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Dynamic population models. (English) Zbl 1151.91007

The Springer Series on Demographic Methods and Population Analysis 17. Dordrecht: Springer (ISBN 978-1-4020-5229-3/hbk). xv, 251 p. (2006).
This monograph is both survey and textbook. With over 100 end of chapter exercises (some with solutions in Appendix B) and a very clear style, it is accessible to graduate students in demography with a good background in linear algebra and differential and difference equations. For experts, it provides a fine survey of the most used mathematical models in demography, with many practical applications.
The 9 chapters fall into 3 main modules. Module 1, chapters 1 through 4, covers basic models with constant rates (mortality, birth, growth). One way to think about this material is in terms of the Leslie matrix, \(A\), which contains all that information in an \(m\)-matrix. Letting \(x\) represent that distributional state of the population, and \(n\) be the maximum age at reproduction, one gets the dynamic system \(x(t+n)= Ax(t)\).
Results concerning the existence of a stationary state, or a dynamic steady state, then follow from the properties of \(A\). An insightful application of the above system is Singapore.
Module 2, chapters 5 and 6, looks at the comparative statics and dynamics of changes in longevity, fertility and divorce, still within the constant rates framework – although now those rates may be affected by one time shocks. This leads to tractable models of demographic transition. In terms of the above equation, think of \(m\) and \(n\) changing, all well as many entries of the matrix \(A\) itself. An insightful application of this material is for Germany, Austria, and Switzerland.
The third module, 7 through 9, drops the constant rate assumption. Here, either there are multiple-age models with rates by age, or multiple-state models, with rates being state-dependent, or both. These models are computationally quite intensive, and indeed raise open questions. For instance, it is not known whether there in general exists a steady state for the multi-age, multi-state model.
I learned a lot from reading this book, and if you are interested in mathematical demography, you will too. With index and over 200 references.

MSC:

91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91D20 Mathematical geography and demography
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