PAPP103 - S07: Demographic models: stable population models

Approximate estimates of stable population growth rate based on generation length

The iterative method is the most accurate way of obtaining an exact solution of Lotka’s equation, and the method of moments usually produces accurate results. Both require extended calculations, and here we examine an alternative, simpler approach for estimating the growth rate, based on various approximations for generation length.

The generation length method makes use of the relationship

$r = \frac{ln (N R R)}{G}$

and allows us to use different assumptions about the value of G, the generation length, based on the shape of the age-specific schedules of fertility (f_x) or reproductivity (f_x l_x) or childbearing (B_x).

The following tables show the range of values produced for r using accurate and approximate methods. All estimates use the data from Kuwait, 1982.

Accurate methods:	r	G
Iteration	0.028662	28.59
Method of Moments	0.028682	28.59

Approximations using estimates of generation length:	G	r	*% error in r*
Reasonable guess	27	0.031147	8.7%
Mean age of *f_x* the fertility schedule – 1	29.22	0.028783	0.4%
Mean age of *f_x* *l_x* reproductivity	30.11	0.02793	2.5%
Mean age of *B_x* childbearing in an observed population	28.64	0.02936	2.4%

The method of moments and iterative method produce very similar values for r. These accurate, methods show us the true value of G, based on the relationship:

$G = \frac{ln (N R R)}{r}$

The estimates of growth rate based on different approximations for generation length show greater variation. In situations where we suspect the accuracy of the available fertility and mortality data it is not worth calculating r using more accurate methods, we can obtain a reasonably close estimate using estimates based on generation length – a quick, “back-of-the-envelope” way of calculating r.