Homotopy methods for counting reaction network equilibria

TL;DR

This paper introduces homotopy-based methods to determine the number of equilibria in complex biochemical reaction networks, accounting for parameter dependencies and network structure.

Contribution

It presents novel homotopy invariance techniques to count equilibria and analyze their dependence on parameters in high-dimensional nonlinear reaction networks.

Findings

Methods can determine when multiple equilibria exist.

Network structure influences equilibrium uniqueness.

Parameter relationships dictate equilibrium multiplicity.

Abstract

Dynamical system models of complex biochemical reaction networks are usually high-dimensional, nonlinear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfied. We describe methods based on homotopy invariance of degree which allow us to determine the number of equilibria for complex biochemical reaction networks and how this number depends on parameters in the model.