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Inserting the last two equations into equation (7) gives the difference equa-
tions for the
tri-variate marginal distributions
. The approximations have to ful-
fill constraints derived from probability theory.
i
1
i
1
p(
i
1
;
i
;
i
+1
;
i
+1
;
i
+2
i
1
In the same manner approximations of different precision can be obtained.
We just discuss the simplest approximation, using
uni-variate marginal distri-
butions
. Here equation (6) is approximated by
(10)
i
1
i
1
i
1
;
i
+1
differ-
ence equations, but these difference equations are nonlinear. It seems very un-
likely that analytical solutions of these equations can be obtained. For
spatially
homogeneous
problems we have
The approximation by uni-variate marginal distributions leads to
. In this case the probabili-
ties do not depend on the locus of the cell. This is the
mean-field limit
known
from statistical physics [15]. With
we obtain the
mean-field equation
2
(11)
The mean-field limit is exact, if the
neighbors
of the cellular automata are
chosen
randomly
for each step. For
and
the equation has stable
attractors at
and
.For
the equation has two stable attractors
at
. Thus the mean-field limit approximation indicates a
bifurcation
for
.
It is also possible to derive a bi-variate approximation for the spatial ho-
mogeneous case. A fairly simple expression for
can be derived if
and
are used as free parameters.
=1jx
i
1
4.2
Numerical Analysis of 1-D Vo ter Model
We now make a comparison of the approximations. The mean-field approxima-
tion predicts a bifurcation at
.Wefirst discuss the case
and
. The mean-field analysis gives
for
and
. The exact Markov chain computa-
tion shows a different picture. The results are displayed in table 1.
for
