The rate of convergence of a generalized stable population |
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Authors: | Marc Artzrouni |
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Institution: | (1) Department of Mathematical Sciences, Clemson University, 29634-1907 Clemson, SC, USA |
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Abstract: | In an age-structured population that grows exponentially, each age groupP
i(t) at periodt is asymptotically equivalent tox
0
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 vi(t)=Pi(t)/pi(t–1) converges tox
0. The age specific rate of convergence is determined by considering a quantityr satisfyingv
i(t)-x
0
¦ 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 tox
0;S
i
=- In 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 contextS
0 is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability. |
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Keywords: | Exponential growth Stable age distribution Generalized stable population Rate of convergence R-factor Population entropy |
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