A functional law of large numbers for a spatial model of Muller's ratchet

Abstract

The spatial Muller's ratchet is a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population. The model is an interacting particle system consisting of particles performing symmetric random walks that reproduce and die with rates that depend on the local number of particles. For each particle, we keep track of the number of deleterious mutations that it carries, and after each birth event, with some positive probability, the offspring particle can acquire an additional mutation that gives it a lower reproduction rate than its parent. We show that, under an appropriate scaling, the process converges weakly to the solution of an infinite system of partial differential equations (PDEs), confirming non-rigorous computations of Foutel-Rodier and Etheridge. Combining the weak convergence with analytical results for the limiting PDE system, we derive quantitative lower and upper bounds on the proportion of particles with mutations that hold with high probability for the particle system. A key obstacle is the absence of uniform bounds on the number of particles per site, together with the presence of infinitely many types of particle and the nonlinear interactions. To address this, we establish a new tightness criterion for interacting particle systems in general Lp spaces based only on local properties of the dynamics.