A Probabilistic Cellular Automaton for Evolution

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A Probabilistic Cellular Automaton for Evolution

Nikolaus Rajewsky, Michael Schreckenberg

To cite this version:

Nikolaus Rajewsky, Michael Schreckenberg. A Probabilistic Cellular Automaton for Evolution. Jour-

nal de Physique I, EDP Sciences, 1995, 5 (9), pp.1129-1134. �10.1051/jp1:1995186�. �jpa-00247124�

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Classification

Physics Abstracts

05.40+j ^05.70Jk 87.10+e

A Probabilistic Cellular Automaton for Evolution

Nikolaus

Rajewsky

and Michael

Schreckenberg

Iustitute for Theoretical Physics, University of Cologne, 50923 KôIn, Germany

(Received

15 May1995, received in final form 30 May 1995, accepted 6 June

1995)

Abstract. We study two variauts of _a binary model for Darwinian Coevolutiou _m _an _ecosys- tem of iuteracting species based _ou _a model recently proposed by Bak and Sueppen

iii.

We present ^results ^from a computer simulation and _an improved mean-field calculation, including

exponents, ^critical points ^and probability of certain spiu configurations.

Recently

a model for Darwinian Coevolution in _a dynamical ecosystem ^of

interacting species

was

proposed by

Bak lc

Sneppen (BS-model) Iii.

Trie

only

mechanisms which _are considered in this model _are mutation and natural selection. In this note, _we will

study

_a

simplified

version of the BS-model and _we will not discuss

questions concerning

the

applicability

to

biological

evolution. The BS-model consists of

L~

_sites

(species) arranged

on a d-dimensional

hypercubic

lattice.

Here,

_we set d

= 1 and _use

periodic boundary

conditions. To each

species

_a random number

(barrier) B~,

0 < B~ < 1

(equally distributed)

is

assigned

in the

beginning.

The

evaluation

proceeds

_as follows: choose the site with trie lowest value B~ <

Bj

for ail

j #

and

assign

new random numbers to1 and its left and

right neighbors.

This choice of the interaction

between the _species _is

quite arbitrary.

Que _can, for

example,

also

study

_a

only

"one-sided"-

version of this model

(assign

_new random numbers

only

to1

and,

_say,

1+1)

_(2]. After _a

transient time the system

organizes

^itself into _a

stationary

state with critical

properties.

The distribution

p(B)

of the barriers converges towards _a

step-function

_m the limit of _an infinite system

(L

_-

cc):

~~~~

^~~~~

~Î

ÎÎÎÎ

Î ÎÎÎ

We _now define _a

probabilistic binary

cellular automaton

(PCA-model)

based _on this

picture

of the

stationary

state of the BS-model

(3,4].

Ail barriers below the threshold _are set to 1

(spin 1),

all

remaining

barriers _are set to 0

(spin 0).

The evaluation

proceeds

as follows:

choose

randomly

_a site1 with spin n~ = 1 and

assign independently

three _new spins _ni, n~-i and n~+i

to1,1-1

and 1+1

il

with

probability

_c, 0 with

probability

1

c), irrespective

^of their former values. We also

study

the one-sided version of this model.

Note that the

global

feature of the

BS-update-rule disappears

but is reflected in the _new control parameter c.

The one-sided _case with the constraint to leave the _spm 1 site which _was chosen for

update

(3)

l130 JOURNAL DE PHYSIQUE I N°9

unchanged

and

only

to

assign

a new

spin

to its _one

neighbor

_was solved in reference [Si. ^There the

stationary

state is trivial:

< nz~ni~.. n~~ ^>= c~

for ail _c and it is

possible

to calculate the

equilibrium

_spm autocorrelation function of the

time-dependent occupation

number fluctuations at _a certain site.

In the _case of _our PCA-models _exists _a critical

probability

c* below which the system _converges towards the

absorbing

state _ni ₌ 0. Above c* the

density

_pi of sites with

spin

1 converges to _some

c-dependent

value.

Figure

1 shows _pi _vs. time for _a system ^of 3 _x 10~ _sites. _The

simulation _was

performed

_on _a SUN workstation. The unit of time _is

simply

the total number of

updates.

At c* _we find

fil ~

t~,

_T < Ù.

Dur results for the PCA'S _are c*

= 0.6353 ~ 0.0001

(two-sided,

in agreement ^with (4]) ^and c* = 0.768 ~ 0.0001

(one-sided).

The exponent T turned out to be the _same for both models:

T = -0.185 ~ 0.004. The two-sided PCA with

parallel dynamics

_was studied in (6] ^and ^is

known _to be in the _same

universality

class _as directed

percolation.

Kinzel finds _a diflerent c*.

In order to compare his exponent ^T' ₌ ^0.157 ^~ ^0.002 ^with our data _one has to rescale the time

properly,

since _our rate of

update

is

coupled

to pi

Ii). Doing this,

_one obtains T'

=

T/(T +1),

which is in

good

agreement ^with our value of _T.

We _tutu _to _an

improved

mean-field

description

of the PCA'S in the

stationary

state. We denote the

probability

to find _a

configuration

_njnj+i _nj+~ of r+1

neighbored

spins with pn,n,~~ _n,~~, for

example

_pli _=< _n~ _n~+1 _>.

pi O

o-1

100000 1e+06 le+07 1e+08 1eW9 1e+10

log(t)

Fig. l. Trie density of 1-spms m a system ^of ³ x 10~ _sites with twc-side update _versus time. Trie time unit is trie total number of updates and trie parameter _is the probability _c.

(4)

We _now write down the flow

equation

for _pi

(two-sided):

fli

₌

~tl13c-1)pi 2piij.

7 characterizes the rate of the

update.

This

equation yields

_pli

"

