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