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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.
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
Appendix: Markov Chains and Classical Results
References
Index
"This book is a welcome and timely addition to the existing literature on rigorous mathematical analysis of various genetics models, with a special focus on a careful understanding of the impact of stochastic effects. The book is written with the potential reader in mind, such that it is a real pleasure to read. It should be interesting to anyone working at the interface of mathematics and population genetics." (Artem S. Novozhilov, Mathematical Reviews, August, 2023)
"The text is written in an easy-to-read style and is suitable for use in various courses, including probability theory, Markov chains, mathematical ecology and population dynamics. The book can be expected to provide several ideas for further investigation of finite population models." (Attila Denes, zbMATH 1507.92072, 2023)
