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
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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
dP
 P
dt
P is the population (number of people)
γ is the growth rate (birth rate – death rate)
Equilibrium : P  0
t
for   0 (unstable)
t
for   0 ( stable)
P  P0e
P  P0e
Linear Model
γ = +1
P  P0e
t
P  P0e
t
for   0
for   0
γ = −1
Logistic Model
dP
 P (1  P )
dt
T wo equilibria :
P  0 (stable for γ  0)
P  1 (stable for   0)
'Carrying capacity'
γ = +1
Attractor
Repellor
Lotka-Volterra Model
dP
 P  PT (people/ predat or)
dt
dT
 T (1  T  P ) (t rees/ prey)
dt
Three equilibria:
T
Coexisting equilibrium
P
η = 4.8
γ = 2.5
Brander-Taylor
Model
Point
Attractor
η = 4.8
γ = 2.5
Brander-Taylor
Model
Basener-Ross Model

dP
P 
 P1 

dt
 T 
(people)
dT
 T (1  T )  P (trees)
dt
Three equilibria:
T
P
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

dP
P 

  P P 1 
dt
  PT 
(people)
dR
 R
  R R 1  
(rat s)
dt
 T
dT
T

(1  T )  P (t rees)
dt 1   R R
Four equilibria:
1.
2.
3.
4.
P=R=0
R=0
P=0
coexistence
ηP = 0.47 ηR = 0.7
γP = 0.1 γR = 0.3
Chaos
Fractal
Return map
γP = 0.1
γR = 0.3
ηR = 0.7
Lyapunov exponent
Period doubling
Bifurcation diagram
γP = 0.1
γR = 0.3
ηR = 0.7
Crisis
Hopf bifurcation
Overconsumption
Reduce harvesting
Eradicate the rats
Conclusions


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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)