Editing Competitive Lotka–Volterra equations (section)

===Matrix organization===
One possible way to incorporate this spatial structure is to modify the nature of the Lotka–Volterra equations to something like a 
[[reaction–diffusion system]].  It is much easier, however, to keep the format of the equations the same and instead modify the interaction matrix.  For simplicity, consider a five species example where all of the species are aligned on a circle, and each interacts only with the two neighbors on either side with strength {{math|''α''<sub>−1</sub>}} and {{math|''α''<sub>1</sub>}} respectively.  Thus, species 3 interacts only with species 2 and 4, species 1 interacts only with species 2 and 5, etc.  The interaction matrix will now be
<math display="block">\alpha_{ij} = \begin{bmatrix}1 & \alpha_1 & 0 & 0 & \alpha_{-1} \\ \alpha_{-1} & 1 & \alpha_1 & 0 & 0 \\ 0 & \alpha_{-1} & 1 & \alpha_1 & 0 \\ 0 & 0 & \alpha_{-1} & 1 & \alpha_1 \\ \alpha_1 & 0 & 0 & \alpha_{-1} & 1 \end{bmatrix}.</math>

If each species is identical in its interactions with neighboring species, then each row of the matrix is just a [[permutation]] of the first row.  A simple, but non-realistic, example of this type of system has been characterized by Sprott ''et al.''<ref>{{cite journal | last1=Sprott | first1=J.C. | last2=Wildenberg | first2=J.C. | last3=Azizi | first3=Yousef | title=A simple spatiotemporal chaotic Lotka–Volterra model | journal=Chaos, Solitons & Fractals | publisher=Elsevier BV | volume=26 | issue=4 | year=2005 | issn=0960-0779 | doi=10.1016/j.chaos.2005.02.015 | pages=1035–1043| bibcode=2005CSF....26.1035S }}</ref>  The coexisting [[equilibrium point]] for these systems has a very simple form given by the [[Inverse element|inverse]] of the sum of the row
<math display="block">\overline{x}_i = \frac{1}{\sum_{j=1}^N \alpha_{ij}} = \frac{1}{\alpha_{-1} + 1 + \alpha_1}.</math>