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  1. Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems Fichter, Lynn S., Pyle, E.J., and Whitmeyer, S.J., 2010, Journal of Geoscience Education (in press)

  2. Boids

  3. MATFA’s Boids

  4. SOC Self-Organized Criticality

  5. Evolution Via Self Organization Self Organized Criticality

  6. Evolution Via Self Organization Self Organized Criticality

  7. 1948-2002 Self-Organized Criticality Per Bak “Complex behavior in nature reflects the tendency of large systems with many components to evolve into a poised, "critical" state, way out of balance, where minor disturbances may lead to events, called avalanches, of all sizes. Most of the changes take place through catastrophic events rather than by following a smooth gradual path. The evolution to this very delicate state occurs without design from any outside agent. The state is established solely because of the dynamical interactions among individual elements of the system: the critical state is self-organized. Self-organized criticality is so far the only known general mechanism to generate complexity.”

  8. Avalanche Behavior The sand pile builds . . . grain . . . by grain . . . by grain . . . by grain . . . by grain . . . by grain . . . by grain . . . by grain . . . Building toward the critical state . . . Where it avalanches building building building avalanche avalanche avalanche Avalanche- a large mass of snow, ice, etc., detached from a mountain slope and sliding or falling suddenly downward. Avalanche- anything like an avalanche in suddenness and overwhelming quantity: an avalanche of misfortunes; an avalanche of fan mail.

  9. Power Law Sand Supply Now, imagine the sand supply follows a power law (or is fractal), with different numbers of grains falling at different times. Avalanches will follow a power law distribution. Earth Temp. curve over the past 400,000 years http://atlas.gc.ca/maptexts/topic_texts/english/images/TemperatureCO2.jpg

  10. Examples of Extreme Avalanches

  11. 1929 stock market crash 1987 stock market crash

  12. Examples of Extreme Avalanches Cascading Power Grids Failures – when a hub is required to carry more than it is capable of carrying, and so crashes, leading to the next hub to crash, etc. North America blackout 2003 North America blackout 1965

  13. Examples of Extreme Avalanches Extinctions Life is a Self Organized Critical phenomena

  14. Stuart Kauffman “The critical point is not, as Stuart Kauffman once described it, “a nice place to be.” So “survival of the fittest” does not imply evolution to a state where everybody is well off. On the contrary, individual species are barely able to hang on - like the grains of sand in the critical sand pile.” Systems are always at the critical point, or if they are not at the critical point they are evolving toward the critical point. That is, the common idea that systems evolve toward equilibrium is a misperception of reality.

  15. Cellular Automata

  16. Cellular Automata and Self Organization Cellular Automata (CA) are simply grids of cells, where the individual cells change states according to a set of rules. The CA may be one dimensional, or linear, like a string of cells in a row (below), or two dimensional, like a checkerboard Local Rules/Global Behavior Optimal Local Rule Set Survival Rules – 2/3 a live cell survives to the next generation if at least 2 but no more than three of the surrounding 8 cells are alive. Less than 2 and it dies of loneliness; more than 3 and it dies of over crowding.- 1 2 3 ? 8 4 Birth Rules – 3/3 a dead cells comes alive the next generation if 3, any 3, of the surrounding 8 cells are also alive. 7 6 5 1 2 3 8 4 Now, as cells are added, which will come alive, which survive? 7 6 5

  17. Cellular Automata and Self Organization

  18. Classifying Cellular Automata Rules Stephen Wolfram Class One - Fixed or Static: Rules that produce dull universes, such as all dead cells or all living cells; e.g. ice. Class Two - Periodic or Oscillatory: Rules that produce stable, repetitive configurations. Class Three - Chaotic: Rules that produce chaotic patterns; e.g. molecules in a gas. Class Four - Complexity: Rules that produce complex, locally organized patterns; e.g. behave like a flowing liquid..

  19. Classifying Cellular Automata Rules Chris Langton

  20. Power-Law Relationships in Cellular Automata Run a random array until it stops, add a live cell at random, run again until it stops. Avalanche size is the number of generations from initiation until it stops. Most avalanches last only one or a few generations; others may last hundreds of generations. Plotted up the avalanches follow a power-law meaning Cellular Automata (with optimum local rules) are Self Organized Critical systems.

