Janine Bolliger 1, Julien C. Sprott 2, David J. Mladenoff 1 1 Department of Forest Ecology & Management, University of Wisconsin-Madison 2 Department of Physics, University of Wisconsin-Madison Self-organized criticality of landscape patterning

Characteristics of SOC Self-organized criticality (SOC) … is manifested by temporal and spatial scale invariance (power laws) is driven by intermittent evolutions with bursts/ avalanches that extend over a wide range of magnitudes may be a characteristic of complex systems

Some definitions of SOC Self-organized criticality (SOC) is a concept to describe emergent complex behavior in physical systems (Boettcher and Percus 2001) SOC is a mechanism that refers to a dynamical process whereby a non-equilibrium system starts in a state with uncorrelated behavior and ends up in a complex state with a high degree of correlation (Paczuski et al. 1996) The HOW and WHY of SOC are not generally understood

SOC is universal Some examples: Power-law distribution of earthquake magnitudes (Gutenberg and Richter 1956) Luminosity of quasars ( in Press 1978) Sand-pile models (Bak et al. 1987) Chemical reactions (e.g., BZ reaction) Evolution (Bak and Sneppen 1993)

Research questions Can landscapes (tree-density patterns) be statistically explained by simple rules? Does the evolution of the landscape show self- organization to the critical state? Is the landscape chaotic?

Township Corner 6 miles 1 mile MN WI IL IA MO IN MI Data: U.S. General Land Office Surveys

Information used for this study U.S. General Land Office Surveys are classified into 5 landscape types according to tree densities (Anderson & Anderson 1975): 1. Prairie (< 0.5 trees/ha*) 2. Savanna(0.5 – 46 trees/ha) 3. Open woodland( trees/ha) 4. Closed forest(> 99 trees/ha) 5. Swamps(Tamaracks only) *ha = hectares = 10,000m 2

Landscape of early southern Wisconsin

Cellular automaton (CA) r 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< r <10). The time-scale is the average life of a cell (~100 yrs) Constraint: The proportions of land types are kept equal to the proportions of the experimental data Conditions: - boundary: periodic and reflecting - initial: random and ordered

Random Initial conditions Ordered

Cluster probabilities 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

Evolving cellular automaton

Temporal evolution (1) Initial conditions = random r = 1 r = 3 r = 10 experimental value

Temporal evolution (2) Initial conditions = ordered r = 1 r = 3 r = 10 experimental value

Fluctuations in cluster probability r = 3 Number of generations Cluster probability

Power law ! Power laws (1/f d ) for both initial conditions; r=1 and r=3 slope (d) = 1.58 r = 3 Frequency Power

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

Spatial variation of the CA Cluster probability

Perturbation test Log(median decay time) Log(perturbation size)

Conclusions Convergence of the cluster probability and the power law behavior after convergence indicate self-organization of the landscape at a critical level Independence of the initial and boundary conditions indicate that the critical state is a robust global attractor for the dynamics There is no characteristic temporal scale for the self-organized state for r = 1 and 3 There is no characteristic spatial scale for the self-organized state Even relatively large perturbations decay (not chaotic)

Where to go from here ? Further analysis: - incorporate deterministic rules - search for percolation thresholds Other applications: - urban sprawl - spread of epidemics - any kind of biological succession … We are interested in collaboration!

Thank you! David Albers Ted Sickley Lisa Schulte This work is supported by a grant of the Swiss Science Foundation for Prospective Researchers by the University of Bern, Switzerland