
\documentstyle[multicol,aps,epsf]{revtex}
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\begin{document}
\title{Species segregation in a model of interacting populations}

\author{L.~Frachebourg$^\ast$, P.~L.~Krapivsky$^{\ast}\dag$, and 
E.~Ben-Naim$\ddag$} 
\address{$^\ast$Center for Polymer Studies and Department of Physics,
Boston University, Boston, MA 02215}
\address{$\dag$Courant Institute of Mathematical Sciences, 
New York University, New York, NY 10012-1185}
\address{$\ddag$The James Franck Institute, The University of Chicago, 
Chicago, IL 60637}
\maketitle
\begin{abstract}

We investigate segregation and spatial organization in a
one-dimensional system of $N$ competing species forming a cyclic food
chain. For $N<5$, the system organizes into single-species domains,
with an algebraically growing typical size.  For $N=3$ and $N=4$, the
domains are correlated and they organize into ``superdomains'' which
are characterized by an additional length scale.  We present scaling
arguments as well as numerical simulations for the leading asymptotic
behavior of the density of interfaces separating neighboring domains.
We also discuss statistical properties of the system such as the
mutation distribution and present an exact solution for the case
$N=3$.

{PACS numbers: 02.50.Ga, 05.70.Ln, 05.40.+j}

{Key Words: species segregation, two-length scaling}

\end{abstract}
\begin{multicols}{2}

\section{INTRODUCTION}

Coarsening underlies numerous natural processes including phase
separation, grain growth, soap bubbles, and species segregation.  It is
generally believed that coarsening systems exhibit dynamical
scaling\cite{bray}, {\it i.e.}, the typical domain size grows
algebraically with time, $\ell(t)\sim t^\alpha$.  The exponent $\alpha$
is usually independent of many details of the system such as the spatial
dimension.  However, little is known on coarsening in systems with more
than two equilibrium phases.  In this study, we investigate species
segregation and reveal the two-length scaling instead of the
ordinary single-length scaling. 

Lotka-Volterra model of interacting populations ``living'' on a
one-dimensional lattice is the simplest system which exhibits species
segregation. The case where $N$ species form a food chain is especially
well suited for studying species segregation.  We assume that every
species plays the role of prey and predator simultaneously.  The food
chain is arranged in a cyclic manner. For example, when $N=3$, $A$ eats
$B$, $B$ eats $C$, and $C$ eats $A$.  ``Eating'' events involve nearest
neighbors and lead to duplication of the winner and elimination of the
loser, corresponding to the following reaction scheme

\begin{equation} 
A+B\to 2A, \quad B+C\to 2B, \quad C+A\to 2C.
\label{Eq-1} 
\end{equation}

Here and throughout this study we restrict ourselves to random and
symmetric initial conditions, where the average initial species
densities are all equal $1/N$. Despite the nonconserving nature of the
process, the average densities remain constant in the thermodynamic
limit.

\section{SEGREGATION IN ONE DIMENSION}

For a large number of species, most pairs of species do not interact
and the system quickly reaches a frozen state.  Previous studies
\cite{bramson,fisch} have mainly concentrated on establishing the
upper bound for $N$ above which the system does not coarsen.  It has
been proved rigorously that the marginal chain length is $N_c=5$
\cite{bramson,fisch}. For $N\geq N_c$ each site quickly reaches a
final frozen state, while for $N<N_c$, the state of each site changes
an infinite number of times. However, theoretical understanding of the
kinetic behavior and the coarsening properties of the system is still
incomplete\cite{fisch,tainaka}. In this study, we illuminate the rich
kinetic behavior of the system by analyzing the density of interfaces
separating different single-species domains.


For $N=2$, this system is equivalent to the voter model\cite{lig,elp}
which can be solved exactly \cite{glauber}. In terms of interfaces,
the $N=2$ model is equivalent to an ensemble of annihilating random
walks.  The system separates into single species domains.  The average
domain size $\ell$ exhibits a diffusive growth law $\ell(t)\sim
\sqrt{t}$.


