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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

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Chaos and Self-Organization in Spatiotemporal Models of Ecology

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Chaos and self organization in spatiotemporal models of ecology

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


Powerpoint version

Collaborators

  • Janine Bolliger

  • Swiss Federal

  • Research Institute

  • David Mladenoff

  • University of

  • Wisconsin - Madison


Outline

Outline

  • Historical forest data set

  • Stochastic cellular automaton model

  • Deterministic cellular automaton model

  • Application to corrupted images


Landscape of early southern wisconsin usa

Landscape of Early Southern Wisconsin (USA)


Stochastic cellular automaton model

Stochastic Cellular Automaton Model


Cellular automaton voter 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


Initial conditions

Initial Conditions

Ordered

Random


Cluster probability

Cluster Probability

  • 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 probabilities 1

r = 1

r = 3

r = 10

Cluster Probabilities (1)

Random initial conditions

experimental

value


Cluster probabilities 2

r = 1

r = 3

r = 10

Cluster Probabilities (2)

Ordered initial conditions

experimental

value


Fluctuations in cluster probability

Fluctuations in Cluster Probability

r = 3

Cluster probability

Number of generations


Power spectrum 1

Power Spectrum (1)

Power laws (1/fa) for both initial conditions; r = 1 and r = 3

Slope: a = 1.58

r = 3

SCALE INVARIANT

Power

Power law !

Frequency


Power spectrum 2

Power Spectrum (2)

No power law (1/fa) for r = 10

r = 10

Power

No power law

Frequency


Powerpoint version

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 

e

Power law: C = D (a fractal) where D is the fractal dimension:

D = log C / log


Powerpoint version

Fractal Dimension (2)

Observed landscape

Simulated landscape


Powerpoint version

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

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

Deterministic Cellular Automaton Model


Why a deterministic model

Why a deterministic model?

  • Randomness conceals ignorance

  • Simplicity can produce complexity

  • Chaos requires determinism

  • The rules provide insight


Model fitness

Model Fitness

Define a spectrum of

cluster probabilities

(from the stochastic

model):

CP1 = 40.8%

CP2 = 27.5%

CP3 = 20.2%

CP4 = 13.8%

3

4

4

2

4

1

2

4

0

3

1

1

3

2

1

2

4

4

4

3

4

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

Update Rules

Truth Table

3

4

4

2

4

1

2

4

0

3

1

1

3

2

1

2

4

4

4

3

4

210 = 1024 combinations

for 4 nearest neighbors

22250 = 10677 combinations

for 20 nearest neighbors

Totalistic rule


Genetic algorithm

Genetic Algorithm

Mom: 1100100101

Pop: 0110101100

Cross: 1100101100

Mutate: 1100101110

Keep the fittest two and repeat


Powerpoint version

Is it Fractal?

Stochastic Model

Deterministic Model

D = 1.666

D = 1.685

0

0

e

e

log C( )

log C( )

-3

-3

e

log

e

0

3

0

log

3


Is it self organized critical

Is it Self-organized Critical?

Slope = 1.9

Power

Frequency


Is it chaotic

Is it Chaotic?


Conclusions

Conclusions

A purely deterministic cellular

automaton model can produce

realistic landscape ecologies

that are fractal, self-organized,

and chaotic.


Application to corrupted images

Application to Corrupted Images


Landscape with missing data

Landscape with Missing Data

Original

Corrupted

Corrected

Single 60 x 60 block of missing cells

Replacement from 8 nearest neighbors


Image with corrupted pixels

Image with Corrupted Pixels

Cassie Kight’s calico cat Callie

Original

Corrupted

Corrected

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

Replacement from 8 nearest neighbors


Summary

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

References


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