Lambda-lookdown model with selection

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The goal of this paper is to study the lookdown model with selection in the case of a population containing two types of individuals, with a reproduction model which is dual to the Λ-coalescent. In particular we formulate the infinite population "Λ-lookdown model with selection". When the measure Λ gives no mass to 0, we show that the proportion of one of the two types converges, as the population size N tends to infinity, towards the solution of a stochastic differential equation driven by a Poisson point process. We show that one of the two types fixates in finite time if and only if the Λ-coalescent comes down from infinity. We give precise asymptotic results in the case of the Bolthausen-Sznitman coalescent. We also consider the general case of a combination of the Kingman and the Λ-lookdown model.