Chaos in EasterIsland Ecology

J. C. SprottDepartment 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 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

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

Pdt

dP

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

)( 0for

)( 0for

0

0

stableePP

unstableePPt

t

0 Pm: Equilibriu

Linear Model

0for

0for

0

0

t

t

ePP

ePP

γ = +1

γ = −1

Logistic Model

)1( PPdt

dP

capacity''Carrying

P

γP

0)for (stable 1

0)for (stable 0

:equilibria Two

Attractor

Repellor

γ = +1

Lotka-Volterra Model

prey) / (trees )1(

predator) / (people

PTTdt

dT

PTPdt

dP

P

T

Three equilibria:

Coexisting equilibrium

η = 4.8γ = 2.5

Brander-TaylorModel

η = 4.8γ = 2.5

Brander-TaylorModel

Point Attractor

Basener-Ross Model

(trees) )1(

(people) 1

PTTdt

dT

T

PP

dt

dP

P

T

Three equilibria:

η = 25γ = 4.4

Basener-RossModel

η = 0.8γ = 0.6

Basener-RossModel

Requiresγ = 2η − 1

Structurallyunstable

Poincaré-Bendixson TheoremIn 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

(trees) )1(1

(rats) 1

(people) 1

PTR

T

dt

dT

T

RR

dt

dR

T

PP

dt

dP

R

R

PP

Four equilibria:1. P = R = 02. R = 03. P = 04. coexistence

ηP = 0.47γP = 0.1

ηR = 0.7γR = 0.3 Chaos

Return map

Fractal

γP = 0.1γR = 0.3ηR = 0.7

Bifurcation diagram

Lyapunov exponent

Period doubling

γP = 0.1γR = 0.3ηR = 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

http://sprott.physics.wisc.edu/

lectures/easter.ppt (this talk)

http://sprott.physics.wisc.edu/chaostsa/

(my chaos book)

sprott@physics.wisc.edu (contact me)