Mathematical Structures in Population Genetics

This monograph systematically reviews recent developments in population genetics. The reader can use this book either to obtain a thorough and comprehensive coverage of the present state of knowledge of the subject, and to learn new methods from it, or else to obtain an elementary introduction to the field.

Klappentext

In the theory of population genetics, fundamental results onits dynamical processes and equilibrium laws have emergedduring the last few decades. This monograph systematicallyreviews these developments, beginning from elementaryexamples and explanations.Mathematically, the main emphasis of the book is theinvestigation of iterations of a quadratic or fractionalquadratic operator in the simplex. By the use of somenon-associative algebra, many results can be obtained inexplicit form eg. the explicit description (Bernsteinproblem) of stationary quadratic operators, and the explicitsolutions of a nonlinear evolutionary equation in theabsence of selection, as well as general theorems onconvergence to equilibrium in the presence of selection.Some of the algebraic theory used is interesting for its ownsake.The reader can use this book either to obtain a thorough andcomprehensive coverage of the present state of knowledge ofthe subject, and to learn new methods from it,or else toobtain an elementary introduction to the field.

Inhalt
1 Introduction to Population Dynamics.- 1.1 Elements of Classical Genetics.- 1.2 General Evolutionary Equations.- 1.3 Two-level Populations.- 2 Elementary Models.- 2.1 The Free Single Locus Population.- 2.2 The Free Two Locus, Two Allele Population.- 2.3 The Dynamic Effects of Sex Linkage.- 2.4 Selection at a Two Allele, Autosomal Locus.- 2.5 Mutation.- 2.6 Appendix: Asymptotic Estimates in One-Dimensional Dynamics.- 3 Algebra Foundations.- 3.1 Nonassociative Algebras. Idempotents and Nilpotents.- 3.2 Nilalgebras.- 3.3 Bark Algebras and Spaces.- 3.4 Bernstein Algebras.- 3.5 Genetic Algebras and Train Algebras.- 3.6 Duplication of an Algebra.- 3.7 Stochastic Spaces.- 3.8 Stochastic Algebras.- 3.9 The Kin of an Evolutionary Algebra. Normalization.- 4 Stationary Gene Structure.- 4.1 Statement of Problem. Examples.- 4.2 Zygote Genotypes. Principle of Gene Inheritance.- 4.3 Elementary Gene Structure (E.G S).- 4.4 Nonnegativity for an Arbitrary S.G.S..- 4.5 Genetic Criterion for E.G S.- 4.6 Nonelementary S.G.S..- 5 The General Bernstein Problem.- 5.1 The Necessity of Mendel's First Law.- 5.2 Theorem on Constant Inheritance.- 5.3 The Nonnegative Projection Associated with an Isolated Idempotent.- 5.4 A Theorem on Two Non-Splitting Types.- 5.5 The Solution of the Bernstein Problem for Exceptional Populations.- 5.6 Small Dimensions.- 5.7 Estimate of the Number of Constant Subpopulations. Ultranormal Bernstein Populations.- 5.8 Appendix. The Proof of Topological Lemma 5.7.2.- 6 Recombination Processes.- 6.1 Linkage Distribution. Chromosome Structures.- 6.2 Evolutionary Equations. Reiersöl Algebra.- 6.3 Structure of the Set of Equilibrium States. Convergence to Equilibrium.- 6.4 Exact Linearization. Evolutionary Spectrum.- 6.5 Explicit Evolutionary Formula.- 6.6 The Rate ofConvergence to Equilibrium.- 6.7 Combining Recombination and Mutation.- 6.8 Appendix. Recombination Among Sex Chromosomes.- 7 Evolution in Genetic Algebras.- 7.1 Reiersöl Algebras are Genetic.- 7.2 Idempotents. Convergence of Trajectories.- 7.3 Exact Linearization. Evolutionary Spectrum.- 7.4 The Explicit Evolutionary Formula.- 8 General Quadratic Evolutionary Operators.- 8.1 The Set of Equilibrium States.- 8.2 Contracting Quadratic Operators.- 8.3 An Example of Irregular Trajectory Behavior.- 9 Selection Dynamics.- 9.1 Evolutionary Selection Equations for an Autosomal Multiallele Locus.- 9.2 Stability of Equilibrium States. Fundamental Theorem.- 9.3 Variance Estimates of Disequilibrium.- 9.4 Convergence to Equilibrium for Selection in an Autosomal Multiallele Locus.- 9.5 Evolutionary Selection Equations in a System of Autosomal Multiallele Loci.- 9.6 Convergence to Equilibrium under Additive Selection for a System of Autosomal Multiallele Loci.- 9.7 Appendix. Characterization of Additive Selection.- A Commentaries and References.