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
Jan 15, 2016
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)
[email protected] (contact me)