We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n = 0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n = 0. In scenario B there is an intermediate repelling point n = n1 between the attracting point n = 0 and another attracting point n = n2 in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasi-stationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasi-stationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multi-step processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.

Theoretical ecologists have long sought to understand how the persistence of populations depends on biotic and abiotic factors. Classical work showed that demographic stochasticity causes the mean time to extinction to increase exponentially with population size, whereas variation in environmental conditions can lead to a power-law scaling. Recent work has focused especially on the influence of the autocorrelation structure ('color') of environmental noise. In theoretical physics, there is a burst of research activity in analyzing large fluctuations in stochastic population dynamics. This research provides powerful tools for determining extinction times and characterizing the pathway to extinction. It yields, therefore, sharp insights into extinction processes and has great potential for further applications in theoretical biology.

Using the weak-noise theory, we evaluate the probability distribution P(H,t) of large deviations of height H of the evolving surface height h(x,t) in the Kardar-Parisi-Zhang equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height H at time t. We argue that the tails of P behave, at arbitrary time t>0, and in a proper moving frame, as -lnP∼|H|^{5/2} and ∼|H|^{3/2}. The 3/2 tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.

We study the short-time behavior of the probability distribution P(H, t) of the surface height h(x = 0, t) = H in the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimension. The process starts from a stationary interface: h(x, t = 0) is given by a realization of two-sided Brownian motion constrained by h(0, 0) = 0. We find a singularity of the large deviation function of H at a critical value H = Hc. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x ↔ −x of optimal paths h(x, t) predicted by the weak-noise theory of the KPZ equation. At |H| ≫ |Hc| the corresponding tail of P(H) scales as − ln P ∼ |H| 3/2 /t 1/2 and agrees, at any t > 0, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of P scales as − ln P ∼ |H| 5/2 /t 1/2 and coincides with the corresponding tail for the sharp-wedge initial condition.

Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics -such as those determining population extinction, fixation or switching between different statesare presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.