# Chaos and Self-Organization in

Τεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 4 χρόνια και 7 μήνες)

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Chaos and Self
-
Organization in
Spatiotemporal Models of Ecology

J. C. Sprott

Department of Physics

University of Wisconsin
-

Presented at the

Eighth International Symposium
on Simulation Science

in Hayama, Japan

on March 5, 2003

Collaborators

Janine Bolliger

Swiss Federal

Research Institute

University of

Wisconsin
-

Outline

Historical forest data set

Stochastic cellular automaton
model

Deterministic cellular automaton
model

Application to corrupted images

Landscape of Early Southern
Wisconsin (USA)

Stochastic Cellular
Automaton Model

Cellular Automaton

(Voter Model)

r

Cellular automaton:

Square array of cells where
each cell takes one of the 6 values representing the
landscape on a 1
-
square mile resolution

Evolving single
-
parameter model:

A cell dies out at

random times and is replaced by a cell chosen
r

(1
<

r
<

10)

Boundary conditions
:

periodic and reflecting

Initial conditions
: random and ordered

Constraint:

The proportions of land types are kept equal to the proportions of
the experimental data

Random

Initial Conditions

Ordered

A point is assumed to be part of a
cluster if its 4 nearest neighbors are
the same as it is.

CP (Cluster probability) is the % of
total points that are part of a cluster.

Cluster Probability

Cluster Probabilities (1)

Random initial conditions

r

= 1

r

= 3

r

= 10

experimental

value

Cluster Probabilities (2)

Ordered initial conditions

r

= 1

r

= 3

r

= 10

experimental

value

Fluctuations in Cluster Probability

r

= 3

Number of generations

Cluster probability

Power Spectrum (1)

Power laws (
1
/f
a
) for both initial conditions; r =
1

and r =
3

Slope:
a

=
1.58

r =
3

Frequency

Power

Power law !

Power Spectrum (2)

Power

Frequency

No power law (
1
/f
a
) for r =
10

r =
10

No power law

Fractal Dimension (1)

= separation between
two points of the same
category (e.g., prairie)

C

= Number of points of
the same category that are
closer than

Power law
: C
=

D
(a fractal) where
D

is the
fractal dimension:

D =
log
C /
log

Fractal Dimension (2)

Simulated landscape

Observed landscape

A Measure of Complexity
for Spatial Patterns

One measure of complexity is the size of the
smallest computer program that can replicate the
pattern.

A GIF file is a maximally compressed image
format. Therefore the size of the file is a lower limit
on the size of the program.

Observed landscape:

6205 bytes

Random model landscape:

8136 bytes

Self
-
organized model landscape:

6782 bytes

(
r

= 3)

Simplified Model

Previous model

6 levels of tree densities

nonequal probabilities

randomness in 3 places

Simpler model

2 levels (binary)

equal probabilities

randomness in only 1 place

Deterministic Cellular
Automaton Model

Why a deterministic model?

Randomness conceals ignorance

Simplicity can produce complexity

Chaos requires determinism

The rules provide insight

Model Fitness

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

4

4

4

Define a spectrum of

cluster probabilities

(from the stochastic

model):

CP
1

= 40.8%

CP
2

= 27.5%

CP
3

= 20.2%

CP
4

= 13.8%

Require that the deterministic model

has the same spectrum of cluster

probabilities as the stochastic model

(or actual data) and also 50% live cells.

Update Rules

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

4

4

4

Truth Table

2
10

= 1024 combinations

for 4 nearest neighbors

2
2250

= 10
677

combinations

for 20 nearest neighbors

Totalistic rule

Genetic Algorithm

Mom: 1100100101

Pop: 0110101100

Cross: 1100101100

Mutate: 1100101110

Keep the fittest two and repeat

Is it Fractal?

Deterministic Model

Stochastic Model

D = 1.666

D = 1.685

0

0

0

0

3

3

-
3

-
3

log

log

log C( )

log C( )

Is it Self
-
organized Critical?

Frequency

Is it Chaotic?

Conclusions

A purely deterministic cellular

automaton model can produce

realistic landscape ecologies

that are fractal, self
-
organized,

and chaotic.

Application to Corrupted
Images

Landscape with Missing Data

Single 60 x 60 block of missing cells

Replacement from 8 nearest neighbors

Original

Corrupted

Corrected

Image with Corrupted Pixels

441 missing blocks with 5 x 5 pixels each and 16 gray levels

Replacement from 8 nearest neighbors

Original

Corrupted

Corrected

Cassie Kight’s calico cat Callie

Summary

Nature is complex

Simple models may
suffice

but

References

http://sprott.physics.wisc.edu/
lectures/japan.ppt

(This talk)

J. C. Sprott, J. Bolliger, and D. J.
Mladenoff, Phys. Lett. A 297, 267
-
271
(2002)

sprott@physics.wisc.edu