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 . 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 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 , the vector converges to the GEM distribution with parameter . In models II and III, arriving types are drawn in a discrete or in a continuous spectrum, respectively. The limits of the vectors are also described.