\infty$ holds. \item[(ii)] There has $\lambda_0>0$, for any $\lambda<\lambda_0$, $\int_{0}^{\infty}f(a)e^{-da}e^{2D(\cosh\lambda-1)a-\lambda ca}da<\infty$ and $$\lim_{\lambda\to \lambda_0-0}\int_{0}^{\infty}f(a)e^{-da} e^{2D(\cosh\lambda-1)a-\lambda ca}da=+\infty.$$ \end{itemize} If case (i) holds, let $\bar{\lambda}=\bar{\lambda}(c)=+\infty$; if case (ii) holds, let $\bar{\lambda}=\bar{\lambda}(c)=\lambda_0$. \end{itemize} Assume that (H1)-(H4) hold. Then \eqref{2.3} has two equilibria $w=0$ and $w=w^+>0$ in $[0,K]$. Denote the characteristic equation of \eqref{2.3} at $w^0:=0$, by $\Delta(\lambda,c)=0$, we have $$\label{2.4} \Delta(\lambda,c)= -c\lambda +D[e^{\lambda}+e^{-\lambda}-2] -d +\frac{b'(0)}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)e^{\lambda l}e^{-\lambda ca}da.$$ where $\frac{1}{2\pi}\sum_{l=-\infty}^{\infty}\beta(a,l)e^{\lambda l} =\exp\{D[e^{-\lambda}+e^{\lambda}-2]a\} =e^{2D(\cosh\lambda -1)a}$ (see \cite{WHW}). Simplify \eqref{2.4} to obtain $$\label{2.5} \Delta(\lambda,c):= -c\lambda +D[e^{\lambda}+e^{-\lambda}-2]-d+ b'(0)\int_{0}^{\infty}f(a)e^{[-d-c\lambda+2D(\cosh\lambda-1)]a}da=0.$$ From \eqref{2.4}-\eqref{2.5}, it is easy to observe the following fact. \begin{lemma}\label{L2.1} If $b$ satisfies {\rm(H2)-(H4)}. Then there exists a unique pair $(c_*,\lambda_*)$ $(c_*>0, \lambda_*>0)$ such that \begin{itemize} \item[(i)] $\Delta(\lambda_*,c_*)=0$, $\frac{\partial}{\partial \lambda}\Delta(\lambda_*,c_*)=0$; \item[(ii)] for $00$; \item[(iii)] for $c>c_*$, the equation $\Delta(\lambda,c)=0$ has two positive real roots $0<\lambda_1 <\lambda_2<\bar{\lambda}$, and there exists $\epsilon_0>0$ such that for any $\epsilon\in(0,\epsilon_0)$ with $0<\lambda_1<\lambda_1+\epsilon<\lambda_2$, we have $\Delta(\lambda_1+\epsilon,c)<0$. \end{itemize} \end{lemma} We rewrite \eqref{2.5} as $$\label{2.6} 1=\frac{1}{\delta+\lambda c} \Big[D(e^{\lambda}+e^{-\lambda})+ b'(0)\int_{0}^{\infty}f(a) e^{-da}e^{2D(\cosh\lambda-1)a-\lambda ca}da \Big]=:L_c(\lambda),$$ where $\delta:=2D+d$. Hence $c_*$ can be represented as $$c_*:=\inf\{c>0:\text{ there exists some }\lambda \in \mathbb{R}_+, \text{ such that } L_c(\lambda)=1\}.$$ From Lemma \ref{L2.1} we have $$L_c(\lambda)>1 \text{ for } \lambda\in (0,\bar{\lambda}), \text{ and } c\in (0,c_*);\quad L_c(\lambda)<1 \text{ for } \lambda\in (\lambda_1,\lambda_2)\text{ and } c>c_*.$$ Now we shall show that $c_*$ is the spreading speed of \eqref{1.1}. Consider the equivalent form \label{2.7} \begin{gathered} \begin{aligned} w_j(t)&=e^{-\delta t}w_j(0)+\int_0^te^{-\delta(t-s)}\{D[w_{j+1}(s)+w_{j-1}(s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(w_{l+j}(s-a)) da\}ds, \quad j\in\mathbb{Z},\; t\geq 0, \end{aligned}\\ w_j(t)=w^o_j(t), \quad j\in \mathbb{Z},\; t\in (-\infty,0], \end{gathered} For any $W^o=\{w_j^o\}_{j\in \mathbb{Z}}$, $w^o_j\in C_{K}^+(-\infty,0]$, $w_j^0(0)>0,\ j\in \mathbb{Z}$, and $T\in [0,\infty]$, define the set $$\Lambda_T=\{W=\{w_j\}_{j\in \mathbb{Z}} : w_j\in C_K^+(-\infty,T),\, w_j(t)=w^o_j(t)\text{ for } t\in(-\infty,0]\},$$ Equip $\Lambda_T$ with the norm $$\|W\|_{\lambda}:=\sup_{t\in [0,T),j\in \mathbb{Z}}|w_j(t)|e^{-\lambda t}.$$ Therefore, $(\Lambda_T,\|\cdot\|_{\lambda})$ is a Banach space. Define the sequence of functions $S^T=\{S^T_j\}_{j\in \mathbb{Z}} \in \Lambda_T$ by $$S_j^T[W](t)=\begin{cases} e^{-\sigma t}w_j(0)+\int_0^t e^{-\sigma (t-s)}\{D[w_{j+1}(s)+w_{j-1}(s)]\\ +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da} \sum_{l=-\infty}^{\infty}\beta(a,l)b(w_{l+j}(s-a))da\}ds,& j\in\mathbb{Z},\; t\geq 0,\\ w^o_j(t), & j\in \mathbb{Z},\; t<0. \end{cases}$$ Then $S_j^T[W](t)$ is continuous in $t\in (-\infty,T)$. \begin{theorem}\label{T2.1} Suppose the initial function $W^o=\{w_j^o\}_{j\in \mathbb{Z}}$ is isotropic on interval $(-\infty,0]$, $w_j^o\in C_{K}^+(-\infty,0]$, $j\in \mathbb{Z}$, and there exists $\bar N\in \mathbb{N}$ such that $\operatorname{supp} W^o(t,\cdot)\subseteq B_{\bar N}$, $t\in (-\infty,0]$. Then for any $c>c_*$, \eqref{2.1} holds; i.e., $\lim_{t\to\infty}\sup\{w_j(t)|\ |j|\geq ct\}=0$. \end{theorem} \begin{proof} Define a sequence of maps by \begin{gather*} W^{(n)}(t)=S^{\infty}[W^{(n-1)}](t)\quad \text{for } n\in \mathbb{N}, \; t\in \mathbb{R},\quad W^{(o)}(t)=\{w_j^{(o)}(t)\}_{j\in\mathbb{Z}},\\ w_j^{(o)}(t)=\begin{cases} w_j^o(t),& t\in (-\infty,0],\\ w_j^o(0),& t\in (0,\infty).