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Population growth and replacement
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701
The interaction of fertility and mortality leads to a consideration of population growth^{1}. Itis convenient to regard a population decline^{2} as negative growth^{3}. A distinction may be drawn between a closed population^{4} in which there is no migration either inwards or outwards and whose growth depends entirely on the difference between births and deaths, and an open population^{5} in which there may be migration. The total growth^{6} of an open population consists of the balance of migration (cf. chapter 8) and natural growth^{7} or natural increase^{7} which is the excess of births over deaths^{8}, sometimes called the balance of births and deaths^{8}.
702
The ratio of total growth in, a given period to the mean population of that period is called the crude rate of increase^{1}. Occasionally this rate is computed with the population at the beginning of the period rather than, with the mean population as a denominator. When population increase over a period of more than one calendar year is studied, the mean annual rate of increase^{2} may be computed. In computing this rate it is sometimes assumed that the population is growing in accordance with the exponential law of growth during the period; such a population is called an exponential population^{3}. Alternatively the total growth may be divided by the number of years and this average growth related to the mean population of the period. The ratio of natural increase (7017) to the average population is called the crude rate of natural increase^{4}. The ratio of the crude birth rate (6301) to the less crude death rate (4014) is called the vital index^{5}, but this measure is no longer much used.
703
It can be shown that when a closed population (7014) is subjected to constant agespecific fertility and mortality rates (6318; 4121) for a sufficiently long period of time, its annual rate of increase will tend to become constant. This constant rate of increase is called the intrinsic rate of natural increase^{1} or true rate of natural increase^{1}, and a population .which has reached this stage is called a stable population^{2}. The proportion of persons in different age groups in such a population will be constant, i.e., the population will have a stable age distribution^{3} which is independent of the initial age distribution^{4} and depends only on the fertility and mortality rates that obtained. Human populations never reach stability in practice, as fertility and mortality rates constantly change, but the computation of a stable population as a model may serve as an index of the growth potential^{5} of a set of agespecific fertility and mortality rates. A stable population in which the rate of natural increase is zero is called a stationary population^{6}, in such a population the numbers in a given age group are equal to the integral of the survivorship function (4313) of the life table taken between the upper and lower age limits of the group. A logistic population^{7} growing in accordance with the logistic law of growth is one in which the rate of growth decreases as a linear function of the population already alive and which will tend asymptotically to an upper limit.
 2. The crude birth and death rates of a stable population, are called the Stable birth rate and the stable death rate respectively.
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