Consider now the $N=3$ case. There are two types of interfaces: right
moving ($AB$, $BC$, and $CA$) and left moving ($BA$, $CB$, and $AC$),
denoted by $R$ and $L$, respectively. The interface dynamics and
consequently, the coarsening kinetics are sensitive to the microscopic
realization of the reaction process. For parallel dynamics (all bonds
are updated simultaneously) only opposite moving interfaces
annihilate, $R+L\to \emptyset$, while for sequential dynamics (bonds
are updated one at a time) interfaces moving in the same direction
react as well, $R+L\to \emptyset$, $R+R\to L$, and $L+L\to R$.  Hence,
for the 3-species model with parallel dynamics the interface reaction
process is equivalent to the well-known two-velocities ballistic
annihilation process\cite{Elskens} and the interface density, $M(t)$,
decays as $t^{-1/2}$. This behavior can be understood by arguing that
in a linear region of size ${\cal L}$, the imbalance between the
number of left and right moving interfaces is of order
$\Delta\sim\sqrt{c_0{\cal L}}$. After a time $t={\cal L}/v_0$ only
this residual fluctuation remains and as a result the concentration
decay $M(t)\sim \Delta/{\cal L}\sim (c_0/v_0t)^{1/2}$ follows.


The above heuristic picture suggests a special domain pattern.  The
system organizes into ballistically growing superdomains.  Each
superdomain contains interfaces moving in the same direction, while
neighboring superdomains are separated by opposite moving interfaces.
Domains inside each superdomain are arranged cyclically ($ABCABC$ or
$CBACBA$).  In addition to the average size of superdomains, one can
define the average distance between two adjacent similar velocity
interfaces. We define these two typical length scales using an
illustrative configuration:

\begin{equation} 
B\overbrace{AABBB\underbrace{CCCC}_{\ell}AAABBCCC}^{\cal L}B.
\label{Eq-2}
\end{equation}

The corresponding coarsening exponents, $\alpha$ and $\beta$, are
defined via $\ell\sim t^{\alpha}$ and ${\cal L}\sim t^{\beta}$,
respectively.  For $N=3$ with parallel dynamics we thus have
$\alpha=1/2$ and $\beta=1$.  Computation of $\ell_n(t)=\langle
x^n\rangle^{1/n}$, the moments of the domain distribution, reveals a
richer spatial structure than that anticipated by Eq.~(\ref{Eq-2}).
Namely, a variety of scales is found\cite{lp}: $\ell_n(t)\sim 1$ when
$n<1/2$ and $\ell_n(t)\sim t^{1-1/2n}$ when $n>1/2$.  However, only
the two extreme scales, the ballistic scale ${\cal O}(t)$ and the
scale ${\cal O}(1)$ characterizing the initial conditions seem to be
important.  All other scales, including the average length of a domain
$\ell=\ell_1$, should be interpreted as effective scales arising as
the result of the competition between the two extreme scales present
in this system.

In the complementary sequential dynamics case, interfaces perform a
biased random walk and thus, the ballistic motion is now supplemented
by diffusion.  The system again organizes into domains of right and
left moving interfaces.  Inside a domain, interfaces moving in the
same direction can now annihilate via a diffusive mechanism, unlike
the case of parallel dynamics (more precisely, collision of say two
right moving interfaces gives birth to a left moving interface which
is soon annihilated with the nearest right moving interface).  Similar
single-species annihilation with convective-diffusive transport has
been investigated in Ref.\cite{brk} where the concentration decay
$M(t)\sim t^{-3/4}$ has been established.  This prediction is
consistent with numerical simulations. The simulations also indicate
that the system slowly approaches the asymptotic behavior $\ell(t)\sim
t^{3/4}$ \cite{lpe}.

The resulting spatial structure is thus similar to the parallel case,
Eq.~(\ref{Eq-2}). However, while the larger length scale remains
unchanged, ${\cal L}\sim t$, the smaller length scale is now a
geometric average of a diffusive and a ballistic scale. We conclude
that the coarsening patterns are characterized by two length scales,
and the coarsening kinetics are sensitive to the details of the
dynamics.  We anticipate that a variety of scales would appear from
the moments $\ell_n(t)$, and they should be interpreted as effective
scales resulting from the competition between the ballistic ${\cal
O}(t)$ and diffusive ${\cal O}(\sqrt{t})$ scales.


