The paper focuses on the methodological foundations of one of the leading trendsin modern mathematical demography, associated with the construction of a generalized theory of human reproduction, which can be applicable to any population open to migration and demonstrating the changing regime of generation replacement.

The 1980s witnessed a qualitative leap in the development of the central problemof mathematical demography – the creation of a generalized r-variable model of population dynamics. Significant success, that has been achieved, in constructing the mostuniversal mathematical model of human reproduction regime and population age structure had a considerable influence on the development of the demographic analysis worldwide and approaches to a good deal of issues within the historical, regional, and other fields of population studies. Unfortunately, this methodological breakthroughwas little noticed by Russian-speaking experts, and even if seen, was hardly enough considered from both theoretical and practical perspectives. To this day, the sectionsof Russian demography textbooks devoted to mathematical modeling do without referenceto the existing generalizations of a stable population theory to the case of a non-stable, time-varying regime of population reproduction, and populations open to migration.

The paper provides examples of use of the generalized model based on the country – specific, the USSR and Russia’s data, which not only demonstrate the capabili tiesof the model, but also expand our knowledge about the demographic history of the country, as well as provide ground for further theoretical and statistical-mathematical studies of patterns of reproduction of any non-demographic collectivities, which members can be distributed on the basis of a parameter similar to age (i.e., a defined duration of existence in a certain state, characteristic of a given collectivity). For the relation to hold, the distribution of numbers and the force of each attrition factor must be continuous functions of age, or its equivalent.