Journal of Probability and Statistics
Volume 2009 (2009), Article ID 714701, 22 pages
Research Article

A Duality Approach to the Genealogies of Discrete Nonneutral Wright-Fisher Models

Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, CNRS-UMR 8089, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France

Received 3 September 2008; Revised 5 November 2008; Accepted 6 November 2008

Academic Editor: Rongling Wu

Copyright © 2009 Thierry E. Huillet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Discrete ancestral problems arising in population genetics are investigated. In the neutral case, the duality concept has been proved of particular interest in the understanding of backward in time ancestral process from the forward in time branching population dynamics. We show that duality formulae still are of great use when considering discrete nonneutral Wright-Fisher models. This concerns a large class of nonneutral models with completely monotone (CM) bias probabilities. We show that most classical bias probabilities used in the genetics literature fall within this CM class or are amenable to it through some “reciprocal mechanism” which we define. Next, using elementary algebra on CM functions, some suggested novel evolutionary mechanisms of potential interest are introduced and discussed.