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

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?

TownshipCorner

6 m

iles

1 mile

MN WI

ILIA

MO IN

MI

Data: U.S. General Land Office Surveys

Information used for this study

U.S. General Land Office Surveys are classifiedinto 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 (46 - 99 trees/ha)4. Closed forest (> 99 trees/ha)5. Swamps (Tamaracks only)

*ha = hectares = 10,000m2

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

Application

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 = 10experimental value

Fluctuations in cluster probability

r = 3

Number of generations

Clu

ste

r p

rob

ab

ility

Power law !

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

slope (d) = 1.58

r = 3

Frequency

Pow

er

Power law ?P

ower

Frequency

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

r = 10

Spatial variation of the CA

0

1

2

3

4

5

0 2 4 6 8 10

Cell size (miles)

Rad

ius

Cluster probability

Perturbation testL

og

(med

ian

dec

ay t

ime)

Log(perturbation size)

y = 1.0094x + 0.0826

R2 = 0.9767

0

1

2

3

4

0 1 2 3 4

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