1 / 36

Self-organization in Forest Evolution

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

genica
Download Presentation

Self-organization in Forest Evolution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. Collaborators • Janine Bolliger • Swiss Federal Research Institute • David Mladenoff • University of Wisconsin - Madison • George Rowlands • University of Warwick (UK)

  3. Outline • Historical forest data set • Stochastic cellular automaton model • Deterministic coupled-flow lattice model

  4. MN WI MI IA IL IN MO Section corner Quarter corner Meander corner 9.6 km 1.6 km Wisconsin surveys conducted between 1832 – 1865

  5. Landscape of Early Southern Wisconsin

  6. Stochastic Cellular Automaton Model

  7. 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

  8. Initial Conditions Ordered Random

  9. 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.

  10. r = 1 r = 3 r = 10 Cluster Probabilities (1) Random initial conditions experimental value

  11. r = 1 r = 3 r = 10 Cluster Probabilities (2) Ordered initial conditions experimental value

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

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

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

  15. 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

  16. Fractal Dimension (2) Observed landscape Simulated landscape

  17. 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)

  18. Deterministic Coupled-flow Lattice Model

  19. Lotka-Volterra Equations • R = rabbits, F = foxes • dR/dt = r1R(1 - R - a1F) • dF/dt = r2F(1 - F - a2R) Intraspecies competition Interspecies competition r and a can be + or -

  20. Types of Interactions dR/dt = r1R(1 - R - a1F) dF/dt = r2F(1 - F - a2R) + a2r2 Prey- Predator Competition - + a1r1 Predator- Prey Cooperation -

  21. Equilibrium Solutions • dR/dt = r1R(1 - R - a1F) = 0 • dJ/dt = r2F(1 - F - a2R) = 0 Equilibria: • R = 0, F = 0 • R = 0, F = 1 • R = 1, F = 0 • R = (1 - a1) / (1 - a1a2), F = (1 - a2) / (1 - a1a2) F R

  22. Stability - Bifurcation r1(1 - a1) < -r2(1 - a2) r1 = 1 r2 = -1 a1 = 2 a2 = 1.9 r1 = 1 r2 = -1 a1 = 2 a2 = 2.1 F R R

  23. Generalized Spatial Lotka-Volterra Equations • Let Si(x,y) be density of the ith species (rabbits, trees, seeds, …) • dSi / dt = riSi(1 - Si - ΣaijSj) ji where S = Sx-1,y + Sx,y-1 + Sx+1,y + Sx,y+1 + aSx,y 2-D grid:

  24. Typical Results

  25. Typical Results

  26. Typical Results

  27. Dominant Species

  28. Fluctuations in Cluster Probability Cluster probability Time

  29. Power Spectrumof Cluster Probability Power Frequency

  30. Fluctuations in Total Biomass Time Derivative of biomass Time

  31. Power Spectrumof Total Biomass Power Frequency

  32. Sensitivity to Initial Conditions Error in Biomass Time

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

  34. Summary • Nature is complex • Simple models may suffice but

  35. http://sprott.physics.wisc.edu/ lectures/forest/ (This talk) sprott@juno.physics.wisc.edu References

More Related