\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 \begin{equation}\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} \end{equation} 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 \begin{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} \end{equation} 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 \begin{equation}\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. \end{equation} 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 \begin{equation}\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. \end{equation} 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 \begin{equation}\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), \end{equation} 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 \begin{equation} \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} \end{equation} 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 \begin{equation}\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}. \end{equation} For $t\geq 0$, by \eqref{2.8} we have \begin{equation}\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} \end{equation} From the above inequality and by induction, we obtain \begin{equation} \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]. \end{equation} 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 \begin{equation}\label{3.1} E^T[\Phi](t)\succ \Phi(t)\quad \text{for } t\ge T, \end{equation} and the solution of \eqref{1.1} satisfies \begin{equation}\label{3.2} W(\bar t+t)\succ\Phi(t)\quad \text{for } t\in (-\infty,T] \end{equation} for some $\bar t\geq 0$. Then \begin{equation}\label{3.3} W(\bar t+t)\succ\Phi(t)\quad \text{for } t\in [0,\infty). \end{equation} \end{lemma} \begin{proof} Let \begin{equation}\label{3.4} t_0=\sup\{t\geq T: W(\bar t+t)\succ\Phi(t)\}\geq T. \end{equation} 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 \begin{equation}\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} \end{equation} \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 \begin{equation}\label{3.6} K_c(h,T,N,\lambda)>1\quad\text{for }\lambda \in \mathbb{R}. \end{equation} \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 $$ 10$, a continuous function $ \omega= \omega(\zeta)$ defined on $[0,\zeta_0]$, and a positive number $\delta_1\in (0,1)$ such that \begin{equation}\label{3.7} \begin{split} &\int_0^T e^{-\delta s}\big\{D[q(m+cs+1)+q(m+cs-1)] \\ &+\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\big\}ds \geq q(m-\delta_1), \end{split} \end{equation} for $m\in \mathbb{Z}$, where $q(y)=q(y; \omega(\zeta),\zeta)$. \end{lemma} \begin{proof} Define \begin{align*} L(\lambda) &=\int_0^T e^{-\delta s}\big\{D[e^{-\lambda(cs+1)} +e^{-\lambda(cs-1) }]\\ &\quad+\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)e^{-\lambda (l+cs+ca)}da\big\}ds, \end{align*} where $T,h,N$ are defined in Lemma \ref{L2.3}. By Lemma \ref{L2.3}, for sufficiently large $N$, \begin{equation}\label{3.8} L(\lambda)=K_c(h,T,N,\lambda)>1\quad\text{for } \lambda \in \mathbb{R}. \end{equation} Let $\lambda=\omega+i\zeta$, then we have $$ L(\lambda)|_{\lambda=\omega+i\zeta}=\operatorname{Re}[L(\lambda)]+i\ \operatorname{Im}[L(\lambda)], $$ where \begin{gather*} \begin{aligned} &\operatorname{Re}[L(\lambda)]\\ &= D\int_0^T e^{-\delta s} \Big\{e^{-\omega(cs+1)}\cos\zeta(cs+1)+ e^{-\omega(cs-1)}\cos\zeta(cs-1)\Big\}ds\\ &\quad +\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)\Big\{\int_0^Te^{-\delta s} e^{-\omega(l+cs+ca)}\cos\zeta(l+cs+ca)ds\Big\}da, \end{aligned}\\ \begin{aligned} &\operatorname{Im}[L(\lambda)]\\ &= -D\int_0^Te^{-\delta s} \Big\{e^{-\omega(cs+1)}\sin\zeta(cs+1)+ e^{-\omega(cs-1)}\sin\zeta(cs-1)\Big\}ds\\ &\quad -\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l) \Big\{\int_0^T e^{-\delta s}e^{-\omega(l+cs+ca)} \sin\zeta(l+cs+ca)ds\Big\}da. \end{aligned} \end{gather*} Since $L''(\lambda)>0$ and $\lim_{|\lambda|\to\infty}L(\lambda)=\infty$ for $\lambda\in \mathbb{R}$, $L(\lambda)$ attains the minimal value at $\lambda=\theta\in \mathbb{R}$. Thus, \begin{align*} L'(\theta)&= -D\int_0^T e^{-\delta s}[(cs+1)e^{-\theta(cs+1)}+(cs-1) e^{-\theta(cs-1)}]ds\\ &\quad -\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l) \big[\int_0^T e^{-\delta s}(l+cs+ca)e^{-\theta(l+cs+ca)}ds\big]da=0. \end{align*} Define a function $H=H(\omega,\zeta)$ by \begin{gather*} H(\omega,\zeta)=\frac{1}{\zeta} \operatorname{Im}[L(\lambda)]\qquad \text{for }\zeta\neq 0,\\ H(\omega,0)=\lim_{\zeta\to 0}H(\omega,\zeta)=L'(\omega). \end{gather*} Obviously $H(\theta,0)=0$ and $\frac{\partial H}{\partial \omega}(\theta,0)=L''(\theta)>0$. By implicit function theorem, there exist $\zeta_1>0$ and continuous function $\omega= \omega(\zeta),\zeta)\in[0,\zeta_1]$ satisfying $\omega(0)=\theta$, and $H( \omega(\zeta),\zeta)=0,~\zeta\in [0,\zeta_1]$. Thus, \begin{equation}\label{3.9} \operatorname{Im}[L(\lambda)]|_{\lambda= \omega(\zeta)+i\zeta}=0,\quad\zeta\in [0,\zeta_1].\end{equation} By \eqref{3.5} and \eqref{3.9}, we have $$ \operatorname{Re}[L(\omega+i\zeta)]|_{\omega=\theta,\zeta=0}=L(\theta)>1. $$ Then there exists $\zeta_2>0$ such that \begin{equation}\label{3.10} \operatorname{Re}[L( \omega(\zeta)+i\zeta)]>1,\quad \zeta\in[0,\zeta_2]. \end{equation} Let $0<\zeta\leq \zeta_0:=\min\{\zeta_1,\zeta_2,\frac{\pi}{N+2c_*T}\}$. For $m\in [0,\frac{\pi}{\zeta}],\ |l|\leq N$ and $a,s\in [0,T]$, $$ -\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}. $$ Thus, \begin{equation}\label{3.11} \sin\zeta(m+l+c(s+a))<0,\quad \text{for } m+l+c(s+a)\in (-\frac{\pi}{\zeta},0) \cup (\frac{\pi}{\zeta},\frac{2\pi}{\zeta}). \end{equation} From the definition of $q(\cdot)$ we obtain \begin{equation}\label{3.12} \begin{aligned} &\int_0^T e^{-\delta s}\{D[q(m+cs+1)+q(m+cs-1)] \\ &\quad +\frac{h}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\}ds\\ &\geq D\int_0^T e^{-\delta s}\Big\{e^{- \omega(\zeta)(m+cs+1)} \sin(\zeta(m+cs+1)) \\ &\quad +e^{- \omega(\zeta)(m+cs-1)} \sin(\zeta(m+cs-1))\Big\}ds\\ &\quad +\frac{h}{2\pi}\int_0^T e^{-\delta s} \int_{0}^{T}f(a)e^{-da}\\ &\quad\times \sum_{|l|\leq N} \beta(a,l)e^{- \omega(\zeta)(m+l+cs+ca)} \sin(\zeta(m+l+cs+ca))da\ ds. \end{aligned} \end{equation} Using $\sin(A+B)=\sin A\cos B+\sin B\cos A$ and \eqref{3.10}-\eqref{3.12}, we have \begin{equation}\label{3.13} \begin{aligned} &\int_0^T e^{-\delta s}\{D[q(m+cs+1)+q(m+cs-1)]\\ &\quad +\frac{h}{2\pi} \int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)q(m+l+cs+ca)da\}ds\\ &\geq e^{- \omega(\zeta)m}\sin(\zeta m)\text{Re}[L(\lambda)]|_{\lambda= \omega(\zeta)+i\zeta}+e^{- \omega(\zeta)m}\cos(\zeta m) \operatorname{Im}[L(\lambda)]|_{\lambda= \omega(\zeta)+i\zeta}\\ &= e^{- \omega(\zeta)m}\sin(\zeta m)=q(m). \end{aligned} \end{equation} Choose $N$ large enough such that $-N+2c_*T<0$, thus \eqref{3.12} and \eqref{3.13} are strict inequalities on $m\in (0,\frac{\pi}{\zeta})$. Moreover, from \eqref{3.11}-\eqref{3.12}, we know that \eqref{3.13} is also a strict inequality for $m=0$ or $m=\frac{\pi}{\zeta}$. In fact, let $ a,s\in [0,T]$, ~$m=\frac{\pi}{\zeta}$ and $l=N$, then $$ 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}]$, \begin{equation}\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} \end{equation} 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, \begin{equation}\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} \end{equation} where \begin{equation}\label{3.16} M=M(\omega,\zeta): =\max\{q(y;\omega,\zeta)|\ 