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\nologo \TagsOnRight \NoBlackBoxes
\pageno=211
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{\hfil Traveling Wave Fronts
\hfil\folio}
\def\leftheadline{\folio\hfil Xingfu Zou
\hfil}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Differential Equations and Computational Simulations III\hfill\break
J. Graef, R. Shivaji, B. Soni, \& J. Zhu (Editors)\hfill\break
Electronic Journal of Differential Equations, Conference~01, 1997.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp 147.26.103.110 or 129.120.3.113 (login: ftp)\bigskip\bigskip} }
\topmatter
\title
Traveling wave fronts in spatially discrete reaction-diffusion equations
on higher dimensional lattices
\endtitle
\thanks
{\it 1991 Mathematics Subject Classifications:} 34B99,34C37, 34K99, 35K57.
\hfil\break\indent
{\it Key words and phrases:} spatially discrete, reaction-diffusion equation,
delay, lattice, \hfil\break\indent
traveling wave front, upper-lower solution.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and
University of North Texas. \hfil\break\indent
Published November 12, 1998. \hfil\break\indent
Supported by an NSERC (Canada) Postdoctoral Fellowship.
\endthanks
\author Xingfu Zou\endauthor
\address
Xingfu Zou \hfil\break
Department of Mathematics and Statistics,
University of Victoria, Victoria, BC, Canada V8W 3P4\hfil\break
Current address: Center for Dynamical Systems and Nonlinear Studies,
Georgia Institute of Technology, Atlanta, GA 30332-0190, USA.
\endaddress
\email xzou\@math.gatech.edu
\endemail
\abstract
This paper deals with the existence of traveling wave fronts of spatially
discrete reaction-diffusion equations with delay on lattices with general
dimension. A monotone iteration starting from an upper solution is
established, and the sequence generated from the iteration is shown to
converge to a profile function. The main theorem is then applied to a
particular equation arising from branching theory.
\endabstract
\endtopmatter
\document
\head 1. Introduction \endhead
Consider the spatially discrete reaction-diffusion equation
$$
u'_\eta (t)=\alpha (\Delta_n u)_\eta +f(u_\eta ), \quad \eta
\in \Omega \subset {\Bbb Z}^n
\tag1.1
$$
where $\alpha >0$ is a constant, and $\Delta_n$ is the standard $n$-dimensional
discrete Laplacian,
$$
(\Delta_n u)_\eta =\big(\sum_{|\xi-\eta|=1}u_{\xi} \big)-2n\,u_\eta \,.
\tag1.2
$$
where $| \cdot |$ denotes the Euclidean norm in ${\Bbb R}^n$.
Systems of differential equations with an underlying
lattice structure (referred as {\it lattice differential equations} in
literature) occur in mathematical models in many scientific disciplines,
and have attracted many mathematicians and scientists from other fields.
We mention here, among the others, materials science [1],
population biology [13,16], pattern recognition [3,4].
For additional references, see the excellent survey papers [5,6,17,20].
In addition to the above motivation for studying
equation (1.1), there are also some theoretical reasons. As indicated
in the title of this paper, Eq.(1.1) is a spatial discretization of the
partial differential equation (reaction-diffusion equation)
$$
\frac{\partial u(t,x)}{\partial t}=\alpha \Delta u(t,x)+f(u(t,x)),
\quad x \in \Omega \in {\Bbb R}^n
\tag1.3
$$
where $\Delta$ is the usual Laplacian with respect to the spatial
variable $x$. Therefore, it is interesting and worthwhile to compare
the dynamics of (1.3) with that of (1.1). It has been noticed that an
anisotropy in directional dependence is often introduced in
discretizing the $n$-dimensional Laplacian for $n\ge 2$, and thus,
spatially discrete equations often exhibit more complicated and richer
dynamics than spatially continuous equations. See, for example,
[1,6,7,22,27].
