A Unified View on Bipartite Species-reaction Graphs and Their Relation to Interaction Graphs and Qualitative Dynamics of Chemical Reaction Networks

Abstract

The Jacobian matrix of a dynamic system and its principal minors play a prominent role in the study of qualitative dynamics and bifurcation analysis. When interpreting the Jacobian as an adjacency matrix of an interaction graph, its principal minors reate to sets of disjoint cycles in this graph and conditions for qualitative dynamic behaviors can be inferred from its cycle structure. The Jacobian of chemical reaction systems decomposes into the product of two matrices, which allows more fine-grained analyses by studying a corresponding bipartite species-reaction graph. Several different bipartite graphs have been proposed and results on injectivity, multistationarity, and bifurcations have been derived. Here, we present a new definition of the species-reaction graph that directly connects the cycle structure with determinant expansion terms, principal minors, and the coefficients of the characteristic polynomial. It encompasses previous graph constructions as special cases. This graph has a direct relation to the interaction graph, and properties of cycles and sub-graphs can be translated in both directions. A simple equivalence relation enables simplified decomposition of determinant expansions and allows simpler and more direct proofs of previous results.