In the $N=4$ case, there are static interfaces denoted by $S$ ($AC$,
$BD$, $CA$, and $DB$), in addition to the right and left moving
interfaces, ($AB$, $BC$, $CD$, $DA$) and ($BA$, $CB$, $DC$, $AD$),
respectively.  For sequential dynamics, interfaces react upon collision
according to $R+L\to \emptyset$, $R+S\to L$, $R+R\to S$, $L+L\to S$, and
$S+L\to R$.  Under the assumption that neighboring interfaces are
uncorrelated, the interface densities evolve according to the following
rate equations

\begin{eqnarray}    
\label{Eq-3}   
\dot R=&-&2R^2-2RL-RS+SL, \nonumber\\
\dot L=&-&2L^2-2RL-SL+RS, \\
\dot S=&R&^2+L^2-RS-SL.   \nonumber
\end{eqnarray}
Solving these equations subject to the initial conditions 
$R(0)=L(0)=S(0)=1/4$ gives
\begin{equation}    
M(t)={1\over 4+4t}, \quad
S(t)={1\over \sqrt{4+4t}}-{1\over 4+4t}.
\label{Eq-4}
\end{equation}
In the above, $M(t)=R(t)=L(t)$ is the density of moving interfaces.
According to the rate equation theory, the typical distance between
two static interfaces, $\ell \sim t^{1/2}$, grows slower than the
distance between two moving interfaces, ${\cal L}\sim t$.  A nontrivial
spatial organization occurs in which large ``superdomains'' contain
many domains of alternating noninteracting ($AC$ or $BD$) species.
Similar to the $N=3$ case, there are two relevant growing length
scales as in the following illustration
\begin{equation}
B\overbrace{AACCCAAA\underbrace{CCCC}_{\ell}
AACCAAACCC}^{{\cal L}}D. 
\label{Eq-5}
\end{equation} 
Numerical simulations agree qualitatively with this picture.
However, the quantitative predictions for the coarsening exponents
fail.  

In the following, we use heuristic arguments to obtain the exponent
values.  Numerical simulations indicate that parallel and sequential
dynamics are asymptotically equivalent and thus, we restrict ourselves
to the former simpler case.  The annihilation reaction $R+L\to
\emptyset$ is supplemented by the exchange reaction $R+S\to L$ and
$L+S\to R$.  According to the rate theory as well as the simulations
$M(t)\ll S(t)$, and thus, we assume an alternating spatial structure of
``empty'' regions (with no more than one moving interface) and
``stationary'' regions (with many stationary interfaces inside any such
region).  If the interface densities obey scaling, then the size of the
empty and the stationary regions should be comparable.  The typical size
of an empty or a stationary region is therefore of the order of
$M^{-1}$.  The typical number of stationary interfaces inside a
stationary region is of the order of $S/M$.  The evolution proceeds as
follows: a moving interface hits the least stationary particle and
bounces back.  Then this interface hits the least stationary particle of
the neighboring stationary region, and bounces back again.  This
``zig-zag'' process continues until one of these stationary regions
``melts'', thereby giving birth to a larger empty region.  If there is a
moving particle inside merging empty region, the two moving particles
quickly annihilate. Otherwise, the moving particle continues to
eliminate stationary interfaces.  The typical time $\tau$ for a
stationary region to melt is $\tau=M^{-1}\times S/M=S/M^2$.  This
melting time $\tau$ is also the typical time for annihilation of a
moving interface and thus,

\begin{equation}    
\dot M\sim -{M\over \tau}\sim -{M^3\over S}.
\label{Eq-6}
\end{equation} 
Using $M(t)\sim {\cal L}(t)^{-1}\sim t^{\alpha}$ and $S(t)\sim
{\ell(t)}^{-1}\sim t^{\beta}$, the exponent relation $2\beta-\alpha=1$
emerges. A second independent exponent relation, $\alpha+\beta=1$, will
be presented in the discussion of the mutation distribution below.
Combining these  two relations we find that $\alpha=1/3$ and
$\beta=2/3$. These values are in good agreement with parallel as well
as sequential simulations. 

