Volume 12 - Article 3 | Pages 51–76
Intrinsically dynamic population models
|Date received:||07 Jun 2004|
|Date published:||10 Mar 2005|
|Keywords:||dynamic models, dynamic population models, eigenvalues, Leslie matrices, population momentum|
|Updated Items:||Erratum: Varrious small mathematical corrections were made on pages 60, 61, 64, and 65 on March 10, 2006.|
Intrinsically dynamic models (IDMs) depict populations whose cumulative growth rate over a number of intervals equals the product of the long term growth rates (that is the dominant roots or dominant eigenvalues) associated with each of those intervals. Here the focus is on the birth trajectory produced by a sequence of population projection (Leslie) matrices.
The elements of a Leslie matrix are represented as straightforward functions of the roots of the matrix, and new relationships are presented linking the roots of a matrix to its Net Reproduction Rate and stable mean age of childbearing. Incorporating mortality changes in the rates of reproduction yields an IDM when the subordinate roots are held constant over time. In IDMs, the birth trajectory generated by any specified sequence of Leslie matrices can be found analytically.
In the Leslie model with 15 year age groups, the constant subordinate root assumption leads to reasonable changes in the age pattern of fertility, and equations (27) and (30) provide the population size and structure that result from changing levels of net reproduction. IDMs generalize the fixed rate stable population model. They can characterize any observed population, and can provide new insights into dynamic demographic behavior, including the momentum associated with gradual or irregular paths to zero growth.
Robert Schoen - Pennsylvania State University, United States of America
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