  We provide a theoretical analysis of some autocatalytic reaction networks
exhibiting the phenomenon of discretely induced transitions (DITs). The family
of networks that we address includes the celebrated Togashi and Kaneko model.
We prove positive recurrence, finiteness of all moments, and geometric
ergodicity of the models in the family. For some parameter values, we find the
analytic expression for the stationary distribution, and discuss the effect of
volume scaling on the stationary behavior of the chain. We find the exact
critical value of the volume for which DITs disappear.
