The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP_i(t) at periodt is asymptotically equivalent tox₀^t 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 v_i(t)=P_i(t)/p_i(t–1) converges tox₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv_i(t)-x₀¦ r^t whent is large;R_i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev_i(t) converges tox₀;S_i=- In R_i 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 contextS₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.