Pair Formation in Age-Structured Populations

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.