Journal of Mathematical Biology

, Volume 78, Issue 3, pp 549–577 | Cite as

Impact of demography on extinction/fixation events

  • Camille CoronEmail author
  • Sylvie Méléard
  • Denis Villemonais


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.


Population dynamics and population genetics Demography and extinction Allelic fixation Diffusion processes Path integrability Diffusion absorption 

Mathematics Subject Classification

60J60 92D10 92D40 



This work was partially funded by the Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MnHn-FX and also supported by public Grants as part of the “Investissement d’avenir” Project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and reference ANR-10-CAMP-0151-02, FMJH, and by the Mission for Interdisciplinarity at the CNRS.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance
  2. 2.CMAP, CNRSEcole PolytechniquePalaiseauFrance
  3. 3.IECL, UMR 7502, CNRS, Vandœuvre-lès-Nancy et Inria, TOSCA teamUniversité de LorraineVillers-lès-NancyFrance

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