A survey of methods for deciding whether a reaction network is multistationary

TL;DR

This survey reviews methods developed over 40 years to determine when reaction networks with mass-action kinetics can have multiple steady states, emphasizing recent results on network embedding relations and minimal multistationary networks.

Contribution

It introduces new results on embedding-minimal multistationary networks and provides novel constructions of infinite such network families.

Findings

Embedding relations connect multistationarity between networks.

Not all embedded networks share steady state properties.

New constructions of infinite minimal multistationary networks.

Abstract

Which reaction networks, when taken with mass-action kinetics, have the capacity for multiple steady states? There is no complete answer to this question, but over the last 40 years various criteria have been developed that can answer this question in certain cases. This work surveys these developments, with an emphasis on recent results that connect the capacity for multistationarity of one network to that of another. In this latter setting, we consider a network $N$ that is embedded in a larger network $G$, which means that $N$ is obtained from $G$ by removing some subsets of chemical species and reactions. This embedding relation is a significant generalization of the subnetwork relation. For arbitrary networks, it is not true that if $N$ is embedded in $G$, then the steady states of $N$ lift to $G$. Nonetheless, this does hold for certain classes of networks; one such class is that of fully open networks. This motivates the search for embedding-minimal multistationary networks: those networks which admit multiple steady states but no proper, embedded networks admit multiple steady states. We present results about such minimal networks, including several new constructions of infinite families of these networks.