Time-resolved extinction rates of stochastic populations

Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasistationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover-renewal and removal-is much slower than all other processes. In this case there is a time-scale separation in the system which enables one to introduce a short-time quasistationary extinction rate W-1 and a long-time quasistationary extinction rate W-2, and to develop a time-dependent theory of the transition between the two rates. It is shown that W-1 and W-2 coincide with the extinction rates when the population turnover is absent and present, but very slow, respectively. The exponentially large disparity between the two rates reflects fragility of the extinction rate in the population dynamics without turnover.