We all know that traveling wave solutions play an important role in
understanding the dynamics of the PDE (1.3). Naturally, we expect that
traveling wave solutions be also an important class of solutions for
(1.1). For (1.3), a traveling wave solution takes the form $u(t,x)=
\phi(\sigma\cdot x+ct)$ for some function $\phi: {\Bbb R} \to {\Bbb R}$
where $\sigma \in {\Bbb R}^n$ is a unit vector representing the
direction of motion of the wave, and $c>0$ is the wave
speed. Note that both the wave profile function and the wave
speed $c$ are
unknown. By substituting the traveling wave formula into (1.3), we
arrive at a second order ordinary differential equation
$$
c\phi'(\xi)=\alpha \phi''(\xi)+f(\phi(\xi)), \quad \xi \in {\Bbb R}
\tag1.4
$$
where $\xi=\sigma \cdot x+ct$ is the moving coordinate. Usually, one
imposes the boundary conditions
$$
\phi(- \infty)=q_-, \quad \phi(\infty)=q_+
\tag1.5
$$
to obtain a traveling wave {\it front} that represents a transition
from one equilibrium to another in applications.
Observe that (1.4) is {\it independent}
of the dimension $n$ and the direction $\sigma$.
Analogously, for the discrete reaction-diffusion equation (1.1) we can
also look for traveling wave solutions of the form $u_\eta (t)=
y(\sigma \cdot \eta+ct)$, where $\sigma=(\sigma_1,\sigma_2,\cdots,
\sigma_n) \in {\Bbb R}^n$ and $c>0$ are as before.
Now substitution of $u_\eta (t)=y(\sigma \cdot \eta+ct)$ into (1)
yields the difference-differential equation
$$
cy'(s)=\alpha \sum_{k=1}^n [y(s+\sigma_k)+y(s-\sigma_k)]
-2 \alpha n y(s)+f(y(s)), \quad s \in {\Bbb R}
\tag1.6
$$
where $s=\sigma \cdot \eta +ct$. Just as for PDE case, one also
imposes the boundary conditions
$$
y(- \infty)=q_- ,\quad y(\infty)=q_+
\tag1.7
$$
for Eq.(1.6). One notices
that in contrast to the second order ordinary differential equation
(1.4), the difference-differential equation (1.6) is a genuinely infinite
dimensional problem. Moreover, it depends on
the dimension $n$ as well as the direction $\sigma$ and involves
not only retarded but also advanced arguments.
While a great deal is known [10]
about differential equations with retarded arguments, very little
of any general theory has addressed the so-called ``mixed'' type
equation (1.6) in which both forward $s+\sigma_k$ and backward
$s-\sigma_k$ shifts of the argument $s$ appear. It was not until
recently, a systematic study of the general
theory of such mixed equations and of the global structure of the
solutions was initiated in [18,19].
There have been many arguments and evidences that time delay always
exists in reality and should be taken into consideration in modeling.
See, for example, [8,9,10,15,21]. For this reason, we incorporate a
discrete delay into (1.1) and (1.3) to consider, respectively,
$$
u'_\eta (t)=\alpha (\Delta_n u)_\eta
+f(u_\eta (t), u_\eta (t-\tau)), \quad \eta
\in {\Bbb Z}^n
\tag1.8
$$
and
$$
\frac{\partial u(t,x)}{\partial t}=\alpha \Delta u(t,x)
+f(u(t,x),u(t-\tau,x)),
\quad x \in {\Bbb R}^n.
\tag1.9
$$
The corresponding wave equations for (1.8) and (1.9) become, respectively,
$$
cy'(s)=\alpha \sum_{k=1}^n [y(s+\sigma_k)+y(s-\sigma_k)]
-2 \alpha n y(s)+f(y(s),y(s-c\tau)),\,\,\,\, s \in {\Bbb R}
\tag1.10
$$
and
$$
c\phi'(\xi)=\alpha \phi''(\xi)+f(\phi(\xi),\phi(\xi-c\tau)),
\quad \xi \in {\Bbb R}.
\tag1.11
$$
Here, (1.11) is an ordinary differential equation with only retarded
argument, but (1.10) is again a {\it mixed} equation.
In this paper, we deal with the existence of traveling wave fronts of
the delayed lattice differential equations (1.8). We mention that for
$\tau=0$, existence results are established in [11,12, 24-26],
for one dimensional lattice ($n=1$)
using comparison and continuation methods. In [2]
the existence of traveling wave fronts is explored, for two dimensional
($n=2$) lattice differential equations with some idealized nonlinearities
by considering differential inclusion.
