References & Citations
Mathematics > Probability
Title: Reduction of the effective population size in a branching particle system in the moderate mutation-selection regime
(Submitted on 26 Apr 2024 (v1), last revised 29 Apr 2024 (this version, v2))
Abstract: We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with positive drift $\beta\in(0,1)$, branching rate 1/2, killed at $L(\beta)>0$, and reflected at 0. The killing boundary $L(\beta)$ is chosen so that the total population size is approximately constant, proportional to $N\in\mathbb{N}$. This branching system is interpreted as a population accumulating deleterious mutations.
We prove that, when the typical width of the cloud of particles is of order $c\log(N)$, $c\in(0,1)$, the demographic fluctuations of the system converge to a Feller diffusion on the time scale $N^{1-c}$. In addition, we show that the limiting genealogy of the system comprises only binary mergers and that these mergers are concentrated in the vicinity of the reflective boundary. This model is a version of the branching Brownian motion with absorption studied by Berestycki, Berestycki and Schweinsberg to describe the effect of natural selection on the genealogy of a population accumulating beneficial mutations. In the latter case, the genealogical structure of the system is described by a Bolthausen-Sznitman coalescent on a logarithmic time scale. In this work, we show that, when the population size in the fittest class is mesoscopic, namely of order $N^{1-c}$, the genealogy of the system is given by a Kingman coalescent on a polynomial time scale.
Submission history
From: Julie Tourniaire [view email][v1] Fri, 26 Apr 2024 16:46:10 GMT (288kb,D)
[v2] Mon, 29 Apr 2024 15:26:52 GMT (288kb,D)
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