### Synopses & Reviews

Population genetics occupies a central role in a number of important biological and social undertakings. It is fundamental to our understanding of evolutionary processes, of plant and animal breeding programs, and of various diseases of particular importance to mankind. This is the first of a planned two-volume work discussing the mathematical aspects of population genetics, with an emphasis on the evolutionary theory. This first volume draws heavily from the author's classic 1979 edition since the material in that edition may be taken, to a large extent, as introductory to the contemporary theory. It has been revised and expanded to include recent topics that follow naturally from the treatment in the earlier edition, e.g., the theory of molecular population genetics and coalescent theory. This book will appeal to graduate students and researchers interested in theoretical population genetics and evolution. Reviews of the first edition: Ewens book will be an important reference to anyone interested in the mathematical aspects of population genetics, not only to those actually doing it, but also to anyone trying to bridge the now substantial gap between theoretical and experimental population genetics. Woodrow Setzer, Quarterly Review of Biology, 1980 This book is an excellent combination of an introduction to population genetics theory for a mathematically sophisticated reader, together with a survey of current work in the field. Stanley Sawyer, SIAM Review, 1980

#### Review

From reviews of the 1979 edition: "Here we have perhaps the most articulate of the many fine Australian population geneticists bringing us up to date on the mathematical aspects of his field." -B. S. Weir, William Neal Reynolds Professor of Statistics and Genetics, Director, Bioinformatics Research Center, North Carolina State University "This book is an excellent source to learn the field for a mathematician or mathematically sophisticated reader." -SIAM Review "An important reference to anyone interested in the mathematical aspects of population genetics, not only to those actually doing it, but to anyone trying to bridge the now substantial gap between theoretical and experimental population genetics." -The Quarterly Review of Biology From the reviews of the second edition: "It is the first of a planned two-volume sequence treating mathematical aspects of population genetics theory and its applications to evolution. ... The presentation is very clear and the author confers many of his deep insights to the reader. Therefore, this new edition has very good prospects to serve as the most important introductory text to this active field of research ... ." (R. Bürger, Monatshefte für Mathematik, Vol. 145 (1), 2005) From the reviews of the second edition: "This book is an extensively revised and expanded second edition ... . It presents the principles of mathematical population genetics with an emphasis on evolutionary theory. ... Ewens presentation bridges marvellously mathematics and biology. The author effectively copes with the problem that mathematicians want to see technical details, while biologists do not like formalism." (Martin Möhle, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 85 (1), 2005) From the reviews of the second edition: "The book concentrates on the mathematical aspects of population genetics at the graduate or research level. ... an excellent summary of the most important results, and very welcome in view of a vast scattered literature. I particularly like the many interesting connections that are made ... . Another highlight is an extra chapter on Moran model ... . Ewens' account of mathematical population genetics is unique ... . I am very happy to see this second edition in print ... ." (Ellen Baake, Mathematical Biosciences, Vol. 197, 2005) "This is an excellent book on population genetics and evolution placing the emphasis on mathematical and statistical aspects of the theory. ... the author successfully connects classical prospective theory with the current retrospective viewpoint of population genetics. ... this is an exciting and significant book which reflects the author's enthusiasm and experience in the field through many decades. It should be read by graduate students and researchers interested in mathematical aspects of population genetics ... ." (Günther Karigl, Zentralblatt MATH, Vol. 1060, 2005) "This book is in a series of texts specializing in interdisciplinary applied mathematics and is scheduled as the first volume of two devoted to population genetics by the same author; it is the second edition of the book first published in 1979. ... This book will be of most use to postgraduate researchers ... . the book under review admirably sets the scene by including a discussion of the broad theories of population dynamics." (Tony Crilly, The Mathematical Gazette, Vol. 89 (516), 2005)

#### Synopsis

Population genetics occupies a central role in a number of important biological and social undertakings. It is fundamental to our understanding of evolutionary processes, of plant and animal breeding programs, and of various diseases of particular importance to mankind.

This is the first of a planned two-volume work discussing the mathematical aspects of population genetics, with an emphasis on the evolutionary theory. This first volume draws heavily from the author's classic 1979 edition, which appeared originally in Springer's Biomathematics series. It has been revised and expanded to include recent topics which follow naturally from the treatment in the earlier edition, e.g., the theory of molecular population genetics.

This book will appeal to graduate students and researchers in mathematical biology and other mathematically-trained scientists looking to enter the field of population genetics.

#### Synopsis

This is the first of a planned two-volume work discussing the mathematical aspects of population genetics with an emphasis on evolutionary theory. This volume draws heavily from the author's 1979 classic, but it has been revised and expanded to include recent topics which follow naturally from the treatment in the earlier edition, such as the theory of molecular population genetics.

