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,...) of relative
abundances of surviving families converges a.s. In the first model, the limit
is the GEM distribution with parameter theta/b.Comment: 24 pages, 3 figure