Stochastic modeling of density-dependent diploid populations and extinction vortex

TL;DR

This paper models the genetic evolution of diploid populations, revealing how deleterious mutations accumulate and lead to an extinction vortex, using stochastic processes and fixation probability analysis.

Contribution

It introduces a novel stochastic model for density-dependent diploid populations, analyzing mutation fixation and extinction dynamics without prior studies addressing this complexity.

Findings

Fixation probability of non-neutral alleles computed.

Deleterious mutation fixation rate increases with existing fixed mutations.

Identification of the extinction vortex phenomenon.

Abstract

We model and study the genetic evolution and conservation of a population of diploid hermaphroditic organisms, evolving continuously in time and subject to resource competition. In the absence of mutations, the population follows a 3-type nonlinear birth-and-death process, in which birth rates are designed to integrate Mendelian reproduction. We are interested in the long term genetic behaviour of the population (adaptive dynamics), and in particular we compute the fixation probability of a slightly non-neutral allele in the absence of mutations, which involves finding the unique sub-polynomial solution of a nonlinear 3-dimensional recurrence relationship. This equation is simplified to a 1-order relationship which is proved to admit exactly one bounded solution. Adding rare mutations and rescaling time, we study the successive mutation fixations in the population, which are given by the jumps of a limiting Markov process on the genotypes space. At this time scale, we prove that the fixation rate of deleterious mutations increases with the number of already fixed mutations, which creates a vicious circle called the extinction vortex.