Geometric singular perturbation theory meets cell biology: new challenges and advances

Many important biological processes evolve on different time scales and therefore consist of slow and fast components. Differential equations involving variables evolving on widely different time scales yield rich dynamics and notoriously hard mathematical questions. Geometric methods and dynamical systems theory play important roles in the study of such so-called slow-fast systems. During the last decades geometric singular perturbation theory (GSPT) and the blow-up method have become powerful tools for analyzing low-dimensional slow-fast systems in standard form and have been successfully used in many areas of mathematical biology. However, GSPT of mathematical models arising in molecular cell biology is much less established. The main reason for this seems to be that the corresponding models typically do not have an obvious slow-fast structure of the standard form. Nevertheless, many of these models exhibit some form of hidden slow-fast dynamics, which can be utilized in the analysis.

In this talk I will survey recent advances in GSPT beyond the standard form in the context of prototypical examples from cell biology. I will show that geometric methods based on the blow-up method provide a systematic approach to problems of this type.