A net reproductive number for periodic matrix models

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.