Relation between the convective field and the stationary probability distribution of chemical reaction networks

We investigate the relation between the stationary probability distribution of chemical reaction systems and the convective field derived from the chemical Fokker-Planck equation (CFPE). This convective field takes into account the drift term of the CFPE and the reaction bias introduced by the diffusion term. For one-dimensional systems, fixed points and bifurcations of the convective field correspond to extrema and phenomenological bifurcations of the stationary probability distribution as long as the CFPE is a good approximation to the stochastic dynamics. This provides an efficient way to calculate the effect of system size on the number and location of probability maxima and their phenomenological bifurcations in parameter space. For two-dimensional systems, we study models that have saddle-node and Hopf bifurcations in the macroscopic limit. Here, the existence of two stable fixed points of the convective field correlates either with two peaks of the stationary probability distribution, or with a peak and a shoulder. In contrast, the system-size dependence of the parameter values for a Hopf bifurcation of the convective field is not reflected by the onset of a crater-shaped probability distribution, as the expected crater tends to be washed out for smaller system sizes.