\end{cases} \end{gather*} Then $W^{(o)}(t)$ is isotropic on $\mathbb{R}$, and $\operatorname{supp} W^{(o)}(t,\cdot)\subset B_{\bar N}$ for $t\in\mathbb{R}$. Similarly to \cite[Theorem 3.1]{LW}, we obtain a convergent sequence in $\Lambda_{\infty}$, which is denoted as $\{W^{(n)}(t)\}$,~$t\in[0,\infty)$. Let $$W(t)=\begin{cases} \lim_{n\to\infty}W^{(n)}(t), & t\in [0,\infty),\\ W^{(o)}(t),& t\in (-\infty,0].\end{cases}$$ By Lebesgue's dominated convergence theorem, \eqref{2.7} has a solution $W\in \Lambda_{\infty}$, which is isotropic on $\mathbb{R}$. For any $c_1>c_*$, let $c_2\in (c_*,c_1)$. By the assumption on $W^{(o)}$, we choose proper $N\in \mathbb{N}$ such that $$\label{2.8} w_j^{(o)}(t)e^{\lambda( j-c_2 t)}\leq Ke^{\lambda N}\quad \text{for }t\geq 0,\;\lambda>0,\; j\in\mathbb{Z}.$$ For $t\geq 0$, by \eqref{2.8} we have \label{2.9} \begin{aligned} & w_j^{(1)}(t)e^{\lambda(j-c_2 t)}\\ &=e^{-(\delta+\lambda c_2)t}\Big\{ w_j^{(o)}(0)e^{\lambda j}+\int_0^t e^{\delta s}D[w_{j+1}^{(o)}(s)e^{\lambda(j+1)} e^{-\lambda}+w_{j-1}^{(o)}(s)e^{\lambda(j-1)} e^{\lambda}]ds\\ &\quad +\frac{1}{2\pi}\int_0^t e^{\delta s}\int_{0}^{\infty}f(a)e^{-da} \sum_{l=-\infty}^{\infty} \beta(a,l)b(w_{l+j}(s-a))e^{\lambda(j+l)}e^{-\lambda l} da\ ds\Big\} \\ &\leq e^{-(\delta+\lambda c_2)t}\Big\{Ke^{\lambda N}+D\int_0^t Ke^{\lambda N}e^{(\delta+\lambda c_2)s}( e^{-\lambda}+ e^{\lambda})ds \\ &\quad +b'(0)\Big(\int_0^{\infty}f(a)e^{-da}e^{2D(cosh\lambda-1)a}da \Big) Ke^{\lambda N} \int_0^t e^{(\delta+\lambda c_2)s}ds\Big\} \\ &=e^{-(\delta+\lambda c_2)t}Ke^{\lambda N} \Big\{1+\big[D(e^{-\lambda}+e^{\lambda})\\ &\quad + b'(0)\int_0^{\infty}f(a)e^{-da}e^{2D(cosh\lambda-1)a}da \big]\int_0^t e^{(\delta+\lambda c_2)s}ds\Big\}\\ &\leq Ke^{\lambda N}[1+ L_{c_2}(\lambda)]. \end{aligned} From the above inequality and by induction, we obtain $$\label{2.10} w_j^{(n)}(t)e^{\lambda(j-c_2 t)}\leq Ke^{\lambda N}[1+ L_{c_2}(\lambda)+\dots +( L_{c_2}(\lambda))^n].$$ Noting $-d+b'(0)\int_0^{\infty}f(a)e^{-da}da>0$, we have $L_c(0)>1$ for $c>0$. Since $L_c(\lambda)=1$ has two roots for $c>c_*$, we can choose $\lambda >0$ such that $L_{c_2}(\lambda)<1$ for $c_2> c_*$. Clearly the right side of \eqref{2.10} is uniformly bounded for $n$, thus for every $j\in \mathbb{Z}$, $$w_j(t)\leq \frac{Ke^{\lambda N}}{1- L_{c_2}(\lambda)} e^{\lambda(c_2t-j)}\text{ for } t\geq 0.$$ Since $W$ is isotropic, we have $$w_j(t)\leq \frac{Ke^{\lambda N}}{1-L_{c_2}(\lambda)} e^{\lambda(c_2t-|j|)}\text{ for }t\geq 0;$$ thus, $$\sup\{w_j(t)|\ |j|\geq c_1t\}\leq \frac{Ke^{\lambda N}}{1- L_{c_2}(\lambda)} e^{\lambda(c_2-c_1)t}\to 0\text{ as }t\to \infty.$$ Hence we obtain $\lim_{t\to\infty}\sup\{w_j(t)|\ |j|\geq c_1t\}=0$, $c_1> c_*$. \end{proof} \section{The spreading speed and minimal speed} For $\Phi\in M_{\infty}$, $t\ge T>0$, $j\in\mathbb{Z}$, we define the mapping on $M_{\infty}= \{\Phi=\{\phi_j\}_{j\in \mathbb{Z}}: \phi_j\in C_{K}^+(\mathbb{R})\}$ by \begin{align*} E^T_j[\Phi](t)&:=\int_0^Te^{-\delta s}\{D[\phi_{j+1}(t-s)+\phi_{j-1}(t-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(\phi_{l+j}(t-s-a))da\}ds. \end{align*} \begin{lemma}\label{L2.2} Suppose $\Phi\in M_{\infty}$ and satisfies the following conditions: \begin{itemize} \item[(i)] for any $t'>0$, there exists an $N=N(t')\in\mathbb{N}$ such that for any $t\in [0,t']$,~$supp\Phi(t,\cdot)\subset B_N$; \item[(ii)] if $\{(t_n,j_n)\}_{n=1}^{\infty}\subset \mathbb{R}_+\times\mathbb{Z}$, $j_n\in \operatorname{supp}\Phi(t_n,\cdot)$, and $\lim_{n\to\infty}(t_n,j_n)=(t_0,j_0)$, then $j_0\in supp\Phi(t_0,\cdot)$. \end{itemize} For such $\Phi$, assume that $$\label{3.1} E^T[\Phi](t)\succ \Phi(t)\quad \text{for } t\ge T,$$ and the solution of \eqref{1.1} satisfies $$\label{3.2} W(\bar t+t)\succ\Phi(t)\quad \text{for } t\in (-\infty,T]$$ for some $\bar t\geq 0$. Then $$\label{3.3} W(\bar t+t)\succ\Phi(t)\quad \text{for } t\in [0,\infty).