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 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 A. Markov Chains and Classical Results References Index.