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Chaos in Easter Island Ecology J. C. Sprott Department of Physics University of Wisconsin – Madison Presented at the Chaos and Complex Systems Seminar in Madison, WI on January 25, 2011

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Easter Island

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Chilean palm (Jubaea chilensis)

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Easter Island History 400-1200 AD? First inhabitants arrive from Polynesia 1722 Jacob Roggevee (Dutch) visited Population: ~3000 1770’s Next foreign visitors 1860’s Peruvian slave traders Catholic missionaries arrive Population: 110 1888 Annexed by Chilie 2010 Popular tourist destination Population: 4888

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Things should be explained as simply as possible, but not more simply. −Albert Einstein

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All models are wrong; some models are useful. −George E. P. Box

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Linear Model P is the population (number of people) γ is the growth rate (birth rate – death rate)

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Linear Model γ = +1 γ = −1

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Logistic Model

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Attractor Repellor γ = +1

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Lotka-Volterra Model P T Three equilibria: Coexisting equilibrium

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η = 4.8 γ = 2.5 Brander-Taylor Model

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η = 4.8 γ = 2.5 Brander-Taylor Model Point Attractor

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Basener-Ross Model P T Three equilibria:

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η = 25 γ = 4.4 Basener-Ross Model

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η = 0.8 γ = 0.6 Basener-Ross Model Requires γ = 2η − 1 Structurally unstable

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Poincaré-Bendixson Theorem In a 2-dimensional dynamical system (i.e. P,T), there are only 4 possible dynamics: 1. Attract to an equilibrium 2. Cycle periodically 3. Attract to a periodic cycle 4. Increase without bound Trajectories in state space cannot intersect

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Invasive Species Model Four equilibria: 1. P = R = 0 2. R = 0 3. P = 0 4. coexistence

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η P = 0.47 γ P = 0.1 η R = 0.7 γ R = 0.3 Chaos

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Return map Fractal

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γ P = 0.1 γ R = 0.3 η R = 0.7 Bifurcation diagram Lyapunov exponent Period doubling

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γ P = 0.1 γ R = 0.3 η R = 0.7 Hopf bifurcation Crisis

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Overconsumption

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Reduce harvesting

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Eradicate the rats

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Conclusions Simple models can produce complex and (arguably) realistic results. A common route to extinction is a Hopf bifurcation, followed by period doubling of a limit cycle, followed by increasing chaos. Systems may evolve to a weakly chaotic state (“edge of chaos”). Careful and prompt slight adjustment of a single parameter can prevent extinction.

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References http://sprott.physics.wisc.edu/ lectures/easter.ppt (this talk) http://sprott.physics.wisc.edu/ lectures/easter.ppt http://sprott.physics.wisc.edu/chaostsa/ (my chaos book) http://sprott.physics.wisc.edu/chaostsa/ sprott@physics.wisc.edu (contact me) sprott@physics.wisc.edu

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