Singular perturbations and scaling

TL;DR

This paper analyzes the properties and applicability of singular perturbation scaling transformations in chemical reaction networks, distinguishing between standard and nonstandard scenarios, and providing conditions for their validity and potential for algorithmic determination.

Contribution

It clarifies the conditions under which scaling transformations lead to valid singular perturbation reductions, including new insights into nonstandard scenarios and parameter-dependent systems.

Findings

Standard scenario aligns with classical QSS reduction.

Nonstandard scenario requires scaling for valid reduction.

Algorithmic determination of all possible scalings is feasible.

Abstract

Scaling transformations involving a small parameter ({\em degenerate scalings}) are frequently used for ordinary differential equations that model (bio-) chemical reaction networks. They are motivated by quasi-steady state (QSS) of certain chemical species, and ideally lead to slow-fast systems for singular perturbation reductions, in the sense of Tikhonov and Fenichel. In the present paper we discuss properties of such scaling transformations, with regard to their applicability as well as to their determination. Transformations of this type are admissible only when certain consistency conditions are satisfied, and they lead to singular perturbation scenarios only if additional conditions hold, including a further consistency condition on initial values. Given these consistency conditions, two scenarios occur. The first (which we call standard) is well known and corresponds to a classical quasi-steady state (QSS) reduction. Here, scaling may actually be omitted because there exists a singular perturbation reduction for the unscaled system, with a coordinate subspace as critical manifold. For the second (nonstandard) scenario scaling is crucial. Here one may obtain a singular perturbation reduction with the slow manifold having dimension greater than expected from the scaling. For parameter dependent systems we consider the problem to find all possible scalings, and we show that requiring the consistency conditions allows their determination. This lays the groundwork for algorithmic approaches, to be taken up in future work. In the final section we consider some applications. In particular we discuss relevant nonstandard reductions of certain reaction-transport systems.