Chaos%20in%20Low-Dimensional%20Lotka-Volterra%20Models%20of%20Competition

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Chaos in Low-Dimensional Lotka-Volterra Models of
Competition

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented at the
• UW Chaos and Complex System Seminar
• on February 3, 2004

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Collaborators

• John Vano
• Joe Wildenberg
• Mike Anderson
• Jeff Noel

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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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Logistic Differential Equation

• dR/dt rR(1 R)

Nonlinear saturation
R
Exponential growth
rt
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Multispecies Lotka-Volterra Model

• Let xi be population of the ith species
  (rabbits, trees, people, stocks, )
• dxi / dt rixi (1 - S aijxj )
• 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
1 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
1
P(a) e-a
P(a)
a -LOG(RND)
mean 1
0
a
0
5
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Typical Time History
15 species
xi
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
xi
Time
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A Deterministic Chaotic Solution


Largest Lyapunov exponent ?1 ? 0.0203
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Time Series of Species
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Strange Attractor
Attractor Dimension DKY 2.074
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Route to Chaos
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Homoclinic Orbit
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Self-Organized Criticality
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Extension to High Dimension(Many Species)
1 x 0 0
x 1 x 0
x x 1 x
x 0 x 1
1
2
3
4
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Future Work

1. Is chaos generic in high-dimensional LV systems?
2. What kinds of behavior occur for spatio-temporal
   LV competition models?
3. Is self-organized criticality generic in
   high-dimension LV systems?

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Summary

• Nature is complex
• Simple models may suffice

but
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References

• http//sprott.physics.wisc.edu/lectures/lvmodel.pp
  t (This talk)
• http//sprott.physics.wisc.edu/chaos/lvmodel/pla.d
  oc (Preprint)
• sprott_at_physics.wisc.edu