  We study the asymptotic behaviors of stochastic cell fate decision between
proliferation and differentiation. We propose a model of a self-replicating
Langevin system, where cells choose their fate (i.e. proliferation or
differentiation) depending on local cell density. Based on this model, we
propose a scenario for multi-cellular organisms to maintain the density of
cells (i.e., homeostasis) through finite-ranged cell-cell interactions.
Furthermore, we numerically show that the distribution of the number of
descendant cells changes over time, thus unifying the previously proposed two
models regarding homeostasis: the critical birth death process and the voter
model. Our results provide a general platform for the study of stochastic cell
fate decision in terms of nonequilibrium statistical mechanics.
