Chaos in Low-Dimensional Lotka-Volterra Models of Competition 9/19/2018 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

Let R = # of rabbits dR/dt = bR - dR = rR Rabbit Dynamics r = b - d 9/19/2018 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

Logistic Differential Equation dR/dt = rR(1 – R) Nonlinear saturation R Exponential growth rt

Multispecies Lotka-Volterra Model 9/19/2018 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

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

Choose ri and aij randomly from an exponential distribution: 1 P(a) = e-a P(a) a = -LOG(RND) mean = 1 a 5

Typical Time History 15 species xi 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 xi Time

A Deterministic Chaotic Solution Largest Lyapunov exponent: 1  0.0203

Time Series of Species

Strange Attractor Attractor Dimension: DKY = 2.074

Route to Chaos

Homoclinic Orbit

Self-Organized Criticality

Extension to High Dimension (Many Species) 1 x 1 2 4 3

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?

Simple models may suffice 9/19/2018 Summary Nature is complex Simple models may suffice but

9/19/2018 References http://sprott.physics.wisc.edu/lectures/lvmodel.ppt (This talk) http://sprott.physics.wisc.edu/chaos/lvmodel/pla.doc (Preprint) sprott@physics.wisc.edu