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 focus on stochastic models and discuss two topics.  In the first, I will provide a large class of interesting nonlinear models for which the stationary distribution can be solved for exactly.  Such results are particularly useful in multi-scale settings, where the  dynamics of a fast subsystem are typically averaged over via the stationary distribution.  The second topic will pertain to an interesting class of models that exhibits fundamentally different long term behaviour when modelled deterministically or stochastically (the ODEs predict stability, the stochastic model goes extinct with probability one).  Recent work pertaining to the distribution of the stochastic model on compact time intervals (similar to the study of its quasi-stationary distribution) resolves the apparent discrepancy between the modelling choices.  The work presented in this talk falls into the broad category of Chemical Reaction Network Theory (CRNT).