# J. C. Wildenberg, J. A. Vano, and J.C. Sprott

Departments of Physics and Mathematics, University of Wisconsin, Madison, WI 53706, USA
April 19, 2005

Chaos and periodicity can occur in Lotka-Volterra systems that have been modified to include spatial dependence.  One method for doing this is to arrange the interaction matrix such that the species interact as though part of a ring (explained in another publication).  The eigenvalues for these systems are easy to calculate and exhibit some interesting properties.  The behavior of the eigenvalues for the system

dxi/dt = xi(1 - axi-2 - xi - bxi+1)        (1)

is shown here.

A more realistic system

dxi/dt = xi(1 - 0.451xi-2 - 0.505xi-2 - xi - 0.852xi+1 - 0.237xi+2)        (2)

exhibits chaos for N = 100 (where 1 < i < N) and has eigenvalues in a distorted trefoil shape.

This shape is maintained through changes in the number of species in the ring.  This animation shows how the eigenvalues fill in the shape as the number of species is increased.  If we multiply the off-diagonal elements by a bifurcation parameter s, we can see in this animation how the shape of the trefoil expands as s increases.

Line Systems

Line systems can be created from a ring in many ways.  These line systems may have similar dynamics to the ring system, but the eigenvalues are drastically different.  For N = 100 there are obvious differences in the eigenvalues for the line system (below) and the ring system for Eq. (1). Animating how the eigenvalues change with N reveals that the overall shape doesn't change (as with the ring system) but the new eigenvalues fill in the shape as N increases.

An extremely long line is indistinguishable from a ring for the species far from the ends.  This animation shows that, for N > 155, the eigenvalues for the line systems slowly approach those of the ring system.