Introduction
Stable population models were introduced in session PAPP103_S07 
. They provide a precise mathematical description of the relationship between the fertility and mortality schedules of a closed population and its age structure in the long run. Lotka demonstrated that such populations “forget their past” and develop an age structure that is entirely determined by their vital rates.
Stable population models are a useful tool for comparative statics. That is to say, they can be used to investigate the long-run implications for the age structure of populations of different levels and patterns of fertility and mortality.
Although one can gain important insights from the comparison of stable population models, real populations almost always have a history of changing vital rates and therefore age structures that continually change. In other words, the growth rate of most populations varies both between age groups and over time.
It was not till the start of the 1980s, that Bennett and Horiuchi realised that the relationships between fertility, mortality, growth and age structure described by Lotka 70 years earlier could be generalized to non-stable populations with variable growth rates. In essence, as Preston and Coale demonstrated in 1982, growth affects the age structure of a population in exactly the same way as mortality or any other decrement, such as migration.
Mathematical models derived from this insight are called generalized stable population models and methods of estimation and analysis based on it are termed variable growth rate methods or, more concisely, variable r methods.
Generalized stable population models provide a precise mathematical description of how age structure, age-specific growth, and age-specific vital rates are related at a particular point in time in any population. Thereby, they provide the basis for a series of methods for demographic estimation.
Whenever information is available on a population’s age structure and growth and on all the in-flows and out-flows that affect it except one, it is possible to estimate the unknown set of age-specific rates. For example, the approach can be used to estimate a population's age-specific mortality from data on the size of the population by age collected in two successive censuses.
This session covers both the theory underlying generalized stable population models and several variable r methods of estimation based on it.