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Lines of descent in a Moran model with frequency-dependent selection and mutation

Ellen Baake, Luigi Esercito, Sebastian Hummel

Abstract

We study ancestral structures for the two-type Moran model with mutation and frequency-dependent selection under the nonlinear dominance or fittest-type-wins scheme. Under appropriate conditions, both lead, in distribution, to the same type-frequency process. Reasoning through the mutations on the ancestral selection graph (ASG), we develop the corresponding killed and pruned lookdown ASG and use them to determine the present and ancestral type distributions. To this end, we establish factorial moment dualities to the Moran model and a relative. We extend the results to the diffusion limit and present applications for finite population size as well as moderate and weak selection.

Lines of descent in a Moran model with frequency-dependent selection and mutation

Abstract

We study ancestral structures for the two-type Moran model with mutation and frequency-dependent selection under the nonlinear dominance or fittest-type-wins scheme. Under appropriate conditions, both lead, in distribution, to the same type-frequency process. Reasoning through the mutations on the ancestral selection graph (ASG), we develop the corresponding killed and pruned lookdown ASG and use them to determine the present and ancestral type distributions. To this end, we establish factorial moment dualities to the Moran model and a relative. We extend the results to the diffusion limit and present applications for finite population size as well as moderate and weak selection.

Paper Structure

This paper contains 26 sections, 25 theorems, 120 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The ancestral selection graph
  4. Type distribution via a killed ASG with multiple branching
  5. Siegmund duality
  6. Ancestral type distribution
  7. Diffusion limit
  8. The kASG in the diffusion limit
  9. The pLD-ASG in the diffusion limit
  10. Applications
  11. Fixation probability under moderate selection, and the expected number of potential influencers
  12. Stationary and ancestral type distributions under FTW: some observations
  13. Details of the two selection models
  14. Graphical construction with type and ancestry propagation
  15. Connecting nonlinear dominance with fittest-type-wins
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let $(\widehat{s}_m)_{m>0}$ and $(s_m)_{m>0}$ be two sequences in $\mathbb R_+$ satisfying $0 < \sum_{m>0} \widehat{s}_m m < \infty$, $0 < \sum_{m>0} s_m m < \infty$, and $(\widehat{s}_m)_{m>0}$ is non-increasing. Let $\widehat{Y}$ and $Y$ be the MoMo processes with DOM and FTW scheme and with sel

Figures (11)

  • Figure 1: An untyped realisation of the Moran interacting particle system with nonlinear dominance; $t$ is the time increment.
  • Figure 2: A selective event of order $1$ (left) and of order $2$ (center) in the DOM model. The continuing line is indicated by co, the checking line by ch, the incoming line by i, and the descendant line by d. On the right, all the possible configurations in a selective event of order $2$ that lead to an unfit descendant are depicted; type 0 in dark green, type 1 in light brown.
  • Figure 3: The realisation of Fig. \ref{['fig:untypedmomo']}, but now with types (type 0 in dark green, type 1 in light brown).
  • Figure 4: In red, the ASG for one of the individuals in Fig. \ref{['fig:untypedmomo']}, but now for the FTW model. Notice that arrows that start from lines in the ASG and hit individuals not in the current graph are not relevant for the types in our initial sample. Grey dotted line, black arrow, and red arrow indicate absolute time, forward time increment, and backward time increment, respectively.
  • Figure 5: All individuals in the population share a common ancestor in the sufficiently distant past.
  • ...and 6 more figures

Theorems & Definitions (60)

  • Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3: Factorial moment duality
  • Corollary 2.4: Representation absorption probabilities
  • proof
  • Lemma 2.5: Siegmund duality,siegmund1976equivalence
  • Remark 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Proposition 2.9
  • ...and 50 more