PAPP103 - S09: Models of populations with variable growth rates

Summary

Generalized stable population relationships provide a theoretical account of how a population’s age structure, growth and vital rates are interrelated at any point in time.
Using notation established already, the age structure of any population is a function of current births, growth, net emigration and mortality:

P_{x} = B_{0} e^{- \sum_{a = 0}^{x - 1} r_{a} - 0.5 r_{x}} e^{- \sum_{a = 0}^{x - 1} e_{a} - 0.5 e_{x}} \frac{L_{x}}{l_{0}}

This generalization of stable population theory provides the basis for a range of methods for demographic estimation. These methods are used to estimate life tables for mortality and other decrements that remove members of a population from that population.
Dividing prevalence by incidence to calculate expected time spent as a member of a population produces biased estimates in any population that is not stationary. In all other populations, an adjustment must be made for growth to obtain an unbiased estimate.
All such variable growth rate methods require two successive sets of data on the population of interest, so that one can calculate its age-specific rates of growth.
Variable growth rate methods are particularly valuable when statistics are lacking on either the event of interest or the population at risk, so that one cannot calculate a life table starting from the duration-specific rates of exit from the population.
The basis of the methods is to make an adjustment for age-specific growth that either converts the population’s actual age structure into the equivalent stationary age structure, L_x, or converts counts of one form of exit from the population by age into the equivalent life table counts of events, d_x. Once either of these columns of a life table has been estimated, the other columns of the life table can be calculated from it.