Population-size-dependent branching processes
Summary: In a recent paper [the author, ``Coupling and population dependence in branching processes'' (to appear)] 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 \(m(n)\geq m>1\), then essentially \(\sum^\infty_{n=1}\{m(n)- m\}<\infty\) 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 [\textit{F. C. Klebaner}, J. Appl. Probab. 21, 40-49 (1984; Zbl 0544.60073)], Markov branching [\textit{F. C. Klebauer}, ibid. 31, No. 3, 614-625 (1994; Zbl 0819.60064)], and a binary splitting tumor model [\textit{M. Gyllenberg} and \textit{G. F. Webb}, J. Math. Biol. 28, No. 6, 671-694 (1990; Zbl 0744.92026)], which all require only something like \(\sum^\infty_{n=1}\{m(n)= m\}/n<\infty\). This note studies such latter criteria more generally.
scientific article; zbMATH DE number 988083