The asymmetric ancestral graph

Speaker: Jose Luis Perez 
Affiliation: CIMAT, Mexico

Abstract

In this talk, we study the meaning of having a selective disadvantage in the general context of populations with skewed reproduction mechanisms. For instance, if the reproduction events of individuals of one type are less frequent or have a smaller size than those of the other type. For this, we will consider a pair of populations such that its ancestry lies in the universality class of a Lambda coalescent. Our approach consists of constructing a Moran model in which individuals of the two populations compete, instead of a single Moran model for each population, allowing us to define a frequency process associated with one of the types in the population. We introduce the partial order of adaptation in the space of finite measures on [0,1] which characterizes the fact that one of the types in the population has a selective advantage over the other one. This result is a consequence of a duality result that extends the known duality relation for lambda coalescents with a classical selection, by means of the construction of the Lambda ancestral graph, to those with skewed reproduction mechanisms. We then study the scaling limits of our models to extend the partial order of adaptation to a more general setting. Finally, we derive a Griffiths representation for the generator of the frequency process to obtain semi-explicit expressions for the probability of fixation of one of the types in the population.