As in the previously discussed $N=3$ case the spatial organization
occurs on two different length scales. On the other hand, for $N=4$, the
coarsening kinetics are independent of the details of the dynamics, in
contrast with the $N=3$ behavior.

For the 5-species cyclic Lotka-Volterra model, it is known that
the system approaches a frozen state\cite{bramson,fisch}.
Nevertheless, it is useful to consider the interface dynamics for the
$N=5$ case, where there are two types of stationary interfaces, $S_R$
($AC,BD,CE,DA,EB$) and $S_L$ ($AD,BE,CA,DB,EC$), in addition to the
right and left moving interfaces, $R$ ($AB,BC,CD,DE,EA$) and $L$
($BA,CB,DC,AD,AE$).  The reaction process is symbolized by $R+L\to
\emptyset$, $R+S_L\to L$, $R+S_R\to S_L$, $S_R+L\to R$, $S_L+L\to
S_R$, $R+R\to S_R$, and $L+L\to S_L$. It is straightforward to
generalize the rate equations (\ref{Eq-3}) to this case as well, and
we merely quote the results.  According to these equations, the static
interfaces approach a final nonzero value $S(t)\to S_{\infty}$, and
the mobile interfaces decay exponentially, $M(t)\sim
\exp(-S_{\infty}t)$, as $t\to\infty$.  

Interestingly, the rate equations correctly predict the marginal
number of species $N_c=5$.  However, for the 5-species cyclic
Lotka-Volterra model (similarly to the previously discussed cases of
$N=2,3,4$) only the qualitative predications of the rate equations are
correct, but the quantitative predictions fail.  To derive a correct
asymptotic behavior we first observe that the density of mobile
interfaces rapidly decreases while the density of stationary
interfaces remains finite.  This allows us to ignore collisions
between mobile interfaces in the large time limit.  We should estimate
the survival probability of a mobile interface in a sea of stationary
ones.  There are two reactions in which moving interfaces survive
although they change their type, $R+S_L\to L$ and $L+S_R\to R$.  Thus,
a moving interface is long lived in the following environment: $\cdots
S_RS_RS_RS_RMS_LS_LS_LS_L\cdots$.  Clearly, in such configurations the
zig-zag reaction process takes place.  The moving interface travels to
the right during a time $t_0=(c_0v_0)^{-1}$, eliminates a stationary
interface and travels to the left a time of order $2t_0$, eliminates
an interface and travels back to the right, {\it etc}. Thus, to
eliminate $N_s$ interfaces, the moving interface should spend a time
of order $t\simeq t_0\sum_{i=1}^{N_s}i=t_0N_s(N_s+1)/2$.  Therefore,
the number of stationary interfaces $N_s(t)$ eliminated by a moving
interface scales with time as $N_s(t)\sim \sqrt{c_0v_0t}$.  Special
configuration of length $N_s$ are encountered with probability
$\propto \exp(-N_s)$, and thus, the density of moving interfaces
exhibits a stretched exponential decay, $M(t)\propto
\exp(-\sqrt{c_0v_0t})$.  Hence, the approach towards the frozen state
is slowed down due to spatial fluctuations.


It was pointed out recently that nontrivial behavior underlies
low-activity or persistent sites in coarsening systems \cite{dbg,kbr}.
It is useful to consider the mutation distribution, $P_n(t)$, defined
as the fraction of sites that have mutated (changed their state)
exactly $n$ times during the time interval $[0:t]$.  Mutation kinetics
and coarsening kinetics are intimately related.  Let $\langle
n(t)\rangle=\sum_n nP_n(t)\sim t^{\nu}$ be the average number of
mutations.  Since every motion of an interface contributes to an
increase in the number of mutations in one site, the mutation rate
equals the density of moving interfaces, $d\langle
n(t)\rangle/dt=M(t)$. Using $M(t)\sim t^{-\mu}$ one has $\nu=1-\mu$.
In the closely related voter model \cite {lig,elp,glauber}
(corresponding to $N=2$), it was found that the mutation distribution
$P_n(t)$ obeys scaling \cite{elp}

