Exact steady-state distributions of multispecies birth-death-immigration processes: effects of mutations and carrying capacity on diversity

TL;DR

This paper derives exact and asymptotic steady-state distributions for multispecies birth-death-immigration models, analyzing effects of mutations and carrying capacity on diversity, with implications for ecological and evolutionary systems.

Contribution

It provides the first full high-dimensional stochastic process analysis and exact solutions for models including carrying capacity and mutations, advancing understanding of species diversity.

Findings

Species richness best distinguishes different models in the fast immigration limit.

Carrying capacity through birth rate breaks detailed balance in neutral models.

Analytic and approximate results for diversity indices and species distributions.

Abstract

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.