~~~ _~~

pi in the station- 2

ary state. A

simple

mean-field ansatz would factorize _pli into pi pi and therefore lead to

pi "

(3c 1)/2

with _a critical

point

at c =

1/3.

This is the _same result _as for the

original

BS-model.

Figure

2 shows _pi

(c),pli(c),..,piiiiii (ci (note

that _pli...i

(r sites)

is

equal

to

< ni ni+i n~+~-i

>).

The mean-field solution is dotted. It is clear that this mean-field solution becomes _exact for _c _- 1.

We _now discuss the

special

_case c = 1. The

stationary

solution in this _case is trivial and the transient _can be

exactly

^solved. To do

this,

it is convenient to examine

strings

of1 sites with n~ = and to denote the

probability

for such _a

string

with _zi. The _case _zo

corresponds

to pli It is easy ^to find the flow equation ^for _zi

by considering

the

possible gains

and losses in _one

time-step, here done for the one-sided model:

ii =

7(zi+i zi) Il # 0)

and

do

='f(zi).

The rate of

update

₇ in _our model is

given by 1/(Lpi) (L=system size)

and therefore time-

dependent.

The strategy ^is now to solve these

equations

for _a system

consisting only

^of

strings

_zi with _one certain ₌

la

in tue

beginning.

The solution for _a

general starting configuration

is then obtained

by taking

the _average _over the

string-configuration

in the

beginning

since the

strings

are not

spatially

correlated. The solution for the mentioned

special starting configuration

is

~~~~°~

(lo

1)l~

~~~~

with

t

rit)

=

/~f(t')dt'

As _an

example,

_one _can calculate

pi(t)

for _a

starting configuration

which is

given by

pi

Ii

₌

0)

₌

p°

and

poli

₌

0)

₌ 1

p°.

