  Stochastic models that incorporate birth, death and immigration (also called
birth-death and innovation models) are ubiquitous and applicable to many
research topics such as quantifying species sizes in ecological populations,
describing gene family sizes, modeling lymphocyte evolution in the body, and
modeling the evolution of firm sizes. Many of these applications involve the
immigration of new species into the system. We develop the full
high-dimensional stochastic process associated with multispecies
birth-death-immigration processes and present a number of exact and asymptotic
results for the steady-state solutions to these types of processes. We
considered models that include carrying capacity and random mutations and find
analytic and approximate results for the statistics of the total number of
individuals, the total number of species, the species size distribution, and
various diversity indices. Our results include a rigorous analysis of the
behavior of these systems in the fast immigration limit which shows that of the
different diversity indices, the species richness is best able to distinguish
different types of birth-death-immigration models. Surprisingly, we also find
that detailed balance is preserved in all of our neutral
birth-death-immigration models, except for those that incorporate carrying
capacity through the birth rate.

Populations and Evolution;Probability