1. Predator-Prey Dynamics for Rabbits, Trees, & Romance
11/10/2018
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented to
International Conference on Complex Systems
in Nashua, NH
on May 10, 2002

2. Lotka-Volterra Equations
11/10/2018
Lotka-Volterra Equations
R = rabbits, F = foxes
dR/dt = r1R(1 - R - a1F)
dF/dt = r2F(1 - F - a2R)
r and a can be + or -

3. Types of Interactions - - dR/dt = r1R(1 - R - a1F)
dF/dt = r2F(1 - F - a2R) +
a2r2
Prey-
Predator
Competition - +
a1r1
Predator-
Prey
Cooperation -

4. Equilibrium Solutions
11/10/2018
Equilibrium Solutions
dR/dt = r1R(1 - R - a1F) = 0
dF/dt = r2F(1 - F - a2R) = 0
Equilibria:
R = 0, F = 0
R = 0, F = 1
R = 1, F = 0
R = (1 - a1) / (1 - a1a2), F = (1 - a2) / (1 - a1a2) F R

5. 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.

6. Alternate Spatial Lotka-Volterra Equations
11/10/2018
Alternate Spatial Lotka-Volterra Equations
Let Si(x,y) be density of the ith species (rabbits, trees, seeds, …)
dSi / dt = riSi(1 - Si - ΣaijSj)
ji
where
S = Sx-1,y + Sx,y-1
+ Sx+1,y + Sx,y+1 + aSx,y
2-D grid:

7. 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

8. Typical Results

9. Dominant Species

10. Fluctuations in Cluster Probability
Time

11. Power Spectrum of Cluster Probability
Frequency

12. Sensitivity to Initial Conditions
Error in Biomass
Time

13. Results Most species die out Co-existence is possible
Densities can fluctuate chaotically
Complex spatial patterns spontaneously arise
One implies
the other

14. 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

15. Romantic Styles dR/dt = rR(1 - R - aJ) a - r - + + Cautious lover
Narcissistic
nerd - + r
Eager
beaver
Hermit -

16. Pairings - Stable Mutual Love
Cautious
Lover
Narcissistic
Nerd
Eager
Beaver
Hermit
Narcissistic
Nerd
46%
67% 5% 0%
Eager
Beaver
67%
39% 0% 0%
Cautious
Lover 5% 0% 0% 0%
Hermit 0% 0% 0% 0%

17. 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

18. Simple models may suffice
11/10/2018
Summary
Nature is complex
Simple models may suffice
but

19. 11/10/2018
References
lectures/iccs2002/ (This talk)