Large fluctuations in stochastic population dynamics: momentum-space calculations

Abstract

Momentum-space representation provides an interesting perspective on the theory of largefluctuations in populations undergoing Markovian stochastic gain–loss processes. Thisrepresentation is obtained when the master equation for the probability distribution of thepopulation size is transformed into an evolution equation for the probability generatingfunction. Spectral decomposition then yields an eigenvalue problem for a non-Hermitianlinear differential operator. The ground-state eigenmode encodes the stationary distributionof the population size. For long-lived metastable populations which exhibit extinction orescape to another metastable state, the quasi-stationary distribution and the mean time toextinction or escape are encoded by the eigenmode and eigenvalue of the lowest excitedstate. If the average population size in the stationary or quasi-stationary state is large, thecorresponding eigenvalue problem can be solved via the WKB approximationamended by other asymptotic methods. We illustrate these ideas in several modelexamples.