Spatial Fleming-Viot Models with Selection and Mutation

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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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.

• 60J70,92D15,92D25,60J80,60J85.
• Dual processes
• Emergence of rare mutants
• Fleming-Viot processes
• Mutation-selection
• Spatial migration

• 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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