Uniqueness of feasible equilibria for mass action law (MAL) kinetic systems

TL;DR

This paper investigates the conditions for the uniqueness and stability of equilibria in polynomial ODE models of kinetic systems, introducing a flux-based framework and new criteria for equilibrium characterization and complex balancing.

Contribution

It develops a canonical flux-based representation of feasible equilibria and links monotonic functions to classical CRNT results, advancing understanding of equilibrium uniqueness.

Findings

Characterization of feasible equilibria via strictly stable matrices

Identification of conditions for complex balancing and equilibrium uniqueness

Potential support for an alternative proof of the deficiency one theorem

Abstract

This paper studies the relations among system parameters, uniqueness, and stability of equilibria, for kinetic systems given in the form of polynomial ODEs. Such models are commonly used to describe the dynamics of nonnegative systems, with a wide range of application fields such as chemistry, systems biology, process modeling or even transportation systems. Using a flux-based description of kinetic models, a canonical representation of the set of all possible feasible equilibria is developed. The characterization is made in terms of strictly stable compartmental matrices to define the so-called family of solutions. Feasibility is imposed by a set of constraints, which are linear on a log-transformed space of complexes, and relate to the kernel of a matrix, the columns of which span the stoichiometric subspace. One particularly interesting representation of these constraints can be expressed in terms of a class of monotonous decreasing functions. This allows connections to be established with classical results in CRNT that relate to the existence and uniqueness of equilibria along positive stoichiometric compatibility classes. In particular, monotonicity can be employed to identify regions in the set of possible reaction rate coefficients leading to complex balancing, and to conclude uniqueness of equilibria for a class of positive deficiency networks. The latter result might support constructing an alternative proof of the well-known deficiency one theorem. The developed notions and results are illustrated through examples.