Speciations and Extinctions in a Self-Organizing Critical Model of Tree-Like Evolution

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Speciations and Extinctions in a Self-Organizing Critical Model of Tree-Like Evolution

M. Kramer, N. Vandewalle, M. Ausloos

To cite this version:

M. Kramer, N. Vandewalle, M. Ausloos. Speciations and Extinctions in a Self-Organizing Criti- cal Model of Tree-Like Evolution. Journal de Physique I, EDP Sciences, 1996, 6 (4), pp.599-606.

�10.1051/jp1:1996232�. �jpa-00247205�

Speciations and Extinctions in

_a

Self-Organizing ^Critical Model of Tree-Like Evolution

M. Kramer

(*),

N. Vandewalle

(**)

and M. Ausloos

(***)

S.U.P.R.A.S., Institut de

Physique

85, Sart

Tilman,

Universit6 de

Libge,

B-4000

Libge, Belgium

(Received

4 December 1994, revised 15 December1995 and

accepted

5

January 1996)

PACS.87.10.+e General, theoretical and mathematical

biophysics

PACS.05.40.+j

Fluctuation

phenomena,

random _processes and Brownian motion

Abstract. We

study analytically

_a

simple

model of _a

self-organized

critical evolution. The model considers both extinction and speciation events

leading

to the

growth

of

phylogenetic-like

trees.

Through

_a mean-field like

theory,

_we

study

the evolution of the local

configurations

for

the tree leaves. The fitness

threshold,

below which life

activity

takes

place through

avalanches of all sizes is calculated. The transition between

speciating (evolving)

and dead trees is obtained and is in agreement with numerical simulations.

Moreover,

this theoretical work suggests that

the structure of the tree is

strongly dependent

_on the extinction

strength.

R4sum4. Nous 4tudions

analytiquement

_un modAle

simple

d'4volution

auto-organis6e

cri- tique. Le modAle considAre des extinctions et des

sp4ciations

conduisant h _{une croissance} d'arbres

phylog4n4tiques.

Nous 4tudions ici par une th40rie de

champ

_moyen l'4volution des

configura-

tions des extr4mit4s de l'arbre. Le seuil critique de "fitness" en-dessous

duquel

des

explosions

d'activit6

biologique

de toutes tailles _se produisent est calculi. La transition entre arbres _crois-

sants et arbres 6teints est

4galement

obtenue _en accord

avec les simulations. En outre, ce travail

th60rique suggAre

_que la structure des arbres

g6n6r6s d6pend

fortement du paramAtre d'extinc- tions.

1. Introduction

Biological

evolution is not considered _{as a} continuous process but is

thought

to be like _a succession of bursts of life

activity separated by long periods

of

quiescence

_{on a}

geological

timescale [1]. ^{Such bursts} seem to have all sizes

including catastrophic

_ones like the Cambrian

explosion

some 570 million years ago.

In order to mimic the

intermittency

_{[2] of} an

evolving ecosystem,

^{Bak and coworkers have} introduced _an

interesting

model of evolution

[3,4].

This model

(BS)

considers _an ecosystem made of N species

arranged

_{on a} d-dimensionnal

hypercubic

lattice. A fitness

f~,

which is

a random number between _zero and _{one, is}

assigned

to the I

species.

The

higher

the fitness is, the _more

adapted

is the _{species to} its environment

(the ecology).

At each

step

^{of the}

(*)e-mail

address:

kramer©gw.unipc.ulg.ac.be (**)e-mail

address:

vandewal©gw.umpc.ulg.ac.be

(***)

Author for the

correspondence (e-mail

address:

ausloosflgw.unipc.ulg.ac.be)

©

Les

kditions

_de

_Physique

₁₉₉₆

600 JOURNAL DE

PHYSIQUE

I N°4

Y,

Z+

Y,

~

Y ^~,

Zo

Fig.

1. A small part ^of a

phylogenetic-like

tree. The species _are labelled "y" ^and "z" with sublabels

I'), (°)

and

(+) according

to the different local

configurations

_as discussed in the text.

evolution,

the

species having

the minimum fitness value is selected and is

supposed

to make _a so-called

adaptative

_move

by receiving

a new random fitness value. Because of interactions _m the

ecosystem,

^this

adaptative

_{move is}

supposed

to affect also the fitness values of

neighbounng species.

