GERAND William2008/2009

Institut Supérieur des Matériaux et Mécaniques avancés44, avenue F.A. Bartholdi, 72000 - LE MANS

Téléphone : 33 (0) 243 21 40 00 ; E-mail : ismans@ismans.fr ; http://www.ismans.fr

ISMANS

Collaborators:

LIZE FlorianLETANG BaStien

LAURENT VincentM. WANG

Interests of the Bak-Sneppen model:

Evolution theories: dynamics systems theory of evolutionClimatology: Used to study extreme climatic events

Epidemiology: In the fractal growth modelEconomy: Same events than in evolution theories

Objectives:

Determinate the informational entropy : disorder of the system.Introduce variations in the original model to see what happened in the model.

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I- Introduction to the BAK-SNEPPEN model

II- Information theory

III- New sympatrics evolutions of the BAK-SNEPPEN model

IV- Allopatrics theories of the BAK-SNEPPEN model

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Species evolutions with a probabilistic approach.

“Fitness” transcript the adaptability rate of a specie and defined by:0 < f < 1

Fitness discrete of the model in classes is necessary.

The value of the size of the intervals is called p.

Principle:

Picture 1: Iterative process of the Bak-Sneppen model.

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In a probabilistic universe Ω (Ω=N).N: number of species in the model.p: number of intervals in the model.ni: number of species in the interval i.

Probability is defined by : Pi = ni/N (1)

The informal entropy permit to determine the disorder of the system: S(N,p)= - Σ Pi.ln(Pi). (2)

Informational entropy:

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1- Bak-Sneppen model.

2- Generalized Bak-Sneppen models:1.1- The generalized Bak-Sneppen model1.2- The generalized cannibal model

3- Kauffmann models:2.1- The Kauffmann Bak-Sneppen model2.2- The generalized Kauffmann model

4- The Gould-Phan model

5- Evaluating of a new sort of choice

6- Use of the co-death phenomenon

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2- Bak-Sneppen model:

Picture 2: Probability evolution in the Bak-Sneppen model (N=50, p=20, m=20000)

Picture 3: Informational entropy in the Bak-Sneppen model (N=2000, p=20, m=50000)

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2- Generalized Bak-Sneppen models:

Wanderwalle-Ausloos (1996):

"If the distance of interaction k is finite size, it seems clear that some species are excluded from the overall and no longer modify their fitness."

Any evolution of a specie in a food chain is sufficient to create an adaptation of all other species.

Creation of a co evolution vector (k) defined by a maximum:

(5)1

2max

Nk

Picture 4: Evolution of the co evolution vector.

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Impacted: 2k+1 species.Here k=3

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2.1- The generalized Bak-Sneppen model:

Picture 5: Probability evolution in the generalized Bak-Sneppen model in function of k, (N=100, p=10, m=1000) 8

This model used a co evolution factor defined by: max;2: kiFor ifff i

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2.1- The generalized Bak-Sneppen model:

9Picture 6: Informational entropy of the generalized Bak-Sneppen model in function of k, (N=100, p=10, m=1000)

This model used a co evolution factor defined by: max;2: kiFor ifff i

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2.2- The generalized cannibal model:

Cannibal model: only an increase of fitness.

Picture 7: Informational entropy of the generalized cannibal model k=25, (N=100, p=10, m=1000)

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co evolution factor :

max;2: kiFor

ifff i

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Analysis :

Generalized Bak-Sneppen model: Not realistic.

Generalized cannibal model:

In a first approximation: tends to validate the Wanderwalle-Ausloos theory. Not realistic.

Need to increase p, m and N to confirm our conclusions.

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3- Kauffmann models:

Stuart Kauffmann:

"Any alteration results in an avalanche effect (or domino effect). In an intermediate situation on the border of chaos, with moderate interactions, only some disturbances associated cascading changes that can trigger massive avalanches, similar to mass extinctions. When the system is at the border of chaos, the changes follow a scaling law. "

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Kauffmann-Bak-Sneppen model: Generalized Kauffmann model:

0: Bak-Sneppen model.random

1: Generalized Bak-Sneppen model.

0: small krandom

1: large k

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3.1- Kauffmann-Bak-Sneppen model:No real effect when

k increase.

Picture 8: Informational entropy of the Kauffmann-Bak-Sneppen model in

function of k, (N=100, p=10, m=1000)

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0: Bak-Sneppen model.random

1: Generalized Bak-Sneppen model.

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3.2- Generalized Kauffmann model:

Picture 9: Probability evolution in thegeneralized Kauffmann model in function of k, (N=100, p=10, m=1000)

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Increase k = Increase the co evolution effect

0: small krandom

1: large k

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Analysis :Kauffmann-Bak-Sneppen model:

Presence of surviving areas for small to mid fitness species. Surviving probabilities decrease for high fitness species. Seems tends to an equilibrium.

