Self-organized criticality of landscape patterning Janine Bolliger1, Julien C. Sprott2, David J. Mladenoff1 1 Department of Forest Ecology & Management, University of Wisconsin-Madison 2 Department of Physics, University of Wisconsin-Madison

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?

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

Information used for this study U.S. General Land Office Surveys are classified into 5 landscape types according to tree densities (Anderson & Anderson 1975): Prairie (< 0.5 trees/ha*) Savanna (0.5 – 46 trees/ha) Open woodland (46 - 99 trees/ha) Closed forest (> 99 trees/ha) Swamps (Tamaracks only) *ha = hectares = 10,000m2

Landscape of early southern Wisconsin

Cellular automaton (CA) 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) r 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

Initial conditions Random 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

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

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

Fluctuations in cluster probability Number of generations

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

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

Spatial variation of the CA Cluster probability

Log(median decay time) Log(perturbation size) 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