$$ \end{lemma} \begin{proof} Let $$\label{3.4} t_0=\sup\{t\geq T: W(\bar t+t)\succ\Phi(t)\}\geq T.$$ If $t_0<\infty$, since $W(t)$ is non-negative, there exists $\{(t_n,j_n)\}_{n=1}^{\infty}$ such that \begin{itemize} \item[(a)] $t_n\downarrow t_0$, $n\to\infty$, \item[(b)] $j_n\in \operatorname{supp}\Phi(t_n,\cdot)$, \item[(c)] $w_{j_n}(\bar t+t_n)\leq \phi_{j_n}(t_n)$. \end{itemize} By assumption (i), $\{j_n\}$ must be bounded. Thus $\{j_n\}$ is composed of finite integers and contains a convergent sub-sequence, which is a constant sequence $\{j_0\}$. From (b) and (c), we know that $j_0\in supp\Phi(t_0,\cdot)$ and $w_{j_0}(\bar t+t_0)\leq \phi_{j_0}(t_0)$. For $t_0\geq T$ and $\bar t\geq 0$, from \eqref{2.7} and \eqref{3.4} we have \begin{align*} w_{j_0}(\bar t+t_0) &\geq \int_0^T e^{-\delta s}\{D[w_{j_0+1}(\bar t+t_0-s) +w_{j_0-1}(\bar t+t_0-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(w_{j_0+l}(\bar t+t_0-s-a))da\}ds\\ &\geq \int_0^T e^{-\delta s}\{D[\phi_{j_0+1}(t_0-s) +\phi_{j_0-1}(t_0-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(\phi_{l+j_0}(t_0-s-a))da\}ds\\ &= E^T_{j_0}[\Phi](t_0)>\phi_{j_0}(t_0). \end{align*} Since $w_{j_0}(\bar t+t_0)\leq \phi_{j_0}(t_0)$, the above inequality is a contradiction. Thus we have $t_0=\infty$. \end{proof} Define $K_c=K_c(h,T,N,\lambda)$ by \label{3.5} \begin{aligned} &K_c(h,T,N,\lambda)\\ &= \int_0^T e^{-(\delta+\lambda c)s}\{D[e^{-\lambda} +e^{\lambda }]+\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)e^{\lambda l-\lambda ca}da\}ds\\ &=\frac{1-e^{-(\delta+\lambda c)T}}{\delta+\lambda c}\{D[e^{-\lambda} +e^{\lambda}]+\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)e^{\lambda l-\lambda ca}da\}. \end{aligned} \begin{lemma}\label{L2.3} For any $c\in (0,c_*)$, there exist $h\in (0,b'(0)),T>0$ and $N\in\mathbb{N}$ such that $$\label{3.6} K_c(h,T,N,\lambda)>1\quad\text{for }\lambda \in \mathbb{R}.$$ \end{lemma} \begin{proof} From the definition of $K_c(h,T,N,\lambda)$, we have $$K_c(h,T,N,-\lambda)\geq K_c(h,T,N,\lambda),\quad \lambda\geq 0.$$ We claim that $$K_c(h,T,N,\lambda)>1 \quad \text{for }\lambda\geq 0.$$ We first show that there exist $N_0>0,\lambda_0>0,h_0\in (0,b'(0))$ and $T_0>0$ such that $$K_c(h,T,N,\lambda)>1\quad\text{for }\lambda\geq \lambda_0,\; N\geq N_0,\; h\geq h_0,\; T\geq T_0.$$ However, we can choose proper $N_0>0$ and $h_0\in(0,b'(0))$ such that for all $T>0$, $N\geq N_0$ and $h\geq h_0$, $$\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{l=-N}^{N}\beta(a,l)e^{\lambda (l-ca)}da>0$$ holds uniformly for $\lambda\geq 0$. Since $$\lim_{\lambda\to\infty}\frac{e^{\lambda}}{\lambda c_*+\delta}=\infty,$$ we can choose $T_0>0$ and $\lambda_0>0$ such that \begin{gather*} 1-e^{-(\lambda c+\delta)T}\geq 1-e^{-\delta T}\geq 1-e^{-\delta T_0}>0, \\ \frac{D}{\lambda c+\delta}(1-e^{-\delta T_0})e^{\lambda}> \frac{D}{\lambda_0 c_*+\delta}(1-e^{-\delta T_0})e^{\lambda_0}\geq 1, \end{gather*} for $T\geq T_0$, $\lambda\geq \lambda_0$. For any $N\geq N_0,T\geq T_0,h\geq h_0$ and $\lambda\geq \lambda_0$, we have $$K_c(h,T,N,\lambda)> \frac{D}{\lambda_0 c_*+\delta}(1-e^{-\delta T_0})e^{\lambda_0}\geq 1.$$ If \eqref{3.6} is not true, there exist $\{h_n\},\{T_n\},\{\lambda_n\},\{N_n\}$ such that $h_n\uparrow b'(0)$, $T_n\uparrow \infty$, $N_n\uparrow\infty$, $\{\lambda_n\}\subset [0,\lambda_0]$ and $$K_c(h_n,T_n,N_n,\lambda_n)\leq 1, \quad n=1,2,\dots.