Self-organization in Forest Evolution

transcript

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

1.6 km

Section corner

Quarter corner

Meander corner

ILIAMO IN

Wisconsin surveys conducted between 1832 – 1865

Landscape of Early Southern Wisconsin

Stochastic Cellular

Automaton Model

Cellular Automaton(Voter Model)

• 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

Random

Initial ConditionsOrdered

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 = 10

experimental value

Cluster Probabilities (2)Ordered initial conditions

r = 10experimental value

Fluctuations in Cluster Probability

Number of generations

Power Spectrum (1)

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

Slope: = 1.58

Frequency

SCALE INVARIANT

Power law !

Power Spectrum (2)Po

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)

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:

Stability - Bifurcationr1(1 - a1) < -r2(1 - a2)

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

Typical Results

Dominant Species

Fluctuations in Cluster Probability

Power Spectrumof Cluster Probability

Frequency

Fluctuations in Total BiomassTi

Power Spectrumof Total Biomass

Frequency

Sensitivity to Initial Conditions

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

References

http://sprott.physics.wisc.edu/ lectures/forest/ (This talk)

sprott@juno.physics.wisc.edu

Self-organization in Forest Evolution

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