Recently (1.8) was studied, [22], with $n=1$
but with general delay. In the remainder of this paper,
we follow the idea of upper and lower solutions in
[22] to study the existence of traveling wave fronts of
(1.8) with general dimension $n$. The rest of this paper is organized as
follows. In section 2, we establish an iteration scheme starting
from an upper solution, and prove that the iteration converges to a
solution of (1.10) and (1.7) provided that the upper solution
is properly chosen. In Section 3, we apply the main theorem established
in Section 2 to a particular equation arising from branching theory.
By analyzing the corresponding characteristic equation, we are able to
construct the required ordered pair of upper and lower solutions, and thus,
claim the existence of traveling wave fronts with large velocity.
\heading { 2. Monotone Iteration} \endheading
We have seen in Section 1 that the existence of traveling wave fronts of
(1.8) is equivalent to the existence of solutions of (1.10) and (1.7).
Without loss of generality, we can assume $q_-=0$ and $q_+=q>0$, because
other cases can be reduced to such a case simply by a translation. So, in
what follows, we look for solutions of (1.10) and (1.7) with $q_-=0$ and
$q_+=q>0$, i.e., solutions of
$$
cy'(s)=\alpha \sum_{k=1}^n [y(s+\sigma_k)+y(s-\sigma_k)]
-2 \alpha n y(s)+f(y(s),y(s-c\tau)), \quad s \in {\Bbb R}
\tag2.1
$$
and
$$
y(- \infty)=0,\quad y(\infty)=q.
\tag2.2
$$
It is obvious that if (2.1)-(2.2) has a solution, then $0$ and $q$ must be
zeros of the nonlinear function $f$. Thus, it is natural to make the
following assumption.
\parindent=20pt
\itemitem{\bf (A1)} $f$ is continuous and $f(0,0)=0=f(q,q)$ and
$f(u,u) \ne 0$ for $u \in (0,q)$.
\parindent=12pt
Moreover, in order to get the monotonicity of our iteration, we need the
following quasi-monotonicity condition for $f$.
\parindent=20pt
\itemitem{\bf (A2)} There exists a $\beta >0$ such that for any $u_1$,
$u_2$, $v_1$ and $v_2$ with $0 \le u_1 \le u_2 \le q$ and $0 \le v_1
\le v_2 \le q$, one has
$$
f(u_2,v_2)-f(u_1,v_1)+\beta (u_2-u_1) \ge 0\,.
$$
\parindent=12pt
Define the set of profiles by
$$
\Gamma=\left\{ \aligned &\rho:\,{\Bbb R} \to [0,q],\quad
\rho \quad \text{is continuous and nondecreasing},\\
&\lim_{t \to -\infty}\rho(t)=0 ,\quad \text{and} \quad
\lim_{t \to \infty}\rho(t)=q.
\endaligned
\right\}
$$
Also define $H_{\beta}:\,C({\Bbb R};{\Bbb R}) \to C({\Bbb R};{\Bbb R})$ by
$$
H_{\beta}(\rho)(t)=f(\rho(t),\rho(t-c\tau))+\beta \rho(t)
+\alpha \sum_{k=1}^n [\rho(t+\sigma_k)+\rho(t-\sigma_k)],\,\,\,\,t\in {\Bbb R}.
\tag2.3
$$
Then $H_{\beta}$ has the following properties.
\proclaim{\bf Proposition 2.1} Assume (A1) and (A2) are satisfied.
\itemitem{(i)} If $\rho$ is in $\Gamma$, then
$H_{\beta}(\rho)(t) $ is nondecreasing, and
$\lim_{t \to -\infty}H_{\beta}(\rho)(t)=0$ and
$\lim_{t \to \infty}H_{\beta}(\rho)(t)=(\beta +2n \alpha)q$;
\itemitem{(ii)} $H_{\beta}(\psi)(t) \le H_{\beta}(\phi)(t) $
for $t \in {\Bbb R}$, if
$\psi,\,\,\,\phi \in C({\Bbb R}, {\Bbb R})$ with $0\le \psi \le \phi \le q$.