$$ Since $\{\lambda_n\}$ is bounded, we choose a convergent sub-sequence $\{\lambda_{n_k}\}$. Obviously $\{\lambda_{n_k}\}$ has a finite limit, denotes as $\tilde{\lambda}$. By Fatou's lemma, we have 10, a continuous function  \omega= \omega(\zeta) defined on [0,\zeta_0], and a positive number \delta_1\in (0,1) such that $$\label{3.7} \begin{split} &\int_0^T e^{-\delta s}\big\{D[q(m+cs+1)+q(m+cs-1)] \\ &+\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\big\}ds \geq q(m-\delta_1), \end{split}$$ for m\in \mathbb{Z}, where q(y)=q(y; \omega(\zeta),\zeta). \end{lemma} \begin{proof} Define \begin{align*} L(\lambda) &=\int_0^T e^{-\delta s}\big\{D[e^{-\lambda(cs+1)} +e^{-\lambda(cs-1) }]\\ &\quad+\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)e^{-\lambda (l+cs+ca)}da\big\}ds, \end{align*} where T,h,N are defined in Lemma \ref{L2.3}. By Lemma \ref{L2.3}, for sufficiently large N, $$\label{3.8} L(\lambda)=K_c(h,T,N,\lambda)>1\quad\text{for } \lambda \in \mathbb{R}.$$ Let \lambda=\omega+i\zeta, then we have L(\lambda)|_{\lambda=\omega+i\zeta}=\operatorname{Re}[L(\lambda)]+i\ \operatorname{Im}[L(\lambda)], where \begin{gather*} \begin{aligned} &\operatorname{Re}[L(\lambda)]\\ &= D\int_0^T e^{-\delta s} \Big\{e^{-\omega(cs+1)}\cos\zeta(cs+1)+ e^{-\omega(cs-1)}\cos\zeta(cs-1)\Big\}ds\\ &\quad +\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)\Big\{\int_0^Te^{-\delta s} e^{-\omega(l+cs+ca)}\cos\zeta(l+cs+ca)ds\Big\}da, \end{aligned}\\ \begin{aligned} &\operatorname{Im}[L(\lambda)]\\ &= -D\int_0^Te^{-\delta s} \Big\{e^{-\omega(cs+1)}\sin\zeta(cs+1)+ e^{-\omega(cs-1)}\sin\zeta(cs-1)\Big\}ds\\ &\quad -\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l) \Big\{\int_0^T e^{-\delta s}e^{-\omega(l+cs+ca)} \sin\zeta(l+cs+ca)ds\Big\}da. \end{aligned} \end{gather*} Since L''(\lambda)>0 and \lim_{|\lambda|\to\infty}L(\lambda)=\infty for \lambda\in \mathbb{R}, L(\lambda) attains the minimal value at \lambda=\theta\in \mathbb{R}. Thus, \begin{align*} L'(\theta)&= -D\int_0^T e^{-\delta s}[(cs+1)e^{-\theta(cs+1)}+(cs-1) e^{-\theta(cs-1)}]ds\\ &\quad -\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l) \big[\int_0^T e^{-\delta s}(l+cs+ca)e^{-\theta(l+cs+ca)}ds\big]da=0. \end{align*} Define a function H=H(\omega,\zeta) by \begin{gather*} H(\omega,\zeta)=\frac{1}{\zeta} \operatorname{Im}[L(\lambda)]\qquad \text{for }\zeta\neq 0,\\ H(\omega,0)=\lim_{\zeta\to 0}H(\omega,\zeta)=L'(\omega). \end{gather*} Obviously H(\theta,0)=0 and \frac{\partial H}{\partial \omega}(\theta,0)=L''(\theta)>0. By implicit function theorem, there exist \zeta_1>0 and continuous function \omega= \omega(\zeta),\zeta)\in[0,\zeta_1] satisfying \omega(0)=\theta, and H( \omega(\zeta),\zeta)=0,~\zeta\in [0,\zeta_1]. Thus, $$\label{3.9} \operatorname{Im}[L(\lambda)]|_{\lambda= \omega(\zeta)+i\zeta}=0,\quad\zeta\in [0,\zeta_1].$$ By \eqref{3.5} and \eqref{3.9}, we have \operatorname{Re}[L(\omega+i\zeta)]|_{\omega=\theta,\zeta=0}=L(\theta)>1. $$Then there exists \zeta_2>0 such that $$\label{3.10} \operatorname{Re}[L( \omega(\zeta)+i\zeta)]>1,\quad \zeta\in[0,\zeta_2].$$ Let 0<\zeta\leq \zeta_0:=\min\{\zeta_1,\zeta_2,\frac{\pi}{N+2c_*T}\}. For m\in [0,\frac{\pi}{\zeta}],\ |l|\leq N and a,s\in [0,T],$$ -\frac{\pi}{\zeta}<-N\leq l\leq m+l+cs+ca\leq m+l+2cT < N+2c_*T +\frac{\pi}{\zeta}\leq \frac{2\pi}{\zeta}. Thus, $$\label{3.11} \sin\zeta(m+l+c(s+a))<0,\quad \text{for } m+l+c(s+a)\in (-\frac{\pi}{\zeta},0) \cup (\frac{\pi}{\zeta},\frac{2\pi}{\zeta}).$$ From the definition of q(\cdot) we obtain \label{3.12} \begin{aligned} &\int_0^T e^{-\delta s}\{D[q(m+cs+1)+q(m+cs-1)] \\ &\quad +\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\}ds\\ &\geq D\int_0^T e^{-\delta s}\Big\{e^{- \omega(\zeta)(m+cs+1)} \sin(\zeta(m+cs+1)) \\ &\quad +e^{- \omega(\zeta)(m+cs-1)} \sin(\zeta(m+cs-1))\Big\}ds\\ &\quad +\frac{h}{2\pi}\int_0^T e^{-\delta s} \int_{0}^{T}f(a)e^{-da}\\ &\quad\times \sum_{|l|\leq N} \beta(a,l)e^{- \omega(\zeta)(m+l+cs+ca)} \sin(\zeta(m+l+cs+ca))da\ ds. \end{aligned} Using \sin(A+B)=\sin A\cos B+\sin B\cos A and \eqref{3.10}-\eqref{3.12}, we have \label{3.13} \begin{aligned} &\int_0^T e^{-\delta s}\{D[q(m+cs+1)+q(m+cs-1)]\\ &\quad +\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\}ds\\ &\geq e^{- \omega(\zeta)m}\sin(\zeta