  21. Evolution of Ripples A cellular automata model Mechanics of Wind Ripple Stratigraphy Author(s): Spencer B. Forrest and Peter K. Haff Source: Science, New Series, Vol. 255, No. 5049 (Mar. 6, 1992), pp. 1240-1243 Published by: American Association for the Advancement of Science

  22. Cellular Automata Examples

  23. Bak-Sneppen Ecosystem Model

  24. Modeling an Evolutionary System The Bak-Sneppen evolutionary model is an “ecosystem” in which the fitness of each “species” changes because of its relationships with other “species”, following two simple rules Generation 1 Threshold fitness the highest level the lowest fitness species has reached Rule One - find the species with the lowest fitness and randomly change its fitness. Rule Two - at the same time the lowest fit species is changed, also randomly change the fitness of the species to the immediate left and right.

  25. Modeling an Evolutionary System The Bak-Sneppen evolutionary model is an “ecosystem” in which the fitness of each “species” changes because of its relationships with other “species”, following two simple rules Generation 2 Threshold fitness the highest level the lowest fitness species has reached Rule One - find the species with the lowest fitness and randomly change its fitness. Rule Two - at the same time the lowest fit species is changed, also randomly change the fitness of the species to the immediate left and right.

  26. Modeling an Evolutionary System The Bak-Sneppen evolutionary model is an “ecosystem” in which the fitness of each “species” changes because of its relationships with other “species”, following two simple rules Threshold fitness the highest level the lowest fitness species has reached An avalanche is a cascade of fitness changes below the threshold (i.e. all the blinking dots below the line) between one rise of the threshold fitness and the next rise. Rule One - find the species with the lowest fitness and randomly change its fitness. Rule Two - at the same time the lowest fit species is changed, also randomly change the fitness of the species to the immediate left and right.

  27. Modeling an Evolutionary System We do not expect random processes to result in an organized outcome. Does any interesting behavior emerge from this simple system? Run Bak-Sneppen

  28. ? What Are the Implications ? • Watch the species above the threshold. How stable are they? • How much are they able to change on their own? • How much do they contribute to raising the threshold line to the next level? P 2. Get personal. Pick out one species above the threshold line and identify with it; imagine it is you. Who is likely to be conservative; liking things just the way they are? Who is likely to be liberal; wanting things to change? Q • How safe are you in this avalanche prone world? • How much control do you have over your destiny? Why or why not? • Are there any innocent victims? • Is there any way to protect yourself in such a world? 3. Is there any part of this ecosystem that is isolated from the rest, sitting in a protected niche, independent and self sufficient?

  29. Modeling an Evolutionary System Dynamics of the Bak-Sneppen Evolutionary Model Activity pattern for the Bak-Sneppen model. Time begins at an arbitrary time after the model has self-organized at the critical state near the 0.66 threshold. Species are arranged along the horizontal axis (from -20 to +20). Each circle indicates a time a given species undergoes a mutation. For example, at about time 2000 species -7 through +7 are undergoing mutations; by time 4000 activity has shifted to -20 to -10. That is, there is an avalanche in that portion of the ecosystem. As the avalanches move to other species the activity circles move to those other species, and species that are not mutating do not have activity circles for that time span.

  30. Modeling an Evolutionary System Dynamics of the Bak-Sneppen Evolutionary Model Graph showing the climb of the threshold fitness for the whole ecosystem with time. Threshold fitness is the highest fitness the least fit species has attained. A step up to a new threshold occurs only when all species climb above the old threshold, thus ending an avalanche. As the graph shows this takes progressively more time as the threshold fitness rises. Note that the rise in ecosystem fitness is punctuational, or behaves like a Self Organized Critical sandpile.

  31. Modeling an Evolutionary System Dynamics of the Bak-Sneppen Evolutionary Model The Devil's staircase shows the accumulated activity of one species. Horizontal lines are times of stasis. Vertical jumps are mutations; note these come in bundles over short time intervals (are punctuational). In reality there are many more mutation steps than shown. One can think of the number of changes as representing the amount of physical change in the animal, such as size. The Self Organized Criticality (aka "punctuated equilibrium“) nature of the curve is evident in the long times of stasis followed by jumps in activity.

  32. Universality Properties of Complex Evolutionary Systems Power Law Relationships – Bak-Sneppen Avalanche sizes in the Bak-Sneppen model Mutation frequency in the Bak-Sneppen model

  33. Stuart Kauffman “The critical point is not, as Stuart Kauffman once described it, “a nice place to be.” So “survival of the fittest” does not imply evolution to a state where everybody is well off. On the contrary, individual species are barely able to hang on - like the grains of sand in the critical sand pile.” Maybe there is no “cause” to disasters and extinctions Maybe disasters (avalanches) are just part of the dynamic of evolution.