\endproclaim
\demo{\bf Proof} The two limits in (i) are obvious. Fix $t \in
{\Bbb R}$ and $s>0$. Using (A2), we get
$$
\aligned
&H_{\beta}(\rho)(t+s)-H_{\beta}(\rho)(t) \\
&=f(\rho(t+s),\rho(t+s-c \tau))-f(\rho(t), \rho(t-c \tau))
+\beta[\rho(t+s)-\rho(t)]\\
&\,\,\,\,+\alpha \sum_{k=1}^n[\rho(t+s+\sigma_k)-\rho(t+\sigma_k)]
+\alpha \sum_{k=1}^n[\rho(t+s-\sigma_k)-\rho(t-\sigma_k)]\\
& \ge 0.
\endaligned
$$
This proves (i). As for (ii), it is just an immediate consequence of (A2).
This completes the proof.
\enddemo
Denote $\mu = \beta +2n \alpha$ and rewrite (2.1) as
$$
c\frac{d}{dt}y(t)=\,-\mu y(t)+H_{\beta}(y)(t).
\tag2.4
$$
It is easy to verify that $y:\,{\Bbb R} \to [0,q]$ is a solution of (2.4)
with $\lim _{t\to -\infty }y(t)=0$ if and only if it solves the
following integral equation
$$
y(t)=e^{-\frac{\mu t}{c}} \int_{- \infty}^t\,\frac{1}{c}
e^{\frac{\mu s}{c}}H_{\beta}(x)(s)\,ds.
\tag2.5
$$
\demo{\bf Definition 2.1} $\rho \in C({\Bbb R}, {\Bbb R})$ is called
an {\it upper solution} of (2.1) if it is
differentiable almost everywhere,
and satisfies
$$
c\frac{d}{dt}\rho (t) \ge
\alpha \sum_{k=1}^n[\rho(t+\sigma_k)+\rho(t-\sigma_k)]-2n \alpha \rho(t)+
f(\rho(t),\rho(t-c\tau))
\tag2.6
$$
a.e. on ${\Bbb R}$. {\it Lower solutions} of (2.1) can be similarly defined
by reversing the inequality in (2.6).
\enddemo
We now establish an iteration that generates a monotone sequence. In order
to start the iteration, let us first assume that there exist
an upper solution $\overline{\rho}(t)$ that is in $\Gamma$ and a lower solution
$\underline{\rho}(t)$ (not necessarily in $\Gamma$) of (2.1)
with $0 \le \underline{\rho}(t) \le \overline{\rho}(t) \le q$ for $t \in {\Bbb R}.$ We assume
$\underline{\rho}$ is a nontrivial lower solution (that is , $\underline{\rho} \not \equiv 0$
on ${\Bbb R}$). It is easy to verify that $y_1:\,
{\Bbb R} \to {\Bbb R}$ given by
$$
y_1(t)=e^{-\frac{\mu t}{c}} \int_{-\infty}^t \frac{1}{c}
e^{\frac{\mu s}{c}}H_{\beta}(\overline{\rho})(s)\,ds,\quad t
\in {\Bbb R}
\tag2.7
$$
is a well defined $C^1$- function. Some of the important properties of
$y_1$ are formulated as follows:
\proclaim{\bf Proposition 2.2} The function $y_1$ defined by (2.7) satisfies
\itemitem{(i)} $\frac{d}{dt}y_1(t) \ge 0$ for $t\in {\Bbb R}$;
\itemitem{(ii)} $\underline{\rho}(t) \le y_1(t) \le \overline{\rho}(t)$ for $t \in {\Bbb R}$;
\itemitem{(iii)}$\lim_{t \to -\infty}
y_1(t)=0$ and $\lim_{t \to + \infty}y_1(t) =q.$
\endproclaim
\demo{\bf Proof} Using the monotonicity of $\overline{\rho}$ and (i)
of Proposition 2.1, we get
$$
\aligned
\frac{d}{dt}y_1(t)&=\,-\frac{\mu}{c} e^{-\frac{\mu t}{c}}\int_{-\infty}^t
e^{\frac{\mu s}{c}}H_{\beta}(\overline{\rho})(s)\,ds+\frac{1}{c}H_{\beta}(\overline{\rho})(t)\\
&=\,-\frac{\mu}{c} e^{-\frac{\mu t}{c}}\int_{-\infty}^t
e^{\frac{\mu s}{c}}H_{\beta}(\overline{\rho})(s)\,ds
+\frac{\mu}{c} e^{-\frac{\mu t}{c}}
\int_{-\infty}^t e^{\frac{\mu s}{c}}H_{\beta}(\overline{\rho})(t)\,ds\\
&=\,\frac{\mu}{c} e^{-\frac{\mu t}{c}}\int_{-\infty}^t
e^{\frac{\mu s}{c}}[H_{\beta}(\overline{\rho})(t)-
H_{\beta}(\overline{\rho})(s)]\,ds \ge 0.