m)\text{Re}[L(\lambda)]|_{\lambda= \omega(\zeta)+i\zeta}+e^{- \omega(\zeta)m}\cos(\zeta m) \operatorname{Im}[L(\lambda)]|_{\lambda= \omega(\zeta)+i\zeta}\\ &= e^{- \omega(\zeta)m}\sin(\zeta m)=q(m). \end{aligned} Choose N large enough such that -N+2c_*T<0, thus \eqref{3.12} and \eqref{3.13} are strict inequalities on m\in (0,\frac{\pi}{\zeta}). Moreover, from \eqref{3.11}-\eqref{3.12}, we know that \eqref{3.13} is also a strict inequality for m=0 or m=\frac{\pi}{\zeta}. In fact, let  a,s\in [0,T], ~m=\frac{\pi}{\zeta} and l=N, then m+l+c(s+a)>\frac{\pi}{\zeta}. $$Similarly, if m=0 and l=-N, then m+l+c(s+a)<-N+2c_*T<0. Thus for both cases, we have$$ q(m+l+cs+ca)=0 \quad \text{and}\quad \sin(\zeta(m+l+cs+ca))<0, which means \eqref{3.13} is a strict inequality for m=0 or m=\frac{\pi}{\zeta}. Then for any m\in [0,\frac{\pi}{\zeta}], \label{3.14} \begin{aligned} &\int_0^Te^{-\delta s}\{D[q(m+cs+1)+q(m+cs-1)]\\ &+\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\}ds>q(m). \end{aligned} If m\not\in [0,\frac{\pi}{\zeta}], \eqref{3.14} still holds since q(m)=0. From the above discussion, we know that \eqref{3.14} holds for m\in \mathbb{R}, then \eqref{3.7} follows from the continuity consideration. \end{proof} Consider the family of functions, \label{3.15} \begin{aligned} R(y;\omega,\zeta,\gamma):&=\max_{\eta\geq -\gamma} q(y+\eta;\omega,\zeta)\\ &=\begin{cases} M, & y\leq \gamma+\rho,\\ q(y-\gamma;\omega,\zeta),& \gamma+\rho\leq y\leq \gamma+ \frac{\pi}{\zeta},\\ 0,& y\geq \gamma+\frac{\pi}{\zeta}, \end{cases} \end{aligned} where $$\label{3.16} M=M(\omega,\zeta): =\max\{q(y;\omega,\zeta)|\ 0\leq y\leq\frac{\pi}{\zeta}\}.$$ We assume M attain the maximum at \rho=\rho(\omega,\zeta). The following lemma gives a sub-solution of \eqref{1.1}. \begin{lemma}\label{L2.5} Let c\in (0,c_*) be given, then there exist T>0,\zeta>0,\omega \in \mathbb{R},\vartheta>0 and \sigma_0>0 such that for \sigma \in (0,\sigma_0) and t\geq T, there holds $$\label{3.17} E^T[\sigma\Phi](t)\succ\sigma\Phi(t)\quad\text{for } t\geq T,$$ where \Phi(t)=\{\phi_j(t)\}_{j\in\mathbb{Z}},\phi_j(t) =R(|j|;\omega,\zeta,\vartheta+ct). \end{lemma} \begin{proof} Let h\in(0,b'(0)),T>0,N>0 be chosen such that K_c(h,T,N,\lambda)>1 for \lambda \in \mathbb{R}. By Lemma \ref{L2.4}, we can choose \zeta>0, \omega= \omega(\zeta) and \delta_1\in (0,1) such that \eqref{3.7} holds. Let \sigma_h be the smallest positive root of the equation b(w)=hw, then b(w)>hw for w\in (0,\sigma_h). Choose \sigma_0\in (0,\sigma_hM^{-1}), where M is defined in \eqref{3.16}. For \sigma\in (0,\sigma_0) and t\geq T, we have \label{3.18} \begin{aligned} E_j^T[\sigma\Phi](t) &= \int_0^T e^{-\delta s}\big\{D\sigma[\phi_{j+1}(t-s)+ \phi_{j-1}(t-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty}\beta(a,l) b(\sigma\phi_{j+l}(t-s-a))da\big\}ds\\ \geq& \int_0^T e^{-\delta s}\big\{D\sigma[\phi_{j+1}(t-s)+ \phi_{j-1}(t-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)b(\sigma\phi_{j+l}(t-s-a))da\big\}ds. \end{aligned} For any given \vartheta>0, we consider two cases. Case (i) |j|\leq \vartheta+\rho+c(t-2T)-N. For |l|\leq N, a, s\in [0,T], then |l+j|\leq \vartheta+\rho+c(t-2T)\leq \vartheta+\rho+c(t-s-a) Since the definition of E_j^T[\Phi](t) and b(\sigma\phi_{j+l}(t-s-a))=b(\sigma M)>h\sigma M, we have \label{3.19} \begin{aligned} E_j^T[\sigma\Phi](t) &\geq \big\{2D\sigma M+ \frac{1}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{l=-N}^{N}\beta(a,l) b(\sigma M)da\big\}\int_0^Te^{-\delta s}ds\\ &>\sigma MK_c(h,T,N,0)>\sigma M. \end{aligned} Case (ii) \vartheta+\rho+c(t-2T)-N\leq |j|\leq \frac{\pi}{\zeta}+\vartheta+ct. Let |l|\leq N,~t\geq T. If \vartheta\geq \frac{N^2}{2\delta_1}-\rho+cT +N  (\delta_1 is defined in Lemma \ref{L2.4}), then \begin{align*} |l+j| &=(l^2+2lj+j^2)^{1/2}\leq |j|+\frac{lj}{|j|}+\frac{l^2}{2|j|}\\ &\leq |j|+\frac{lj}{|j|}+\frac{N^2}{2|j|}\\ &\leq|j|+\frac{lj}{|j|}+\frac{N^2}{2(\vartheta+\rho-cT-N)}\leq|j| +\frac{lj}{|j|}+\delta_1. \end{align*} Since \phi_j(t) is non-decreasing for |j|, by \eqref{3.18} we obtain \begin{align*} & E_j^T[\sigma\Phi](t)\\ &\geq \int_0^T e^{-\delta s}\Big\{D\sigma[\max_{\eta\geq -\vartheta-c(t-s)}q(|j|+1+\delta_1+\eta)\\ &\quad +\max_{\eta\geq -\vartheta-c(t-s)} q(|j|-1+\delta_1+\eta)]\\ & \quad +\frac{h\sigma}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)\max_{\eta\geq -\vartheta-c(t-s-a)}q(|j|+l+\delta_1+\eta)da\Big\}ds\\ &= \sigma\int_0^T e^{-\delta s}\Big\{D[\max_{\eta\geq -\vartheta-ct}q(|j|+1+cs+\delta_1+\eta)\\ &\quad +\max_{\eta\geq -\vartheta-ct} q(|j|-1+cs+\delta_1+\eta)]\\ & \quad +\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)\max_{\eta\geq -\vartheta-ct}q(|j|+l+cs+ca+\delta_1+\eta)da \Big\}ds\\ \geq &\sigma \max_{\eta\geq -\vartheta-ct}q(|j|+\eta). \end{align*} Combining (i) and (ii), we obtain \eqref{3.17} and complete the proof. \end{proof} The proof of the following lemma is similar to \cite[Lemma 5.5]{WHW}, and hence is omitted. \begin{lemma}\label{L2.6} Assume that W=\{w_j\}_{j\in \mathbb{Z}} is a solution of \eqref{1.1}, and the following conditions hold: \begin{itemize} \item[(i)] W^o=\{w_j^o\}_{j\in \mathbb{Z}} is isotropic on (-\infty,0],  w_j^o\in C_{K}^+(-\infty,0]; \item[(ii)] there exists N_1\in \mathbb{N} such that \operatorname{supp} W^o(t,\cdot)\subset B_{N_1} for t\in (-\infty,0], w_j^o(0)>0 for j|\leq N_1. \end{itemize} Then there exists t_0>0 such that w_j(t)>0 for t\in [t_0,\infty), j\in\mathbb{Z}. \end{lemma} \begin{lemma}\label{L2.7} Let \{Q_n(t,N)\} be defined by Q_1(t,N)\equiv a\in (0,w^+), $$\label{3.20} \begin{split} Q_{n+1}(t,N)\ &=\frac{1}{\delta} \Big[2D Q_n(t,N) \\ &\quad +\frac{1}{2\pi}\int_{0}^{T}\{f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)da\}b(Q_n(t,N))\Big] (1-e^{-\delta t}) \end{split}$$ for n=1,2,\dots. Then for \epsilon>0, there exist \bar t(\epsilon), \bar N(\epsilon), \bar T(\epsilon)  and \bar n(\epsilon) such that for any T\geq \bar{T}(\epsilon),t\geq \bar t(\epsilon),N\geq \bar N(\epsilon) and n\geq \bar n(\epsilon), Q_n(t,N)\geq w^+-\epsilon. $$\end{lemma} \begin{proof} Since \begin{gather*} \frac{2D w^++b(w^+)\tilde f(d) }{\delta}=w^+,\quad \delta=2D+d,\quad \tilde f(d)=\int_{0}^{\infty}f(a)e^{-da}da, \\ 0(2D+d)w, for 00, we have$$ \sup \Big\{\frac{2Dw+\tilde f(d) b(w)}{(2D+d)w}|\ 01. $$Let \tilde f_T(d)=\int_{0}^{T}f(a)e^{-da}da. Choose large enough \alpha(\epsilon)<1, ~\bar T=\bar T(\epsilon)  such that for 0(2D+d)w. Define the sequence:$$ M_1\equiv a ,\quad M_{n+1}= \frac{\alpha(\epsilon)}{\delta}\Big[2DM_n +\tilde f_{{T}}(d) b(M_n)\Big]\text{ for } n\geq 2. $$Obviously, \begin{itemize} \item[(i)] if 0w^+-\epsilon, then$$ M_{n+1}> \frac{\alpha(\epsilon)}{\delta}\Big[2D(w^+-\epsilon) +\tilde f_{{T}}(d) b(w^+-\epsilon)\Big]\geq w^+-\epsilon. $$\end{itemize} Now we show that M_n>w^+-\epsilon for sufficiently large n. If that is not true, we can assume that M_n\leq w^+-\epsilon holds for all n. By (i), we know that \lim_{n\to \infty} M_n=M\leq w^+-\epsilon exists and satisfies$$ M=\frac{\alpha(\epsilon)}{\delta}[2DM+ b(M)\tilde f_{{T}}(d) ]. which is a contraction to \eqref{3.21}. Thus there exists \bar n(\epsilon)>0 such that M_n>w^+-\epsilon for any n>\bar n(\epsilon). Let T\geq \bar{T}=\bar{T}(\epsilon). We choose \bar t=\bar t(\epsilon) and \bar N=\bar N(\epsilon) such that 1-e^{-\delta \bar t(\epsilon)}\geq \alpha(\epsilon) and $$\label{3.22} \frac{1}{2\pi}(1-e^{-\delta \bar t(\epsilon)}) \int_{0}^{{T}}\{f(a)e^{-da}\sum_{|l|\leq \bar{N}} \beta(a,l)\}da \geq \alpha(\epsilon)\tilde f_{{T}}(d).