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

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.

Figures (2)

• Figure 1: Coupling between two realisations of the spatial Muller's ratchet $(\eta^{(1)}(t))_{t \geq 0}$ and $(\eta^{(2)}(t))_{t \geq 0}$, started from initial configurations $\eta^{(1)}(0) = \boldsymbol{e}^{(0)}_0$ and $\eta^{(2)}(0) = \boldsymbol{e}^{(0)}_0 + \boldsymbol{e}^{(0)}_1$. Then, by letting $\tau$ denote the first time at which the type-$0$ particle dies or reproduces in either process, with positive probability either the type-$0$ particle in $\eta^{(1)}$ reproduces at time $\tau$, or the type-$0$ particle in $\eta^{(2)}$ dies at time $\tau$.
• Figure 2: Representation of some of the birth and death events in the coupling described in this section. Different colours represent different particle classes but do not distinguish between types (i.e. the number of mutations). Class $0$ particles are shown in white; class $1$ and class $1*$ particles are shown in dark and light blue, respectively; and class $2$ and class $2*$ particles are shown in red and pink, respectively. Dual pairs (a class $1*$ and a class $2*$ particle) are circled in yellow. Death events occurring exclusively in $(\eta^{n,(1)}(t))_{t \geq 0}$ are marked with dark blue crosses, while those occurring exclusively in $(\eta^{n,(2)}(t))_{t \geq 0}$ are marked with red crosses. Arrows indicate which infected particle caused a transmission event, coloured according to the class of the transmitting particle. The figure illustrates four types of events: (a) Transmission without recovery. The presence of an extra class $1$ particle increases the death rate in $\eta^{n,(2)}$ relative to $\eta^{n,(1)}$, causing a susceptible particle to die in $\eta^{n,(2)}$, but not in $\eta^{n,(1)}$. We say that the class $1$ particle 'transmits' the infection to a susceptible particle, which then becomes class $1$. (b) Transmission with partial recovery. An extra class $2$ particle increases the death rate in $\eta^{n,(2)}$ relative to $\eta^{n,(1)}$. As a result, the class $2$ particle transmits the infection to a susceptible particle, which becomes class $1$. However, a 'partial recovery' then occurs: the newly infected particle becomes class $1*$, forming a dual pair with the transmitting particle, which becomes class $2*$. (c) Death of a partially recovered particle with reinfection. The presence of an extra class $2$ particle increases the death rate in $\eta^{n,(1)}$ relative to $\eta^{n,(2)}$, causing a class $1*$ particle to die, but not its class $2*$ dual pair particle. Since there are no class $1$ particles at the deme when the death occurs, 'reinfection' takes place: the class $2*$ dual pair particle becomes class $2$. (d) Death of a partially recovered particle with replacement. Here, an extra class $2$ particle causes the death of a class $2*$ particle, but not its class $1*$ dual pair particle. One of the class $2$ particles is then chosen uniformly at random (indicated by the pink circle) to become a new class $2*$ particle, replacing the particle that died in its dual pair. We will show that transmission without recovery and reinfection events only occur at demes where the number of particles is sufficiently small (see Lemma \ref{['Paper01_lem:cutoffconseq']}).

Theorems & Definitions (73)

• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 4.1: The Stone-Weierstrass theorem
• Theorem 4.2
• proof
• Proposition 5.1
• Definition 5.2
• Lemma 5.3