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The Quasispecies Equation and Classical Population Models

  • This book provides an in-depth study of the Eigen's Quasispecies equation in the context of population models.

  • The case of the Wright-Fisher model is treated in detail, other classical population models are also discussed.

  • Contains a foreword by Michel Benaïm.

Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 102)

  • 1611 Accesses

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eBook USD 119.00
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  • ISBN: 978-3-031-08663-2
  • Instant PDF download
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Hardcover Book USD 159.99
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Table of contents (28 chapters)

  1. Front Matter

    Pages i-x
  2. Introduction

    • Raphaël Cerf, Joseba Dalmau
    Pages 1-5
  3. Finite Genotype Space

    1. Front Matter

      Pages 7-8
    2. The Quasispecies Equation

      • Raphaël Cerf, Joseba Dalmau
      Pages 9-12
    3. Non-Overlapping Generations

      • Raphaël Cerf, Joseba Dalmau
      Pages 13-22
    4. Overlapping Generations

      • Raphaël Cerf, Joseba Dalmau
      Pages 23-30
    5. Probabilistic Representations

      • Raphaël Cerf, Joseba Dalmau
      Pages 31-38
  4. The Sharp Peak Landscape

    1. Front Matter

      Pages 39-40
    2. Long Chain Regime

      • Raphaël Cerf, Joseba Dalmau
      Pages 41-45
    3. Error Threshold and Quasispecies

      • Raphaël Cerf, Joseba Dalmau
      Pages 47-51
    4. Probabilistic Derivation

      • Raphaël Cerf, Joseba Dalmau
      Pages 53-63
    5. Summation of the Series

      • Raphaël Cerf, Joseba Dalmau
      Pages 65-71
    6. Error Threshold in Infinite Populations

      • Raphaël Cerf, Joseba Dalmau
      Pages 73-80
  5. Error Threshold in Finite Populations

    1. Front Matter

      Pages 81-82
    2. Phase Transition

      • Raphaël Cerf, Joseba Dalmau
      Pages 83-86
    3. Computer Simulations

      • Raphaël Cerf, Joseba Dalmau
      Pages 87-93
    4. Heuristics

      • Raphaël Cerf, Joseba Dalmau
      Pages 95-101
    5. Shape of the Critical Curve

      • Raphaël Cerf, Joseba Dalmau
      Pages 103-105
    6. Framework for the Proofs

      • Raphaël Cerf, Joseba Dalmau
      Pages 107-113
  6. Proof for Wright–Fisher

    1. Front Matter

      Pages 115-116

About this book

This monograph studies a series of mathematical models of the evolution of a population under mutation and selection. Its starting point is the quasispecies equation, a general non-linear equation which describes the mutation-selection equilibrium in Manfred Eigen’s famous quasispecies model. A detailed analysis of this equation is given under the assumptions of finite genotype space, sharp peak landscape, and class-dependent fitness landscapes. Different probabilistic representation formulae are derived for its solution, involving classical combinatorial quantities like Stirling and Euler numbers.

It is shown how quasispecies and error threshold phenomena emerge in finite population models, and full mathematical proofs are provided in the case of the Wright–Fisher model. Along the way, exact formulas are obtained for the quasispecies distribution in the long chain regime, on the sharp peak landscape and on class-dependent fitness landscapes.

Finally, several other classical population models are analyzed, with a focus on their dynamical behavior and their links to the quasispecies equation. 

This book will be of interest to mathematicians and theoretical ecologists/biologists working with finite population models.

Keywords

  • Analysis of Quasispecies equation
  • Manfred Eigen quasispecies model
  • Population models book
  • Mutation-Selection equilibrium analysis
  • Markov chains mathematical biology
  • Wright-Fisher model
  • Long Chain Regime
  • Galton-Watson model biology
  • Population Genetics
  • Moran-Kingman model
  • Raphael Cerf math
  • Joseba Dalmau

Authors and Affiliations

  • LMO, Université Paris-Sud, CNRS, Université Paris-Saclay, Orsay, France, DMA, Ecole Normale Supérieure, CNRS, PSL Research University, Paris, France

    Raphaël Cerf

  • NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, Shanghai, China

    Joseba Dalmau

About the authors


Bibliographic Information

Buying options

eBook USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-031-08663-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book USD 159.99
Price excludes VAT (USA)