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of evolutionary games in [10]. In the very next future we will extend the method
presented here to evolutionary games.
3
The Nonlinear Vo ter Model
We consider a model of two species (or two opinions). For the spatial distribu-
tion we assume a one-dimensional stochastic cellular automaton (SCA) defined
by a circle of
cells. Each cell
is occupied by one individual, thus each cell is
characterized by a discrete value
.Weassume a circular connection.
Therefore we set
and
. The state of cell
at time
is defined by the states of cells
,
,
at time
. The state transitions of
n
+1
the voter model depend only on
. This class
of automata is called
totalistic
. For the stochastic voter model the transitions are
defined as follows.
i
1
i
1
k
(
t
)
(
t
)+
x
i
(
t
)+
x
i+1
(
t
)
3
k
(
t
)
p
(
x
(
t
+1) = 1j
k
(
t
))
2
1
0
denotes the transition probability given
.
is a small stochastic
disturbance parameter. The model is defined by
one speaks of pos-
itive frequency dependent invasion. This model is also called the
majority vote
model
, because the individuals join the opinion of the majority in the neighbor-
hood. For
.If
p
(
x
=1j
k
)
the model is called a negative frequency dependent invasion
process. In this case the minority opinion has more weight. The deterministic
cellular automata are given by
>
0
:
5
. The voter model has been
intensively investigated by micro simulations. The reader is referred to [3].
and
0
;
4
Exact Analysis of CA by Markov Processes
The nonlinear voter model, as any cellular automata, can be seen as a Markov
process. We just sketch the Markov process analysis. Let
denote a binary vector representing the state of a dynamical system at time
.
(
x
The vector at time
is denoted by
. We assume
.
t
+1
Definition 1
Let
denote the probability of
in the population at gen-
eration
defines a uni-variate marginal
distribution. Bi-variate marginal distributions are defined by
.Then
p
(
x
;t
)
p(
x
;t)
x
;X
. Conditional distributions are defined for
i
1
p
(
x
;x
i
;t
))
by
,where
y
and
are disjoint sub-vectors of
.
p
(
x
;t
)
p
(
z
)
x
;X
i
1
i
1
