Mathematical methods have been applied successfully to population genet ics for a long time. Even the quite elementary ideas used initially proved amazingly effective. For example, the famous Hardy-Weinberg Law (1908) is basic to many calculations in population genetics. The mathematics in the classical works of Fisher, Haldane and Wright was also not very complicated but was of great help for the theoretical understanding of evolutionary pro cesses. More recently, the methods of mathematical genetics have become more sophisticated. In use are probability theory, stochastic processes, non linear differential and difference equations and nonassociative algebras. First contacts with topology have been established. Now in addition to the tra ditional movement of mathematics for genetics, inspiration is flowing in the opposite direction, yielding mathematics from genetics. The present mono grapll reflects to some degree both patterns but especially the latter one. A pioneer of this synthesis was S. N. Bernstein. He raised-and partially solved- -the problem of characterizing all stationary evolutionary operators, and this work was continued by the author in a series of papers (1971-1979). This problem has not been completely solved, but it appears that only cer tain operators devoid of any biological significance remain to be addressed. The results of these studies appear in chapters 4 and 5. The necessary alge braic preliminaries are described in chapter 3 after some elementary models in chapter 2.
Table of Contents1 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 of Convergence 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.