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

dx_{i}/dt = x_{i}(1 - ax_{i}_{-2} - x_{i} - bx_{i}_{+1})
(1)

is shown here.

A more realistic system

dx_{i}/dt = x_{i}(1 - 0.451*x*_{i-2} - 0.505*x*_{i-2} - x_{i} - 0.852x_{i}_{+1}* -
*0.237*x*_{i+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.