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Mathematical structures in population genetics. (Matematicheskie struktury v populyatsionnoj genetike). (Russian) Zbl 0593.92011
Kiev: ”Naukova Dumka”. 296 p. (1983).

The monograph presents a systematic and rigorous study of mathematical (mainly algebraic) structures arising in deterministic models of population genetics. Besides an introduction into the genetical dynamics of populations and their primary models, it covers the following two main themes: (1) explicit description of stationary evolution operators and (2) convergence to an equilibrium state within discrete-time dynamic models.

The former was developed from a well-known fact posed by the Hardy- Weinberg law: an equilibrium among zygotes is set up already in the first generation of the progeny. This ”stationarity principle” was first postulated by S. N. Bernstein as a condition to be satisfied by any simple mechanism of inheritance in a population. In 1922-1924 Bernstein described all the stationary evolution operators for a population consisting of $n=3$ genotypes and discovered several interesting examples for particular values of $n>3·$

The author has investigated the Bernstein problem for an arbitrary n. From the observation that there exist stationary evolution operators which do not permit a genetic interpretation, he has come to the idea of a stationary genetic structure and hence to the problem of explicit description of all such structures, which compose a genetically meaningful core of the Bernstein problem. This essential part of the problem has been completely solved and significant progress in the general Bernstein problem has been achieved by the author as a result of his elaboration of a profound algebraic apparatus, namely, the theory of stationary (in particular, conservative) algebras. Chapters 3-5 are devoted to these results.

The second theme also goes back to the simplest models such as the one for selection in an autosomal locus with two alleles, for example. One of the main problems here is that of convergence to an equilibrium state. For a pure recombination process in a system of autosomal loci this problem was already solved in the 40s, though only in the 60s did O. Reiersöl propose an algebraic language capable of describing these relatively tedious systems in fairly clear and elegant terms.

By developing this approach the author obtains an explicit solution (for arbitrary initial conditions) to the equations of a recombination process in an autosomal multilocal population with any linkage distribution arising from random crossing-overs (Chapter 6). The further developments concern the representation of evolution processes in genetic algebras (Chapter 7) and some general and specific properties of quadratic evolution operators (Chapter 8).

The problem of convergence to an equilibrium becomes much more complex in populations experiencing selective pressures. The author has solved it in the following principal cases: (1) an autosomal multi-allele locus with any (even continual) set of equilibrium states and (2) the additive selection in a system of several autosomal multi-allele loci (Chapter 9). This progress is due to the application of the fairly sophisticated technique of analyzing relaxation processes that was developed previously by the author together with others in order to analyze certain processes of minimization.

A number of sections of the monograph can be used for giving special courses in contemporary mathematics as well as for individual graduate and post-graduate work. In particular, chapter 3 contains a systematic account for the theory of nonassociative baric algebras, which, although it was created as far back as the 30s, has been, up to now, exposed only in journal literature.

In general, the monograph presents a considerable contribution to that branch of modern mathematics which has recently been called ”population genetics”.

##### MSC:
 92D10 Genetics 92-02 Research monographs (appl. to natural sciences) 17D92 Genetic algebras