The evolution _process

generated by

^this model _can be

regarded

as a

particular

random walk in _a random medium. The interests of this model _are

i)

^{that from}

simple rules,

it

generates

intermittent avalanches of all _sizes, and

it)

that the _process is

recognized

to be

self-organized

critical

[3],

1e. to reach

asymptotically

an

apparently

critical

steady-state (the

so-called "evolution to the

edge

of chaos"

).

^{The latter}

dynamics

_is of

physical

interest [5] since

various natural

systems

have _a

tendency

to reach _a critical

self-organized

state like

sandpiles

_[6],

earthquakes iii, high-Tc superconductors [8],..

More

recently,

_we have extended the BS model for _an evolution

allowing

for _a non-conservation of the number of _{species m} the ecosystem

[9-11].

The tree-like evolution model considers

speci-

ation and extinction events

[11].

^Each

speciation corresponds

to the differentiation of _a

species

made of _a

large

_group of individuals _{m a}

population.

The differentiation _is considered to lead to _a

branching

event. The formation of _a tree-like structures is obtained with patterns m

which the different

species

are located _on the leaves

(see Fig. 1).

The "distance"

dmn

between two species

(m

and

n)

is the number of

segments

^{of the} tree needed to connect _one

species (m)

to another

in).

A fitness f~ ^{which is} a scalar random number between zero and _one is

associated to each I

living

_species. At each Monte-Carlo step, ^{the less}

adapted

_{species, i.e.} the

living j species having

the _minimum fitness

fj

is selected. This _{species is} assumed to become extinct with _{an a}

priori probability exp(- fj /r)

_or to

give

two differentiated _species with _a

probability (1- exp(- fj /r>).

The latter

speciation

event leads to two

living

_species which

both _{receive a new}

arbitrary (uncorrelated here>

^fitness value. Of _course, extinct species do

not

participate

^further to the evolution _process. The parameter r is

hereby

called the

strength

of the extinctions

[11].

^{We also consider} the presence of interactions between the

species,

such that the fitness _{f~ of} all the

species

which _are

separated by

_an

arbitrary

distance d~j from the

branching

_or

dying

_species

j

less than _a short _range k _are also

updated

with _{a new} random number. This rule mimics the drastic

change

of the "fitness

landscape"

_[12] within _a range k due _{to an} evolution event. No correlation _is introduced here between fitness values from

ancestors to

offsprings,

_or between

offsprings.

These effects could be examined in further work.

.6

.4

cl 1.2

fi

'

I-O t$

~ O-B

~/~

~j~

~'~

r=0 25

~' O.4

r=0.45 0.2

O-O

O-O O.2 O.4 O.6 O-B I-O

f

Fig.

2. The distribution

n(fmm)

of the minimum fitness values

fmm

selected

during

the tree-like evolution _process. Three different values of the extinction

strength

parameter r are illustrated.

Moreover,

_we consider herein the _case k

= 2 which is of

biological

_interest since the

majority

of

living species (the

so-called

"specialists"

_{[13] occupy narrow}

ecological

niches and _are

only

sensitive to their

genetically

nearest

neighbounng species [13].

In the next

section,

_we will recall the _most

striking

numerical results. In Section

3,

we

theoretically investigate

this evolution

through

an

original study

of the local

configurations

for the leaves of the tree. The critical threshold

fc(r)

below which the life

activity

takes

place

is calculated and the transition between

evolving

and dead trees _is obtained in

agreement

^with the numerical results of Section 2.

Finally,

_a conclusion will be drawn

outlining

the interest of

these results and

suggesting

new

developments.

2. The

Striking

Numerical Results

Let _us recall the most

important

features

presented by

the above model _as found from numerical work

[11].

^Without extinctions

(for

_r

=

0),

the model reduces to a

previously

studied model _[9]

for which the

growth

^of trees never stops. As _r

increases,

the extinction events becomes _more and _more

frequent

such that the tree

stops

to grow above _a certain value _rc. This critical value

was

numerically

found to be _{rc =} 0.48 + 0.01

[11].

For 0 < _r < rc, the evolution process reaches

always

_an apparent

steady-state.

The latter

is characterized

by

_a

step-like

distribution of fitness values between _a value

fc

and _1. This characterizes the state of the

majority

of the _species in the

ecosystem.

^{On the other}

hand,

the fitness values

fmjn

of the selected _{species are} distributed between 0 and

fc.

The latter distribution

n( fmjn)

is illustrated in

Figure

2 for three different values of _r. One should note that the distribution

n( fmin)

_seems to be the _sum of _a

rectangular

and _a

triangular

distribution.

