Multistationarity of Reaction Networks with One-Dimensional Stoichiometric Subspaces

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 provide a necessary condition for a network to admit multistationarity in terms of the stoichiometric coefficients, which can be described by “arrow diagrams”. This necessary condition is not sufficient unless 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.