Markov chain approach to the distribution of ancestors in species of biparental reproduction

TL;DR

This paper introduces a Markov chain model to analyze the distribution of ancestors in sexually reproducing species, addressing limitations of previous models and offering a new approach to genealogical tree reconstruction.

Contribution

It develops a Markov chain framework with dilation and gauge invariance to better estimate ancestor distributions in species with biparental reproduction.

Findings

Markov chain effectively models ancestor distribution.

Dilation of sample space helps resolve maximum ancestor count.

Gauge invariance corrects distribution estimates.

Abstract

We studied how to obtain a distribution for the number of ancestors in species of sexual reproduction. Present models concentrate on the estimation of distributions repetitions of ancestors in genealogical trees. It has been shown that is not possible to reconstruct the genealogical history of each species along all its generations by means of a geometric progression. This analysis demonstrates that it is possible to rebuild the tree of progenitors by modeling the problem with a Markov chain. For each generation, the maximum number of possible ancestors is different. This brings huge problems for the resolution. We found a solution through a dilation of the sample space, although the distribution defined there takes smaller values respect to the initial problem. In order to correct the distribution for each generation, we introduced the invariance under gauge (local) group of dilations. These ideas can used to study the interaction of several processes and provide a new approach on the problem of the common ancestor. In the same direction, this model also provide some elements that can be used to improve models of animal reproduction.