When _{r > rc,} the process is limited in "time" _since the tree

growth stops.

Figure

3

presents fc

as a function of the extinction

strength

r. For _r

=

0,

_one ^finds the

particular

and non-trivial value

fc

= 0.445 + 0.010 [9] which _was

theoretically predicted

in _a

previous

work

[10].

^For r close to rc, the threshold

fc

reaches 1 because the most

adapted

species only

_can resist

against

mutation.

However,

the

fc(r)

_curve falls to zero for _r close to

602 JOURNAL DE PHYSIQUE I N°4

o

o-B

O.6

O.4

0.2

o-o

o-o r

Fig.

3. The threshold

fc

_{as a} function of the extinction

strength

_r. The continuous _curve is the

ong obtained theoretically with the relations

(2)

and

(6).

Data points from reference [11].

rc in

Figure

3. This _is understood _as

resulting

from the finite-size effect _occurring close to rc.

Above _rc, the critical threshold

fc

is rather

meaningless

_since it is defined for

living

_species

only

and for _{r > rc} all

species

are after _a finite "time" accumulated at

f

₌ 1 and die.

The existence of such _a threshold _means that the whole life

activity self-organizes

for

0 < _r < rc and takes

place

below

fc. Physically,

the attractor of this

self-organization

is

thus _{a state} where all

species

have _a fitness above _or

equal

to

fc.

This _state is however unstable

since the

species

with the minimum value

la species

close to

fc)

will then be selected.

Thus,

some species

having

_a fitness below

fc

_{can appear} later and _can also induce _some

perturbation

for the attractor of fitness distribution.

Then,

the _new

f~'s

are chosen until the unstable state

is

again

reached. This defines _an avalanche _{as a} connected _sequence of

biological activity (spe-

ciations and

extinctions)

below the threshold

fc,

_one observes _an

mtermittency

of avalanches of all _sizes. The size-distribution of avalanches

n(s)

is

numerically

found to be _a power law

n(s>

_r~ s~~ with _an

exponent

T =

3/2

which _seems to be

independent

of _r for 0 < _r < rc

[11].

This "no-scale" _or power law distribution _is the

signature

^of

self-organized critically

_[5]

driving

this evolution process.

Beside the

self-organized

critical

dynamics

of this process, ^it is of interest to observe the

structure of the growing trees. The trees _were

numerically

found to be scale invariant for

0 < _r < rc

[11].

^{This is another}

signature

^of

critically. However,

the fractal dimension

Df

of

the trees _was

thought

to be

dependent

of the

parameter

r

[11].

^{This result} was

unexpected

and is _m contrast with classical

phase

transitions for which the value of the critical exponents

does not

change

with _any

tuning

^of the relevant parameter. In

fact, Df

_was

numerically

found to decrease

continuously

^from 2

(for

_r

=

0)

towards for _{r ~ rc}

[11].

^This

r-dependence

is illustrated in

Figure

4 where three simulated trees _are drawn for three different values of the extinction

strength

_r. Each tree contains 2000

branching points.

^The variation of the _tree

structure from _a two-dimensional

growth

for _r

= 0 towards _a

filamentary-like

gro1&>th ^for r m r~

is well marked.

In the

following section,

_we

theoretically investigate

how the system ^evolves

through

local

configurations

and _we extract theoretical values for

f~(r>

and _rc.

00 r=0.

Fig.

4. Three

phylogenetic-like

trees simulated for three different values of the extinction

strength:

r = 0.001, _r

= 0.25 and _r

= 0.45 close to _rc. Each tree _is made of 2000

branching

points.

3. Theoretical

Investigations

For the k

= 2 process, ^two different

types

^{of local}

configurations

_{can occur} on the leaves.

They

_are labelled

"y"

and "z" in

Figure

1. The so-called y-species are

coupled by

_pairs, while the

z-species

are somewhat

lonely.

These different

types

^of

species

lead to different

speciation

processes.

Indeed,

the

speciation

^of a

y-species gives

_{a new pair} of

y-species branching

out the selected

species.

^This can also affect the nearest

living neighbour species

since the distance between the

speciating species (the

selected

one>

^and the direct

neighbouring y-species

is

equal

to k

= 2. The latter becomes _a z-species

(see

_an

example

of

y-speciation

_m

Fig.

