Power-law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species

TL;DR

This paper develops determinant-based criteria to determine when networks of interacting species cannot exhibit multiple steady states, focusing on power-law kinetics and extending previous methods for analyzing multistationarity.

Contribution

It introduces simple, computationally feasible determinant criteria for ruling out multistationarity in networks with power-law and general kinetics, extending prior graphical and mass-action analyses.

Findings

Criteria are derived from the Jacobian determinant.

Applicable to power-law and general kinetics.

Criteria are simple, tractable, and implementable.

Abstract

We present determinant criteria for the preclusion of non-degenerate multiple steady states in networks of interacting species. A network is modeled as a system of ordinary differential equations in which the form of the species formation rate function is restricted by the reactions of the network and how the species influence each reaction. We characterize families of so-called power-law kinetics for which the associated species formation rate function is injective within each stoichiometric class and thus the network cannot exhibit multistationarity. The criterion for power-law kinetics is derived from the determinant of the Jacobian of the species formation rate function. Using this characterization we further derive similar determinant criteria applicable to general sets of kinetics. The criteria are conceptually simple, computationally tractable and easily implemented. Our approach embraces and extends previous work on multistationarity, such as work in relation to chemical reaction networks with dynamics defined by mass-action or non-catalytic kinetics, and also work based on graphical analysis of the interaction graph associated to the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graph