Large fluctuations in stochastic population dynamics: momentum space calculations

TL;DR

This paper explores the use of momentum-space representation and spectral decomposition to analyze large fluctuations in stochastic population dynamics, providing a new perspective on calculating extinction times and distributions.

Contribution

It introduces a momentum-space approach combined with spectral methods and WKB approximation to analyze large fluctuations in Markovian population models, offering novel analytical tools.

Findings

Eigenvalue problem characterizes stationary and quasi-stationary distributions.

WKB approximation effectively solves the eigenvalue problem for large populations.

Illustrative models demonstrate the method's applicability.

Abstract

Momentum-space representation renders an interesting perspective to theory of large fluctuations in populations undergoing Markovian stochastic gain-loss processes. This representation is obtained when the master equation for the probability distribution of the population size is transformed into an evolution equation for the probability generating function. Spectral decomposition then brings about an eigenvalue problem for a non-Hermitian linear differential operator. The ground-state eigenmode encodes the stationary distribution of the population size. For long-lived metastable populations which exhibit extinction or escape to another metastable state, the quasi-stationary distribution and the mean time to extinction or escape are encoded by the eigenmode and eigenvalue of the lowest excited state. If the average population size in the stationary or quasi-stationary state is large, the corresponding eigenvalue problem can be solved via WKB approximation amended by other asymptotic methods. We illustrate these ideas in several model examples.