$$ Then Q_1(t,N)=a\geq M_1 for t\geq \bar t(\epsilon),T\geq {\bar{T}(\epsilon)} and N\geq \bar N(\epsilon). By \eqref{3.22} we obtain \begin{align*} &Q_{n+1}(t,N)\\ &\geq \frac{1}{\delta} (1-e^{-\delta \bar t(\epsilon)})\Big[2DQ_n(t,N)+\frac{b(Q_n(t,N))}{2\pi} \int_{0}^{T}\{f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)\}da \Big]\\ & >\frac{1}{\delta}\Big[2D\alpha(\epsilon)Q_n(t,N) +\alpha(\epsilon)\tilde f_{T}(d)b(Q_n(t,N))\Big]\\ &=\frac{\alpha(\epsilon)}{\delta}\Big[2DQ_n(t,N) +\tilde f_{T}(d)b(Q_n(t,N))\Big]. \end{align*} Using monotonicity of b, we have Q_n(t,N)\geq M_n\geq w^+-\epsilon for n>\bar n(\epsilon). \end{proof} \begin{theorem}\label{T2.2} Assume all the conditions for W^o in Lemma \ref{L2.6} are satisfied. Then for any c\in (0,c_*), there holds \lim_{t\to\infty}\inf\{w_j(t): |j|\leq ct\}\geq w^+. $$\end{theorem} \begin{proof} Let c_1\in (0,c_*),~c_2\in (c_1,c_*). From Lemma \ref{L2.5}, there exist T>0,\zeta >0,\omega \in \mathbb{R},\vartheta>0 and \sigma_0>0 such that for \sigma \in (0,\sigma_0) บอ~t\geq T,$$ E^T[\sigma\Phi](t)\succ\sigma\Phi(t), $$where \Phi(t)=\{\phi_j(t)\}_{j\in \mathbb{Z}}, \phi_j(t):=R(|j|;\omega, \zeta,\vartheta+c_2 T). We can assume T\geq \bar{T}, and \bar{T} is defined in Lemma \ref{L2.7}. From Lemma \ref{L2.6}, there exists t_0>0 such that$$ w_j(t)>0\quad \text{for }t\in [t_0,t_0+T],\ j\in \mathbb{Z}. $$Since \Phi(t) is a bounded function, we can choose \sigma_1\in (0,\sigma_0) such that$$ \sigma_1M\sigma_1\phi_j(t)\quad \text{for } t\in [0,T],\ j\in \mathbb{Z}. Using the comparison principle (Lemma \ref{L2.2}), we have $$\label{3.23} w_j(t_0+t)>\sigma_1\phi_j(t)\quad \text{for }t\in [0,\infty),\; j\in \mathbb{Z}.$$ From \eqref{3.23} and definition of \phi_j(t), we have $$\label{3.24} w_j(t_0+t)\geq \sigma_1 M, \quad t\geq 0,\ |j|\leq \rho+\vartheta+c_2t.$$ By \eqref{2.7}, we have \label{3.25} \begin{aligned} w_j(t_0+t)&\geq \int_0^te^{-\delta s}\{ D[w_{j+1}(t_0+t-s)+w_{j-1}(t_0+t-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)b(w_{l+j}(t_0+t-s-a))da\} ds. \end{aligned} Let a=\sigma_1M=Q_1(t,N), and Q_n(t,N) be defined in Lemma \ref{L2.7}. From \eqref{3.24}-\eqref{3.25}, we have by induction w_j(t_0+t)\geq Q_n(t,N),\quad t\geq 0,|j|\leq \rho+\vartheta+c_2t-n(N+T). $$For any \epsilon >0, we choose \bar t(\epsilon), \bar T(\epsilon), \bar N(\epsilon) and \bar n(\epsilon) such that $$\label{3.26} w_j(t)\geq w^+-\epsilon,\quad t\geq t_0+\bar t(\epsilon),\quad |j|\leq \rho+\vartheta+c_2(t-t_0)-\bar n(\epsilon)(\bar N(\epsilon) +\bar T(\epsilon)).$$ Define$$ t_1:=\max\Big\{t_0+\bar t(\epsilon),\frac{\bar n(\epsilon) [\bar N(\epsilon)+\bar T(\epsilon)]+c_2t_0-\rho-\vartheta}{c_2-c_1}\Big\}. $$Since c_2>c_1 and \eqref{3.26}, we obtain$$ w_j(t)\geq w^+-\epsilon\quad\text{for }t\geq t_1,|\; j|\leq c_1t. Then \eqref{2.2} holds. \end{proof} The following theorem shows the relation between the minimal wave speed and the spreading speed. \begin{theorem}\label{T3.1} Assume {\rm (H1)--(H4)} are satisfied. Then lattice system \eqref{1.1} admits two equilibria, W=0 and W=w^+>0. Further, for c\geq c_*, Equation \eqref{1.1} has a monotone traveling wave satisfying $$\label{3.27} \lim_{s\to-\infty}\phi(s)=0,\quad \lim_{s\to\infty}\phi(s)=w^+.$$ For c\in(0, c_*), \eqref{1.1} has no monotone traveling wave satisfying \eqref{3.27}. \end{theorem} \begin{proof} From \cite[Theorem 5.1]{LW}, we have that \eqref{1.1} admits monotone traveling wave satisfying \eqref{3.27} for c> c_*, thus we only need to claim the case as c=c_*. Choose a sequence \{c_n\}\in (c_*,c_*+1] such that c_{n+1}>c_n and \lim_{n\to \infty}c_n=c_*. Then the wave equation \eqref{2.1} admits a wavefront connecting 0 with w^+, say \phi_n(j+c_nt), which has the speed c_n. It is easy to see 0< \phi_n(j+c_nt)< w^+, and \label{3.28} \begin{aligned} c\phi_n'(s) &= D[\phi_n(s+1) +\phi_n(s-1) -2\phi_n(s)] - d\phi_n(s)\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(\phi_n(s+l-c_na))da. \end{aligned} Since \eqref{3.28} is a homogeneous system, from the basis theory of differential equation, we know that a traveling wave of \eqref{3.28} is still another traveling wave after sliding. Without generality, we assume \phi_n(0)=\frac{w^+}{2}. Differentiating \eqref{3.28} with respect to s, we obtain \label{3.29} \begin{aligned} c\phi_n''(s) &= D[\phi'_n(s+1) +\phi'_n(s-1) -2\phi'_n(s)] - d\phi'_n(s)\\ &\quad+\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)\frac{db}{dw}(\phi_n(s+l-c_na))\phi_n'(s+l-c_na)da. \end{aligned} From \eqref{3.28} and 