\endaligned
$$
Applying the L' Hospital's rule, we get
$$
\aligned
&\lim_{t \to -\infty}y_1(t)
=\lim_{t \to -\infty}\frac{\frac{1}{c}e^{\frac{\mu t}{c}}
H_{\beta}(\overline{\rho})(t)}{\frac{\mu}{c} e^{\frac{\mu t}{c}}}
= \lim_{t \to -\infty}\frac{1}{\mu}H_{\beta}(\overline{\rho})(t)=0;\\
&\lim_{t \to \infty}y_1(t)
=\lim_{t \to \infty}\frac{\frac{1}{c}e^{\frac{\mu t}{c}}
H_{\beta}(\overline{\rho})(t)}{\frac{\mu}{c} e^{\frac{\mu t}{c}}}
= \lim_{t \to -\infty}\frac{1}{\mu}H_{\beta}(\overline{\rho})(t)
=\frac{(\beta+2n \alpha)q}{\mu}=q.
\endaligned
$$
The inequality $\underline{\rho}(t) \le y_1(t) \le \overline{\rho}(t)$ for $t \in {\Bbb R}$
follows from the definition of $y_1$, the upper solution and the
monotonicity $H_{\beta}(\overline{\rho})(t) \ge H_{\beta}(\underline{\rho})(t)$ for $t \in {\Bbb R}.$
This completes the proof.
\enddemo
Note that by (ii) of Proposition 2.1, we have
$$
\aligned
&c\frac{d}{dt}y_1(t)\\
&=\, -\mu y_1(t)+H_{\beta}(\overline{\rho})(t)\\
&\ge \,- \mu y_1(t)+H_{\beta}(y_1)(t)\\
&=f(y_1(t), y_1(t-c\tau))
+\alpha \sum_{k=1}^n[y_1(t+\sigma_k)+y_1(t-\sigma_k)]-2n \alpha y_1(t),
\quad t \in {\Bbb R}.
\endaligned
$$
Therefore, $y_1$ is also an upper solution of (2.1) and is in $\Gamma$. Thus,
we can repeat the above process for the pair $(y_1,\, \underline{\rho})$ to obtain
another upper solution
$$
y_2(t)=\,\frac{1}{c}e^{- \frac{\mu t}{c}}\int_{- \infty}^t
e^{\frac{\mu s}{c}} H_{\beta}(y_1)(s)\,ds, \quad t \in {\Bbb R}.
\tag2.8
$$
Inductively, we can define
$$
y_n(t)=\,\frac{1}{c}e^{- \frac{\mu t}{c}}\int_{- \infty}^t
e^{\frac{\mu s}{c}} H_{\beta}(y_{n-1})(s)\,ds, \quad t
\in {\Bbb R},\,\,\,n \ge 2
\tag2.9
$$
and obtain:
\proclaim{\bf Proposition 2.3} The above sequence is well-defined and
satisfies
\itemitem{(i)} $\frac{d}{dt}y_n(t) \ge 0$ for $ t \in {\Bbb R}$;
\itemitem{(ii)} $\lim_{t \to -\infty}y_n(t)=0,\,\,\,\,\,
\lim_{t \to +\infty}y_n(t)=q$;
\itemitem{(iii)} $\underline{\rho}(t) \le y_n(t) \le y_{n-1}(t) \le
\overline{\rho}(t)$ for $t\in {\Bbb R}$ and $n \ge 2.$
\endproclaim
The monotonicity (iii) in the above result ensures the existence of
$$
y(t)= \lim_{n \to \infty}y_n(t), \quad t \in {\Bbb R}.
\tag2.10
$$
Clearly, the limit function $y:\, {\Bbb R} \to {\Bbb R}$ is nondecreasing.
