Multistability of Reaction Networks with One-Dimensional Stoichiometric Subspaces

TL;DR

This paper investigates the stability and maximum number of positive steady states in reaction networks with one-dimensional stoichiometric subspaces, providing conditions that relate the total number of steady states to the number of stable ones.

Contribution

It establishes a relationship between the maximum number of positive steady states and stable steady states, including conditions for odd and even counts in one-dimensional networks.

Findings

Maximum stable states are half of the total steady states for even N.

For odd N, the maximum stable states depend on specific network conditions.

Provides criteria to determine stability counts based on the number of steady states.

Abstract

For the reaction networks with one-dimensional stoichiometric subspaces, we show the following results. (1) If the maximum number of positive steady states is an even number N, then the maximum number of stable positive steady states is N/2. (2) If the maximum number of positive steady states is an odd number N, then we provide a condition on the network such that the maximum number of stable positive steady states is (N-1)/2 if this condition is satisfied, and this maximum number is (N+1)/2 otherwise.