Mathematical Structures in Population Genetics

In the theory of population genetics, fundamental results on its dynamical processes and equilibrium laws have emerged during the last few decades. This monograph systematically reviews these developments, beginning from elementary examples and explanations. Mathematically, the main emphasis of the book is the investigation of iterations of a quadratic or fractional quadratic operator in the simplex. By the use of some non-associative algebra, many results can be obtained in explicit form eg. the explicit description (Bernstein problem) of stationary quadratic operators, and the explicit solutions of a nonlinear evolutionary equation in the absence of selection, as well as general theorems on convergence to equilibrium in the presence of selection. Some of the algebraic theory used is interesting for its own sake. 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.

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.

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.