Moreover, we claim
\proclaim{\bf Proposition 2.4} $y :{\Bbb R} \to {\Bbb R} $ obtained from
(2.7), (2.8), (2.9) and (2.10) is a solution of the asymptotic
boundary value problem (2.1)-(2.2).
\endproclaim
\demo{\bf Proof} Applying the Lebesgue's Dominated Convergence
Theorem to (2.9), we can establish
$$
y(t)=\frac{1}{c}e^{-\frac{\mu t}{c}}\int_{- \infty}^t\,
e^{\frac{\mu s}{c} }H_{\beta}(y)(s)\, ds
\tag2.11
$$
from which it follows that $y$ satisfies (2.1). $\lim_{t \to -\infty}
y(t)=0$ is obvious since $0 \le \underline{\rho}(t) \le y(t) \le \overline{\rho}(t)$
and $\overline{\rho} \in \Gamma$.
It remains to show
that $\lim_{t \to \infty}y(t) =q$.
Note that $y$ is nondecreasing and bounded. So $y^*:=
\lim_{t \to \infty}y(t)\le q$ exists. Taking limit as $t\to \infty $
in (2.1), we get $f(y^*,y^*)=0$. On the other hand, we have $y_n(t)
\ge \underline{\rho}(t)$ for $n \ge 1$ and $t \in {\Bbb R}$. Therefore,
$y(t) \ge \underline{\rho}(t)$ and hence $y^* \ge \sup_{t \in {\Bbb R}}\underline{\rho}(t) >0.$
Consequently, in view of (A1), we must have $y^* =q$. This completes
the proof.
\enddemo
Summarizing the above propositions, we have
\proclaim{\bf Theorem 2.5} Assume (A1) and (A2) are satisfied. Suppose (2.1)
has an upper solution $\overline{\rho}$ in $\Gamma$ and a non-trivial lower solution
$\underline{\rho}$ (not necessarily in $\Gamma$) satisfying
\parindent=20pt
\itemitem{\bf (H1)} $0 \le \underline{\rho}(t) \le
\overline{\rho}(t) \le q, \quad t \in {\Bbb R}$.
\parindent=12pt
\noindent
Then, (2.1)-(2.2) has a solution in $\Gamma$, that is, (1.8) has a traveling wave
front.
\endproclaim
\demo{\bf Remark 2.1} In the proof of Theorem 2.5, the assumption
$f(r,r,)\ne 0$ for $r \in (0,q)$ in (A1) is used only in proving
$\lim_{t \to \infty}y(t)=q$. Therefore, any replacement that ensures
$\lim_{t \to \infty}y(t)=q$ will not change the conclusion of Theorem 2.5.
So, we have
\enddemo
\proclaim{\bf Theorem 2.5$^*$} Assume $f$ is continuous and (A2)
is satisfied. Suppose (2.1)
has an upper solution $\overline{\rho}$ in $\Gamma$ and a non-trivial lower solution
$\underline{\rho}$ (not necessarily in $\Gamma$) satisfying (H1) and
\parindent=20pt
\itemitem{{\bf (A1)}$^*$} $f(u,u) \ne 0$ for $u \in (m,q)$, where $m=\sup_{t \in
{\Bbb R}} \underline{\rho}(t)$.
\parindent=12pt
\noindent
Then, (2.1)-(2.2) has a solution in $\Gamma$, that is, (1.8) has a traveling wave
front.
\endproclaim
\demo{\bf Remark 2.2} In (1.8), we just incorporated a single discrete delay.
The approach used in Section 2 is also applicable to lattice differential
equations with general delay, i.e., equations of the form
$$
u'_\eta (t)=\alpha (\Delta_n u)_\eta +f((u_\eta )_t), \quad \eta
\in {\Bbb Z}^n
\tag2.12
$$
where $f: C([-\tau,0];{\Bbb R}) \to {\Bbb R}$ and $(u_\eta )_t \in
C([-\tau,0];{\Bbb R})$ is defined by $(u_\eta )_t(\theta)
=u_\eta (t+\theta)$ for $\theta \in [-\tau, 0]$. In such a general case,
the quasi-monotonicity condition (A2) should be replaced by
\parindent=20pt
\itemitem{{\bf (A2)}$^*$} There exists a $\beta >0$ such that for any
$\phi,\,\,\,\,\psi \in C([-\tau,0];{\Bbb R})$ with $0 \le \phi \le \psi \le q$,
one has
$$
f(\psi)-f(\phi)+\beta[\psi(0)-\phi(0)] \ge 0.
$$
\parindent=12pt
\noindent
Moreover, the monotonicity condition (A2) ((A2)$^*$) can be relaxed to some
extent, but as a cost, the requirements on the ordered pair of upper and lower
solutions will be more restrictive. For the details of this idea,
see [22,23].
