1. 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

2. 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

3. 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

4. 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)

5. 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?

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

7. 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 ( trees/ha)
Closed forest (> 99 trees/ha)
Swamps (Tamaracks only)
*ha = hectares = 10,000m2

8. Landscape of early southern Wisconsin

9. 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

10. Initial conditions
Random
Ordered

11. 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

12. Evolving cellular automaton

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

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

15. Fluctuations in cluster probability
Number of generations

16. 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

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

18. Spatial variation of the CA
Cluster probability

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

20. 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)

21. 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!

22. 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