### Table of Contents

Contents Preface Introduction 1 Historical Background 1.1 Biometricians, Saltationists and Mendelians 1.2 The Hardy-Weinberg Law 1.3 The Correlation Between Relatives 1.4 Evolution 1.4.1 The Deterministic Theory 1.4.2 Non-Random-Mating Populations 1.4.3 The Stochastic Theory 1.5 Evolved Genetic Phenomena 1.6 Modelling 1.7 Overall Evolutionary Theories 2 Technicalities and Generalizations 2.1 Introduction 2.2 Random Union of Gametes 2.3 Dioecious Populations 2.4 Multiple Alleles 2.5 Frequency-Dependent Selection 2.6 Fertility Selection 2.7 Continuous-Time Models 2.8 Non-Random-Mating Populations 2.9 The Fundamental Theorem of Natural Selection 2.10 Two Loci 2.11 Genetic Loads 2.12 Finite Markov Chains 3 Discrete Stochastic Models 3.1 Introduction 3.2 Wright-Fisher Model: Two Alleles 3.3 The Cannings (Exchangeable) Model: Two Alleles 3.4 Moran Models: Two Alleles 3.5 K-Allele Wright-Fisher Models 3.6 Infinitely Many Alleles Models 3.6.1 Introduction 3.6.2 The Wright-Fisher In.nitely Many Alleles Model 3.6.3 The Cannings In.nitely Many Alleles Model 3.6.4 The Moran In.nitely Many Alleles Model 3.7 The Effective Population Size 3.8 Frequency-Dependent Selection 3.9 Two Loci 4 Diffusion Theory 4.1 Introduction 4.2 The Forward and Backward Kolmogorov Equations 4.3 Fixation Probabilities 4.4 Absorption Time Properties 4.5 The Stationary Distribution 4.6 Conditional Processes 4.7 Diffusion Theory 4.8 Multi-dimensional Processes 4.9 Time Reversibility 4.10 Expectations of Functions of Di.usion Variables 5 Applications of Diffusion Theory 5.1 Introduction 5.2 No Selection or Mutation 5.3 Selection 5.4 Selection: Absorption Time Properties 5.5 One-Way Mutation 5.6 Two-Way Mutation 5.7 Diffusion Approximations and Boundary Conditions 5.8 Random Environments 5.9 Time-Reversal and Age Properties 5.10 Multi-Allele Diffusion Processes 6 Two Loci 6.1 Introduction 6.2 Evolutionary Properties of Mean Fitness 6.3 Equilibrium Points 6.4 Special Models 6.5 Modifier Theory 6.6 Two-Locus Diffusion Processes 6.7 Associative Overdominance and Hitchhiking 6.8 The Evolutionary Advantage of Recombination 6.9 Summary 7 Many Loci 7.1 Introduction 7.2 Notation 7.3 The Random Mating Case 7.3.1 Linkage Disequilibrium, Means and Variances 7.3.2 Recurrence Relations for Gametic Frequencies 7.3.3 Components of Variance 7.3.4 Particular Models 7.4 Non-Random Mating 7.4.1 Introduction 7.4.2 Notation and Theory 7.4.3 Marginal Fitnesses and Average Effects 7.4.4 Implications 7.4.5 The Fundamental Theorem of Natural Selection 7.4.6 Optimality Principles 7.5 The Correlation Between Relatives 7.6 Summary 8 Further Considerations 8.1 Introduction 8.2 What is Fitness? 8.3 Sex Ratio 8.4 Geographical Structure 8.5 Age Structure 8.6 Ecological Considerations 8.7 Sociobiology 9 Molecular Population Genetics: Introduction 9.1 Introduction 9.2 Technical Comments 9.3 In.nitely Many Alleles Models: Population Properties 9.3.1 The Wright-Fisher Model 9.3.2 The Moran Model 9.4 In.nitely Many Sites Models: Population Properties 9.4.1 Introduction 9.4.2 The Wright-Fisher Model 9.4.3 The Moran Model 9.5 Sample Properties of In.nitely Many Alleles Models 9.5.1 Introduction 9.5.2 The Wright-Fisher Model 9.5.3 The Moran Model 9.6 Sample Properties of In.nitely Many Sites Models 9.6.1 Introduction 9.6.2 The Wright-Fisher Model 9.6.3 The Moran Model 9.7 Relation Between In.nitely Many Alleles and Infinitely Many Sites Models 9.8 Genetic Variation Within and Between Populations 9.9 Age-Ordered Alleles: Frequencies and Ages 10 Looking Backward in Time: The Coalescent 10.1 Introduction 10.2 Competing Poisson and Geometric Processes 10.3 The Coalescent Process 10.4 The Coalescent and Its Relation to Evolutionary Genetic Models 10.5 Coalescent Calculations: Wright-Fisher Models 10.6 Coalescent Calculations: Exact Moran Model Results 10.7 General Comments 10.8 The Coalescent and Human Genetics 11 Looking Backward: Testing the Neutral Theory 11.1 Introduction 11.2 Testing in the Infinitely Many Alleles Models 11.2.1 Introduction 11.2.2 The Ewens and the Watterson Tests 11.2.3 Procedures Based on the Conditional Sample Frequency Spectrum 11.2.4 Age-Dependent Tests 11.3 Testing in the Infinitely Many Sites Models 11.3.1 Introduction 11.3.2 Estimators of è 11.3.3 The Tajima Test 11.3.4 Other "Tajima-like" Testing Procedures 11.3.5 Testing for the Signature of a Selective Sweep 11.3.6 Combining Infinitely Many Alleles and In.nitely Many Sites Approaches 11.3.7 Data from Several Unlinked Loci 11.3.8 Data from Unlinked Sites 11.3.9 Tests Based on Historical Features 12 Looking Backward in Time: Population and Species Comparisons 12.1 Introduction 12.1.1 The Reversibility Criterion 12.2 Various Evolutionary Models 12.2.1 The Jukes-Cantor Model 12.2.2 The Kimura Model and Its Generalizations 12.2.3 The Felsenstein Models 12.3 Some Implications 12.3.1 Introduction 12.3.2 The Jukes-Cantor Model 12.3.3 The Kimura Model 12.4 Statistical Procedures Appendix A: Eigenvalue Calculations Appendix B: Significance Levels for ˆ F Appendix C: Means and Variances of ˆ F References Author Index Subject Index