The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

Abstract

Using graphical methods based on a `lookdown' and pruned version of the {\em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.

minor improvements in presentation; Electron. Commun. Probab., in press

References in corpus (4)

• The fixation line in the $\Lambda$-coalescent
• Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution
• Lambda-lookdown model with selection
• The common ancestor process revisited

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• A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent
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• General selection models: Bernstein duality and minimal ancestral structures
• Moran models and Wright--Fisher diffusions with selection and mutation in a one-sided random environment
• Solving the selection-recombination equation: Ancestral lines under selection and recombination
• On the stationary distribution of the block counting process for population models with mutation and selection
• On the boundary classification of $\Lambda$-Wright-Fisher processes with frequency-dependent selection