Chaos and self organization in spatiotemporal models of ecology
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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

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 Ecology

  • Janine Bolliger

  • Swiss Federal

  • Research Institute

  • David Mladenoff

  • University of

  • Wisconsin - Madison


Outline
Outline Ecology

  • Historical forest data set

  • Stochastic cellular automaton model

  • Deterministic cellular automaton model

  • Application to corrupted images




Cellular automaton voter model
Cellular Automaton Ecology(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 Ecology

Ordered

Random


Cluster probability
Cluster Probability Ecology

  • 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 Ecology = 1

r = 3

r = 10

Cluster Probabilities (1)

Random initial conditions

experimental

value


Cluster probabilities 2

r Ecology = 1

r = 3

r = 10

Cluster Probabilities (2)

Ordered initial conditions

experimental

value


Fluctuations in cluster probability
Fluctuations in Cluster Probability Ecology

r = 3

Cluster probability

Number of generations


Power spectrum 1
Power Spectrum (1) Ecology

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) Ecology

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

r = 10

Power

No power law

Frequency


Fractal Dimension (1) Ecology

 = 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


Fractal Dimension (2) Ecology

Observed landscape

Simulated landscape


A Measure of Complexity for Spatial Patterns Ecology

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 Ecology

  • 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



Why a deterministic model
Why a deterministic model? Ecology

  • Randomness conceals ignorance

  • Simplicity can produce complexity

  • Chaos requires determinism

  • The rules provide insight


Model fitness
Model Fitness Ecology

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 Ecology

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 Ecology

Mom: 1100100101

Pop: 0110101100

Cross: 1100101100

Mutate: 1100101110

Keep the fittest two and repeat


Is it Fractal? Ecology

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? Ecology

Slope = 1.9

Power

Frequency



Conclusions
Conclusions Ecology

A purely deterministic cellular

automaton model can produce

realistic landscape ecologies

that are fractal, self-organized,

and chaotic.



Landscape with missing data
Landscape with Missing Data Ecology

Original

Corrupted

Corrected

Single 60 x 60 block of missing cells

Replacement from 8 nearest neighbors


Image with corrupted pixels
Image with Corrupted Pixels Ecology

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 Ecology

  • Nature is complex

  • Simple models may suffice

but


References

http://sprott.physics.wisc.edu/ lectures/japan.ppt Ecology (This talk)

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

[email protected]

References


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