A two-type Bellman--Harris process initiated by a large number of particles

2013/11/05 by Vladimir Vatutin, Vatutin, Vladimir, Alexander Iksanov +4
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Diffusion and Search Dynamics #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1311.1060

submitted, 42 pages

arxiv created 2013/11/05 · openalex publication_date 2013/11/05 · arxiv updated 2013/11/06 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We investigate a two-type critical Bellman--Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order o(t^-2); the tail of the life-length distribution of the second type particles is regularly varying at infinity with index -β, β∈ (0,1]; at time t=0 the process starts with a large number N of the second type particles and no particles of the first type. It is shown that the time axis 0≤ t<∞ splits into several regions whose ranges depend on β and the ratio N/t within each of which the process at time t exhibits asymptotics (as N,t→ ∞) which is different from those in the other regions.

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