  In large but finite populations, weak demographic stochasticity due to random
birth and death events can lead to population extinction. The process is
analogous to the escaping problem of trapped particles under random forces.
Methods widely used in studying such physical systems, for instance,
Wentzel-Kramers-Brillouin (WKB) and Fokker-Planck methods, can be applied to
solve similar biological problems. In this article, we comparatively analyse
applications of WKB and Fokker-Planck methods to some typical stochastic
population dynamical models, including the logistic growth, endemic SIR,
predator-prey, and competitive Lotka-Volterra models. The mean extinction time
strongly depends on the nature of the corresponding deterministic fixed
point(s). For different types of fixed points, the extinction can be driven
either by rare events or typical Gaussian fluctuations. In the former case, the
large deviation function that governs the distribution of rare events can be
well-approximated by the WKB method in the weak noise limit. In the later case,
the simpler Fokker-Planck approximation approach is also appropriate.