0< \phi_n(j+c_nt)< w^+, there exists M_1,M_2 such that |\phi_n'(s)|\leq M_1,~|\phi_n''(s)|\leq M_2 for s\in \mathbb{R}. Thus \phi_n and \phi'_n are uniformly bounded, equsi-continuous in \mathbb{R}. According to Arzela-Ascoli theorem, there has a sub-sequence of c_n, still denoted as c_n, such that \phi_n(s) and \phi'_n(s) are convergent to limits in every bounded and closed subset in \mathbb{R}. We denote the limits as \phi_*(s),\phi'_*(s) respectively. Let n\to \infty in \eqref{3.28}. By Lebesque's dominated convergence theorem, we have \label{3.30} \begin{aligned} c\phi_*'(s) &= D[\phi_*(s+1) +\phi_*(s-1) -2\phi_*(s)] - d\phi_*(s)\\ &\quad +\frac{1}{2\pi}\int_{0}^{\infty}f(a)e^{-da}\sum_{l=-\infty}^{\infty} \beta(a,l)b(\phi_*(s+l-c_*a))da. \end{aligned} Hence \phi_*(j+c_*t) is the traveling wavefront of \eqref{1.1} with speed c_* satisfying \eqref{3.1}. Now we prove \eqref{1.1} admits no traveling wavefront for c_1\in(0,c_*). Suppose that is not true, and system \eqref{1.1} has monotone traveling wave \phi(s)=\phi(j+c_1t) satisfying \eqref{3.27}. Thus there exists s_1>0 such that \phi(s)>\frac{w^+}{2} for s\geq s_1. Choose proper initial function: w_j^o(t)=\phi(j+c_1t), t\in (-\infty,0], and \{w_j^o(t)\}_{j\in \mathbb{Z}}\in C_K^+(-\infty,0]. Let \{w_j(t)=\phi(j+c_1t)\}_{j\in \mathbb{Z}} be a solution of \eqref{1.1} with initial value w_j^o(t). Noting \{w_j^o(t)\}_{j\in \mathbb{Z}} satisfying conditions in Theorem \ref{T2.2}, we have \lim_{t\to\infty}\inf\{w_j(t)|\ |j|\leq ct\} =\lim_{t\to\infty}\inf\{\phi(j+c_1t)\ |j|\leq ct\}\geq w^+\;\text{ for } c\in (0,c_*). $$Choose c_2\in(c_1,c_*), j=-c_2t, then$$ \phi(j+c_1t)=\phi((c_1-c_2)t) \geq w^+ \quad \text{for }t\geq t_1. $$Let t\to\infty, we have$$ \lim_{t\to\infty}\inf\{\phi(j+c_1t)\ | j=-c_2t\} =\lim_{t\to\infty}\inf\{\phi((c_1-c_2)t)\} \geq w^+,  which leads to a contradiction to the first equality in \eqref{3.27}. Hence \eqref{1.1} admits no monotone traveling wave for $c_1\in(0,c_*)$. \end{proof} \begin{thebibliography}{00} \bibitem{AW1} D. G. Aronson, H. F. Weinberger; \emph{Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics} (J. A. Goldstein, ed.), 5-49, Lecture Notes in Mathematics, 446 (1975), Springer-Verlag, Heidelberg/Berlin. \bibitem{CMS} S. N. Chow, J. Mallet-Paret, W. X. Shen; \emph{Traveling waves in lattice dynamical systems}, J. Differ. Equations 149 (1998) 248-291. \bibitem{HLR} J. H. Huang, G. Lu, S. G. Ruan; \emph{Traveling wave solutions in delayed lattice differential equations with partial monotonicity}, Nonlinear Anal. TMA. 60 (2005) 1331-1350. \bibitem{HLZ} J. H. Huang, G. Lu, X. F. Zou; \emph{Existence of traveling wavefronts of delayed lattice differential equations}, J. Math. Anal. Appl. 298 (2004) 538-558. \bibitem{LW} J. X. Ling, P. X. Weng; \emph{Extinction and wavefront in an age-structured population model on infinite patches with distributed maturation delay}, J. Systems Science and Complexity 18 (2005) 464-477. \bibitem{LZ} X. Liang, X.-Q. Zhao; \emph{Asymptotic speeds of spread and traveling waves for monotone semiflows with applications}, Commun. Pure Appl. Math. 60 (2007) 1-40. Erratum: 61 (2008) 137-138. \bibitem{MZ} S. W. Ma, X. F. Zou; \emph{Propagation and its failure in a lattice delayed differential equation with global interaction}, J. Differ. Equations 212 (2005) 129-190. \bibitem{Mallet2} J. Mallet-Paret; \emph{Traveling waves in spatially discrete dynamical systems of diffusive type}. Dynamical Systems(ed. T.W. Macki \& P. Zecca), Lecture Notes in Math. 1822 (2003) 231-298. Berlin: Springer-Verlag. \bibitem{Th2} H. R. Thieme; \emph{Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread}. J. Math. Biology 8 (1979) 173-187. \bibitem{WHW} P. X. Weng, H. X. Huang, J. H. Wu; \emph{Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction}. IMA J. Appl. Math. 68 (2003) 409-439. \bibitem{Widder} D. V. Widder; \emph{The Laplace Transform}, Princeton University Press, 1941. \end{thebibliography} \end{document}