Chaos and Self-Organization in Spatiotemporal Models of Ecology J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Eighth International Symposium on Simulation Science in Hayama, Japan on March 5, 2003

Collaborators Janine Bolliger Swiss Federal Research Institute David Mladenoff University of Wisconsin - Madison

Outline n Historical forest data set n Stochastic cellular automaton model n Deterministic cellular automaton model n Application to corrupted images

Landscape of Early Southern Wisconsin (USA)

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) Boundary conditions : periodic and reflecting Initial conditions : random and ordered Constraint: The proportions of land types are kept equal to the proportions of the experimental data

Random Initial Conditions Ordered

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 Probability

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)

Simplified Model n Previous model u 6 levels of tree densities u nonequal probabilities u randomness in 3 places n Simpler model u 2 levels (binary) u equal probabilities u randomness in only 1 place

Deterministic Cellular Automaton Model

Why a deterministic model? n Randomness conceals ignorance n Simplicity can produce complexity n Chaos requires determinism n The rules provide insight

Model Fitness Define a spectrum of cluster probabilities (from the stochastic model): CP 1 = 40.8% CP 2 = 27.5% CP 3 = 20.2% CP 4 = 13.8% Require that the deterministic model has the same spectrum of cluster probabilities as the stochastic model (or actual data) and also 50% live cells.

Update Rules Truth Table 2 10 = 1024 combinations for 4 nearest neighbors = combinations for 20 nearest neighbors Totalistic rule

Genetic Algorithm Mom: Pop: Cross: Mutate: Keep the fittest two and repeat

Is it Fractal? Deterministic ModelStochastic Model D = 1.666D = log   log C( )  

Is it Self-organized Critical? Frequency Power Slope = 1.9

Is it Chaotic?

Conclusions A purely deterministic cellular automaton model can produce realistic landscape ecologies that are fractal, self-organized, and chaotic.

Application to Corrupted Images

Landscape with Missing Data Single 60 x 60 block of missing cells Replacement from 8 nearest neighbors OriginalCorruptedCorrected

Image with Corrupted Pixels 441 missing blocks with 5 x 5 pixels each and 16 gray levels Replacement from 8 nearest neighbors OriginalCorruptedCorrected Cassie Kight’s calico cat Callie

Multispecies Lotka- Volterra Model with Evolution

Let S i ( x,y ) be density of the i th species (trees, rabbits, people, …) dS i / dt = r i S i (1 - S i - Σ a ij S j ) Choose r i and a ij from a Poisson random distribution (both positive) Replace species that die with new ones chosen randomly Multispecies Lotka-Volterra Model with Evolution jiji

Evolution of Total Biomass Time Biomass

Conclusions n Competitive exclusion eliminates most species. n The dominant species is eventually killed and replaced by another. n Evolution is punctuated rather than continual.

Summary n Nature is complex n Simple models may suffice but

References n lectures/japan.ppt (This talk) lectures/japan.ppt n J. C. Sprott, J. Bolliger, and D. J. Mladenoff, Phys. Lett. A 297, (2002) J. C. Sprott, J. Bolliger, and D. J. Mladenoff, Phys. Lett. A 297, (2002) n