Impact of demography on extinction/fixation events

• Published in: Journal of mathematical biology Vol. 78; no. 3; pp. 549 - 577
• Main Authors: Coron, Camille, Méléard, Sylvie, Villemonais, Denis
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2019; Springer Nature B.V; Springer
• Subjects: Alleles; Applications of Mathematics; Demography; Diffusion; Dimensional changes; Dynamics; Extinction; Fixation; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Population; Population genetics; Population number; Probability; Rescaling; Scaling; Stochasticity; 92D10; Demography and extinction; 60J60; Population dynamics and population genetics; Diffusion processes; Allelic fixation; Path integrability; 92D40; Diffusion absorption; population dynamics and population genetics; diffusion processes; demography and extinction; diffusion absorption; allelic fixation; one-dimensional diffusion processes; path integrability; population dynamics; extinction and allelic fixation
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-018-1283-1

In this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright–Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov’s tranform.