The stationary and quasi-stationary properties of neutral multi-type branching process diffusions

TL;DR

This paper analyzes the stationary and quasi-stationary behaviors of multi-type branching process diffusions, providing new results for sub-critical cases and linking to Wright-Fisher models.

Contribution

It derives the first-order quasi-stationary distribution for sub-critical processes and connects these results to Wright-Fisher sampling distributions.

Findings

Quasi-stationary distributions collapse onto eigenvector rays in supercritical and critical cases.

New first-order approximation for sub-critical quasi-stationary distribution.

Sampling distribution matches Wright-Fisher diffusion to first order in mutation rates.

Abstract

The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Pedagogical derivations are given for known results that the limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with discrete multi-type branching processes. For the sub-critical process the previously unsolved quasi-stationary distribution is obtained to first order in the overall mutation rate, which is assumed to be small. The sampling distribution over allele types for a sample of given finite size is found to agree to first order in mutation rates with the analogous sampling distribution for a Wright-Fisher diffusion with constant population size.