Some Developments in Mathematical Demography

Abstract

The use of mathematical models in population studies has played a decisive rule so that demography has been developed as a substantive science. In particular the stable population theory by Alfred Lotka and life table analysis had kept their distinguished status as central dogma in demographic analysis until the end of 60th. However, through the 70th, Lotka model was extended to multistate models, nonlinear models and others, and the diversification of mathematical model analysis made it possible to expand the ability of demographic analysis extreamly. Through those mathematical studies, McKendrick-Von Foerster differential equation has played a most important rule. The formula of Lotka model by the differential equation has been paid much attention by many mathematician and biologists, by whom a general theory of structured population dynamics has been developed during 80th. On the other hand, Preston, Coale and other demographers have found practical applications of McKendrick equation to indirect estimation method. The rapid accumulation of mathematical studies for human populations have strongly motivated mathematical demography to be identified as an independent field of study in the late 80th. Though stable population theory was the first attempt to answer the fundamental question what is the reproductivity of human population, it has been pointed out that there exists an unsolved problem known as "two-sex problem". In order to resolve the two-sex problem, we have to consider nonlinear bisexual models which take into account pair formation phenomena. Although those nonlinear two-sex models are known to be very difficult to analyze, recently some results about those models have been shown by several authors. It is clear that to develop two-sex theory is one of the most important future challenge in mathematical demography. Though during the past decade it seems that mathematical demography has been establishing itself as a discipline, in order to fertilize mathematical demography much more in future, it would be needed to try to communicate with other related fields of study, and to have epistemological concerns in common with population problems in the real world.