Chaos in LowDimensional LotkaVolterra Models of Competition - PowerPoint PPT Presentation

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Chaos in LowDimensional LotkaVolterra Models of Competition

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Vector of growth rates ri. Matrix of interactions aij. Number of species N ... Typical Time History (with Evolution) Time. xi. 15 species. 15 species ... – PowerPoint PPT presentation

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Title: Chaos in LowDimensional LotkaVolterra Models of Competition


1
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

2
Collaborators
  • John Vano
  • Joe Wildenberg
  • Mike Anderson
  • Jeff Noel

3
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

4
Logistic Differential Equation
  • dR/dt rR(1 R)

Nonlinear saturation
R
Exponential growth
rt
5
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
6
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
7
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
8
Typical Time History
15 species
xi
Time
9
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.

10
Typical Time History (with Evolution)
15 species
15 species
xi
Time
11
A Deterministic Chaotic Solution


Largest Lyapunov exponent ?1 ? 0.0203
12
Time Series of Species
13
Strange Attractor
Attractor Dimension DKY 2.074
14
Route to Chaos
15
Homoclinic Orbit
16
Self-Organized Criticality
17
Extension to High Dimension(Many Species)
1
2
3
4
18
Future Work
  • Is chaos generic in high-dimensional LV systems?
  • What kinds of behavior occur for spatio-temporal
    LV competition models?
  • Is self-organized criticality generic in
    high-dimension LV systems?

19
Summary
  • Nature is complex
  • Simple models may suffice

but
20
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
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