On Discrete Stable Population Theory

SUMMARY In studying discrete stable population theory, appeal is made to the fact that the matrix A of birth and survival rates has a positive maximal eigenvalue, associated with which is a positive eigenvector. Although this appeal is sometimes made without justification, in other cases it has been stated that a sufficient condition for lim At to exist is that there be a 'kernel' of at least two adjacent age groups with positive fertility rates. In this paper, those properties of A which influence the limiting behavior of its powers are examined in some detail; in particular, it is shown that all such matrices are irreducible, and that a necessary and sufficient condition for A to be primitive is that the greatest common divisor of the indices of age groups at which fertility is positive be one.