Coalescent point process of branching trees in varying environment

TL;DR

This paper studies the genealogy of populations in changing environments using coalescent point processes, introduces a new Markov process for reconstruction, and identifies conditions for independence in offspring distributions.

Contribution

It proposes a finite-information Markov process to reconstruct genealogies in varying environments, addressing limitations of previous models.

Findings

Counterexample showing non-Markov property of previous process

Introduction of a new Markov process $(B_i)$ for genealogy reconstruction

Identification of i.i.d. property of $(A_i)$ in lineal fractional offspring distributions

Abstract

Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals in the same generation reproduce independently, forward in time, with the same offspring distribution but potentially changing among generations. In other words, the reproduction is driven by a Galton-Watson process in a varying environment. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i, i\geq 1)$, where $A_i$ is the coalescent time between individuals $i$ and $i+1$. In general, this process is not Markov. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We present a counterexample where we show that their process does not have the Markov property. The main contribution of this work is to propose a vector valued Markov process $(B_i,i\geq 1)$, that reach the goal to reconstruct the genealogy, with finite information for every $i$. Additionally, when the offspring distributions are lineal fractional, we show that the variables $(A_i, i\geq 1)$ are independent and identically distributed.