We 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 Lambda-coalescent. In particular we formulate the infinite population “Lambda-lookdown model with selection”. When the measure Lambda gives no mass to 0, we show that the proportion of one of the two types converges, as the population size tends to infinity, towards the solution to a stochastic differential equation driven by a Poisson point process (the same equation, in case without selection, already appeared in the work of Bertoin and Le Gall). We show that one of the two types fixates in finite time if and only if the Lambda-coalescent comes down from infinity.
We give precise asymptotic results in the case of the Bolthausen-Sznitman coalescent. In particular we obtain in that case an explicit formula for the probability that one of the two alleles fixates, which is different from the classical one for the Wright-Fisher model.
This is joint work with my former phd student Boubacar Bah.