Graph theoretic approach to the stability analysis of steady state chemical reaction networks

Abstract

A form of graph theory is developed which makes it possible to write the Routh-Hurwitz stability conditions for any network of chemical reactions as sums of graphs. These sums, which must all be positive for stability, can contain negative terms only through two mechanisms: first, by having an odd number of certain types of cycles in formally positive graphs, or second, by having an even number of these cycles and at least one cycle in formally negative graphs. The set of graphs in each stability inequality may be represented as a set of points which defines a convex polytope in a higher dimensional space. For large parameter values only the vertices of this polytope affect the stability of the network. For each vertex corresponding to a graph which is a negative term in a stability inequality there is a convex coneshaped contribution to the network's unstable region. For large parameter values, this region is the union of the interiors of these convex cones in parameter space whose boundaries are hyperplanes, and the equation of each hyperplane is determined by the graphs of a pair of adjacent vertices of the polytope. The full region of instability is the projection from near the origin of certain faces of a dual polytope: faces which correspond to negative vertices of the original polytope. It is proven that, if the reaction network has a sufficiently rich set of graphs, the vertices of the polytope are all maximum overlap graphs from the diagonal term of the Hurwitz determinant. In other cases, the polytope is truncated and vertices in the truncated region can make the network unstable only if they are negative through the second mechanism mentioned above.