Chemical reactions involving several reactants are described by systems of polynomial ordinary differential equations. The individual concentrations and the reaction rates of the reactions typically have very different orders of magnitudes which causes dynamics on very different time-scales, i.e. some concentrations change very slowly, some very fast, and others with intermediate speeds. This makes the numerical simulation of such processes challenging but allows mathematical analysis by decomposing the total process in subprocesses taking place on these well separated time-scales. For problems where these different time-scales are controlled by a single small parameter a well developed powerful toolbox known as geometric singular perturbation theory (GSPT) exists. For problems where the multi-scale behavior depends on several parameters such a theory does not exist. In this project we aim at developing a novel approach for such problems. A key technique to deal with the multi-parameter singular structure are blow-ups in parameter- and variable space, which allows to adapt existing results from GSPT to these more complicated problems. As applications we consider mathematical models for important processes in biology, e.g. the cell cycle and gene regulatory networks.

Project leader: Peter SZMOLYAN (E101-01), Project member: Lukas BAUMGARTNER