Perpetual integrals convergence and extinctions in population dynamics

TL;DR

This paper investigates the convergence of perpetual integrals in one-dimensional diffusion processes and applies these findings to understand allelic fixation and population extinction in multi-type population dynamics.

Contribution

It provides a new simple proof of a known integrability criterion, extends it to non-homogeneous processes, and applies it to analyze population dynamics and extinction times.

Findings

Explicit bounds for moments of integrals

Characterization of population proportions before extinction

Extension of integrability criterion to non-homogeneous processes

Abstract

In this article we use a criterion for the integrability of paths of one-dimensional diffusion processes from which we derive new insights on allelic fixation in several situations. This well known criterion involves a simple necessary and sufficient condition based on scale function and speed measure. We provide a new simple proof for this result and also obtain explicit bounds for the moments of such integrals. We also extend this criterion to non-homogeneous processes by use of Girsanov's transform. We apply our results to multi-type population dynamics: using the criterion with appropriate time changes, we characterize the behavior of proportions of each type before population extinction in different situations.