The sequential loss of allelic diversity

TL;DR

The paper investigates the sequential loss of allelic diversity in neutral population models, showing that in the Moran binary case the forward extinction process $\hat{A}^N_t$ has the same law as the backward-time block-counting process $A^N_t$ of the $N$-Kingman coalescent, linking allele survival to ancestral coalescence via the Look-Down framework. It extends the comparison to block frequencies at jump times, proving that while one-dimensional marginals coincide, the full block processes generally differ under $\Lambda$-coalescents with multiple mergers; it introduces a $\Lambda$-urn model and a coupon-collection interpretation for haplotype blocks and uses the Look-Down construction to connect forward and backward dynamics. The results identify precise conditions under which Dirichlet-type limits govern block frequencies and establish a mechanism for nondegenerate limits via $\mathcal{D}^{\Lambda}_k$, as well as a criterion for the absence of simultaneous extinctions. Altogether, the work provides a unifying perspective on extinction paths in population genetics across Moran, Kingman, and $\Lambda$-Fleming-Viot settings, clarifying when forward-extinction and backward-coalescent descriptions align and when they diverge, with implications for the structure of ancestral versus haplotype blocks and a robust urn-model framework for future investigations.

Abstract

This paper gives a new flavor of what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population with constant size $N$, we assume that each individual at time $0$ carries a distinct type, or allele. We consider the joint dynamics of these $N$ alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time $t$. We call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the $N$-Kingman coalescent. Stimulated by this result, we investigate: (1) whether it extends to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times; (2) whether it extends to the general case of $Λ$-Fleming-Viot processes.

Figures (1)

• Figure 1: In this example: $N=6$, $k=4$ so that $R^{N,k}=4$. The value of the fixation $L^{k-1}$ (orange line) at time $\hat{T}^{N,k}$ only depends on the type of arrow at the jumps of $L^{k}$ (blue curve). $L^{k-1}$ only jumps at the first jump of $L^k$.

Theorems & Definitions (28)

• Theorem 2.1
• Theorem 2.2: The ancestral block process, Kingman K
• Remark 1
• Theorem 2.3: The haplotype block process
• Remark 2
• Remark 3
• Theorem 3.1
• proof
• Proposition 3.2
• Theorem 3.3: Labbé L, Theorem 1.6