The Boundary Reproduction Number for Determining Boundary Steady State Stability in Chemical Reaction Systems

We introduce the boundary reproduction number, adapted from the next generation matrix method, to assess whether an infusion of species will persist or become exhausted in a chemical reaction system. Our main contributions are as follows: (a) we show how the concept of a siphon, prevalent in Petri nets and chemical reaction network theory, identifies sets of species that may become depleted at steady state, analogous to a disease-free boundary steady state; (b) we develop an approach for incorporating biochemically motivated conservation laws, which allows the stability of boundary steady states to be determined within specific compatibility classes; and (c) we present an effective heuristic for decomposing the Jacobian of the system that reduces the computational complexity required to compute the stability domain of a boundary steady state. The boundary reproduction number approach significantly simplifies existing parameter-dependent methods for determining the stability of boundary steady states in chemical reaction systems and has implications for the capacity of critical metabolites and substrates in metabolic pathways to become exhausted.