The quenched structured coalescent for diploid population models on finite graphs with large migrations and uneven offspring distributions

TL;DR

A new model for the evolution of a diploid structured population backwards in time that allows for large migrations and uneven offspring distributions is described and the annealed and quenched scaling limits coincide if and only if these large migrations and uneven offspring distributions are absent.

Abstract

In this work we describe a new model for the evolution of a diploid structured population backwards in time that allows for large migrations and uneven offspring distributions. The model generalizes both the mean-field model of Birkner et al. [\textit{Electron. J. Probab.} 23: 1-44 (2018)] and the haploid structured model of Möhle [\textit{Theor. Popul. Biol.} 2024 Apr:156:103-116]. We show convergence, with mild conditions on the joint distribution of offspring frequencies and migrations, of gene genealogies conditional on the pedigree to a time-inhomogeneous coalescent process driven by a Poisson point process $Ψ$ that records the timing and scale of large migrations and uneven offspring distributions. This quenched scaling limit demonstrates a significant difference in the predictions of the classical annealed theory of structured coalescent processes. In particular, the annealed and quenched scaling limits coincide if and only if these large migrations and uneven offspring distributions are absent. The proof proceeds by the method of moments and utilizes coupling techniques from the theory of random walks in random environments. Several examples are given and their quenched scaling limits established.

Figures (2)

• Figure 1: Here we see two demes containing $4$ individuals each so that $N(1) = N(2) = 4$. One individual from deme $1$, marked in red, migrates to deme $2$, and two individuals from deme $2$, marked in blue, migrate to deme $1$, so that $N^*(1)(0) = 5$ and $N^*(2)(0) = 3$. The two demes then experience some reproductive event, with the parental relationships tracked by two edges emanating from a child to their two parents. For example, the couple $(1,2)$ have a single child, the second individual in blue, and so $\mathcal{V}_{1,2}^1 =1$. At the same time the couple $(2,4)$ have two children together, the third and fifth individuals after migration. Therefore $\mathcal{V}_{2,4}^1 = 2$. The third individual in deme $2$ has no offspring, so $\mathcal{V}^2_{3,j} = 0$ for all $j$.
• Figure 2: In the figure we see a discretized torus $\mathbb{Z}^2 /\langle L_1,L_2\rangle_\mathbb{Z}$ with $L_1 = 7$ and $L_2 = 5$. At each point on the discrete torus, drawn as a solid ball, is a deme of the population. Emanating from each deme are its nearest neighor edges. A point $w^*$, colored orange, and a radius $r$ is chosen uniformly at random. The ball of radius $r\rho$ centered at $w^*$ is drawn in red. A point $v^*$ is then selected uniformly at random from that ball and drawn in cyan. All of the demes within the ball of radius $r\rho$ from $w^*$ then send a Beta-distributed proportion of their population to $v^*$. These migrations are shown in cyan.

Theorems & Definitions (66)

• Definition 2.1: The pedigree
• Definition 2.2: Ancestral line
• Definition 2.3: Ancestral process
• Remark 3.5
• Remark 3.6
• Remark 3.8
• Remark 3.9
• Remark 3.11
• Definition 3.13
• Theorem 3.14