One has tuen

~~~~~ ~

whicu

yields

Pi

Ii)

"1

Il

P°)e~~~~~~~

Tuis _is _an

integro-diflerential equation

for

pi(t).

Tue

corresponding

diflerential

equation

is

directly

solvable and _one obtains

finally

pi(t)

+

In(1 pi(t))

₌ ^~~~ +

p°

+

In(1 p°).

(5)

0.9 Pi o

Pii +

Piii °

0.8 Piiii X

Piiiii ~

Piiiiii * 0.7

0.6

o

à 0.5

~ o o

o°o o

0.3 o

o +

+

~+ ^~

+

~+~

O.1

0

o.6 o.65 0.7 0.75 0.8 0.85 0.9 0.95

~ c

pl O

Pll +

Plll °

0.8 Piiii X

Piiiii ~

Piiiiii * 0.7

0.6

à 0.5

o 0.4

O ^~

0.3 °

~ +

O-z +

+ o

o

O.1 ^°

0

0.75 0.8 0.85 0.9 0.95

~)

^c

Fig. 2.

a)

The statiouary values of pi, pli,

:., piiiiii as a fuuctiou of _c for _a system of 3 _x 10~ _sites

(two-sided).

Trie simple mean-field solution for _pi

is dotted, trie improved mean-field approximation

is drawn _in fuit fines;

b)

The _same for trie one-sided variant of this model.

We _now return to _our

general

model _c < 1 and

apply

trie

(2,1)-cluster approximation,

_as

described in reference

I?i.

^Tuis means that _we

decompose

ail clusters into two-site-clusters

(6)

overlapping just

_one site:

1+~-2

fl"<"<+i:.n,+~ #

~

~"?"?+ifl"J+inj+2

j=z

fInj+1

Tue

stationary

flow

equation

for _poo,

0 ₌

il c)~pi

₊

Il c)poli

_cpooi

becomes wituin tuis

approximation

° _~~ ~~~~~ ^~ i~

~)~°)~~

^CPoi ⁺

cl~li,

wuicu

implies tuat, using

relations like Pol " Pi Pii,

gc2 _2c ₁

j~

>

c*).

Pi #

7c 1

Tuis leads to c* ₌

il

₊

/À)

m

0.46,

wuicu is _an improvement

compared

_to the

simple mean-field-approximltion. _Figure

2 shows this

improved

solution. Data and calculation agree

very

nicely

^for

higher

values of _c.

Figure

2 also shows that the mean-field cluster approximation is _very

good

for

calculating

expectation values hke < n~ n~+i n~+~-i ^>. The data of the simulation coincide

nicely

with the law

pli

i(r sites)

=

(~~

₂

~)~ ^~pi

We

give

the

corresponding

results for the one-sided PCA:

4c~ c

~~ 3c 1

and

c* = (1 +

Vi)

_m _0.6404,

and tue numerical law

pli...i(r sites)

=

(2c 1)~~~pi.

Again,

the

simple

mean-field solution

yields

tue _same result _as for the one-sided BS-model, wuicu is _c

=

2/3.

To calculate the critical

point

_more

precisely,

one could

simply

try a

(3,2)-cluster approximation

and _so

forth,

but

unfortunately

the dimension of the system ^of equations _grows

exponentially,

besides the dilficulties

arising

in

writing

down the correct flow equations. We tried another

ansatz: one can

imagine

to map ^a system of L spms in trie

stationary

state of _a PCA-model onto

a system ^of L'

=

L/3

_spms

by performing

a sort of

block-spin-transformation

with

majority-

rule

pi

₌ _prit +

2poii

+ plot One expects

pi

_rz _pi in the

vicinity

of c*.

Tjis

^ansatz

^togetuer

witu tue flow

equation

of _poo and tue

(2,1)-type approximation

_prit ₌ ^~~~

yields

c* ₌ 1, ^the pi

trivial critical

point,

and c*

=

iii

+

À~)

m

0.69,

whicu is much better tuan tue

(2,1)-

36

cluster-approximation

result.

(7)

Tue _same metuod used for tue one-sided PCA leads to _a non-trivial critical point

~~

6

~~ ~

~~

'~

~'~~~'

In summary, we did _a computer simulation of two

closely

cellular automata for coevolution. We find tue _same exponent of the time

decay

of tue

density

of sites witu spin 1 _as for directed

percolation

but diflerent critical

points.

The critical

points

_can be

approximated

by

_means of _a

simple

renormalization argument. ^{We find} a numerical law for the

uigher

cluster

probabilities

and

give

the results of _an

improved

mean-field

theory

which is very

good

^for these

probabilities.

References

[il

Bak P. and Sneppen K., Phgs. Rev. Lent. 71 (1993) 4083; Flyvbjerg H., Bak P. and Sneppen K., Phgs. Rev. Lent. 71

(1993)

4087.

[2] ^Vabô R., Rajewsky N., Kertész J. and Schreckenberg M., _in preparation.

[3] Rajewsky N. and Schreckenberg M., Verhandl. DPG Vi 29

(1994)

955.

[4] ^Jovanovié B., Buldyrev S-V-, Havbn S. and Stanley H-E-, Phgs. Rev. E 50

(1994)

2403.

[5] ^Jàckle ^J. ^and Eisinger S., ^Z. Phgs. ^B 84 (1991) 115.

[6] ^Kinzel W., Z. Phgs. B 58

(1985)

229.

[7] Schreckenberg M., Schadschneider A., Nagel K. and Ito N., Phgs. Rev. E 51

(1995)

2939.