Stochastically bistable growth and decay in the Togashi-Kaneko model

The two-state Togashi-Kaneko model demonstrates how, at finite system sizes, autocatalysis can lead to noise-induced bistability between the cellular concentrations of different molecular species. Here, we show that, in the biologically relevant scenario of species-dependent export rates, the nascent stochastic switching between molecular species also drives a concomitant switching between periods of growth or decay in the total population size. We demonstrate this behavior using stochastic simulations, as well as the numerical integration of a Fokker-Planck equation that approximates the finite system-size limit. By combining Piecewise Deterministic Markov and linear-noise approximations, we further find analytic expressions for the stationary distributions of the different molecular species when stochastic switching is faster than the dynamics in the total population size. We envisage that other models in the voter class-including spin systems, flocking, and opinion dynamics-may also exhibit aperiodic growth and decay in population size, as well as be amenable to similar techniques.