Competition with Evolution in Ecology and Finance

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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

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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)
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Rabbit Dynamics

• Let R of rabbits
• dR/dt bR - dR

rR
r b - d
Birth rate
Death rate

• r gt 0 growth
• r 0 equilibrium
• r lt 0 extinction

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Multispecies Lotka-Volterra Model

• Let Si be population of the ith species
  (rabbits, trees, people, viruses, )
• dSi / dt riSi (1 - S aijSj )
• Parameters of the model
• Vector of growth rates ri
• Matrix of interactions aij
• Number of species N

N
j1
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Parameters of the Model
Growth rates
Interaction matrix
r1 r2 r3 r4 r5 r6
1 a12 a13 a14 a15 a16 a21 1 a23 a24 a25 a26 a31
a32 1 a34 a35 a36 a41 a42 a43 1 a45 a46 a51
a52 a53 a54 1 a56 a61 a62 a63 a64 a65 1
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Choose ri and aij randomly from an exponential
distribution
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P(x) e-x
P(x)
x -LOG(RND)
mean 1
0
x
0
5
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Typical Time History
15 species
Si
Time
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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.

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Typical Time History (with Evolution)
15 species
15 species
Si
Time
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Evolution of Total Biomass
32 species
Biomass
Time
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Evolution of Biodiversity
32 species
Biodiversity ()
Time
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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.

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Application to Finance
SP500
Gaussian
Model
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Volatility
SP500
Gaussian
Model
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Digression The Bell Curve
Aka Normal distribution or Gaussian distribution

"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)
-x2/2
P(x) e
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What have mathematicians proved?

• The Gaussian distribution is special because of
  the Central Limit Theorem. Imagine the following
  experiment
• Create a population with a known distribution
  (which does not have to be Gaussian).
• Randomly pick many samples from that population,
  and calculate the means of these samples.
• 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.

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N 1
N 2
N 4
N 100
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N 1
N 2
N 4
N 8
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Mean
Variance
First moment ltxP(x)gt
Second moment ltx2P(x)gt
-1 0 1
0.5
1
2
Skewness
Kurtosis
Third moment ltx3P(x)gt
Fourth moment ltx4P(x)gt - 3
0.5 0 -0.5
3/2 (leptokurtic)
0
-2/3 (platykurtic)
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Kurtosis of Various Time Series
Time Series Kurtosis
EKG (2000 heartbeat intervals) -0.43
Physics 103 final exam -0.26
Light from variable white dwarf star -0.24
Deterministic CA forest model -0.07
Stochastic CA forest model 0
Daily temperature change (1980-1989) 0.57
EEG (8 seconds) 0.61
Daily SP500 change (1975-1987) 2.13
Plasma magnetic fluctuations 3.19
Eye movement data (Aks) 3.96
Pound/ daily exchange rate (1971-2003) 4.35
Yen/ daily exchange rate (1973-2003) 4.77
DM/ daily exchange rate (1973-1987) 6.51
Competition model with evolution ???
2.86
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Is the Model SOC?
EVENT SIZE
RANK
Zipfs law of language
Current model
Scale invariance also observed in city
populations, earthquake sizes, stock prices,
Internet connectivity, Web links, Moon craters,
asteroid sizes,
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Is the Model Chaotic?
Total Biomass versus time
1 perturbation to one of 15 species added here
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Summary

• Nature is complex
• Simple models may suffice

but
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References

• http//sprott.physics.wisc.edu/lectures/darwin.ppt
  (This talk)
• sprott_at_physics.wisc.edu