Peter Jagers^1,2

Copyright © 1996 Peter Jagers. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In a recent paper [7] a coupling method was used to show that if population size, or more generally population history, influence upon individual reproduction in growing, branching-style populations disappears after some random time, then the classical Malthusian properties of exponential growth and stabilization of composition persist. While this seems self-evident, as stated, it is interesting that it leads to neat criteria via a direct Borel-Cantelli argument: If m(n) is the expected number of children of an individual in an n-size population and m(n)≥m>1, then essentially ∑n=1∞{m(n)−m}<∞ suffices to guarantee Malthusian behavior with the same parameter as a limiting independent-individual process with expected offspring number m. (For simplicity the criterion is stated for the single-type case here.)

However, this is not as strong as the results known for the special cases of Galton-Watson processes [10], Markov branching [13], and a binary splitting tumor model [2], which all require only something like ∑n=1∞{m(n)−m}/n<∞.

This note studies such latter criteria more generally. It is dedicated to the memory of Roland L. Dobrushin.