# Age structure and the stable age distribution

In the next sections, we are ready to begin examining the interactions between the age distribution of a population and its life table. As stated previously, the actual age distribution of a population has potentially dramatic effects on population growth in the short term. The age distribution of a population is defined as the proportions of the population belonging to various age categories at a given point in time. The proportion belonging to a given age category, x, is calculated by dividing the number of individuals in that age category by the total population size, N, producing cx:

N V nx cx is the proportion of the population belonging to an age category, x, and nx equals the number of individuals in that age category.

Whenever survivorship and fertility remain constant for long enough, a population will converge on a particular age distribution, known as the stable age distribution, which is unique for each combination of survivorship and fertility. Once this stable age distribution is achieved, the age distribution no longer changes unless and until survivorship or fertility change in the life table. Furthermore, the population will grow or decline at the steady rate, X (unless the r-value = 0, in which case the population is unchanging and X = 1), and each age class will change at the same rate as the population as a whole. If a population has a stable age distribution, X is easy to calculate, since Nt+1/Nt = X. Since r = ln X (Eqn. 1.13), it is also simple to calculate r.

The stable age distribution itself can be calculated from the survivorship column of the life table. In order to predict the stable age distribution, however, it is also necessary to know the value of r as well as the survivorship function, lx. Since the Euler equation requires the knowledge of fertility (mx), we actually must know both survivorship and fertility. The formula for predicting the stable age distribution is as follows:

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