Self organized criticality of landscape patterning
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Self-organized criticality of landscape patterning. 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. Characteristics of SOC.

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Self-organized criticality of landscape patterning

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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










6 miles

1 mile

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)

  • 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



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

r = 1

r = 3

r = 10

Temporal evolution (1)

Initial conditions = random



r = 1

r = 3

r = 10

Temporal evolution (2)

Initial conditions = ordered



Fluctuations in cluster probability

r= 3

Cluster probability

Number of generations

Power law !

Power laws (1/f d) for both initial conditions; r=1 and r=3

slope (d) = 1.58

r = 3



Power law ?

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)


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

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