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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
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

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
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

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
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
Data: U.S. General Land Office Surveys










6 miles

1 mile

information used for this study

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

cellular automaton ca
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
Initial conditions



cluster probabilities
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
temporal evolution 1

r = 1

r = 3

r = 10

Temporal evolution (1)

Initial conditions = random



temporal evolution 2

r = 1

r = 3

r = 10

Temporal evolution (2)

Initial conditions = ordered



fluctuations in cluster probability
Fluctuations in cluster probability

r= 3

Cluster probability

Number of generations

power law
Power law !

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

slope (d) = 1.58

r = 3



power law1
Power law ?

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


r = 10


spatial variation of the ca
Spatial variation of the CA

Cluster probability

perturbation test
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
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
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