Determining “small parameters” for quasi-steady state

For a parameter-dependent system of ordinary differential equations we present a systematic approach to the determination of parameter values near which singular perturbation scenarios (in the sense of Tikhonov and Fenichel) arise. We call these special values Tikhonov–Fenichel parameter values. The principal application we intend is to equations that describe chemical reactions, in the context of quasi-steady state (or partial equilibrium) settings. Such equations have rational (or even polynomial) right-hand side. We determine the structure of the set of Tikhonov–Fenichel parameter values as a semi-algebraic set, and present an algorithmic approach to their explicit determination, using Groebner bases. Examples and applications (which include the irreversible and reversible Michaelis–Menten systems) illustrate that the approach is rather easy to implement.