1. A case study for self-organized criticality and complexity in forest landscape ecology
Janine Bolliger
Swiss Federal Research Institute WSL/FNP,
Birmensdorf, Switzerland

2. Acknowledgements People Funding Julien C. Sprott David J. Mladenoff
David J. Albers
Monica G. Turner
Forest Landscape
Ecology Laboratory at UW Madison
Heike Lischke
Funding
Wisconsin DNR
USGS – BRD
US Forest Service
University of Wisconsin
Swiss Science Foundation

3. 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

4. Points of view fire soil disease Living trees Dead trees
Landscape pattern with and without biotic units (e.g., trees)
Observation
fire
soil
disease
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
Living trees
Dead trees
Endogeneous models

5. 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?

6. Landscape of early southern Wisconsin

7. 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 (1<r<10). r
Conditions: - boundary: periodic and reflecting
- initial: random and ordered

8. Initial conditions
Random
Ordered

9. 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

10. Evolving cellular automaton: Self-organization due to internal dynamics
Animation

11. Comparison between simulated and observed landscape
Fractal dimension
Cluster probability
Measure for complexity (algorithmic)

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

13. Is there any particular temporal scale?
Initial conditions = random
experimental
value
r = 1
r = 3
r = 10

14. Is there any particular temporal scale?
Initial conditions = ordered
r = 1
r = 3
r = 10
experimental
value

15. Fluctuations in cluster probabilities
Cluster probability
Number of generations

16. Is the temporal variation universal? (1)
Power laws (1/f d) for r=1 and r=3
Power law !
slope (d) = 1.58
r = 3
Power
SCALE INVARIANT
Frequency

17. Is the temporal variation universal? (2)
No power law (1/f d) for r = 10
r = 10
Power
Power law ?
Frequency

18. 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: bytes
Random model landscape: bytes
Self-organized model landscape:
Radius = bytes

19. 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

20. 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)

21. 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