0\leq y\leq\frac{\pi}{\zeta}\}. \end{equation} 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 \begin{equation}\label{3.17} E^T[\sigma\Phi](t)\succ\sigma\Phi(t)\quad\text{for } t\geq T, \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\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} \end{equation} 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^+)$, \begin{equation}\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} \end{equation} 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. $$ Let $\tilde f_T(d)=\int_{0}^{T}f(a)e^{-da}da$. Choose large enough $\alpha(\epsilon)<1, ~\bar T=\bar T(\epsilon) $ such that for $0(2D+d)w. \end{equation} Define the sequence: $$ 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) ]. $$ which is a contraction to \eqref{3.21}. Thus there exists $\bar n(\epsilon)>0$ such that $M_n>w^+-\epsilon$ for any $n>\bar n(\epsilon)$. Let $T\geq \bar{T}=\bar{T}(\epsilon)$. We choose $\bar t=\bar t(\epsilon)$ and $\bar N=\bar N(\epsilon)$ such that $1-e^{-\delta \bar t(\epsilon)}\geq \alpha(\epsilon)$ and \begin{equation}\label{3.22} \frac{1}{2\pi}(1-e^{-\delta \bar t(\epsilon)}) \int_{0}^{{T}}\{f(a)e^{-da}\sum_{|l|\leq \bar{N}} \beta(a,l)\}da \geq \alpha(\epsilon)\tilde f_{{T}}(d). \end{equation} Then $Q_1(t,N)=a\geq M_1$ for $t\geq \bar t(\epsilon),T\geq {\bar{T}(\epsilon)}$ and $N\geq \bar N(\epsilon)$. By \eqref{3.22} we obtain \begin{align*} &Q_{n+1}(t,N)\\ &\geq \frac{1}{\delta} (1-e^{-\delta \bar t(\epsilon)})\Big[2DQ_n(t,N)+\frac{b(Q_n(t,N))}{2\pi} \int_{0}^{T}\{f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)\}da \Big]\\ & >\frac{1}{\delta}\Big[2D\alpha(\epsilon)Q_n(t,N) +\alpha(\epsilon)\tilde f_{T}(d)b(Q_n(t,N))\Big]\\ &=\frac{\alpha(\epsilon)}{\delta}\Big[2DQ_n(t,N) +\tilde f_{T}(d)b(Q_n(t,N))\Big]. \end{align*} Using monotonicity of $b$, we have $Q_n(t,N)\geq M_n\geq w^+-\epsilon$ for $n>\bar n(\epsilon)$. \end{proof} \begin{theorem}\label{T2.2} Assume all the conditions for $W^o$ in Lemma \ref{L2.6} are satisfied. Then for any $c\in (0,c_*)$, there holds $$ \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}. $$ Using the comparison principle (Lemma \ref{L2.2}), we have \begin{equation}\label{3.23} w_j(t_0+t)>\sigma_1\phi_j(t)\quad \text{for }t\in [0,\infty),\; j\in \mathbb{Z}. \end{equation} From \eqref{3.23} and definition of $\phi_j(t)$, we have \begin{equation}\label{3.24} w_j(t_0+t)\geq \sigma_1 M, \quad t\geq 0,\ |j|\leq \rho+\vartheta+c_2t. \end{equation} By \eqref{2.7}, we have \begin{equation}\label{3.25} \begin{aligned} w_j(t_0+t)&\geq \int_0^te^{-\delta s}\{ D[w_{j+1}(t_0+t-s)+w_{j-1}(t_0+t-s)]\\ &\quad +\frac{1}{2\pi}\int_{0}^{T}f(a)e^{-da}\sum_{|l|\leq N} \beta(a,l)b(w_{l+j}(t_0+t-s-a))da\} ds. \end{aligned} \end{equation} Let $a=\sigma_1M=Q_1(t,N)$, and $Q_n(t,N)$ be defined in Lemma \ref{L2.7}. From \eqref{3.24}-\eqref{3.25}, we have by induction $$ w_j(t_0+t)\geq Q_n(t,N),\quad t\geq 0,|j|\leq \rho+\vartheta+c_2t-n(N+T). $$ For any $\epsilon >0$, we choose $\bar t(\epsilon), \bar T(\epsilon), \bar N(\epsilon)$ and $\bar n(\epsilon)$ such that \begin{equation}\label{3.26} w_j(t)\geq w^+-\epsilon,\quad t\geq t_0+\bar t(\epsilon),\quad |j|\leq \rho+\vartheta+c_2(t-t_0)-\bar n(\epsilon)(\bar N(\epsilon) +\bar T(\epsilon)). \end{equation} Define $$ 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 \begin{equation}\label{3.27} \lim_{s\to-\infty}\phi(s)=0,\quad \lim_{s\to\infty}\phi(s)=w^+. \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\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} \end{equation} 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. 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