The community matrix in three species community models
The explicit function of the community matrix of a three dimensional Lotka-Volterra model is delineated by a set of necessary and sufficient conditions for a positive equilibrium to be asymptotically stable. In the special case that the community matrix is quasi weakly diagonally dominant, it is shown that a positive determinant for the community matrix is not only necessary but is also sufficient for stability. The results are specific to three dimensional models and do not extend to communities of dimension greater than three.