A net reproductive number for periodic matrix models

Abstract

We give a definition of a net reproductive number R 0 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 R 0 in the autonomous case. The value of R 0 determines whether the population goes extinct (R 0<1) or persists (R 0>1). We discuss the biological interpretation of this definition and derive formulas for R 0 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 R 0 with another definition given recently by Bacaër.