Introduction.
In population theory birth and death can be modeled, to a certain extent, by linear equations, but the formation of pairs is a nonlinear phenomenon. Separation of pairs can again be described by linear equations. In particular in human demography it is a long-standing problem by what type of equation pair formation shall be described, in other words, to find an appropriate marriage function. From the work of Kendall (1949), Keyfitz (1972), McFarland (1972), Parlett (1972), Pollard (1973), Fredrickson (1971), Staroverov (1977), Pollak (1987) it is quite clear that mass action kinetics is not appropriate, the marriage law must be homogeneous of degree one. Several special laws have been proposed, e.g. the harmonic mean and the minimum law, but there seems to be no law which is rigorously derived from a microscopic description of the pair formation process. On the other hand ist is obvious what the general properties of a marriage function should be (Fredrickson 1971, see properties 1), 2), 3) below). For such functions we derive a theory of pair formation, that continues and in some sense completes earlier work on this topic. In (Hadeler et al. 1988) we have developed an approach to homogeneous evolution equations, and this theory provides the appropriate framework for pair formation models. In fact we have given a complete analysis of the existence conditions for equilibria and of the global stability problem for these models, for continuous time and in the absence of age structure. Apart from the immediate application to huma~ demography pair formation models for bisexual populations are necessary prerequisites for the modeling of the spread of sexually transmitted diseases (Dietz and Hadeler 1987). Also there is experimental work on pair formation in animals, in particular in Drosophila and also theoretical research related to such experiments (B.Wallace 1985,1987, Vasco and Richardson 1985). General problems of sex and evolution have been discussed by Williams (1975) and by Karlin and Lessard (1986). In the models which contain only three one-dimensional variables for singles of both sexes and pairs it is not quite clear whether the newly recruited individuals should be interpreted as newborns or as those who enter the sexually active phase of their life. Therefore a more realistic description should comprise age structure. Mating models for age structured populations have been proposed by most of the authors cited above, in particular Staroverov (1977) has formulated very general models for continuous time, however without much analysis. Keyfitz (1972) has applied such models to demographic data, for several choices of marriage functions.