*(English)*Zbl 1218.60073

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 $({P}_{1},{P}_{2},\cdots )$ 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 $({P}_{1},{P}_{2},\cdots )$ 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 $({P}_{1},{P}_{2},\cdots )$ are also described.

##### MSC:

60J80 | Branching processes |

60G55 | Point processes |

92D25 | Population dynamics (general) |

60J85 | Applications of branching processes |

60F15 | Strong limit theorems |

92D40 | Ecology |