Macroscopic behavior of heterogenous populations with fast random life histories

In this paper, we consider the large population limit of an age and characteristic-structured stochastic population model evolving according to individual birth, death and fast characteristics changes during life. Both the large population framework and the fast characteristics changes assumption are motivated by demographic patterns of human populations at the scale of a given country. When rescaling the population process, and under some invariance assumption about the characteristics changes dynamics, the classical determin-istic transport-renewal McKendrick-Von Foerster equation appears, that describes the time evolution of the age pyramid driven by equivalent birth and death rates. The proof follows the work of Méléard and Tran (2012) and Gupta et al. (2014) in which analogous mathematical issues are encountered. We further prove that the sequence of processes taking track of the characteristics distribution is not tight even in the presence of age-independent demographic rates. To illustrate the use of the limiting model, a set of computable invariant distributions is given, as well as numerical implementation of equivalent birth and death rates which mimics real demographic data. These results highlight the fact that characteristics changes frequencies are crucial to understand aggregate demographic rates at the macroscopic scale