chaos and self-organization in spatiotemporal models of ecology j. c. sprott department of physics...

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 ConditionsOrdered

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 = 10experimental value

Fluctuations in Cluster Probability

r = 3

Number of generations

Clu

ster

pro

babi

lity

Power Spectrum (1)

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

Slope: = 1.58

r = 3

Frequency

Pow

er

SCALE INVARIANTPower law !

Power Spectrum (2)Po

wer

Frequency

No power law (1/f) 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 landscapeObserved 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 11

11

22

2 23

3

3

3 44

44

44

44

Define a spectrum ofcluster probabilities(from the stochasticmodel):

CP1 = 40.8%

CP2 = 27.5%

CP3 = 20.2%

CP4 = 13.8% Require that the deterministic modelhas the same spectrum of clusterprobabilities as the stochastic model(or actual data) and also 50% live cells.

Update Rules

0 11

11

22

2 23

3

3

3 44

44

44

44

Truth Table

210 = 1024 combinations for 4 nearest neighbors

22250 = 10677 combinationsfor 20 nearest neighbors

Totalistic rule

Genetic Algorithm

Mom: 1100100101Pop: 0110101100

Cross: 1100101100

Mutate: 1100101110

Keep the fittest two and repeat

Is it Fractal?Deterministic ModelStochastic Model

D = 1.666 D = 1.685

00

0 0

33-3-3

loglog

log

C( )

log

C( )

Is it Self-organized Critical?

Frequency

Pow

er

Slope = 1.9

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 cellsReplacement from 8 nearest neighbors

Original Corrupted Corrected

Image with Corrupted Pixels

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