The intrinsic rate of natural increase and the dominant root of the projection matrix

Abstract

The intrinsic rate of natural increase, introduced by A. J. Lotka, is that which would ultimately be attained by a population under a fixed regime of survivorship and fertility. It is one of the ways of judging and comparing the implications of age-specific rates as among countries and among moments of time. A number of ways are available for calculating the rate, whose interrelations throw light on the process by which a population grows. Eleven different ways are described in the paper; it is possible to group them into two families, the members of one being related to the projection of a population through time, the other being associated with Lotka's integral equation. Association of the intrinsic rate with the projection matrix suggests its calculation through the latent roots of the matrix; not onlythe dominant root which gives the intrinsic rate is meaningful in demographic terms, but other roots as well. Slight differences in numerical results are analysed as due to different finite approximations to essentially the same continuous process. A specimen of the computer programme by which the methods were applied is included, and actual results of computation are shown for 33 countries, intrinsic rates of birth, death, and natural increase being given.