Predator-Prey Dynamics for Rabbits, Trees, & Romance
 J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to International Conference on Complex Systems in Nashua, NH on May 10, 2002

Lotka-Volterra Equations
 R = rabbits, F = foxes dR/dt = r1R(1 - R - a1F) dF/dt = r2F(1 - F - a2R)

Types of Interactions

Equilibrium Solutions
 dR/dt = r1R(1 - R - a1F) = 0 dF/dt = r2F(1 - F - a2R) = 0

Coexistence
 With N species, there are 2N equilibria, only one of which represents coexistence. Coexistence is unlikely unless the species compete only weakly with one another. Diversity in nature may result from having so many species from which to choose. There may be coexisting “niches” into which organisms evolve. Species may segregate spatially.

Alternate Spatial Lotka-Volterra Equations

Features of the Model
 Purely deterministic (no randomness) Purely endogenous (no external effects) Purely homogeneous (every cell is equivalent) Purely egalitarian (all species obey same equation) Continuous time

Typical Results

Dominant Species

Fluctuations in Cluster Probability

Power Spectrum
of Cluster Probability

Sensitivity to Initial Conditions

Results
 Most species die out Co-existence is possible Densities can fluctuate chaotically Complex spatial patterns spontaneously arise

Romance
(Romeo and Juliet)
 Let R = Romeo’s love for Juliet Let J = Juliet’s love for Romeo Assume R and J obey Lotka-Volterra Equations Ignore spatial effects

Romantic Styles

Pairings - Stable Mutual Love

Love Triangles
 There are 4-6 variables Stable co-existing love is rare Chaotic solutions are possible But…none were found in LV model Other models do show chaos

Summary
 Nature is complex Simple models may suffice

References
 http://sprott.physics.wisc.edu/ lectures/iccs2002/  (This talk) sprott@juno.physics.wisc.edu