The quasispecies equation and classical population models
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 a…
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- Full title :The quasispecies equation and classical population models / Raphaël Cerf, Joseba Dalmau.
Summary
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.
Read More ↓Edition Statements
- Publication Date:2022.
Publication Statements
- Publisher(s):Springer
- Place of Publication:Cham, Switzerland
- Language:eng
- Publication Date:2022.
The Physical Object
- Description:1 online resource (x, 242 pages) : : illustrations.
Edition Identifiers
- ISBN:9783031086632, 3031086635
- DOI:10.1007/978-3-031-08663-2
- oclc:on1338559990
- LC Classification:QA273
- Dewey Decimal:519.2
Table of Contents
1. Introduction -- Part I.Finite Genotype Space -- 2. The Quasispecies equation -- 3. Non-Overlapping Generations -- 4. Overlapping Generations -- 5. Probabilistic Representations -- Part II. The Sharp Peak Landscape -- 6. Long Chain Regime -- 7. Error Threshold and Quasispecies -- 8. Probabilistic Derivation -- 9. Summation of the Series -- 10. Error Threshold in Infinite Populations -- Part III. Error Threshold in Finite Populations -- 11.Phase Transition -- 12. Computer Simulations -- 13. Heuristics -- 14. Shape of the Critical Curve -- 15. Framework for the Proofs -- Part IV. Proof for Wright-Fisher -- 16. Strategy of the Proof -- 17. The Non-Neutral Phase M -- 18. Mutation Dynamics -- 19. The Neutral Phase N -- 20. Synthesis -- Part V. Class-Dependent Fitness Landscapes -- 21. Generalized Quasispecies Distributions -- 22. Error Threshold -- 23. Probabilistic Representation -- 24. Probabilistic Interpretations -- 25. Infinite Population Models -- Part VI. A Glimpse at the Dynamics -- 26. Deterministic Level -- 27. From Finite to Infinite Population -- 28. Class-Dependent Landscapes -- A. Markov Chains and Classical Results -- References -- Index.
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