General selection models: Bernstein duality and minimal ancestral structures

Abstract

$\Lambda$-Wright--Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate frequency-dependent selection. A decomposition of the drift allows us to approximate the solution of the stochastic differential equation by a sequence of Moran models. The genealogical structure underlying the Moran model leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundaries; this extends a recent result by González Casanova and Spanò. An intriguing feature of the construction is that the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones.

48 pages, 11 figures

References in corpus (4)

• Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution
• Duality for spatially interacting Fleming-Viot processes with mutation and selection
• Lines of descent in the deterministic mutation-selection model with pairwise interaction
• The sequential loss of allelic diversity

Cited by in corpus (3)

• Moran models and Wright--Fisher diffusions with selection and mutation in a one-sided random environment
• Lines of descent in the deterministic mutation-selection model with pairwise interaction
• Quasi-equilibria and click times for a variant of Muller's ratchet