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Chaos in Low-Dimensional Lotka-Volterra Models of Competition

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. Collaborators. John Vano Joe Wildenberg Mike Anderson Jeff Noel. Rabbit Dynamics.

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Chaos in Low-Dimensional Lotka-Volterra Models of Competition

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  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 > 0 growth • r = 0 equilibrium • r < 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 - Σ aijxj ) • Parameters of the model: • Vector of growth rates ri • Matrix of interactions aij • Number of species N N j=1

  6. Parameters of the Model Growth rates Interaction matrix 1 a12a13a14a15a16 a21 1 a23a24a25a26 a31a32 1a34a35a36 a41a42a43 1a45a46 a51a52a53a54 1a56 a61a62a63a64a65 1 1 r2 r3 r4 r5 r6

  7. Choose riand 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 4 3

  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. http://sprott.physics.wisc.edu/lectures/lvmodel.ppt (This talk) http://sprott.physics.wisc.edu/chaos/lvmodel/pla.doc (Preprint) sprott@physics.wisc.edu References

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