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The solution of time-dependent population models

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  • Nan Li
  • Shripad Tuljapurkar

Abstract

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
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    Citations

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    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. Robert Schoen, 2009. "The metastable birth trajectory," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 21(25), pages 759-764.
    3. Feichtinger, Gustav & Wrzaczek, Stefan, 2024. "The optimal momentum of population growth and decline," Theoretical Population Biology, Elsevier, vol. 155(C), pages 51-66.
    4. Robert Schoen, 2005. "Intrinsically dynamic population models," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 12(3), pages 51-76.
    5. 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.
    6. David N. Koons & Randall Holmes & James B. Grand, 2006. "Population inertia and its sensitivity to changes in vital rates or initial conditions," MPIDR Working Papers WP-2006-040, Max Planck Institute for Demographic Research, Rostock, Germany.

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