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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

slide2

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
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

slide14

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

slide15

Fractal Dimension (2)

Observed landscape

Simulated landscape

slide16

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
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

slide23

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

conclusions
Conclusions

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

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