Chaos and Self-Organization in Spatiotemporal Models of Ecology J. C. Sprott Department of Physics...
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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
Transcript of 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)
