Representation and approximation of large population age distributions using poisson random measures

We study the asymptotics of a class of age structured population models when the initial population is larger and larger and the reproduction regime is properly scaled. Results of Kurtz and Wang are generalised, mainly with extensions of ideas originating from Kurtz. Newborn individuals have life-spans which are i.i.d. and independent of everything else. The initial populations are supposed to have age structures as well as residual life-span distributions that stabilise asymptotically. The infinitesimal probabilities of reproduction of a brood of size k at time t are supposed to depend smoothly on the age structure among living individuals at all timepoints preceding t. The 'age structure' is then thought of as an object which is scaled by a parameter of the order of the size of the initial population. By a multidimensional random clock argument, it is possible to relate such a population to a countable set of time homogeneous Poisson processes, one for each brood of size k, with points that carry k independent life-span marks for the siblings corresponding to it. The equations expressing this relation are quite complicated. First, in theorems 1 and 1', law-of-large-numbers arguments applied to the mentioned Poisson processes yield deterministic equations for an age distribution evolution, which approximates the actual random one asymptotically. This is done in great generality, and restrictions assumed are natural. Under the assumption that no brood sizes are larger than some k0 and some further regularity, then white noise type approximations of the marked Poisson processes are demonstrated to be relevant for the actual fluctuations of the age-structure around its limiting stable evolution.