PAPP103 - S10: Models of populations with variable growth rates

Application 1 (cont.)

To calculate age-specific growth rates for a population, one needs two cross-sectional counts of the population so that one can measure growth in between them. Moreover, one can only measure growth for the age groups on which the data are collected, not for exact ages. Thus, for empirical work, r_a must be redefined as the growth rate in each age group and calculated from the change in the size of the population aged a between the two enumerations. If the enumerations take place h years apart:

$$r_a=\frac{\ln {{\displaystyle P}}_a(t+h)-\ln {{\displaystyle P}}_a(t)}{h}$$

To be consistent with the assumption of constant exponential growth, the adjustment is made to the geometrical average of the two counts of the population from which the growth rates are calculated:

$$P_x(\overline{t})=\sqrt{P_x(t+h)\cdot {}_{}P{}_x(t)}$$

Once one has calculated both the age-specific growth rates and the average population in the period between the two counts of the population:

$$\frac{L_x}{l_0}=\frac{P_x\left(\overline{t}\right)}{I}{e}^{{\displaystyle \sum }_{a=0}^{x-1}{r}_a+0.5r_x}$$

Note that, as the age-specific growth rates in the exponential term used to adjust the average population now refer to age groups falling between x and x+1, they are cumulated from birth up to the mid-point of each age group.