Chaos in Easter Island Ecology

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Chaos in EasterIsland 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
• 1770s
• Next foreign visitors
• 1860s
• 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) ? is the
growth rate (birth rate death rate)
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Linear Model
? 1
? -1
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Logistic Model
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? 1
Attractor
Repellor
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Lotka-Volterra Model
Three equilibria
T
Coexisting equilibrium
P
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? 4.8 ? 2.5 Brander-Taylor Model
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Point Attractor
? 4.8 ? 2.5 Brander-Taylor Model
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Basener-Ross Model
Three equilibria
T
P
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? 25 ? 4.4 Basener-Ross Model
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? 0.8 ? 0.6 Basener-Ross Model
Requires ? 2? - 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
• Attract to an equilibrium
• Cycle periodically
• Attract to a periodic cycle
• Increase without bound
• Trajectories in state space cannot intersect

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Invasive Species Model

• Four equilibria
• P R 0
• R 0
• P 0
• coexistence

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?P 0.47 ?P 0.1
?R 0.7 ?R 0.3
Chaos
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Fractal
Return map
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?P 0.1 ?R 0.3 ?R 0.7
Lyapunov exponent
Period doubling
Bifurcation diagram
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?P 0.1 ?R 0.3 ?R 0.7
Crisis
Hopf bifurcation
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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/chaostsa/ (my
  chaos book)
• sprott_at_physics.wisc.edu (contact me)