Stochastic models for the population growth of the sexes

Abstract

SUMMARY In this paper we present stochastic models to describe the changes that take place with time in the numbers of females and males in the various age-intervals in the population. We first consider the case where females, or males, are marriage dominant, i.e. where the joint distribution of the numbers of boys and girls born at time t depends upon the numbers of females, or males, in the various age-intervals in the immediately preceding time-period. We then consider the case where neither females nor males are marriage-dominant, i.e. where the joint distribution of the numbers of boys and girls born at time t depends upon both the numbers of females and males in the various age-intervals in the immediately preceding time-period. Methods are given for calculating the expected values, variances, and covariances of the number of females and males in each age-interval at time t, and asymptotic results are presented for the case where t -* so. The application of these results is illustrated with data on the fertility and mortality conditions in the U.S. in 1965. The literature on two-sex stochastic models of population growth starts with articles by Kendall (1949) and Goodman (1953). Kendall outlined some of the analytic difficulties of a study of the stochastic aspects of two-sex population growth, and for simplicity limited his discussion of two-sex stochastic models to the special case where (a) each birth is equally likely to add a new female or a new male to the population, (b) the death-rate for females is equal to that for males, and (c) the birth- and death-rates per person are constants that are independent of the age of the person, and also independent of other relevant variables. Although Goodman's analysis was not limited by (a) and (b), it was limited by (c), as was all subsequent analytic work by other authors. One of the models of Goodman (1953) was a stochastic model where one sex is marriage-dominant, i.e. where the joint distribution of the numbers of girls and boys born at time t depends only upon the number of the dominant sex in the population in the immediately preceding time-period; he calculated the means, variances, and covariances of the numbers of females and males in the population at time t. A similar analysis of this model was made by Joshi (1954); see also Barucha-Reid (1960, pp. 175-9), Bailey (1964, pp. 119-20), and Keyfitz (1968). Lamens (1957) supposed that the birth- and death-rates may be functions of time, but he did not allow for possible dependence of these rates on the age-composition of the population, nor for the possibility that neither females nor males are marriage-dominant. Here we shall deal with stochastic models where the birth- and death-rates for females and for males may depend upon the age-composition of the population. First, say, the females will be assumed marriage-dominant, and then the more general situation will be considered. The above articles dealt mainly with some of the stochastic aspects of two-sex populationgrowth. Some of the deterministic aspects have been discussed in the demographic and bio