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Abstract

We present a symbolic algorithmic approach that allows to compute invariant
manifolds and corresponding reduced systems for differential equations modeling
biological networks which comprise chemical reaction networks for cellular
biochemistry, and compartmental models for pharmacology, epidemiology and
ecology. Multiple time scales of a given network are obtained by scaling, based
on tropical geometry. Our reduction is mathematically justified within a
singular perturbation setting using a recent result by Cardin and Teixeira. The
existence of invariant manifolds is subject to hyperbolicity conditions, which
we test algorithmically using Hurwitz criteria. We finally obtain a sequence of
nested invariant manifolds and respective reduced systems on those manifolds.
Our theoretical results are generally accompanied by rigorous algorithmic
descriptions suitable for direct implementation based on existing off-the-shelf
software systems, specifically symbolic computation libraries and
Satisfiability Modulo Theories solvers. We present computational examples taken
from the well-known BioModels database using our own prototypical
implementations.