A case study for self-organized criticality and complexity in forest landscape ecology

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A case study for self-organized criticality and
complexity in forest landscape ecology
Janine Bolliger
Swiss Federal Research Institute WSL/FNP,
Birmensdorf, Switzerland
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Acknowledgements
Funding Wisconsin DNR USGS BRD US Forest
Service University of Wisconsin Swiss Science
Foundation
People Julien C. Sprott David J.
Mladenoff David J. Albers Monica G.
Turner Forest Landscape Ecology
Laboratory at UW Madison Heike Lischke
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Goals

• Understand spatial and temporal features of
  ecosystems
• Predict spatial and temporal features of
  ecosystems
• Determine how much of the ecosystem complexity is
  a result of variations in external conditions and
  how much is a natural consequence of internal
  interactions

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Points of view
Landscape pattern with and without biotic units
(e.g., trees)
Observation

• Effects of specific environmental proces- ses
  on the observed pattern (autecology)
• Externally imposed heterogeneity
• Detailed model parameters

Exogeneous models

• Variation and feedback between biotic units
  creates pattern (synecology)
• Spontaneous symmetry braking and self-
  organization
• Simple model parameters

Endogeneous models
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Research questions

• Can the landscape pattern be statistically
  explained by simple rules?
• Does the evolution of the landscape show symmetry
  breaking and self-organization?
• Are the simulations sensitive to perturbations?

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Landscape of early southern Wisconsin
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Cellular automaton (CA)

• Cellular automaton square array of cells where
  each cell takes one of the n values
  representing the landscape
• Evolving single-parameter model a cell dies out
  at random times and is replaced by a cell
  chosen randomly within a circular radius r
  (1ltrlt10).

• Conditions - boundary periodic and
  reflecting
• - initial random and ordered

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Initial conditions
Ordered
Random
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Smallest unit of organization 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

Center point is part of cluster
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Evolving cellular automaton Self-organization
due to internal dynamics
Animation
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Comparison between simulated and observed
landscape

• Fractal dimension
• Cluster probability
• Measure for complexity (algorithmic)

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Is there any particular spatial scale?
Simulated landscape
Observed landscape
SCALE INVARIANT
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Is there any particular temporal scale?

• Initial conditions random

experimental value
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Is there any particular temporal scale?

• Initial conditions ordered

experimental value
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Fluctuations in cluster probabilities
r 3
Cluster probability
Number of generations
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Is the temporal variation universal? (1)

• Power laws (1/f d) for r1 and r3

Power law !
slope (d) 1.58 r 3
Power
SCALE INVARIANT
Frequency
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Is the temporal variation universal? (2)

• No power law (1/f d) for r 10

r 10
Power
Power law ?
Frequency
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Measure for complexity of landscape pattern
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 Radius 3 6782 bytes
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Tests for simulation robustness
Data set - Proportional variation for input
data ( 20, 50 ) Cellular automaton -
Initial conditions (random, ordered) - Boundary
conditions (periodic, reflecting) - Sensitvity
to perturbations - Rule variations
(uncorrelated, correlated) Model results are
robust towards these tests
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Summary Simulated versus experimental landscapes

• Power-law behavior across spatial and temporal
  scales
• Power laws are footprints of self-organization to
  a critical state
• Self-organized criticality is a universal
  phenomenon
• Earthquakes (Gutenberg and Richter 1957)
• Sand-pile models (Bak et al. 1987)
• Plasma transport (Carreras, et al. 1996)
• Forest fires (Bak, et al. 1990)
• Rainforests (Sole and Manrubia 1997)
• Stock prices (Mandelbrot 1997)
• Traffic jams (Nagel and Herrmann 1993
• Biological evolution (Bak and Sneppen 1993)

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Conclusions for modeling complex forest landscapes

• External spatial heterogeneity may not be
  required for aspects of spatio-temporal
  diversity
• Homogenous systems far from equilibrium
  spontaneously break symmetry and self-organize
• The resulting spatio-temporal patterns are
  scale-invariant
• Thus it may not be necessary to model accurately
  the biological processes when performing
  landscape simulations