_5>. On the other

hand,

the

speciation

of _a

z-species

cannot affect _any other species since the distance

between the

speciating z-species

and _any other

species

is

always greater

than k

= 2.

The

geometrical configurations

of the

tree-leaves,

both _g- and _{z-species, can} be

put

into three different classes: _I> the class with

living species having

a fitness value less than

fc

such that these

species

could be selected

by

the process:

they

_are

active, ii)

the class with

living species having

_a fitness value

larger

than

fc

such that

they

cannot be selected for

speciation

or extinction:

they

_are

inactive,

and

iii)

the dead

species.

For further

developments,

let _us call

y' (z'>

the active

species, y° (z°)

the inactive species and

y+ (z+>

the dead

species.

To describe the evolution _process

only

9 different

configurations

of _tree leaves _as shown in

Figure

6 have to be considered.

They correspond

to 6 _pairs:

y'y', y'y°, y'y+, y°y°, y°y+

and

y+y+,

_as well _as to 3

z-configurations: z',

z° and z+ One should note that 5

configurations

do not

participate

m the evolution _since

they

_are dead _or screened from the _process. These blocked

configurations

_are the

y+y+, y°y°

and

y°y+

_{pairs as} well _as the z° and

z+ species.

604 JOURNAL DE

PHYSIQIJE

I N°4

Y,

Y Z,

Fig.

5. An

example

of evolution event occurring on a

y'-type

leaf

(on

the t.op of the small

branch).

This

example

is _a speciation

leading

to the formation of _{a new}

y°y' pair

and to _a z'-species. The

probability

of _occurrence of the latter event is

ii p)(1 fc) If.

_{The state of the}

_z°-species

on the

bottom of the branch _remains

unchanged

_since the distance from the latter _species to the speciating

one is 3

(> k).

,-~

_W

/~/f

^Y°

_N/ ^/

^Y+

_{, ~/ Y~'j~z° "j~z+}

. . ,

blocked

(b)

^{~~ ~~}

~ ~

Y~

~'~

~'

~ ~/ ~ ~/

,. .,

N/

evolving

Fig.

6. Th~ 9

locally

different

configurations

which _are sufficient to describe the evolution process

m our mean-field like theoretical

development.

One _can

distinguish

between the

(a)

blocked and the

(b) evolving configurations

with either _{one or} two active species.

By

_comparison to

equations ii),

_one

can

easily

_see how the

(b)

cases can lead to _new

(b)-like configurations.

0lhe

(b) configurations leading

to

(a)

_{ones are} irrelevant for

describing

the life _process since such

(a) configurations

do not evolve anymore.

For _{a mean} field-like

approximation,

we can assume that all the active

species speciate

_or become extinct in _a

single

mean-field time

step

t. One should note that the latter time _is not

equiv~lent

to the Monte-Carlo time. In order to

keep

_some essence of the _{process, we} also consider the restriction that for the

particular g'g' configuration only

_{one species} speciates or

becomes extinct. A

possible

evolution of _a

g'g' configuration

_is illustrated in

Figure

5 _as an

example.

The latter

particular example

_{is a}

speciation leading

to a new

g°y'

_pair and to _a z'

species.

Let _p be the

probability

that _a chosen

species

^becomes extinct

(+).

One should

note that the

probability

to have _{a nevz}

living

and active species

I')

is

ii p) f~

while the

probability

tn have _{a new}

living

and inactive _species

(°)

is

(1- pill f~).

One _can write

the mean-field evolution for the number of

configurations participating

to the _process

through

recurrence relations:

Ny,y, it

+

II

=

ii P)f/iNy,y, it)

_{+ Ny,~O}

it)

₊

_Ny,y+ it)

₊

N~,it)i ila) Nyoy, (t

+

1)

₌

2(1 P)fc(I fc)iNy,y, (t)

₊

_Ny,yO it)

₊

_Ny,~+ (t)

+

N~, (t)i lib) N~+y, (t

+

I)

=

PfciN~,~, it)

+ N~,~O

it)i ilc)

N~, (t

₊

i)

=

ii pj f~jN~,~, it)

+

N~,~a jt)j (id)

where the factors _are

probabilities

for

creating

such

configurations.

One _can find that the above

system

^satisfies a

steady-state

solution

only

for

il-P)if/-f/-2f~)+1=0 ₁₂₎

The

probability

_p is _a function of both _r and

f~.

One should remark that the _{case p}

= 0 is

equivalent

to consider

only speciations (r

=

0)

without extinctions.

