Spatial Fleming-Viot Models with Selection and Mutation

✍ Scribed by Donald A. Dawson, Andreas Greven (auth.)

• Publisher: Springer International Publishing
• Year: 2014
• Tongue: English
• Leaves: 866
• Series: Lecture Notes in Mathematics 2092
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xvii
Introduction....Pages 1-10
Mean-Field Emergence and Fixation of Rare Mutants in the Fisher–Wright Model with Two Types....Pages 11-38
Formulation of the Multitype and Multiscale Model....Pages 39-53
Formulation of the Main Results in the General Case....Pages 55-104
A Basic Tool: Dual Representations....Pages 105-145
Long-Time Behaviour: Ergodicity and Non-ergodicity....Pages 147-159
Mean-Field Emergence and Fixation of Rare Mutants: Concepts, Strategy and a Caricature Model....Pages 161-165
Methods and Proofs for the Fisher–Wright Model with Two Types....Pages 167-375
Emergence with M ≥ 2 Lower Order Types (Phases 0,1,2)....Pages 377-714
The General ( M , M )-Type Mean-Field Model: Emergence, Fixation and Droplets....Pages 715-780
Neutral Evolution on E 1 After Fixation (Phase 3)....Pages 781-786
Re-equilibration on Higher Level E 1 (Phase 4)....Pages 787-810
Iteration of the Cycle I: Emergence and Fixation on E 2 ....Pages 811-828
Iteration of the Cycle II: Extension to the General Multilevel Hierarchy....Pages 829-837
Winding-Up: Proofs of the Theorems 3–11....Pages 839-839
Back Matter....Pages 841-858

✦ Subjects

Probability Theory and Stochastic Processes; Evolutionary Biology