Population growth in discrete time: a renewal equation oriented survey

B. Boldin, O. Diekmann*, J. A.J. Metz

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.

Original languageEnglish
JournalJournal of Difference Equations and Applications
Early online date13 Oct 2023
Publication statusE-pub ahead of print - 13 Oct 2023


  • basic reproduction number
  • growth rate
  • next-generation matrix
  • renewal equation
  • reproductive value
  • stable distribution
  • Structured-population model


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