An experimental study of weak ergodicity in human populations

Abstract

In demography, weak ergodicity is usually taken to mean the tendency of an age distribution to “forget” its past, or, equivalently, the essential dependence of current age distribution on only a recent history of age-specific vital rates. We have been interested in providing a more complete mathematical description of the processes underlying weak ergodicity, and decided to begin our inquiry with an experimental approach. Our initial work has been with the discrete formulation of population dynamics, using 10 × 10 matrices for closed populations of human females as experimental data. The findings of our initial experiments are that a sequence of products of human population projection matrices converges towards the sequence defined by its positive, or Perron-Frobenius, eigenstructure, and that, when appropriately defined, sequences both of left and of right eigenvectors converge to constant vectors: left eigenvector sequences converge “forward” from a fixed point in time, while right eigenvector sequences converge “backward” from a given time point. In practice, this means that proportional age structure can be determined from a knowledge only of recent or “late” vital rates, while the relative contribution of initial age structure to eventual population size can be determined with a knowledge of only “early” vital rates in the sequence. For 10 × 10 human population projection matrices, products of 30 or more matrices appear always to contain closely convergent sequences of right and left eigenvectors, and adequate approximations to limiting values usually can be made using about 20 matrices for each limiting eigenvector. Because convergence is not monotone, and because patterns of convergence are specific to the structure of each product, it appears that stronger results will require strong conditions on the sequence of matrices in the product.