Population growth in discrete time: a renewal equation oriented survey

Traditionally, population models distinguish individuals on the basisof their current state. Given a distribution, a discrete time modelthen specifies (precisely in deterministic models, probabilistically instochastic models) the population distribution at the next time point.The renewal equation alternative concentrates on newborn individ-uals and the model specifies the production of offspring as a functionof age. This has two advantages: (i) as a rule, there are far fewer birthstates than individual states in general, so the dimension is oftenlow; (ii) it relates seamlessly to the next-generation matrix and thebasic reproduction number. Here we start from the renewal equationfor the births and use results of Feller and Thieme to characterizethe asymptotic large time behaviour. Next we explicitly elaboratethe relationship between the two bookkeeping schemes. This allowsus to transfer the characterization of the large time behaviour totraditional structured-population models.

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