Analysis of a stochastic chemical system close to a SNIPER bifurcation
of its mean-field model

A framework for the analysis of stochastic models of chemical systems for
which the deterministic mean-field description is undergoing a saddle-node
infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs
for example in the modelling of cell-cycle regulation. It is shown that the
stochastic system possesses oscillatory solutions even for parameter values for
which the mean-field model does not oscillate. The dependence of the mean
period of these oscillations on the parameters of the model (kinetic rate
constants) and the size of the system (number of molecules present) is studied.
Our approach is based on the chemical Fokker-Planck equation. To get some
insights into advantages and disadvantages of the method, a simple
one-dimensional chemical switch is first analyzed, before the chemical SNIPER
problem is studied in detail. First, results obtained by solving the
Fokker-Planck equation numerically are presented. Then an asymptotic analysis
of the Fokker-Planck equation is used to derive explicit formulae for the
period of oscillation as a function of the rate constants and as a function of
the system size.