WKB theory of large deviations in stochastic populations

TL;DR

This paper reviews the application of WKB approximation methods to analyze large deviations in stochastic population dynamics, including extinction, fixation, and invasion processes across various biological scales.

Contribution

It synthesizes recent advances in applying dissipative WKB techniques to evaluate probabilities and times of rare events in stochastic populations, covering well-mixed, networked, and spatial models.

Findings

WKB methods effectively estimate extinction and fixation probabilities.

Analysis includes effects of demographic and environmental noise.

Spatial models reveal fluctuations in invasion speeds.

Abstract

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 states - are 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.