A multispecies birth-death-immigration process and its diffusion approximation

TL;DR

This paper introduces a complex multispecies birth-death-immigration process on a lattice, analyzes its behavior using generating functions, and explores its diffusion approximation, revealing new mathematical connections and distributions.

Contribution

It extends birth-death-immigration models to a multi-ray lattice and derives new distributions, diffusion approximations, and mathematical series related to hypergeometric functions.

Findings

Stationary distribution is a zero-modified negative binomial.

Diffusion process has a gamma-type transient density with a stationary limit.

Derived a closed form for permutations with fixed components and a new series for polylogarithm.

Abstract

We consider an extended birth-death-immigration process defined on a lattice formed by the integers of $d$ semiaxes joined at the origin. When the process reaches the origin, then it may jumps toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth-death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process, which involves a diffusion process with linear drift and infinitesimal variance on each ray. It possesses a gamma-type transient density admitting a stationary limit. As a byproduct of our study, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function expressed in terms of the Gauss hypergeometric function.