Killing versus catastrophes in birth-death processes and an application to population genetics

TL;DR

This paper explores the mathematical relationship between birth-death processes with killing and catastrophes, providing new insights and tools for analyzing population genetics models such as ancestral selection graphs.

Contribution

It establishes a novel connection between absorption probabilities and stationary tails in birth-death processes with killing and catastrophes, using duality and decomposition techniques.

Findings

Derived relationships between absorption probabilities and stationary tails.

Applied theoretical results to population genetics models.

Provided new analytical tools for ancestral process analysis.

Abstract

We establish connections between the absorption probabilities of a class of birth-death processes with killing, and the stationary tail of a related class of birth-death processes with catastrophes. The major ingredients of the proofs are a decomposition of the dynamics of these processes, a Feynman--Kac type relationship for Markov chains with reset and rebirth, and the concept of Siegmund duality, which allows us to invert the relationship between the processes. We apply our results to a pair of ancestral processes in population genetics, namely the killed ancestral selection graph and the pruned lookdown ancestral selection graph, in a finite population setting and its diffusion limit.