Birth-death and immigration models, with mutations and carrying capacity.

Stochastic models that incorporate birth, death and immigration (also called birth-death and innovation models) are quite general and are used in different application areas such as the description species sizes in ecological population, gene family size, lymphocyte evolution in the body, and the evolution of business firm sizes. These models allow the immigration of new species into the system. These individuals then undergo birth and death process. We review and present a number of new results for the high-dimensional steady-state solutions to these types of models. Models that include carrying capacity and random mutations are treated and we find exact statistical descriptions for the total number of individuals, the total number of species, the species size distribution and various diversity indices. We also present a rigorous analysis of the behavior of these systems in the fast immigration limit.