Janine Bolliger

Swiss Federal Research Institute WSL/FNP,

Birmensdorf, Switzerland

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

Funding

Wisconsin DNRUSGS – BRD

US Forest ServiceUniversity 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

Acknowledgements

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

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

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

Points of view

Living trees

Dead trees

Exogeneous models

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

Endogeneous models

fire

soildisease

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?

Landscape of early southern Wisconsin

Cellular automaton (CA)

r

• 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).

• Conditions: - boundary: periodic and reflecting

- initial: random and ordered

Random

Initial conditions

Ordered

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

Evolving cellular automaton: Self-organization due to internal dynamics

Animation

Comparison between simulated and observed landscape

• Fractal dimension

• Cluster probability

• Measure for complexity (algorithmic)

Is there any particular spatial scale?

Simulated landscapeObserved landscape

SCALE INVARIANT

Initial conditions = random

r = 1

r = 3

r = 10

experimental value

Is there any particular temporal scale?

Initial conditions = ordered

r = 1

r = 3

r = 10experimental value

Is there any particular temporal scale?

Fluctuations in cluster probabilities

r = 3

Number of generations

Clus

ter p

roba

bilit

y

Is the temporal variation universal? (1)

Power laws (1/f d) for r=1 and r=3

slope (d) = 1.58

r = 3

Frequency

Powe

r

SCALE INVARIANT

Power law !

Is the temporal variation universal? (2)Po

wer

Frequency

No power law (1/f d) for r = 10

r = 10

Power law ?

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

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

Tests for simulation robustness

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)

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