\begin{equation}
P_n(t)={1\over \langle n(t)\rangle}\,
\Phi\left({n\over \langle n(t)\rangle}\right). 
\label{Eq-8}
\end{equation} 
The scaling function has the following limiting behaviors
\begin{equation}
\Phi(z)\sim\cases{
z^{\gamma}&$z\ll 1$;\cr
\exp(-z^{\delta})&$z\gg 1$}
\label{Eq-9}
\end{equation}
The behavior in the small argument limit reflects the decay of
persistent sites. For the voter model $P_0(t)\sim t^{-\theta}$ with
$\theta=3/8$ \cite{derrida}.  If this power-law decay holds generally,
then the exponent relation $\theta=\nu(\gamma+1)$ should hold.  The
large $z$ limit describes ultra-active sites. A convenient way to
estimate the fraction of such sites is to consider sites which make of
the order of one mutations per unit time.  At time $t$, the fraction
of these rapidly mutating sites is exponentially suppressed,
$P_t(t)\propto \exp(-t)$.  It is therefore natural to assume a
quasi-exponential form $\Phi(z)\sim \exp(-z^{\delta})$ for the tail of
the scaling distribution, thereby implying an additional exponent
relation $\delta=1/(1-\nu)$.

The mutation distribution can be exactly calculated for the $N=3$ case
with parallel dynamics. Since the initial $\pm v_0$ interface
velocities are uncorrelated, the interface ballistic annihilation
process can be mapped into a random walk problem.  As a result, the
interface density is found from first passage properties of a random
walk.  Similarly, the mutation distribution can be shown to be
equivalent to the probability that the minimum of a $t$-step random
walk is exactly $n$ (for details, see\cite{lpe}).  Using the
definition of Eq.~(\ref{Eq-8}), and the average number of mutations
$\langle n(t)\rangle\sim\sqrt{4t/3}$, the exact scaling distribution
can be written
\begin{equation}
\Phi(z)={4\over \sqrt{\pi}}\,e^{-z^2}{\rm Erf}(z),
\label{Eq-10}
\end{equation}
with $z=n/\langle n(t)\rangle$. The limiting behaviors of this scaling
function agree with the predictions of Eq.~(\ref{Eq-9}), and the
scaling exponents $\theta=1$, $\nu=1/2$, $\delta=2$, and $\gamma=1$,
satisfy the predicted scaling relations. 

The knowledge of the mutation distribution suggests that we can
compute the autocorrelation function, $A(t)=\sum_{n\geq 0}P_{3n}(t)$,
as well.  Making use of the above scaling form is not enough -- one
finds rather trivial long-time behavior, $A(t)\to 1/3$.  To determine
a correction to the final value, a combinatorial approach is somewhat
inconvenient.  Thus, we reconsider the problem by assuming that
interfaces are distributed on a continuous line; the long-time
behavior should be the same both in this continuum version and in the
original lattice version.  The continuum version turns out to be
simpler and we have found {\it exact} cumbersome expressions\cite{lp}
for all $P_n(t)$ in terms of the modified Bessel functions.  This
solution is then used to find that $A(t)-1/3 \simeq -2/(9\pi t)$ as
$t\to\infty$. This implies that the autocorrelation exponent
$\lambda$\cite{huse} defined via $A({\cal L})-1/3 \sim {\cal
L}^{-\lambda}$ is given by $\lambda=1$.  Note that this value agrees
with bounds on $\lambda$, $d/2\leq \lambda\leq d$, proposed by Fisher
and Huse\cite{huse} for phase ordering systems with nonconservative
dynamics in $d$ dimensions.

Having discussed the mutation distribution that quantifies the
temporal history of species replacements we turn to the {\it
age distribution} $P(\tau,t)$ defined as the probability that an
individual which has been born at time $\tau$ and survives up to time
$t$.  The average age of the population is then given by $T=\langle
t-\tau\rangle=\int_0^t d\tau (t-\tau)P(\tau,t)$.  This probability
distribution can be calculated for $N=3$\cite{lp2} and we found that
the the average age of the population grows linearly with time
according to $T\simeq (1-2/\pi)t$.  The age distribution obeys the
scaling form $P(\tau,t)\simeq t^{-1}\Psi(\xi)$ in the limit
$t\rightarrow \infty$ and $\tau \rightarrow \infty$ with $\xi=\tau/t$.
The scaling function reads $\Psi(\xi)=(2/\pi)(1-\xi^2)^{-1/2}$.

