Total variation approximation for quasi-equilibrium distributions, II

TL;DR

This paper explores conditions under which quasi-equilibrium distributions can be identified and approximated in population models, even when true quasi-stationary distributions are non-unique or difficult to compute.

Contribution

It extends previous work by providing conditions for identifying and computing quasi-equilibrium distributions in cases where quasi-stationary distributions are non-unique or non-existent.

Findings

Conditions for quasi-equilibrium distribution identification

Approximation methods for quasi-equilibrium distributions

Applicability to biological population models

Abstract

Quasi-stationary distributions, as discussed by Darroch & Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In an earlier paper, we gave biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute. In this paper, we consider conditions under which the quasi-stationary distribution, if it exists, need not be unique, but an apparent stochastic equilibrium can nonetheless be identified and computed; we call such a distribution a quasi-equilibrium distribution.