IDEAS home Printed from
   My bibliography  Save this article

The solution of time-dependent population models


  • Nan Li
  • Shripad Tuljapurkar


We analyze the dynamics of age-structured population renewal when vital rates make a transition in a finite time interval from arbitrary initial values to any specified final values. The general solution to the renewal equation in such cases is obtained. This solution describes the birth sequence explicitly, and also leads to a general formula for population momentum. We show that the duration of the transition determines the complexity of the solution for the birth sequence. For transitions that are completed in a time smaller than the maximum age of reproduction, we show that the classical Lotka solution found in every textbook also applies, with a small modification, to the time-dependent case. Our results substantially extend previous work that has often focused on instantaneous transitions or on slow and infinitely persistent change in vital rates.

Suggested Citation

  • Nan Li & Shripad Tuljapurkar, 2000. "The solution of time-dependent population models," Mathematical Population Studies, Taylor & Francis Journals, vol. 7(4), pages 311-329.
  • Handle: RePEc:taf:mpopst:v:7:y:2000:i:4:p:311-329
    DOI: 10.1080/08898480009525464

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Booth, Heather, 2006. "Demographic forecasting: 1980 to 2005 in review," International Journal of Forecasting, Elsevier, vol. 22(3), pages 547-581.
    2. Thomas Espenshade & Analia Olgiati & Simon Levin, 2011. "On Nonstable and Stable Population Momentum," Demography, Springer;Population Association of America (PAA), vol. 48(4), pages 1581-1599, November.


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:mpopst:v:7:y:2000:i:4:p:311-329. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Chris Longhurst). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.