Generalized Kauffmann model:

Large variations: Small fitness species survive. Large fitness species disappear.

In a first approximation, tends to validate the Kauffmann hypothesis of limited chaos, because the informational entropy tends to be stable.

Need to increase p, m and N to confirm our conclusions.15

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3- Gould-Phan model:

Gould hypothesis of multivers (1989):

"The historical contingency results in an infinity of virtual worlds which us is one of the possible achievements.“

Denis Phan theory of critical states self-organized (2002):

"Morphogenetic processes are highly constrained, reducing the universe of possibilities which reduces the possibilities of stable forms. The areas of co-evolution should move towards areas of critical meta stable maximum fitness “

Using of temporal functions defined by:

(6)

b is a factor choose to see the function influence.

brandTj )1(

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;1: jFor

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3- Gould-Phan model:

Picture 10: Probability evolution in the Gould-Phan model when b 0, (N=100, p=10, m=1000)

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Increase function’s influence = Typical extremes forms.

brandTj )1(

;1: jFor

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Analysis:

b 0: Extreme fitness species survive.

b +∞: High fitness species not too favorites. Presence of big surviving areas in low to mid-fitness species.

In a first approximation: Gould Phan theory may be validate but “maximum fitness” become “extremes fitness”.

Not a realistic model

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4- New sort of choice:

f < 0 or f > 1:

Impossible events↓

nothing happens to the target specie.

Only concluding in the Gould-PhanModel.

Picture 11: Probability evolution in the Gould-Phan choice model (N=100, p=10, m=1000, b=0.001) 19

Most realistic sympatric model.

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5- Co death phenomenon:

When a secondary targeted specie by k obtain a negative fitness:

Introduction of a co death factor call d. same form than k.

Picture 12: Typical informational entropy evolution in the Kauffmann co death model (N=100, p=10, m=1000)

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Same forms at each large variation

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1- One common specie model

2- Two commons species model

3- Simulation of a simplified ecosystem

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Central food chain.

All cycles possess the same number of specie N and the same radius R

R,3R fractal symmetry

Each intermediate cycle possess two commons species with others intermediate cycles.

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1- One common specie model:

Picture 13: Graphic taking place of the on common specie allopatric theory.

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Central specie model.

All cycles possess the same number of specie N and the same radius R

R,2R fractal symmetry

All species are common to two or more others cycles.

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2- Two commons species model:

Picture 14: Simplified graphic model of the two common species model.

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Seven Bak-Sneppen cycles.

Fitness classes for each cycle: the aquatic environment: f <1 the terrestrial environment: f <0.9 the forest environment: f <0.8 the mid mountain environment: f <0.6 the arid environment: f <0.4 the volcanic environment: f <0.3 the middle sub-plot environment: f <0.2

Inclusion of a seed in the aquatic environment.

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2- Simulation of a simplified ecosystem:

Picture 15: First simulation result.

It is the most realistic result obtained during this study.

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Sympatrics models:

Generalized Back-Sneppen model is not realistic. Generalized cannibal Back-Sneppen model tends to validate the Wanderwalle-Ausloos

theory in a first approximation. Kauffmann model is more realistic than previous models. Generalized Kauffman model tends to validate the Kauffman theory of limited chaos in first

approximation. Gould-Phan model is not realistic but, in a first approximation, the Gould Phan theory may be

validate but we must change “maximum fitness” by “extrema fitness”. The new sort of choice tends to make the Gould-Phan model to a realistic one. The co death is useless in this model, but it make Kauffmann instabilities more similar.

Allopatrics models:

The geometric theory of the Bak-Sneppen model make the best results with the only classical Bak-Sneppen model.

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1.2- The generalized cannibal model:

Base statement: This model used a co evolution wave function defined by:fi

iif

2)1(

Informational entropy of the generalized cannibal model in function of k, (N=100, p=10, m=1000) 10

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1.2- The generalized cannibal model:

Base statement: This model used a co evolution wave function defined by:fi

iif

2)1(

Probability evolutions in the generalized cannibal model in function of k, (N=100, p=10, m=1000) 11

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2.1- Kauffmann-Back-Sneppen model:Base statement: This model used a co evolution wave function defined by:

fiiif

2

)1(

Probability evolutions in the Kauffmann-Back-Sneppen model in function of k, (N=100, p=10, m=1000) 15

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2.2- Generalized Kauffmann model:Base statement: Variable strength of the co evolution wave function.

Informational entropy of the generalized Kauffmann model in function of k, (N=100, p=10, m=1000) 16

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3- Gould-Phan model:

Informational entropy of the Gould-Phan model in function of b, (N=100, p=10, m=1000) 20

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