Lambda-lookdown model with selection

Boubacar Bah , Etienne Pardoux (1)

1 I2M - Institut de Mathématiques de Marseille

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 $\Lambda$-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 $N$ 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 $\Lambda$-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 $\Lambda$-lookdown model.

Domaines: Mathématiques [math]; Probabilités [math.PR]

Boubacar Bah, Etienne Pardoux. Lambda-lookdown model with selection. Stochastic Processes and their Applications, 2015, 125 (3), pp.1089-1126. ⟨10.1016/j.spa.2014.10.014⟩. ⟨hal-01231706⟩