2013/09/13 by Camille Coron, Coron, Camille · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Social Sciences · #60J25 #60J27 #60J60 #60J70 #60J75 #92D40 #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #math.PR #msc:60J25 #msc:60J27 #msc:60J60 #msc:60J70 #msc:60J75 #msc:92D40
paper · pdf · doi:10.48550/arxiv.1309.3405
arxiv created 2013/09/13 · openalex publication_date 2013/09/13 · arxiv updated 2013/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are interested in the long-time behavior of a diploid population with sexual reproduction, characterized by its genotype composition at one bi-allelic locus. The population is modeled by a 3-dimensional birth-and-death process with competition, cooperation and Mendelian reproduction. This stochastic process is indexed by a scaling parameter K that goes to infinity, following a large population assumption. When the birth and natural death parameters are of order K, the sequence of stochastic processes indexed by K converges toward a slow-fast dynamics. We indeed prove the convergence toward 0 of a fast variable giving the deviation of the population from Hardy-Weinberg equilibrium, while the sequence of slow variables giving the respective numbers of occurrences of each allele converges toward a 2-dimensional diffusion process that reaches (0,0) almost surely in finite time. We obtain that the population size and the proportion of a given allele converge toward a generalized Wright-Fisher diffusion with varying population size and diploid selection. Using a non trivial change of variables, we next study the absorption of this diffusion and its long time behavior conditioned on non-extinction. In particular we prove that this diffusion starting from any non-trivial state and conditioned on not hitting (0,0) admits a unique quasi-stationary distribution. We finally give numerical approximations of this quasi-stationary behavior in three biologically relevant cases: neutrality, overdominance, and separate niches.