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

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

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

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

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) 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

Initial conditions Random 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? Observed landscape Simulated landscape SCALE INVARIANT

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

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

Fluctuations in cluster probabilities Cluster probability Number of generations

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

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

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

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

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