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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. - PowerPoint PPT Presentation
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Janine Bolliger1, Julien C. Sprott2, David J. Mladenoff1
1 Department of Forest Ecology & Management, University of Wisconsin-Madison2 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 universalSome examples:
• Power-law distribution of earthquakemagnitudes (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 conditionsOrdered
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)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
ster
pro
babi
lity
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 testLo
g(m
edia
n de
cay
ti me)
Log(perturbation size)
y = 1.0094x + 0.0826R2 = 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