A net reproductive number for periodic matrix models

J. M. Cushing, A. S. Ackleh

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


We give a definition of a net reproductive number R0 for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing-Zhou definition of R0 in the autonomous case. The value of R0 determines whether the population goes extinct (R0<1) or persists (R0>1). We discuss the biological interpretation of this definition and derive formulas for R0 for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R0 with another definition given recently by Bacaër.

Original languageEnglish (US)
Pages (from-to)166-188
Number of pages23
JournalJournal of biological dynamics
Issue number2
StatePublished - Mar 2012

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Ecology


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