Transcript Complete Slide Set

Strategies and Rubrics for Teaching
Chaos and Complex Systems Theories as
Elaborating, Self-Organizing, and
Fractionating Evolutionary Systems
Fichter, Lynn S., Pyle, E.J., and Whitmeyer, S.J., 2010,
Journal of Geoscience Education (in press)
Boids
MATFA’s Boids
SOC
Self-Organized
Criticality
Evolution Via Self Organization
Self Organized Criticality
Evolution Via Self Organization
Self Organized Criticality
Self-Organized Criticality
Per Bak
1948-2002
“Complex behavior in nature reflects the
tendency of large systems with many
components to evolve into a poised, "critical"
state, way out of balance, where minor
disturbances may lead to events, called
avalanches, of all sizes. Most of the changes
take place through catastrophic events rather
than by following a smooth gradual path. The
evolution to this very delicate state occurs
without design from any outside agent. The
state is established solely because of the
dynamical
interactions
among
individual
elements of the system: the critical state is selforganized. Self-organized criticality is so far the
only known general mechanism to generate
Avalanche Behavior
The sand pile builds . . . grain . . . by grain . . .
by grain . . .
by grain . . .
by grain . . .
by grain . . .
by grain . . .
by grain . . .
Building toward the critical state . . .
Where it avalanches
building
building
building
avalanche avalanche avalanche
Avalanche- a large mass of snow, ice, etc., detached from a
mountain slope and sliding or falling suddenly downward.
Avalanche- anything like an avalanche in suddenness and
overwhelming quantity: an avalanche of misfortunes; an avalanche
of fan mail.
Power Law Sand Supply
Now, imagine the sand supply follows a
power law (or is fractal), with different
numbers of grains falling at different times.
Avalanches
distribution.
will
follow
a
power
law
Earth Temp.
curve over the
past 400,000
years
http://atlas.gc.ca/maptexts/topic_texts/english/images/TemperatureCO2.jpg
Examples of Extreme
Avalanches
1929 stock
market crash
1987 stock
market crash
Examples of Extreme
Cascading PowerAvalanches
Grids Failures – when a hub is
required to carry more than it is capable of carrying,
and so crashes, leading to the next hub to crash, etc.
North America blackout 1965
North America blackout 2003
Examples of Extreme
Avalanches
Extinctions
8000
Distribution of Geologic Life Spans
of Fossil Genera
Number of Genera
6000
Note that most genera have a life span of 10
million years or less, and that very few
genera survive over 100 million years. This
is a power law relationship.
4000
2000
0
0
50
100
150
Life Span (millions of years)
A r e m ass ext i n ct i on s d i f f er en t
f r om b ackg r ou n d ?
50
Extinction Intensity For
Phanerozoic Genera
In 106 Time Intervals (stages)
Note that the number of large extinctions are
very infrequent. The largest, the Permian, is
the only one of its size. Smaller extinctions—
<10 %—on the other hand, occurred in over 50
separate geological stages.
30
20
Pe
rm
ian
rd
O
Life is a Self Organized Critical
phenomena
K-
T
ov
ic
ian
an
d
10
Tr
ias
si
c
De
vo
ni
an
Number of Geologic Stages
40
0
0
20
40
60
Percent Extinction
80
100
Stuart Kauffman
“The critical point is not, as Stuart
Kauffman once described it, “a nice place
to be.” So “survival of the fittest” does not
imply evolution to a state where everybody
is well off. On the contrary, individual
species are barely able to hang on - like
the grains of sand in the critical sand pile.”
Systems are always at the critical
point, or if they are not at the critical
point they are evolving toward the
critical
point.
That is, the common idea that systems
evolve
toward
equilibrium
misperception of reality.
is
a
Cellular
Automata
Cellular Automata and Self Organization
Cellular Automata (CA) are simply grids of
cells, where the individual cells change
states according to a set of rules. The CA
may be one dimensional, or linear, like a
string of cells in a row (below), or two
dimensional, like a checkerboard
Local Rules/Global
Behavior
Optimal Local Rule Set
Survival Rules – 2/3 a live cell
1 2 3
survives to the next generation if at least 2 but no
more than three of the surrounding 8 cells are alive.
Less than 2 and it dies of loneliness; more than 3 and
it dies of over crowding.-
8 ? 4
Birth Rules – 3/3
7 6 1
5 2 3
8
4
7 6 5
a dead cells comes
alive the next generation if 3, any 3, of the surrounding
8 cells are also alive.
Now, as cells are added, which
will come alive,
which survive?
Cellular Automata and Self Organization
Classifying Cellular Automata Rules
Stephen Wolfram
Class One - Fixed or Static: Rules that
produce dull universes, such as all dead cells
or all living cells; e.g. ice.
Class Two - Periodic or Oscillatory: Rules
that produce stable, repetitive configurations.
Class Three - Chaotic: Rules that produce chaotic
patterns; e.g. molecules in a gas.
Class Four - Complexity: Rules that produce complex,
locally organized patterns; e.g. behave like a flowing
liquid..
Classifying Cellular Automata Rules
Chris Langton
Power-Law
Relationships in Cellular
Run a random array until it stops,
add a live cell at random, run again until it
Automata
stops. Avalanche size is the number of generations from initiation until it stops.
Most avalanches last only one or a few generations; others may last hundreds of
generations. Plotted up the avalanches follow a power-law meaning Cellular
Automata (with optimum local rules) are Self Organized Critical systems.
