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Zbl 1218.60073
Richard, M.
Limit theorems for supercritical age-dependent branching processes with neutral immigration.
(English)
[J] Adv. Appl. Probab. 43, No. 1, 276-300 (2011). ISSN 0001-8678

The author considers models of branching processes with Poissonian immigration where individuals have inheritable types. New individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rate $b$. First, using spine decomposition, the author relaxes previously known assumptions required for almost-sure convergence of the total population size. In the paper, three models of structured populations (i.e., populations where individuals have certain types) are considered. In each model, the vector $(P_{1}, P_{2},\dots )$ of relative abundances of surviving families converges almost surely. In model I, all immigrants have a different type (not present in the current population). If they arrive at rate $\theta$, the vector $(P_{1}, P_{2},\dots )$ converges to the GEM distribution with parameter $\theta / b$. In models II and III, arriving types are drawn in a discrete or in a continuous spectrum, respectively. The limits of the vectors $(P_{1}, P_{2},\dots )$ are also described.
[Leonid Petrov (Moskva)]
MSC 2000:
*60J80 Branching processes
60G55 Point processes
92D25 Population dynamics
60J85 Appl. of branching processes
60F15 Strong limit theorems
92D40 Ecology

Keywords: branching processes with immigration; mainland-island model; splitting tree; crump-mode-jagers process; spine decomposition; structured population; GEM distribution; biogeography; almost-sure limit theorem

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