206
WILLIAM STREIFER
Bailey’s model and the partial differential equation utilize a function of
age and time, ~ ( a ,
t ) . I assume throughout the paper that age a and
time t are measured in the same units. The function ~ ( a ,
t ) is variously
called a density or distribution function and is not to be confused with
the density of individuals occurring within a specified physical area.
Rather, the density function has the property that
J a ;
t)da
(11)
is the total number of individuals in the population between ages a, and
a2 at time t . It is important to understand that q(a, t ) is not itself equal
to a number of individuals; only its integral can be interpreted thus.
The units of ~ ( a ,
t ) are number of individuals per unit of age. A sample
density function at a particular instant of time is shown in Fig. 1. The
FIG. 1. A sample age distribution function at a particular time t. The shaded
area equals the number of individuals between ages a1 and a2; the total area
under the curve equals the total population.
shaded area under the curve between a, and a2 is equal to the integral;
the total area under the curve is equal to the total population, N ( t ) , and
is given by
N ( t ) = 1, ??(a, t)da
(12)
The function ~ ( a ,
t ) satisfies the partial differential equation (Von
Foerster, 1959)
-+- 817 arl = - D(a, t)q(a, t )
at aa
(13)
where D(a, t ) is the death rate for individuals of age a at time t. The
death rate for the total population is