Then,

the relation

(2)

reduces to the relation found in reference

[10] having

_a

single

root between 0 and 1, I-e- _a threshold at

f~

m 0.445.

In order to find _a relation between

f~

^{and the extinction}

strength

_r, one has to calculate the

probability

_p ^for

extincting

_a selected

species.

In

doing

_{so, one} needs to know the distribution for the whole set of _minimum fitness values

RI fmin),

I-e- the distribution of the selected

species.

Indeed,

_{p is given}

by

~,

p "

/ _il(fmin

eXp

(-

^f~'~~ d

jmin (3)

o T

For the BS

model,

the

n( fmin)

distribution _was found to be linear

vanishing

to _zero at

f

_m

_f~

[3,4]. However,

the

present

^{model shows} a different

RI fmin)

distribution which _seems to be the _sum of _a

triangular

and _a

rectangular

distribution _as noted in Section 2. This difference _is understood from the fact that each

speciation

event in the present model creates _new

offsprings

with _new

uniformly

distributed random fitness values. This

corresponds

to add _a

rectangular (flat)

distribution _on the

triangular

distribution of the BS model.

For _r m r~, the distribution for the minimum fitness values is found to be linear

RIjmin)

"

3/2 jmin _14j

with _a

slope

-1.

Moreover,

for all 0 < _r < r~

values,

the

RI fmin)

distribution

collapses

for

fmin

<

f~(r)

_on the above distribution

3/2 fm;n using

an

appropriate scaling

factor. This

is shown _m

Figure

2. The normalization for 0 <

f~;~

<

f~(r)

of the

RI fm;n)

distribution of

equation (4)

_gives

~~~~~"~~ ~jc~~)i~

^~~~

^~~~"

^~~~~~

^~ _lsl

R(fm~n) =0,

for

fm;n

above

f~

Using

the latter ad hoc distribution

R(fm;n),

_{one can} find the extinction

probability

P -

~j~

i

ji

3 2T) Ii

exP

~

+

2fc

_exP

~

)1 ¹⁶⁾

Using

this definition for _{p m}

equation (2)

and

searching numerically

for the solution of the

resulting

transcendental

equation,

_one finds

f~(r).

The result is drawn _m

Figure

3. This _curve is

roughly

linear in the range of interest.

Moreover,

the

agreement

^with the numerical values

is

quite good

_{as seen} in

Figure

3. Since

f~(r~)

=

1,

one can then deduce the critical extinction

strength

_r~ for _{growing or} dead _trees. Our theoretical estimate gives r~ m 0.495 which is also

m

quite good agreement

^{with the numerical value} r~ = 0.48 + 0.01

[10].

606 JOURNAL DE

PHYSIQUE

I N°4

4. Conclusion

In the

preceding section,

we have described the evolution process for leaf

configurations.

Two

hypotheses

have been made:

i)

correlations _are

neglected

_even

though

_more than _one active

species

is selected at each

step

for

describing

the _process

through

the set of

equations ii),

and

it)

the distribution

RI fm;n)

is ad hoc. The

present

mean-field

approximation

_can thus be considered _to be

vanationally

based.

However,

the

agreement

^is quite remarkable. This method is thus useful for

describing

such

a kind of

self-organized

critical _processes.

A further _comment is however in order. As

pointed

out in Section

2,

the most

striking

result for the tree evolution model is the

unexpected

variation of the fractal dimension

Df

of the

trees with the parameter r. The present theoretical

investigations

cannot

predict

the

global

structure of the

growing

tree since the mean-field time _t cannot be related to any

geometrical

property ^{of the} tree.

However,

our theoretical

investigations

show also that the local structure of the tree

(leaf configurations)

is

strongly dependent

_on the extinction

strength

_r

(see

the set of

Eqs. (1)).

Further work should be made _m order _{to test} whether

Df

_vary with _r.

Acknowledgments

Part of this work _was

financially supported through

the

Impulse Program

_on

High Temperature Superconductors

of the

Belgium

Federal Services for

Scientific,

Technical and Cultural

(SSTC)

Affairs under _contract SU

/02 /013.

This work _was also

supported through

the ARC

(94-99 /174)

contract of the

University

^of

LiAge.

N-V- thanks the

Belgium

Research Funds for

Industry

and

Agriculture (FRIA, Brussels).

A

special _grant

from

FNRS/LOTTO

allowed _us to

perform specific

numerical work.

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