Skip Navigation



Mathematical Medicine and Biology Advance Access published online on September 4, 2009
Mathematical Medicine and Biology, doi:10.1093/imammb/dqp018
This Article
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Perthame, B.
Right arrow Articles by Gauduchon, M.
PubMed
Right arrow PubMed Citation
Right arrow Articles by Perthame, B.
Right arrow Articles by Gauduchon, M.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations

Benoît Perthame{dagger}

Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, F-75005 Paris, France, Institut Universitaire de France and Laboratoire Jacques-Louis Lions, UMR 7598, CNRS, F-75005 Paris, France

Mathias Gauduchon{ddagger}

Laboratoire de Microbiologie, Géochimie et Écologie Marines, UMR 6117, Centre d'Océanologie de Marseille, Université de la Méditerranée—CNRS, Campus de Luminy, Case 901, 13288 Marseille Cedex 09, France

{dagger} Email: benoit.perthame{at}upmc.fr

{ddagger} Email: mathias.gauduchon{at}univmed.fr

Received on January 20, 2009. Revised on April 16, 2009. Accepted on July 27, 2009.

Deterministic population models for adaptive dynamics are derived mathematically from individual-centred stochastic models in the limit of large populations. However, it is common that numerical simulations of both models fit poorly and give rather different behaviours in terms of evolution speeds and branching patterns. Stochastic simulations involve extinction phenomenon operating through demographic stochasticity, when the number of individual ‘units’ is small. Focusing on the class of integro-differential adaptive models, we include a similar notion in the deterministic formulations, a survival threshold, which allows phenotypical traits in the population to vanish when represented by few ‘individuals’. Based on numerical simulations, we show that the survival threshold changes drastically the solution; (i) the evolution speed is much slower, (ii) the branching patterns are reduced continuously and (iii) these patterns are comparable to those obtained with stochastic simulations. The rescaled models can also be analysed theoretically. One can recover the concentration phenomena on well-separated Dirac masses through the constrained Hamilton–Jacobi equation in the limit of small mutations and large observation times.

Keywords: evolution; adaptative dynamics; Hamilton–Jacobi equation; asymptotic analysis; demographic stochasticity; individual-based model; evolutionary branching


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.