# R_0 in continuous-time ecological models structured by age and space

## Abstract

In this work we study the asymptotic behaviour in linear models of population dynamics by means of the basic reproduction number R_0. Our aim is to give a practical approach to the computation of the reproduction number in continuous-time population models structured by age and/or space. The traditional approach to the study of linear continuous-time population dynamics is the computation of the*Malthusian parameter*, i.e. the exponential growth rate of the population. Yet, another equivalent approach is possible which takes the generational viewpoint, [2], [3], [5]. For each system, firstly one has to distinguish between birth terms and the other ones like mortality and transition terms. Then, the

*basic reproduction number*is computed as the spectral radius of the next-generation operator. However, different interpretations of what is a birth event give rise to different expressions and results, [2] and [1].В For infinite-dimensional systems (e.g. PDE), the second approach is always related to an eigenvalue problem with its corresponding eigenfunction [4] whereas, in general, this is not the case for the first approach.

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