Multistationarity of Reaction Networks with One-Dimensional Stoichiometric Subspaces

TL;DR

This paper characterizes when reaction networks with one-dimensional stoichiometric subspaces exhibit multistationarity, identifying key subnetwork structures and providing a complete classification for networks with at least three steady states.

Contribution

It provides a complete characterization of multistationarity in one-dimensional reaction networks, linking it to specific embedded subnetworks and arrow diagrams.

Findings

Networks with multistationarity contain specific one-species subnetworks with certain arrow diagrams.

Networks with at least three positive steady states contain at least three bi-arrow diagrams.

Complete classification of bi-reaction networks with three or more positive steady states.

Abstract

We study the multistationarity for the reaction networks with one-dimensional stoichiometric subspaces, and we focus on the networks admitting finitely many positive steady states. We prove that if a network admits multistationarity, then network has an embedded one-species network with arrow diagram (->,<-) and another with arrow diagram (<-,->). The inverse is also true if there exist two reactions in the network such that the subnetwork consisting of the two reactions admits at least one and finitely many positive steady states. We also prove that if a network admits at least three positive steady states, then it contains at least three bi-arrow diagrams. More than that, we completely characterize the bi-reaction networks that admit at least three positive steady states.