\enddemo
\heading {\bf 3. Applications} \endheading
In this section, we apply Theorem 2.5 to a particular system. Consider
$$
u_\eta '(t)=\alpha (\Delta_{n}u)_\eta +u_\eta (t-\tau)[1-u_\eta (t)],
\quad t \in {\Bbb R},\,\,\, \eta \in {\Bbb Z}^n.
\tag3.1
$$
This is a spatial discretization of
$$
\frac{\partial u}{\partial t}=\alpha \Delta u(t,x)+u(t-\tau)[1-u(t,x)],
\quad t \in {\Bbb R},\,\,\,x \in {\Bbb R}^n,
\tag3.2
$$
which was derived from branching theory in [14]. The corresponding
wave equation of (3.1) is
$$
cy'(t)=\alpha \sum_{k=1}^n[y(t+\sigma_k)+y(t-\sigma_k)-2y(t)]+y(t-c\tau)
[1-y(t)],
\tag3.3
$$
and the boundary conditions for wave fronts are
$$
\lim_{t \to -\infty}y(t)=0, \quad \lim_{t \to \infty}y(t)=1.
\tag3.4
$$
The nonlinear function $f(u,v)=v(1-u)$ is obviously continuous and satisfies
(A1) with $q=1$. For (A2), we have
\proclaim{\bf Lemma 3.1} $f(u,v)=v(1-u)$ satisfies (A2) with $q=1$ and
$\beta=1$.
\endproclaim
\demo{ Proof} Let $u_1$, $u_2$, and $v_2$ be such that
$0\le u_1 \le u_2 \le 1$ and $0 \le v_1 \le v_2 \le 1$. Then
$$
\aligned
f(u_2,v_2)-f(u_1,v_1)&=v_2(1-u_2)-v_1(1-u_1)\\
&=(1-u_1)(v_2-v_1)-v_2(u_2-u_1)\\
&\ge -v_2(u_2-u_1)\ge -(u_2-u_1)
\endaligned
$$
which completes the proof of this lemma.
\enddemo
Let $G(s)$ be defined by
$$
G(s)=\alpha \sum_{k=1}^n\left[e^{s \sigma_k}+e^{-s \sigma_k}-2\right]
+e^{-c\tau s}-cs, \quad s \in {\Bbb R}.
$$
Then,
\proclaim{\bf Lemma 3.2} There exists a $c^*>0$ such that
\itemitem{(i)} when $c0$ for $s \in {\Bbb R}$;
\itemitem{(ii)} when $c=c^*,\,\,\,\, G(s)=0$ has a unique positive solution; and
\itemitem{(iii)} when $c>c^*$, there exist $0 0$ for $s \in (-\infty, s_1)\cup(s_2,\infty)$.
\endproclaim
\demo{Proof} Denote $g(s)=\alpha \sum_{k=1}^n[e^{s \sigma_k}+e^{-s \sigma_k}
-2]$ and $h_c(s)=cs-e^{-c \tau s}$. Then, $G(s)=g(s)-h_c(s)$,
and elementary analysis of
$g(s)$ and $h_c(s)$ (see Figure 1) leads to the conclusion of this lemma.
\enddemo
\vskip0.5cm
\input epsf
\epsfxsize=300pt
\centerline{\epsfbox{fig1.ps}}
\centerline{\bf Figure 1}
\bigskip
Note that $c^*$ depends on the direction $\sigma=(\sigma_1,\sigma_2,\cdots,
\sigma_n)$, dimension $n$ as well as the diffusion coefficient $\alpha$ and
the delay $\tau$.
Using $c^*,\,\,\,s_1$ and $s_2$ in Lemma 3.2, we can construct the required
ordered pair of upper and lower solutions.
