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Advances in Applied Probability 43, 1 (2011) 276-300

Limit theorems for splitting trees with structured immigration and applications to biogeography

Mathieu Richard 1

(2011-03)

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate $\theta$, 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 i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate $b$. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector $(P_1,P_2,\dots)$ of relative abundances of surviving families converges a.s. In the first model, the limit is the GEM distribution with parameter $\theta/b$.

1:	Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
	CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot

Subject

:

Mathematics/Probability

Keyword(s): Splitting tree – Crump-Mode-Jagers process – Spine decomposition – Immigration – Structured population – GEM distribution – Biogeography – Almost-sure limit theorem

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From: Mathieu Richard <>

Submitted on: Friday, 30 July 2010 13:27:38

Updated on: Saturday, 11 June 2011 20:48:30