Department of Mathematics Wisconsin University

If the abundances of the constituent molecules of a biochemical reaction system  are sufficiently high then their concentrations are typically modelled by a coupled set of ordinary differential equations (ODEs).  If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behaviour of the system and stochastic models are used.  In this talk, I will first introduce both the stochastic and deterministic models.  I will then provide theorems that allow us to determine the qualitative behaviour of the underlying mathematical models from easily checked properties of the associated reaction network.  I will present results pertaining to so-called ``complex-balanced'' models and those satisfying ``absolute concentration robustness'' (ACR).  In particular, I will show how  ACR models, which are stable when modelled deterministically, necessarily undergo an extinction event in the stochastic setting.  I will then characterise the behaviour of these models prior to extinction.