Self-organization in Forest Evolution
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented at the
US-Japan Workshop on Complexity Science
in Austin, Texas
on March 12, 2002
Collaborators• Janine Bolliger Swiss Federal Research Institute
• David Mladenoff University of Wisconsin - Madison
• George Rowlands University of Warwick (UK)
Outline
Historical forest data set
Stochastic cellular automaton model
Deterministic coupled-flow lattice model
9.6
km
1.6 km
#
#
#
Section corner
Quarter corner
Meander corner
MN WI
ILIAMO IN
MI
Wisconsin surveys conducted between 1832 – 1865
Landscape of Early Southern Wisconsin
Stochastic Cellular
Automaton Model
Cellular Automaton(Voter Model)
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)
• Constraint: The proportions of land types are kept equal to the proportions of the experimental data
• Boundary conditions: periodic and reflecting
• Initial conditions: random and ordered
Cluster Probability 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.
Cluster Probabilities (1)Random initial conditions
r = 1
r = 3
r = 10
experimental value
Cluster Probabilities (2)Ordered initial conditions
r = 1
r = 3
r = 10experimental value
Fluctuations in Cluster Probability
r = 3
Number of generations
Clu
ster
pro
babi
lity
Power Spectrum (1)
Power laws (1/f) for both initial conditions; r = 1 and r = 3
Slope: = 1.58
r = 3
Frequency
Pow
er
SCALE INVARIANT
Power law !
Power Spectrum (2)Po
wer
Frequency
No power law (1/f) for r = 10
r = 10
No power law
Fractal Dimension (1) = separation between two points of the same category (e.g., prairie)
C = Number of points of the same category that are closer than
Power law: C = D (a fractal) where D is the fractal dimension:
D = log C / log
Fractal Dimension (2)Simulated landscapeObserved landscape
A Measure of Complexity for Spatial Patterns
One measure of complexity is the size of the smallest computer program that can replicate the pattern.
A GIF file is a maximally compressed image format. Therefore the size of the file is a lower limit on the size of the program.
Observed landscape: 6205 bytes
Random model landscape: 8136 bytes
Self-organized model landscape: 6782 bytes (r = 3)
Deterministic Coupled-
flow Lattice Model
Lotka-Volterra Equations
R = rabbits, F = foxes
dR/dt = r1R(1 - R - a1F)
dF/dt = r2F(1 - F - a2R)
Interspecies competitionIntraspecies competition
r and a can be + or -
Types of InteractionsdR/dt = r1R(1 - R - a1F)
dF/dt = r2F(1 - F - a2R)
+
+
-
-
a1r1
a2r2
Competition
Predator-Prey
Prey-Predator
Cooperation
Equilibrium Solutions
dR/dt = r1R(1 - R - a1F) = 0
dJ/dt = r2F(1 - F - a2R) = 0
• R = 0, F = 0
• R = 0, F = 1
• R = 1, F = 0
• R = (1 - a1) / (1 - a1a2), F = (1 - a2) / (1 - a1a2)
Equilibria:
R
F
Stability - Bifurcationr1(1 - a1) < -r2(1 - a2)
F
R R
r1 = 1r2 = -1a1 = 2a2 = 1.9
r1 = 1r2 = -1a1 = 2a2 = 2.1
Generalized Spatial Lotka-Volterra Equations
• Let Si(x,y) be density of the ith
species (rabbits, trees, seeds, …)
• dSi / dt = riSi(1 - Si - ΣaijSj)
2-D grid: S = Sx-1,y + Sx,y-1
+ Sx+1,y + Sx,y+1 + Sx,y
ji
where
Typical Results
Typical Results
Typical Results
Dominant Species
Fluctuations in Cluster Probability
Time
Clu
ster
pro
babi
lity
Power Spectrumof Cluster Probability
Frequency
Pow
er
Fluctuations in Total BiomassTi
me
Der
ivat
ive
of b
iom
ass
Time
Power Spectrumof Total Biomass
Frequency
Pow
er
Sensitivity to Initial Conditions
Time
Erro
r in
Biom
ass
Results
Most species die out
Co-existence is possible
Densities can fluctuate chaotically
Complex spatial patterns arise
Summary
Nature is complex
Simple models may suffice
but