We now return now to the 4-species Lotka-Volterra model.  According to
the above discussion, the zig-zag reactions $R+S\to L$ and $L+S\to R$
dominate over the annihilation reaction $R+L\to \emptyset$ in the
long-time limit.  We therefore consider a simpler solvable case where
a single mobile interface is placed in a regular sea of static
interfaces to evaluate the scaling function $\Phi(z)$ defined in
Eq.~(\ref{Eq-8}). This interface moves one site to the right, two to
the left, three to the right {\it etc.}  Similar to the above
discussion on the survival probability in the $N=5$ case, at time $t$
this interface has eliminated $N_s\sim (t/t_0)^{1/2}$ static
interfaces, with $t_0=(c_0v_0)^{-1}$. The origin is visited $N_s t_0$
times, site $1$ is visited $(N_s-1)t_0$, site $-1$ is visited
$(N_s-2)t_0$, {\it etc.}  This implies that the mutation distribution
is $P_n(t)=\langle n\rangle^{-1}\Phi(n/\langle n\rangle)$, with
$\langle n\rangle\sim N_s t_0$ and $\Phi(z)=1$ for $z<1$ and
$\Phi(z)=0$ for $z>1$. Therefore, $\gamma=0$.  This approximation is
inappropriate for predicting the tail of $\Phi(z)$ which is sensitive
to annihilation of the moving interfaces.  However, in the small $z$
limit the annihilation process should be negligible, and thus
$\gamma=0$.  The fraction of unvisited sites is equivalent
asymptotically to the survival probability of a stationary interface
and thus $\theta=\alpha$.  Using the previously established relations
$\nu=1-\mu$, $\theta=\nu(\gamma+1)$, and $\mu=\beta$, we obtain the
second independent exponent relation $\alpha+\beta=1$ which was used
to obtain the asymptotic behavior of the mobile and the static
interfaces.

\section{EXTENSIONS} 

The cyclic lattice Lotka-Volterra model can be generalized in a number
of directions.  A natural generalization is to higher dimensions.
Two-dimensional case seems to be especially interesting from the point
of view of mathematical biology\cite{murray}.  In the exactly solvable
$N=2$ case (the voter model), coarsening occurs for $d\leq 2$\cite{lig},
for the marginal dimension $d=2$, the density of interfacial bonds
decays logarithmically, $c(t)\sim 1/\ln t$\cite{lp1}, while for $d>2$, no
coarsening occurs and the system reaches a reactive steady state.  In
two dimensions, our numerical simulations indicate that there is no
coarsening, {\it i.e.} the density of reacting interfaces saturates at a
{\it finite} value.  For sufficiently large number of species the
fixation is expected but we could not determine the threshold value, at
least up to $N=10$ we have not seen fixation.

