A Scaling Limit Process for the Age-Reproduction Structure in a Markov Population

#### A. Bose, I. Kaj

2000, v.6, №3, 397-428

ABSTRACT

The purpose of this paper is to introduce a model for a reproducing Markov population, set within terms of a fairly general Markovian framework and with arbitrary offspring mechanism, and study the time evolution of the joint empirical distribution of age and reproduction numbers. The particular objective is to apply a diffusion approximation scaling as the population size grows and establish the existence of a non-degenerate measure-valued Markov process in the scaling limit. In general the limit process is not a superprocess. But it may be characterized by its log-Laplace function being the unique solution of a non-linear integral equation. Under additional assumptions the limit process is recognized as a superprocess, and its martingale characterization is obtained.

Keywords: measure-valued branching,log-Laplace function,diffusion approximation