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Journal of Mathematical Biology

, Volume 71, Issue 5, pp 1211–1242 | Cite as

Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks

  • Sylvain Billiard
  • Régis Ferrière
  • Sylvie Méléard
  • Viet Chi Tran
Article

Abstract

How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effects of the trait are also expected to influence the generation and maintenance of neutral variation. We consider a large population limit with rare mutation, under the assumption that the neutral marker mutates faster than the trait under selection. We prove the convergence of the stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming–Viot Process (SFVP). When restricted to the trait space this process is the Trait Substitution Sequence first introduced by Metz et al. (1996). During the invasion of a favorable mutation, a genetical bottleneck occurs and the marker associated with this favorable mutant is hitchhiked. By rigorously analysing the hitchhiking effect and how the neutral diversity is restored afterwards, we obtain the condition for a time-scale separation; under this condition, we show that the marker distribution is approximated by a Fleming–Viot distribution between two trait substitutions. We discuss the implications of the SFVP for our understanding of the dynamics of neutral variation under eco-evolutionary feedbacks and illustrate the main phenomena with simulations. Our results highlight the joint importance of mutations, ecological parameters, and trait values in the restoration of neutral diversity after a selective sweep.

Keywords

Mutation-selection Measure-valued individual-based model  Neutral diversity Hitchhiking Selective sweeps  Adaptive dynamics  Limit theorems for multi-scale processes  Substitution Fleming–Viot Process 

Mathematics Subject Classification

92D25 60J80 92D15 60J75 

Notes

Acknowledgments

S. B., S. M. and V. C. T. have been supported by the ANR MANEGE (ANR-09-BLAN-0215), the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X. V.C.T. also acknowledges support from Labex CEMPI (ANR-11-LABX-0007-01). R. F. acknowledges support from National Science Foundation Award EF-0623632, the Institut Universitaire de France, and the Agence Nationale de la Recherche (“EVORANGE” Grant).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Sylvain Billiard
    • 1
  • Régis Ferrière
    • 2
    • 3
  • Sylvie Méléard
    • 4
  • Viet Chi Tran
    • 5
  1. 1.Laboratoire EEP, UFR de BiologieUniversité des Sciences et Technologies Lille 1Villeneuve d’Ascq CedexFrance
  2. 2.Eco-Evo-Math International LaboratoryInstitut de Biologie de l’ENSParisFrance
  3. 3.Department of Ecology and Evolutionary BiologyUniversity of ArizonaTucsonUSA
  4. 4.CMAP, Ecole PolytechniqueCNRSPalaiseau CedexFrance
  5. 5.Laboratoire P. Painlevé, UFR de MathématiquesUniversité des Sciences et Technologies Lille 1Villeneuve d’Ascq CedexFrance

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