# STABLE POPULATION THEORY WITH TIME VARYING IMMIGRATION

## Abstract

The effect of immigration has become of some importance. Originally, stable population theory was developed to study so called closed populations (Lotka (1939), Sharpe and Lotka (1911)). Over the past quarter of a century or so open populations have been investigated and the effect of immigration on the long-run behaviour has been examined.

Keyfitz (1971) investigated the long-run effect of emigration on a population while Coale (1972) considered the reduction in fertility needed to counterbalance the effect of a steady stream of immigrants. Epenshade et al. (1982) and Mitra (1983) analysed the consequences of a constant indefinite stream of immigrants while Cerone (1986) extended stable population theory to include a constant stream of immigrants.

Mitra (1990) examined the vital rates and the age structure of a long term stationary population brought about by the effect of a constant stream of immigrants on below replacement fertility regimen of the local population. Smertmann (1990) investigated the rejuvenating force reflecting through the eventual age - structure of the ensuing population under similar conditions as Mitra (1990). Blanchet (1989) examines the possibility of regulating the age - structure of an ensuing population under a constant stream of immigrants.

General immigration models have been developed in the past. Sivamurthy (1982) used a discrete formulation of a Leslie matrix and the population was projected. An integral equation model was developed by Langhaar (1972) which may be solved numerically. In this article however, some simple models shall be developed in order to investigate the effect of time variation of immigration levels. Further, a number of authors including Feichtinger and Steinmann (1992) and Friedlander and Feldmann (1993), have indicated that the adoption of local fertility behaviour by the immigrant population to be unrealistic. A model is developed and analysed that allows for a gradual transition of the fertility behaviour of an immigrant population to that of the local population.

The modelling in this article is done through the use of the Sharpe Lotka single-sex integral population model to determine the eventual population behaviour using the conventional method of Laplace transforms.

The outline of the paper is as follows: Section 2 develops the general model that allows for time varying immigration and vital rates. Expressions for the total birth rate, B(t), the age distribution A(x,t) and the total population numbers N(t) are presented. Section 3 presents and analyses models which allow the immigration behaviour to vary with time. The asymptotic or long-term behaviour of the total births is obtained using some simple but instructive time varying immigrant regimen. A model that allows the immigrant maternity behaviour to gradually approach that of the local population is analysed in Section 4. A numerical example demonstrating the transient behaviour of the total births and its approach towards the predicted asymptotic is demonstrated in Section 5. In Section 6 the asymptotic or large time population numbers and age - distribution are detailed together with the limiting age-density for the models presented in Sections 3 and 4.