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Population-size-dependent branching processes. (English) Zbl 0867.60057

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 [F. C. Klebaner, J. Appl. Probab. 21, 40-49 (1984; Zbl 0544.60073)], Markov branching [F. C. Klebauer, ibid. 31, No. 3, 614-625 (1994; Zbl 0819.60064)], and a binary splitting tumor model [M. Gyllenberg and 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.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F25 \(L^p\)-limit theorems
92D25 Population dynamics (general)
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