Chaos in Low-Dimensional LotkaVolterra Models of Competition
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
Rabbit Dynamics

Let R = # of rabbits

dR/dt = bR - dR = rR
Birth rate Death rate
r=b-d
•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
• Let xi be population of the ith species
(rabbits, trees, people, stocks, …)
N
• dxi / dt = rixi (1 - Σ aijxj )
j=1
• Parameters of the model:
• Vector of growth rates ri
• Matrix of interactions aij
• Number of species N
Parameters of the Model
Growth
rates
1
r2
r3
r4
r5
r6
Interaction matrix
1
a21
a31
a41
a51
a61
a12
1
a32
a42
a52
a62
a13
a23
1
a43
a53
a63
a14
a24
a34
1
a54
a64
a15
a25
a35
a45
1
a65
a16
a26
a36
a46
a56
1
Choose ri and aij randomly from
an exponential distribution:
1
P(a) = e-a
P(a)
a = -LOG(RND)
mean = 1
0
0
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
 1 
0.72

ri  
1.53 


1.27 
0 
 1 1.09 1.52
 0

1
0
.
44
1
.
36

aij  
2.33 0
1
0.47


1 
1.21 0.51 0.35
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 0 0
x 1 x 0
1
x x 1 x
2
x 0 x 1
4
3
Future Work
1.
Is chaos generic in highdimensional LV systems?
2.
What kinds of behavior occur for
spatio-temporal LV competition
models?
3.
Is self-organized criticality generic in
high-dimension LV systems?
Summary

Nature is complex
but

Simple models may
suffice
References



http://sprott.physics.wisc.edu/lectures/
lvmodel.ppt (This talk)
http://sprott.physics.wisc.edu/chaos/lv
model/pla.doc (Preprint)
sprott@physics.wisc.edu