japan

Chaos and Self-Organization in Spatiotemporal Models of Ecology: 

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

Slide2: 

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

Outline: 

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

Landscape of Early Southern Wisconsin (USA): 

Landscape of Early Southern Wisconsin (USA)

Stochastic Cellular Automaton Model : 

Stochastic Cellular Automaton Model

Cellular Automaton(Voter 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) 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

Initial Conditions: 

Random Initial Conditions 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 Probability

Cluster Probabilities (1): 

Cluster Probabilities (1) Random initial conditions experimental value

Cluster Probabilities (2): 

Cluster Probabilities (2) Ordered initial conditions experimental value

Fluctuations in Cluster Probability: 

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

Power Spectrum (1) : 

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

Power Spectrum (2): 

Power Spectrum (2) Power Frequency No power law (1/fa) for r = 10 r = 10 No power law

Slide14: 

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  e Power law: C = D (a fractal) where D is the fractal dimension: D = log C / log 

Slide15: 

Fractal Dimension (2) Simulated landscape Observed landscape

Slide16: 

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: 

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

Deterministic Cellular Automaton Model : 

Deterministic Cellular Automaton Model

Why a deterministic model?: 

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

Model Fitness: 

Model Fitness 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 Define a spectrum of cluster probabilities (from the stochastic model): CP1 = 40.8% CP2 = 27.5% CP3 = 20.2% CP4 = 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: 

Update Rules 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 Truth Table 210 = 1024 combinations for 4 nearest neighbors 22250 = 10677 combinations for 20 nearest neighbors Totalistic rule

Genetic Algorithm: 

Genetic Algorithm Mom: 1100100101 Pop: 0110101100 Cross: 1100101100 Mutate: 1100101110 Keep the fittest two and repeat

Slide23: 

Is it Fractal? Deterministic Model Stochastic Model D = 1.666 D = 1.685 0 0 0 0 3 3 -3 -3 log log e e log C( ) log C( ) e e

Is it Self-organized Critical?: 

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

Is it Chaotic?: 

Is it Chaotic?

Conclusions: 

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

Application to Corrupted Images: 

Application to Corrupted Images

Landscape with Missing Data: 

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

Image with Corrupted Pixels: 

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

Multispecies Lotka-Volterra Model with Evolution: 

Multispecies Lotka-Volterra Model with Evolution

Multispecies Lotka-Volterra Model with Evolution: 

Let Si(x,y) be density of the ith species (trees, rabbits, people, …) dSi / dt = riSi (1 - Si - Σ aijSj ) Choose ri and aij from a Poisson random distribution (both positive) Replace species that die with new ones chosen randomly Multispecies Lotka-Volterra Model with Evolution ji

Evolution of Total Biomass: 

Evolution of Total Biomass Time Biomass

Conclusions: 

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

Summary: 

Summary Nature is complex Simple models may suffice but

References: 

References http://sprott.physics.wisc.edu/ lectures/japan.ppt (This talk) J. C. Sprott, J. Bolliger, and D. J. Mladenoff, Phys. Lett. A 297, 267-271 (2002) sprott@physics.wisc.edu