2012/07/20 by Camille Coron, Coron, Camille
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Social Sciences · #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1207.4920
arxiv created 2012/07/20 · openalex publication_date 2012/07/20 · arxiv updated 2012/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.