Another extension concerns a different interaction rule.  Previously, we
investigated the cyclic Lotka-Volterra model with {\it asymmetric}
interactions.  To learn how the coarsening kinetics depend on the
interaction rules it is useful also to consider a {\it symmetric}
interaction rule where both of the reaction channels $A+B\to 2A$ and
$A+B\to 2B$ are allowed.  In this case, asymptotically exact results can
be obtained.  We discuss only the $N=4$ case since the $N=2$ and $N=3$
cases reduce to the voter model (see\cite{elp}).  Denote the static
interfaces ($AC$, $CA$, $BD$, $DB$) by $S$ and moving interfaces by $M$.
The symmetric ``eating'' rule implies that moving interfaces perform
simple random walks.  Interfaces react according to $M+M\to S$, and
$M+S\to M$ or $M+S\to S$ depending on the local environment.  Moving
interfaces undergo diffusive annihilation and thus, $M(t)\sim t^{-1/2}$.
The fraction of surviving stationary interfaces is proportional
asymptotically to the fraction of sites which have not been visited by
mobile interfaces up to time $t$, $S(t)\sim P_0(t)\sim t^{-3/8}$.  We
should also take into account creation of stationary interfaces by
annihilation of moving interfaces. This process produces new stationary
interfaces with rate of the order $-dM/dt$. Thus, the stationary
interface density satisfies the rate equation $\dot S=\dot P_0-\dot M$.
Combining this equation with $P_0(t)\sim t^{-3/8}$\cite{derrida} and
$M(t)\sim t^{-1/2}$, we find that the surviving interfaces provide the
dominant contribution while those created in the process $M+M\rightarrow
S$ contribute only to a correction of the order $t^{-1/8}$.  Thus a
two-scale structure similar to Eq.~(\ref{Eq-5}), with $\ell\sim t^{3/8}$
and ${\cal L}\sim t^{1/2}$, emerges. Hence, it is seen that the
coarsening exponents can be sensitive to the details of the interaction
rules.

We also looked at the cyclic Lotka-Volterra model with unequal initial
concentrations.  We have seen rich kinetics\cite{lpe} but mention just
one example here: For the 4-species system with one majority species,
say species $A$, and other species having the same concentrations,
$b_0=c_0=d_0$, the system approaches a frozen state with $a_{\rm
frozen}=1/4$ and $c_{\rm frozen}=3/4$. 

\section{SUMMARY}

We investigated coarsening in a one-dimensional model
of competing species.  Interestingly, the coarsening properties of
this model may depend on the details of the dynamics.  The spatial
patterns are characterized by the existence of two characteristic
length scales, the average length of the single-species domains,
$\ell(t)\sim t^{\alpha}$, and the average length of superdomains,
${\cal L}(t)\sim t^{\beta}$ (the corresponding coarsening and mutation
exponents are summarized in Table 1).  

\vspace{.1in}
\centerline{
\begin{tabular}{|l|c|c|c|}
\hline
N&$\alpha$&$\beta$&$\theta$\\ 
\hline
3 (parallel)  &\,1/2\,&\,1\,&\,1\,\\
3 (sequential)&\,3/4\,&\,1\,&\,1\,\\
4             &\,1/3\,&\,2/3\,&\,1/3\,\\
4 (symmetric)  &\,3/8\,&\,1/2\,&\,3/8\,\\
\hline
\end{tabular}}
\vspace{.1in} {\small {\bf Table 1}: Coarsening
($\ell\sim t^{\alpha}$ and ${\cal L}\sim t^{\beta}$) and persistence
($P_0(t)\sim t^{\theta}$) exponents in 1D.}

Thus simple dynamical scaling is violated for the one-dimensional
Lotka-Volterra models.  Violations of scaling have been reported in a
few recent studies of coarsening in one- and two-dimensional
systems\cite{cz,nbm,mg,rb,r,zgg,zz}.  To the best of our knowledge,
however, in previous work violations of dynamical scaling have been
seen only in some systems with vector and more complex order
parameter.  In contrast, Lotka-Volterra models can be interpreted as
systems with {\it scalar} order parameter, although the number of
equilibrium states $N$ generally exceeds two, the characteristic
value for Ising-type systems.

Finally we note that presence of only two length scales exemplifies
the mildest violation of classical single-size scaling.  Generally, if
scaling is violated one expects the appearance of an infinite number
of independent scales, {\it i.e.}, multiscaling\cite{cz}.
Surprisingly, we found no evidence of multiscaling.  Similar
two-length scaling has been observed in the simplest one-dimensional
system with vector order parameter, namely in the XY model\cite{rb},
and in the single-species annihilation with combined diffusive and
convective transport\cite{brk}.  Indications of the three-length
dynamical scaling have been reported in the context of
coarsening\cite{zgg} and chemical kinetics\cite{lr}.

L.~F. is supported by the Swiss NSF, P.~L.~K. is supported in part by a
grant from NSF.  E.~B.~ is supported in part by NSF Award Number
92-08527, and by the MRSEC Program of the NSF under Award Number
DMR-9400379.

\begin{thebibliography}{99}

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