  We consider the dynamics of a population of organisms containing two mutually
inhibitory gene regulatory networks, that can result in a bistable switch-like
behaviour. We completely characterize their local and global dynamics in the
absence of any noise, and then go on to consider the effects of either noise
coming from bursting (transcription or translation), or Gaussian noise in
molecular degradation rates when there is a dominant slow variable in the
system. We show analytically how the steady state distribution in the
population can range from a single unimodal distribution through a bimodal
distribution and give the explicit analytic form for the invariant stationary
density which is globally asymptotically stable. Rather remarkably, the
behaviour of the stationary density with respect to the parameters
characterizing the molecular behaviour of the bistable switch is qualitatively
identical in the presence of noise coming from bursting as well as in the
presence of Gaussian noise in the degradation rate. This implies that one
cannot distinguish between either the dominant source or nature of noise based
on the stationary molecular distribution in a population of cells. We finally
show that the switch model with bursting but two dominant slow genes has an
asymptotically stable stationary density.
