Competition with Evolution in
Ecology and Finance
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented at the
Chaos and Complex Systems
Seminar
in Madison, Wisconsin
on September 23, 2003
Mathematics
To the outsider, mathematics is a strange,
abstract world of horrendous technicality,
full of symbols and complicated
procedures, an impenetrable language
and a black art. To the scientist,
mathematics is the guarantor of precision
and objectivity. It is also, astonishingly,
the language of nature itself. No one who
is closed off from mathematics can ever
grasp the full significance of the natural
order that is woven so deeply into the
fabric of physical reality.
-- Paul Davies (Australian astrobiologist)
Rabbit Dynamics

Let R = # of rabbits

dR/dt = bR - dR = rR
Birth rate Death rate
r=b-d
•r>0
growth
•r=0
equilibrium
•r<0
extinction
Multispecies Lotka-Volterra Model
• Let Si be population of the ith species
(rabbits, trees, people, viruses, …)
N
• dSi / dt = riSi (1 - Σ aijSj )
j=1
• Parameters of the model:
• Vector of growth rates ri
• Matrix of interactions aij
• Number of species N
Parameters of the Model
Growth
rates
r1
r2
r3
r4
r5
r6
Interaction matrix
1
a21
a31
a41
a51
a61
a12
1
a32
a42
a52
a62
a13
a23
1
a43
a53
a63
a14
a24
a34
1
a54
a64
a15
a25
a35
a45
1
a65
a16
a26
a36
a46
a56
1
Choose ri and aij randomly from
an exponential distribution:
1
P(x) = e-x
P(x)
x = -LOG(RND)
mean = 1
0
0
x
5
Typical Time History
15 species
Si
Time
Coexistence




Coexistence is unlikely unless the
species compete only weakly with
one another.
Species may segregate spatially.
Diversity in nature may result from
having so many species from which
to choose.
There may be coexisting “niches” into
which organisms evolve.
Typical Time History (with Evolution)
15 species
15 species
Si
Time
Evolution of Total Biomass
Biomass
32 species
Time
Evolution of Biodiversity
Biodiversity (%)
32 species
Time
Conclusions




Competitive exclusion eliminates
most species.
The dominant species is
eventually killed and replaced by
another.
Species that grow quickly also die
quickly.
Evolution is punctuated rather
than continual.
Application to Finance
S&P500
Gaussian
Model
Volatility
S&P500
Gaussian
Model
Digression – The Bell Curve
Aka: Normal distribution
or
Gaussian distribution
P(x) = e
-x2/2
"Everybody believes in the exponential law of
errors: the experimenters, because they think
it can be proved by mathematics; and the
mathematicians, because they believe it has
been established by observation." (Whittaker
and Robinson, 1967)
What have mathematicians proved?



The Gaussian distribution is special because of the Central
Limit Theorem. Imagine the following experiment:
1.
Create a population with a known distribution (which
does not have to be Gaussian).
2.
Randomly pick many samples from that population, and
calculate the means of these samples.
3.
Draw a histogram of the distribution of the means.
The central limit theorem says that with enough samples, the
distribution of means will follow a Gaussian distribution even
if the population is not Gaussian.
Hence, any set of measurements of a given quantity will be
distributed about the mean in a Gaussian distribution if the
measurements are statistically independent.
N=1
N=2
N=4
N = 100
N=1
N=2
N=4
N=8
Mean
First moment
<xP(x)> -1 0
Variance
1
Second moment
<x2P(x)> 0.5 1
2
Skewness
Third moment
<x3P(x)> 0.5 0 -0.5
Fourth moment
<x4P(x)> − 3
Kurtosis
0
3/2 (leptokurtic)
-2/3
(platykurtic)
Kurtosis of Various Time Series
Time Series
EKG (2000 heartbeat intervals)
Physics 103 final exam
Light from variable white dwarf star
Deterministic CA forest model
Stochastic CA forest model
Daily temperature change (1980-1989)
EEG (8 seconds)
Daily S&P500 change (1975-1987)
Plasma magnetic fluctuations
Eye movement data (Aks)
Pound/$ daily exchange rate (1971-2003)
Yen/$ daily exchange rate (1973-2003)
DM/$ daily exchange rate (1973-1987)
Competition model with evolution
Kurtosis
-0.43
-0.26
-0.24
-0.07
0
0.57
0.61
2.13
3.19
3.96
4.35
4.77
6.51
???
2.86
EVENT SIZE
Is the Model SOC?
RANK
Zipf’s law of language
Current model
Scale invariance also observed in city populations, earthquake sizes, stock
prices, Internet connectivity, Web links, Moon craters, asteroid sizes, …
Is the Model Chaotic?
Total Biomass versus time
1% perturbation to
one of 15 species
added here
Summary

Nature is complex
but

Simple models may
suffice
References


http://sprott.physics.wisc.edu/lectures/
darwin.ppt (This talk)
sprott@physics.wisc.edu