Transcript Generalized Bak-Sneppen model

```ISMANS
Collaborators:
GERAND William
2008/2009
Institut Supérieur des Matériaux et Mécaniques avancés
44, avenue F.A. Bartholdi, 72000 - LE MANS
Téléphone : 33 (0) 243 21 40 00 ; E-mail : [email protected] ; http://www.ismans.fr
LIZE Florian
LETANG BaStien
LAURENT Vincent
M. WANG
ISMANS
Interests of the Bak-Sneppen model:
Evolution theories: dynamics systems theory of evolution
Climatology: Used to study extreme climatic events
Epidemiology: In the fractal growth model
Economy: 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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Principle:
 Species evolutions with a
probabilistic approach.
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.
Picture 1: Iterative process of the Bak-Sneppen model.
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Informational entropy:
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)
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1- Bak-Sneppen model.
2- Generalized Bak-Sneppen models:
1.1- The generalized Bak-Sneppen model
1.2- The generalized cannibal model
3- Kauffmann models:
2.1- The Kauffmann Bak-Sneppen model
2.2- 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."
Impacted: 2k+1 species.
Here k=3
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:

N
k max   1
2
Picture 4: Evolution of the co evolution
vector.
(5)
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2.1- The generalized Bak-Sneppen model:
This model used a co evolution factor defined by: For : i  2; k max

fi 1
2i
fi  fi 
Picture 5: Probability evolution in the generalized Bak-Sneppen model in function of k, (N=100, p=10, m=1000)
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2.1- The generalized Bak-Sneppen model:
This model used a co evolution factor defined by: For : i  2; k max

fi  fi 
Picture 6: Informational entropy of the generalized Bak-Sneppen model in function of k, (N=100, p=10, m=1000)
fi 1
2i
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2.2- The generalized cannibal model:
Cannibal model: only an increase of fitness.
co evolution factor :
For : i  2; k max
fi  fi 

fi 1
2i
Picture 7: Informational entropy of the generalized
cannibal model k=25, (N=100, p=10, m=1000)
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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. "
Kauffmann-Bak-Sneppen model:
0:
Bak-Sneppen model.
1:
Generalized Bak-Sneppen
model.
random
Generalized Kauffmann model:
0:
small k
1:
large k
random
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3.1- Kauffmann-Bak-Sneppen model:
0:
Bak-Sneppen model.
1:
Generalized Bak-Sneppen
model.
random
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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3.2- Generalized Kauffmann model:
Increase k = Increase the co evolution effect
0:
small k
1:
large k
random
Picture 9: Probability evolution in the
generalized Kauffmann model in
function of k, (N=100, p=10, m=1000)
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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.
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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:
For : j  1;
Tj 
 rand (1)
b
(6)
b is a factor choose to see the function influence.
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3- Gould-Phan model:
For : j  1;
 rand (1)
Tj 
b
Increase function’s influence = Typical extremes forms.
Picture 10: Probability evolution in the Gould-Phan
model when b 0, (N=100, p=10, m=1000)
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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-Phan
Model.
Most realistic sympatric model.
Picture 11: Probability evolution in the Gould-Phan choice model
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(N=100, p=10, m=1000, b=0.001)
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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.
Same forms at each large variation
Picture 12: Typical informational entropy evolution in the Kauffmann
co death model (N=100, p=10, m=1000)
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1- One common specie model
2- Two commons species model
3- Simulation of a simplified ecosystem
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1- One common specie model:
 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.
Picture 13: Graphic taking place of the on common specie
allopatric theory.
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2- Two commons species model:
 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.
Picture 14: Simplified graphic model of the two common
species model.
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2- Simulation of a simplified ecosystem:
 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.
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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[email protected]
[email protected]
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1.2- The generalized cannibal model:
Base statement: This model used a co evolution wave function defined by:
f(i
1
)
fi
2
i

Informational entropy of the generalized cannibal model in function of k, (N=100, p=10, m=1000)
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1.2- The generalized cannibal model:
Base statement: This model used a co evolution wave function defined by:
f(i
1
)
fi
2
i

Probability evolutions in the generalized cannibal model in function of k, (N=100, p=10, m=1000)
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2.1- Kauffmann-Back-Sneppen model:
Base statement: This model used a co evolution wave function defined by:
f(i
1
)
fi
2
i

Probability evolutions in the Kauffmann-Back-Sneppen model in function of k, (N=100, p=10, m=1000)
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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)
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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)
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