Existence, uniqueness and moment bounds for a spatial model of Muller's ratchet

TL;DR

This paper extends a spatial Muller's ratchet model to infinite populations, establishing existence, uniqueness, and moment bounds, which are key for proving a law of large numbers in this complex, non-local interaction system.

Contribution

It introduces a construction for an infinite-particle spatial Muller's ratchet model and proves moment bounds and uniqueness despite non-monotonic interactions.

Findings

Constructed the particle system for infinite initial populations.

Established moment bounds on local particle density.

Proved weak convergence and uniqueness of the process.

Abstract

In this article, we consider a generalisation of the spatial Muller's ratchet introduced by Foutel-Rodier and Etheridge. This particle system is a spatial model of an asexual population, with birth and death rates that depend on the local population density. Particles live in discrete demes and migrate to neighbouring demes. Each particle carries some number of mutations (its `type'), and additional mutations can occur during birth events. Mutations are assumed to be deleterious, i.e.~carrying a higher number of mutations results in a lower birth rate. Our main result shows that this interacting particle system can be constructed even when the total initial number of particles is infinite. We also prove moment bounds on the local density of particles; these bounds are a crucial ingredient of the proof of a law of large numbers result for the particle system in the companion article. The construction of the particle system uses a sequence of approximating processes. Proving weak convergence of this sequence of processes is non-trivial because the particle system is non-monotone and interactions are non-local in type space. The uniqueness of the limit relies on a delicate coupling argument.