Ancestral reproductive bias in continuous time branching trees under various sampling schemes

TL;DR

This paper extends results on ancestral reproductive bias from continuous-time branching processes to Bellman-Harris processes, providing a simpler proof and insights into the probabilistic structure of reproduction events.

Contribution

It offers a short proof of ancestral reproductive bias in Bellman-Harris processes and explains differences in bias under various sampling schemes.

Findings

Extended main results to Bellman-Harris processes

Provided a probabilistic interpretation of reproduction rates

Clarified effects of different sampling schemes on ancestral bias

Abstract

Cheek and Johnston (Journal of Mathematical Biology, 2023) consider a continuous-time Bienaym\'e-Galton-Watson tree conditioned on being alive at time $T$. They study the reproduction events along the ancestral lineage of an individual randomly sampled from all those alive at time $T$. We give a short proof of an extension of their main results to the more general case of Bellman-Harris processes. Our proof also sheds light onto the probabilistic structure of the rate of the reproduction events. A similar method will be applied to explain (i) the different ancestral reproduction bias appearing in work by Geiger (Journal of Applied Probability, 1999) and (ii) the fact that the sampling rule considered by Chauvin, Rouault and Wakolbinger (Stochastic Processes and their Applications, 1991) leads to a time homogeneous process along the ancestral lineage.