Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size

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• 作者：Camille Coron
• 关键词：Diploid populations ; Demographic Wright ; Fisher diffusion processes ; Stochastic slow ; fast dynamical systems ; Quasi ; stationary distributions ; Allele coexistence ; 60J10 ; 60J80 ; 60J27 ; 92D25 ; 92D15
• 刊名：Journal of Mathematical Biology
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：72
• 期：1-2
• 页码：171-202
• 全文大小：861 KB
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• 作者单位：Camille Coron (1)
  
  1. Laboratoire de Mathématiques d’Orsay UMR 8628, Université Paris-Sud, Bâtiment 425, 91405, Orsay-Cédex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Biology
  Applications of Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1416

文摘

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. Keywords Diploid populations Demographic Wright-Fisher diffusion processes Stochastic slow-fast dynamical systems Quasi-stationary distributions Allele coexistence