  We study the long-time dynamics in non-Markovian single-population stochastic
models, where one or more reactions are modelled as a stochastic process with a
fat-tailed non-exponential distribution of waiting times, mimicking long-term
memory. We focus on three prototypical examples: genetic switching, population
establishment and population extinction, all with non-exponential 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 WKB
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.
