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

Easter Island

Chilean palm (Jubaea chilensis)

Easter Island History  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

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

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

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

Linear Model γ = +1 γ = −1

Logistic Model

Attractor Repellor γ = +1

Lotka-Volterra Model P T Three equilibria: Coexisting equilibrium

η = 4.8 γ = 2.5 Brander-Taylor Model

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

Basener-Ross Model P T Three equilibria:

η = 25 γ = 4.4 Basener-Ross Model

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

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

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

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

Return map Fractal

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

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

Overconsumption

Reduce harvesting

Eradicate the rats

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.

References  lectures/easter.ppt (this talk) lectures/easter.ppt  (my chaos book)  (contact me)