Editing Competitive Lotka–Volterra equations (section)

===Line systems and eigenvalues===
[[Image:Competitive LV Spatial Eigenvalues.jpg|thumb|right|350px|The eigenvalues of a circle, short line, and long line plotted in the complex plane]]

It is also possible to arrange the species into a line.<ref name=Wildenberg/>  The interaction matrix for this system is very similar to that of a circle except the interaction terms in the lower left and upper right of the matrix are deleted (those that describe the interactions between species 1 and ''N'', etc.).

<math display="block">\alpha_{ij} = \begin{bmatrix}1 & \alpha_1 & 0 & 0 & 0 \\ \alpha_{-1} & 1 & \alpha_1 & 0 & 0 \\ 0 & \alpha_{-1} & 1 & \alpha_1 & 0 \\ 0 & 0 & \alpha_{-1} & 1 & \alpha_1 \\ 0 & 0 & 0 & \alpha_{-1} & 1 \end{bmatrix}</math>

This change eliminates the Lyapunov function described above for the system on a circle, but most likely there are other Lyapunov functions that have not been discovered.

The eigenvalues of the circle system plotted in the [[complex plane]] form a [[trefoil]] shape.  The eigenvalues from a short line form a sideways Y, but those of a long line begin to resemble the trefoil shape of the circle.  This could be due to the fact that a long line is indistinguishable from a circle to those species far from the ends.<ref name="Wildenberg" />