title: Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size.
abstract: We are interested in the long-time behavior of a diploid population with sexual reproduction and randomly varying population size, characterized by its genotype composition at one bi-allelic locus. The population is modeled by a 3-dimensional birth-and-death process with competition, weak 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 individual birth and natural death rates are of order K, the sequence of stochastic processes indexed by K converges toward a new slow-fast dynamics with variable population size. We indeed prove the convergence toward 0 of a fast variable giving the deviation of the population from quasi 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. The population size and the proportion of a given allele converge toward a Wright-Fisher diffusion with stochastically varying population size and diploid selection. We insist on differences between haploid and diploid populations due to population size stochastic variability. Using a non trivial change of variables, we 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 give numerical approximations of this quasi-stationary behavior in three biologically relevant cases: neutrality, overdominance, and separate niches.
authors.affiliation: Laboratoire de Mathématiques d'Orsay UMR 8628, Université Paris-Sud, Bâtiment 425, 91405, Orsay-Cédex, France. camille.coron@math.u-psud.fr.
authors.forename: Camille
authors.initials: C
authors.lastname: Coron
mesh_terms: D000483:Alleles; D000818:Animals; D004171:Diploidy; D005260:Female; D057850:Hermaphroditic Organisms / Q000235:genetics; D016015:Logistic Models; D008297:Male; D055641:Mathematical Concepts; D008954:Models, Biological; D008957:Models, Genetic; D011156:Population Density; D011157:Population Dynamics; D012098:Reproduction; D013269:Stochastic Processes
keywords: Allele coexistence; Demographic Wright-Fisher diffusion processes; Diploid populations; Quasi-stationary distributions; Stochastic slow-fast dynamical systems
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