  The Togashi Kaneko model (TK model), introduced by Togashi and Kaneko in
2001, is a simple stochastic reaction network that displays
discreteness-induced transitions between meta-stable patterns. Here we study a
constrained Langevin approximation (CLA) of this model. The CLA, obtained by
Anderson et al. in 2019, is an obliquely reflected diffusion process on the
positive orthant and hence it respects the constrain that chemical
concentrations are never negative. We show that the CLA is a Feller process, is
positive Harris recurrent, and converges exponentially fast to the unique
stationary distribution. We also characterize the stationary distribution and
show that it has finite moments. In addition, we simulate both the TK model and
its CLA in various dimensions. For example, we describe how the TK model
switches between meta-stable patterns in dimension 6. Our simulations suggest
that, under the classical scaling, the CLA is a good approximation to the TK
model in terms of both the stationary distribution and the transition times
between patterns.
