1. 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

2. Collaborators
John Vano
Joe Wildenberg
Mike Anderson
Jeff Noel

3. 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 > growth
r = equilibrium
r < extinction

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

5. 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

6. 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 a a34 a35 a36 a41 a42 a a45 a46 a51 a52 a53 a a56 a61 a62 a63 a64 a65 1

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

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

20. 9/19/2018
References
(This talk)
(Preprint)