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Spatial Fleming-Viot Models with Selection and Mutation

  • Book
  • © 2014

Overview

Authors:
  1. Donald A. Dawson

    1. Carleton University, School of Mathematics & Statistics, Ottawa, Canada

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  2. Andreas Greven

    1. Department Mathematik, Universität Erlangen-Nürnberg, Mathematisches Institut, Erlangen, Germany

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    You can also search for this author in PubMed   Google Scholar

  • Develops a class of spatial models of a population undergoing mutation, selection and migration
  • Develops new duality methods for multitype population models
  • Develops the McKean-Vlasov limit of exchangeable population models and their entrance laws
  • Identifies mutation-selection equilibria
  • Offers valuable insights into the role of migration in the emergence of rare mutants in spatial Fleming-Viot models
  • Sheds new light on the role of migration in sustaining biodiversity in evolution
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2092)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xvii
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  2. Introduction

    • Donald A. Dawson, Andreas Greven
    Pages 1-10

  3. Mean-Field Emergence and Fixation of Rare Mutants in the Fisher–Wright Model with Two Types

    • Donald A. Dawson, Andreas Greven
    Pages 11-38

  4. Formulation of the Multitype and Multiscale Model

    • Donald A. Dawson, Andreas Greven
    Pages 39-53

  5. Formulation of the Main Results in the General Case

    • Donald A. Dawson, Andreas Greven
    Pages 55-104

  6. A Basic Tool: Dual Representations

    • Donald A. Dawson, Andreas Greven
    Pages 105-145

  7. Long-Time Behaviour: Ergodicity and Non-ergodicity

    • Donald A. Dawson, Andreas Greven
    Pages 147-159

  8. Mean-Field Emergence and Fixation of Rare Mutants: Concepts, Strategy and a Caricature Model

    • Donald A. Dawson, Andreas Greven
    Pages 161-165

  9. Methods and Proofs for the Fisher–Wright Model with Two Types

    • Donald A. Dawson, Andreas Greven
    Pages 167-375

  10. Emergence with M ≥ 2 Lower Order Types (Phases 0,1,2)

    • Donald A. Dawson, Andreas Greven
    Pages 377-714

  11. The General (M, M)-Type Mean-Field Model: Emergence, Fixation and Droplets

    • Donald A. Dawson, Andreas Greven
    Pages 715-780

  12. Neutral Evolution on E 1 After Fixation (Phase 3)

    • Donald A. Dawson, Andreas Greven
    Pages 781-786

  13. Re-equilibration on Higher Level E 1 (Phase 4)

    • Donald A. Dawson, Andreas Greven
    Pages 787-810

  14. Iteration of the Cycle I: Emergence and Fixation on E 2

    • Donald A. Dawson, Andreas Greven
    Pages 811-828

  15. Iteration of the Cycle II: Extension to the General Multilevel Hierarchy

    • Donald A. Dawson, Andreas Greven
    Pages 829-837

  16. Winding-Up: Proofs of the Theorems 3–11

    • Donald A. Dawson, Andreas Greven
    Pages 839-839

  17. Back Matter

    Pages 841-858
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Keywords

About this book

This book constructs a rigorous framework for analysing selected phenomena in evolutionary theory of populations arising due to the combined effects of migration, selection and mutation in a spatial stochastic population model, namely the evolution towards fitter and fitter types through punctuated equilibria. The discussion is based on a number of new methods, in particular multiple scale analysis, nonlinear Markov processes and their entrance laws, atomic measure-valued evolutions and new forms of duality (for state-dependent mutation and multitype selection) which are used to prove ergodic theorems in this context and are applicable for many other questions and renormalization analysis for a variety of phenomena (stasis, punctuated equilibrium, failure of naive branching approximations, biodiversity) which occur due to the combination of rare mutation, mutation, resampling, migration and selection and make it necessary to mathematically bridge the gap (in the limit) between time and space scales.

Authors and Affiliations

  • Carleton University, School of Mathematics & Statistics, Ottawa, Canada

    Donald A. Dawson

  • Department Mathematik, Universität Erlangen-Nürnberg, Mathematisches Institut, Erlangen, Germany

    Andreas Greven

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