Chaos and Self-Organization in Spatiotemporal Models of Ecology

1
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

2
Collaborators

• Janine Bolliger
• Swiss Federal
• Research Institute
• David Mladenoff
• University of
• Wisconsin - Madison

3
Outline

• Historical forest data set
• Stochastic cellular automaton model
• Deterministic cellular automaton model
• Application to corrupted images

4
Landscape of Early Southern Wisconsin (USA)
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Stochastic Cellular Automaton Model
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Cellular Automaton(Voter Model)

• 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 lt r
  lt 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

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Initial Conditions
Ordered
Random
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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.

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Cluster Probabilities (1)

• Random initial conditions

experimental value
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Cluster Probabilities (2)

• Ordered initial conditions

experimental value
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Fluctuations in Cluster Probability
r 3
Cluster probability
Number of generations
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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
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Power Spectrum (2)

• No power law (1/fa) for r 10

r 10
Power
No power law
Frequency
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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 ?
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Fractal Dimension (2)
Simulated landscape
Observed landscape
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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)
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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

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Deterministic Cellular Automaton Model
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Why a deterministic model?

• Randomness conceals ignorance
• Simplicity can produce complexity
• Chaos requires determinism
• The rules provide insight

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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
1
2
4
4
0
1
1
3
3
2
1
2
4
4
3
4
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.
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Update Rules
Truth Table
3
4
4
2
1
2
4
4
0
1
1
3
3
2
1
2
4
4
3
4
4
210 1024 combinations for 4 nearest neighbors
22250 10677 combinations for 20 nearest
neighbors
Totalistic rule
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Genetic Algorithm
Mom 1100100101 Pop 0110101100
Cross 1100101100
Mutate 1100101110
Keep the fittest two and repeat
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Is it Fractal?
Deterministic Model
Stochastic Model
D 1.666
D 1.685
0
0
e
e
log C( )
log C( )
-3
-3
e
log
e
0
0
3
3
log
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Is it Self-organized Critical?
Slope 1.9
Power
Frequency
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Is it Chaotic?
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Conclusions
A purely deterministic cellular automaton model
can produce realistic landscape ecologies that
are fractal, self-organized, and chaotic.
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Application to Corrupted Images
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Landscape with Missing Data
Original
Corrupted
Corrected
Single 60 x 60 block of missing cells Replacement
from 8 nearest neighbors
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Image with Corrupted Pixels
Cassie Kights calico cat Callie
Original
Corrupted
Corrected
441 missing blocks with 5 x 5 pixels each and 16
gray levels Replacement from 8 nearest neighbors
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Multispecies Lotka-Volterra Model with Evolution
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Multispecies Lotka-Volterra Model with Evolution

• Let Si(x,y) be density of the ith species
  (trees, rabbits, people, )
• dSi / dt riSi (1 - Si - S aijSj )
• Choose ri and aij from a Poisson random
  distribution (both positive)
• Replace species that die with new ones chosen
  randomly

j?i
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Evolution of Total Biomass
Biomass
Time
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Conclusions

• Competitive exclusion eliminates most species.
• The dominant species is eventually killed and
  replaced by another.
• Evolution is punctuated rather than continual.

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Summary

• Nature is complex
• Simple models may suffice

but
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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_at_physics.wisc.edu