Escape from a metastable state in non-Markovian population dynamics.

Abstract

We study the long-time dynamics in non-Markovian single-population stochastic models, where one or more reactions are modeled as a stochastic process with a fat-tailed nonexponential distribution of waiting times, mimicking long-term memory. We focus on three prototypical examples: genetic switching, population establishment, and population extinction, all with nonexponential production rates. The system is studied in two regimes. In the first, the distribution of waiting times has a finite mean. Here, the system approaches a (quasi)stationary steady state at long times, and we develop a general Wentzel-Kramers-Brillouin approach for these non-Markovian systems. We derive explicit results for the mean population size and mean escape time from the metastable state of the stochastic dynamics. In this realm, we reveal that for sufficiently strong memory, a memory-induced (meta)stable state can emerge in the system. In the second regime, the waiting time distribution is assumed to have an infinite mean. Here, for bistable systems we find two distinct scaling regimes, separated by an exponentially long time which may strongly depend on the initial conditions of the system.