PAPP103 - S09: Models of populations with variable growth rates

Generalization to open populations

So far in this session, we have derived a set of relationships between the current age structure, the vital rates and the age-specific growth rates of any population. The basis for further generalization of the approach becomes apparent if we express L_x in terms of the age-specific mortality rates that determine it. For single-year age groups, by definition:

$$L_x=e^{-{\displaystyle \sum }_{a=0}^{x-1}{m}_a-0.5m_x}$$

where m_a is the death rate at age a.

Substituting this expression for L_x into the equation for the population aged x gives:

$$P_x=\frac{B_0}{l_0}{e}^{-{\displaystyle \sum }_{a=0}^{x-1}{r}_a-0.5r_x}{e}^{-{\displaystyle \sum }_{a=0}^{x-1}{m}_a-0.5m_x}$$

Thus, generalized stable population theory demonstrates that growth acts on the age structure of a population in the same way as a decrement such as mortality. Multiplying current births by both an adjustment for mortality up to age x and an adjustment for growth up to age x yields the population aged x.

Any other decrement that affects a population can be treated in the same way as mortality and growth. For example, Preston and Coale (1982) demonstrate that in an open population, where e_a is the net emigration rate at age a:

$$P_x=\frac{B_0}{l_0}{e}^{-{\displaystyle \sum }_{a=0}^{x-1}{r}_a-0.5r_x}{e}^{-{\displaystyle \sum }_{a=0}^{x-1}{m}_a-0.5m_x}{e}^{-{\displaystyle \sum }_{a=0}^{x-1}{e}_a-0.5e_x}$$

This makes intuitive sense: the demographic impact on a population of someone leaving it is the same no matter whether they emigrate or die. Thus, all the generalized stable population relationships defined already for closed populations also hold in any open population as long as one adds the age-specific net emigration rates of the population to its age-specific growth rates before adjusting for the latter.

As an example, consider the estimation of mortality from two census enumerations. If the population is subject to significant net inflows or outflows of migrants but net emigration rates ₅e_a can be estimated from other sources of data on the population, one can add them to the age-specific growth rates before adjusting the actual population to get its stationary equivalent. Thus, assuming we are working with five-year age groups ₅P_x, rather than single-year data, the life table can be estimated from the population by age by adjusting for both growth and net migration:

$$\frac{{}_5{{\displaystyle L}}_x}{l_0}=\frac{{}_5{{\displaystyle P}}_x}{B_0}{e}^{5{\displaystyle \sum _{a=0}^{x-5}({}_{5}r{}_a+{}_{5}e{}_a)+2.5({}_5{{\displaystyle r}}_x+{}_5{{\displaystyle e}}_x)}}$$