The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age group pi(t) at period t is asymptotically equivalent to x0t for some positive number x0. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio vi(t) = pi(t)/pi(t-1) converges to x0. The age specific rate of convergence is determined by considering a quantity r satisfying [vi(t)-x0] less than or equal to rt when t is large; Ri = Inf r (over all initial populations, r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequence vi(t) converges to x0; Si = -1n Ri is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this context S0 is compared to the population entropy H, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.