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June 2014 Stochastic modeling of density-dependent diploid populations and the extinction vortex
Camille Coron
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Adv. in Appl. Probab. 46(2): 446-477 (June 2014). DOI: 10.1239/aap/1401369702


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 three-type, nonlinear birth-and-death process, in which birth rates are designed to integrate Mendelian reproduction. We are interested in the long-term genetic behavior of the population (adaptive dynamics), and in particular we compute the fixation probability of a slightly nonneutral allele in the absence of mutations, which involves finding the unique subpolynomial solution of a nonlinear three-dimensional recurrence relationship. This equation is simplified to a one-dimensional 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.


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Camille Coron. "Stochastic modeling of density-dependent diploid populations and the extinction vortex." Adv. in Appl. Probab. 46 (2) 446 - 477, June 2014.


Published: June 2014
First available in Project Euclid: 29 May 2014

zbMATH: 1325.92070
MathSciNet: MR3215541
Digital Object Identifier: 10.1239/aap/1401369702

Primary: 60J10 , 60J80
Secondary: 60J27 , 92D15 , 92D25

Keywords: diploid population , Dirichlet problem , extinction vortex , fixation probability , multidimensional nonlinear recurrence equation , nonlinear birth-and-death process , Population genetics

Rights: Copyright © 2014 Applied Probability Trust


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Vol.46 • No. 2 • June 2014
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