  The MAP kinase cascade is an important signal transduction system in
molecular biology for which a lot of mathematical modelling has been done. This
paper surveys what has been proved mathematically about the qualitative
properties of solutions of the ordinary differential equations arising as
models for this biological system. It focusses, in particular, on the issues of
multistability and the existence of sustained oscillations. It also gives a
concise introduction to the mathematical techniques used in this context,
bifurcation theory and geometric singular perturbation theory, as they relate
to these specific examples. In addition further directions are presented in
which the applications of these techniques could be extended in the future.
