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 n Historical forest data set n Stochastic cellular automaton model n Deterministic coupled-flow lattice model

9.6 km 1.6 km Section corner Quarter corner Meander corner MN WI IL IA MO 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

Random Initial Conditions Ordered

Cluster Probability n A point is assumed to be part of a cluster if its 4 nearest neighbors are the same as it is. n 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 = 10 experimental value

Fluctuations in Cluster Probability r = 3 Number of generations Cluster probability

Power Spectrum (1) Power laws ( 1 /f  ) for both initial conditions; r = 1 and r = 3 Slope:  = 1.58 r = 3 Frequency Power SCALE INVARIANT Power law !

Power Spectrum (2) Power 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 n R = rabbits, F = foxes n dR/dt = r 1 R (1 - R - a 1 F ) n dF/dt = r 2 F (1 - F - a 2 R ) Interspecies competition Intraspecies competition r and a can be + or -

Types of Interactions dR/dt = r 1 R (1 - R - a 1 F ) dF/dt = r 2 F (1 - F - a 2 R ) a1r1a1r1 a2r2a2r2 Competition Predator- Prey Prey- Predator Cooperation

Equilibrium Solutions n dR/dt = r 1 R (1 - R - a 1 F ) = 0 n dJ/dt = r 2 F (1 - F - a 2 R ) = 0 R = 0, F = 0 R = 0, F = 1 R = 1, F = 0 R = (1 - a 1 ) / (1 - a 1 a 2 ), F = (1 - a 2 ) / (1 - a 1 a 2 ) Equilibria: R F

Stability - Bifurcation r 1 (1 - a 1 ) < -r 2 (1 - a 2 ) F RR r 1 = 1 r 2 = -1 a 1 = 2 a 2 = 1.9 r 1 = 1 r 2 = -1 a 1 = 2 a 2 = 2.1

Generalized Spatial Lotka- Volterra Equations Let S i ( x,y ) be density of the i th species (rabbits, trees, seeds, …) dS i / dt = r i S i (1 - S i - Σ a ij S j ) 2-D grid: S = S x- 1, y + S x,y -1 + S x +1, y + S x,y +1 +  S x,y jiji where

Typical Results

Dominant Species

Fluctuations in Cluster Probability Time Cluster probability

Power Spectrum of Cluster Probability Frequency Power

Fluctuations in Total Biomass Time Derivative of biomass Time

Power Spectrum of Total Biomass Frequency Power

Sensitivity to Initial Conditions Time Error in Biomass

Results n Most species die out n Co-existence is possible n Densities can fluctuate chaotically n Complex spatial patterns arise

Summary n Nature is complex n Simple models may suffice but

References n lectures/forest/ (This talk) lectures/forest/ n