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 characterise
the asymptotic large time behaviour. Next we explicitly elaborate the
relationship between the two bookkeeping schemes. This allows us to transfer
the characterisation of the large time behaviour to traditional
structured-population models