Chaos in Low-Dimensional Lotka-Volterra Models of Competition

J. C. Sprott

Department of PhysicsUniversity of Wisconsin - Madison

Presented at the

UW Chaos and Complex System Seminaron February 3, 2004

Collaborators

John Vano

Joe Wildenberg

Mike Anderson

Jeff Noel

Rabbit Dynamics Let R = # of rabbits dR/dt = bR - dR

Birth rate Death rate

= rR

• r > 0 growth

• r = 0 equilibrium

• r < 0 extinction

r = b - d

Logistic Differential Equation dR/dt = rR(1 – R)

R

rt

Exponentialgrowth

Nonlinearsaturation

• 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

Multispecies Lotka-Volterra Model

j=1

N

Parameters of the Model

1r2

r3

r4

r5

r6

1 a12 a13 a14 a15 a16

a21 1 a23 a24 a25 a26

a31 a32 1 a34 a35 a36

a41 a42 a43 1 a45 a46

a51 a52 a53 a54 1 a56

a61 a62 a63 a64 a65 1

Growthrates Interaction matrix

Choose ri and aij randomly from an exponential distribution:

P(a)

a00 5

1 P(a) = e-a

a = -LOG(RND)

mean = 1

Typical Time History

Time

xi

15 species

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)

Time

xi

15 species15 species

A Deterministic Chaotic Solution

27.153.172.01

ir

135.051.021.147.01033.236.144.010052.109.11

ija

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 0x 1 x 0x x 1 xx 0 x 1

1 2

34

Future Work

1. Is chaos generic in high-

dimensional 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

Simple models may suffice

but

References http://sprott.physics.wisc.edu/

lectures/lvmodel.ppt (This talk)

http://sprott.physics.wisc.edu/chaos/lvmodel/pla.doc (Preprint)

sprott@physics.wisc.edu