Chaos and Self-Organization in

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Chaos and Self
-
Organization in
Spatiotemporal Models of Ecology

J. C. Sprott

Department of Physics

University of Wisconsin
-

Madison


Presented at the

Eighth International Symposium
on Simulation Science

in Hayama, Japan

on March 5, 2003

Collaborators



Janine Bolliger

Swiss Federal

Research Institute




David Mladenoff

University of

Wisconsin
-

Madison

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
randomly within a circular radius
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