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Deterministic C haos and Rhythms of Life

Deterministic C haos and Rhythms of Life. Dr. Thomas Caraco. Outline: Population Dynamic Complexity. Background Power of the Calculus, Continuity Nonlinearity & Chaos, Predictability? Single Species Self-Regulating, Discrete-Time Overcompensation, Route to Chaos

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Deterministic C haos and Rhythms of Life

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  1. Deterministic ChaosandRhythmsof Life Dr. Thomas Caraco

  2. Outline: Population Dynamic Complexity Background Power of the Calculus, Continuity Nonlinearity & Chaos, Predictability? Single Species Self-Regulating, Discrete-Time Overcompensation, Route to Chaos General Significance for Ecology/Evolution

  3. Background Newton, Liebniz Deterministic Cause, Effect Calculus, Continuity “Very Similar” Cause  “Very Similar” Effect

  4. Background C Lyell, “Father of Modern Geology” Uniformitarianism: Continuity of Slow Processes C Darwin, From Malthus and Lyell Evolution Slow, Gradual Change

  5. Background Poincaré King Oscar’s Challenge: Prove Solar System Stable 1890: Three or More Interacting Bodies “Unpredictable” Study of Chaos Born

  6. Background Lorenz (1963): Strange Attractor Signature of Chaotic Dynamics Sarkovskii (1964): “Windows” of Order Li & Yorke (1975, Am Math Month 82): “Period three implies chaos” 1970’s to 1990’s “Golden Age”

  7. Chaos: General Significance Biological, Physical and Social Sciences Systems with Nonlinear Dynamics Generator of Chaos and Complexity New Perspective on Law of Causality Very Similar Cause May  Very Different Effect Practical Applications Control Dynamical Systems

  8. Chaos: Ecological Significance Population Regulation before 1975 • Physical Factors  Random Fluctuations • Density Dependence Stabilizing New Perspective on Density Dependence Constancy to Chaotic Complexity Understand Mix Nonlinear, Random Dynamics Challenge: Capacity to Predict

  9. Simple Model, Complex Dynamics R.M. May Logistic Map in Ecology General Paradigm for Emergence of Chaos Distinguish Deterministic Chaos from Stochastic Flux ?

  10. Metric Universalities M. Feigenbaum Feigenbaum Number Feigenbaum Ratios Periodicity Sequence Quantitative Identicality

  11. Finite-Difference Equations Single Species, Unstructured Continuous Density x(t); Discrete (Non-Overlapping) Generations Implies 1-D Map

  12. Annual Life Cycle Individual Reproduction Density-Dependence Self-Regulation Assume Linearity

  13. Quadratic Map: x(t+1) = R {x(t) – [x(t)]2} Increase Fecundity R: (Over)Compensation

  14. Behavior of Map: Dynamics 1 < R < 3 Equilibrium Node Any Initial Density  Same Equilibrium Strongly Compensating Regulation  Simple, Predictable Growth

  15. Dynamics R = 3.3 Bifurcation: Equilibrium 2-Cycle Periodic Dynamics Time Symmetry

  16. Dynamics R = 3.56 Bifurcation: Equilibrium 4-Cycle Increased Complexity

  17. Bifurcation Cascade Period-Doubling Route to Chaos Infinite Number of Bifurcations: Stable Cycles: Lyapunov Exponent Logarithmic Averaging Feigenbaum Point R = 3.56994456… Onset of Deterministic Chaos

  18. Bifurcation Cascade Dk: Range of R: Stable Cycle Has Period k D2k: Range of R: period 2k Feigenbaum’s Number; Universalilty Class Includes Logistic

  19. Deterministic Chaos Bounded Close to Extinction Aperiodic No State Repeats! Not Random! Sensitive Dependence Initial Conditions

  20. Bifurcation Diagram “Route to Chaos” Periodic Windows Universality Strange Attractor

  21. Fractal Behavior Self-Similarity Scale Invariance Repeating Geometry Signature of Chaos

  22. Trainor & Caraco (2006, EER 8) 1-D Map nt+1 = Reproduction x Pr[Survive Starvation, Group Size] x Pr[AvoidPredation] x nt Total Food Density Fixed; Social Acquisition

  23. Trainor & Caraco (2006, EER 8) R = 4, Fixed;  Social Interference, Cooperation

  24. Trainor & Caraco (2006, EER 8) Bifurcation Plot Lyapunov Exponent Non-Cooperative Individuals

  25. G = 5 (k=2)  < 1 G = 20 (k)

  26. Trainor & Caraco (2006, EER 8) Cooperative Individuals

  27. G = 5 (k=1)  > 1 G = 20 (k=0)

  28. Real Populations Chaotic? Within Populations Favor Faster Growth Complex Dynamics, Fluctuations  Extinction Among Populations Dynamic Stability  Persistence Evolve to Edge of Chaos?

  29. Real Populations Remove Random “Error” “Reconstruct Map” Test for Divergence Lack of Data; Require Lengthy Records Costantino et al. 1997. Science 275:389-391. Ellner & Turchin. 1995. Amer. Nat. 145:343-375. Olsen & Schaffer. 1990. Science 249:499-504.

  30. Ellner & Turchin, 1995

  31. Lessons from Simple Nonlinearities Universalities: Stability  Complexity Equilibrium  Chaos Small Parameter Changes  Qualitative Change in Behavior Chaos: Small Change in State  Quantitative Divergence of Systems

  32. Lessons from Simple Nonlinearities Chaos: Emergence of Fractal Order Break Symmetry of Past & Future Non-Random Behavior, Correlations Ecological Complexity  Loss Predictability End of lessons

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