\proclaim{\bf Lemma 3.3} Assume $ c>c^*$ and $s_1$ be as in Lemma 3.2. Then,
$\overline{\rho}(t)=\min\{e^{s_1t},1\}$ is in $\Gamma$ with $q=1$ and is an upper
solution of (3.3).
\endproclaim
\demo{Proof} $\overline{\rho} \in \Gamma$ is obvious. For $t >0$,
$\overline{\rho}(t)=1$, and
$$
\aligned
&\alpha \sum_{k=1}^n[\overline{\rho}(t+\sigma_k)+\overline{\rho}(t-\sigma_k)-2\overline{\rho}(t)]
+\overline{\rho}(t-c\tau)[1-\overline{\rho}(t)]\\
&=\alpha \sum_{k=1}^n[\overline{\rho}(t+\sigma_k)+\overline{\rho}(t-\sigma_k)-2]\\
&\le \alpha \sum_{k=1}^n[1+1-2]=0=c\overline{\rho}'(t).
\endaligned
$$
For $t <0$, $\overline{\rho}(t)=e^{s_1t}$, and
$$
\aligned
&\alpha \sum_{k=1}^n[\overline{\rho}(t+\sigma_k)+\overline{\rho}(t-\sigma_k)-2\overline{\rho}(t)]
\overline{\rho}(t-c\tau)[1-\overline{\rho}(t)]\\
&=\alpha \sum_{k=1}^n\left[\overline{\rho}(t+\sigma_k)+\overline{\rho}(t-\sigma_k)
-2e^{s_1t}\right]+e^{s_1(t-c\tau)}\left(1-e^{s_1t}\right)\\
&\le \alpha \sum_{k=1}^n\left[e^{s_1(t+\sigma_k)}+e^{s_1(t-\sigma_k)}-
2e^{s_1t}\right]+e^{s_1(t-c\tau)}\left(1-e^{s_1t}\right)\\
&=e^{s_1t}\left[\alpha \sum_{k=1}^n\left(e^{s_1\sigma_k}+
e^{-s_1\sigma_k}-2]\right)
+e^{-s_1c\tau}\left(1-e^{s_1t}\right)\right]\\
&\le e^{s_1t}\left[\alpha \sum_{k=1}^n\left(e^{s_1\sigma_k}+
e^{-s_1\sigma_k}-2]\right)+e^{-s_1c\tau} \right]\\
&=e^{s_1t}(cs_1)=c\overline{\rho} '(t).
\endaligned
$$
This completes the proof.
\enddemo
\proclaim{\bf Lemma 3.4} Assume $c>c^*$ and let $s_1$ and $s_2$ be as in
Lemma 3.2. Let $r>0$ be such that $r0$ is sufficiently large.
\endproclaim
\demo{Proof} Let $t_0<0$ be such that $Me^{rt_0}=1$. For $t >t_0,\,\,\,\,
\underline{\rho}(t)=0$, and
$$
\aligned
&\alpha \sum_{k=1}^n[\underline{\rho}(t+\sigma_k)+\underline{\rho}(t-\sigma_k)-2\underline{\rho}(t)]
+\underline{\rho}(t-c\tau)[1-\underline{\rho}(t)]\\
&=\alpha \sum_{k=1}^n[\underline{\rho}(t+\sigma_k)+\underline{\rho}(t-\sigma_k)]
+\underline{\rho}(t-c\tau)\\
&\ge 0=c\underline{\rho} '(t).
\endaligned
$$
For $t e^{s_1t}\left[
cs_1-c(s_1+r)Me^{rt}-Me^{rt}G(s_1+r)-e^{rt}e^{-s_1c\tau}\right]\\
&=c\underline{\rho}'(t)
+e^{(s_1+r)t}\left[-MG(s_1+r)-e^{-s_1c\tau}\right] \\
&\ge c\underline{\rho}'(t),
\endaligned
$$
provided $M \ge \frac{e^{-s_1c\tau}}{-G(s_1+r)}$. This completes the proof.
\enddemo
Combining the above lemmas with Theorem 2.5, we obtain
\proclaim{\bf Theorem 3.5} For each $c>c^*$ where $c^*$ is as in Lemma 3.2,
(3.1) has a traveling wave front with velocity $c$.
\endproclaim
\vskip 8mm
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\enddocument
\bye