Population-size-dependent branching processes. (1996)

• Record Type: Journal Article
• Authors: Jagers, Peter
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 9:Number 4(1996)
• Page Start: 449
• Page End: 457
• Publication Date: 1996
• Subjects: branching processes -- population dynamics -- cell kinetics; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953396000391
• Languages: English
• ISSNs: 1048-9533

Abstract

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.