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

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 $\theta/b$.