On the theory of stable populations: A new and elementary proof of the theorems under weaker assumptions

Abstract

Expositions and elementary proofs are given for the basic theorems of stable population theory: That a population subjected to vital rates (not necessarily constant over time) satisfying certain postulates will eventually “forget” its original age distribution and take on one (not necessarily constant over time) which depends only on its history of agespecific vital rates, a process called “weak ergodicity.” That consequently the subsequent birth, death, and growth rates (none of these necessarily constant over time) depend only on the history of age-specific vital rates and not on the original age distribution. And, in particular, that these results apply to the special case, herein called “classic” stable population theory, in which the age-specific vital rates are constant over time, and in which after the “forgetting” takes place the subsequent age distribution and birth, death, and growth rates all become constant. This formulation of the theory differs from previous ones in two respects: First, the postulates required are weaker, and hence the theorems more general, than previously; in particular, this formulation permits the highest age of childbearing to change from cohort to cohort, which is important for populations practicing contraception. Second, none of the advanced mathematics used in previous formulations is needed; only the manipulation of sums and inequalities from high school algebra and the concept of “limit” from freshman calculus are required.

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Correspondence to David D. McFarland.

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McFarland, D.D. On the theory of stable populations: A new and elementary proof of the theorems under weaker assumptions. Demography 6, 301–322 (1969). https://doi.org/10.2307/2060399

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Keywords

  • Vital Rate
  • Survival Ratio
  • Crude Rate
  • Remote Past
  • Weak Ergodicity