The phenomenon of discreteness-induced transitions is highly stochastic dependent dynamics observed in a family of autocatalytic chemical
reaction networks including the acclaimed Togashi Kaneko model. These
reaction networks describe the behaviour of several different species interacting with each other, and the counts of species concentrate in different
extreme possible values, occasionally switching between them. This phenomenon is only observed under some regimes of rate parameters in the
network, where stochastic effects of small counts of species takes effect.
The dynamics for networks in this family is ergodic with a unique
stationary distribution. While an analytic expression for the stationary distribution in the special case of symmetric autocatalytic behaviour
was derived by Bibbona, Kim, and Wiuf, not much is known about it in
the general case. Here we provide a candidate distribution for reaction
networks when the autocatalytic rates are different. It was inspired by
a model in population genetics, the Moran model with genic selection,
which shares many similar reaction dynamics to our autocatalytic networks. We show that this distribution is stationary when autocatalytic
rates are equal, and that it is close to stationary when they are not equal.