Evolution of Ripples
A cellular automata model
Mechanics of Wind Ripple Stratigraphy
Author(s): Spencer B. Forrest and Peter K. Haff
Source: Science, New Series, Vol. 255, No. 5049 (Mar. 6, 1992), pp. 1240-1243
Published by: American Association for the Advancement of Science
Cellular Automata Examples
Bak-Sneppen
Ecosystem
Model
Modeling an Evolutionary System
Generation 1
The Bak-Sneppen evolutionary
model is an “ecosystem” in which
the fitness of each “species”
changes because of its
relationships with other “species”,
following two simple rules
Threshold fitness the highest level the lowest fitness
species has reached
Rule One - find the species with the lowest fitness and
randomly change its fitness.
Rule Two - at the same time the lowest fit species is
changed, also randomly change the fitness of the species to
the immediate left and right.
Modeling an Evolutionary System
Generation 2
The Bak-Sneppen evolutionary
model is an “ecosystem” in which
the fitness of each “species”
changes because of its
relationships with other “species”,
following two simple rules
Threshold fitness the highest level the lowest fitness
species has reached
Rule One - find the species with the lowest fitness and
randomly change its fitness.
Rule Two - at the same time the lowest fit species is
changed, also randomly change the fitness of the species to
the immediate left and right.
Modeling an Evolutionary System
The Bak-Sneppen evolutionary
model is an “ecosystem” in which
the fitness of each “species”
changes because of its
relationships with other “species”,
following two simple rules
Threshold fitness the highest level the lowest fitness
species has reached
An avalanche is a cascade of fitness changes below the threshold (i.e. all the blinking dots below
the line) between one rise of the threshold fitness and the next rise.
Rule One - find the species with the lowest fitness and
randomly change its fitness.
Rule Two - at the same time the lowest fit species is
changed, also randomly change the fitness of the species to
the immediate left and right.
Modeling an Evolutionary System
We do not expect
random processes
to result in an
organized
outcome. Does
any interesting
behavior emerge
from this simple
system?
Run Bak-Sneppen
? What Are the Implications ?
Q
•
•
3.
1. Watch the species above the
threshold. How stable are they?
• How much are they able to
change on their own?
• How much do they contribute to
P
raising the threshold line to the
next level?
2. Get personal. Pick out one species
Who is likely to be
above the threshold line and identify
conservative; liking things
with it; imagine it is you.
just the way they are?
• How safe are you in this avalanche
Who is likely to be liberal;
prone world?
wanting things to
• How much control do you have
change?
over your destiny? Why or why
Are there any innocent victims? not?
Is there any way to protect yourself in such a world?
Is there any part of this ecosystem that is isolated from the rest,
sitting in a protected niche, independent and self sufficient?
Modeling an Evolutionary System
Dynamics of the Bak-Sneppen Evolutionary Model
Time (generation steps)
Activity Patter n
Species
Activity pattern for the Bak-Sneppen
model. Time begins at an arbitrary time
after the model has self-organized at the
critical state near the 0.66 threshold.
Species are arranged along the
horizontal axis (from -20 to +20). Each
circle indicates a time a given species
undergoes a mutation. For example, at
about time 2000 species -7 through +7
are undergoing mutations; by time 4000
activity has shifted to -20 to -10. That is,
there is an avalanche in that portion of
the ecosystem. As the avalanches move
to other species the activity circles move
to those other species, and species that
are not mutating do not have activity
circles for that time span.
Modeling an Evolutionary System
Dynamics of the Bak-Sneppen Evolutionary Model
Ecosyst em Thr eshold Fit ness
St epwise r ising t hr eshold fit ness
Threshold
Fitness
Graph showing the climb of the
threshold fitness for the whole
ecosystem with time. Threshold
fitness is the highest fitness the
least fit species has attained. A
step up to a new threshold occurs
only when all species climb above
the old threshold, thus ending an
avalanche. As the graph shows
this takes progressively more time
as the threshold fitness rises.
Generations
Note that the rise in ecosystem
fitness is punctuational, or
behaves like a Self Organized
Critical sandpile.
Modeling an Evolutionary System
Dynamics of the Bak-Sneppen Evolutionary Model
The Devil's St air case
The fat e of individual species
st asis
punct uat ional
change
Generations
The Devil's staircase shows the
accumulated
activity
of
one
species. Horizontal lines are times
of stasis.
Vertical jumps are
mutations; note these come in
bundles over short time intervals
(are punctuational). In reality there
are many more mutation steps than
shown.
One can think of the
number of changes as representing
the amount of physical change in
the animal, such as size. The Self
Organized
Criticality
(aka
"punctuated equilibrium“) nature of
the curve is evident in the long
times of stasis followed by jumps in
activity.
Universality
Properties of Complex Evolutionary Systems
Power Law Relationships – Bak-Sneppen
Avalanche sizes in
the Bak-Sneppen
model
Mutation
frequency in the
Bak-Sneppen
model
Stuart Kauffman
“The critical point is not, as Stuart
Kauffman once described it, “a nice place
to be.” So “survival of the fittest” does not
imply evolution to a state where everybody
is well off. On the contrary, individual
species are barely able to hang on - like
the grains of sand in the critical sand pile.”
Maybe there is no “cause” to
disasters and extinctions
Maybe disasters (avalanches) are
just part of the dynamic of
evolution.