Looking forward and backward in time - the concept of duality and applications to population genetics

Population genetics applications are often interested in the descendants and the ancestry of a sample of genes or individuals. This leads to so-called descendant processes X, which characterize the evolution of the considered population forward in time, and ancestral processes Y describing the population backward in time. The processes X and Y are in many cases related in a certain sense, called duality, which turns out to be a powerful tool to study the behaviour of the population.

This talk will give some examples of dual processes arising in interacting particle systems and mathematical population genetics. The presentation will focus on a class of haploid population models with exchangeable reproduction law. The corresponding processes X and Y are dual for fixed population size N and for the diffusion limits as N tends to infinity. The Wright-Fisher diffusion X and its dual counterpart, the coalescent process Y (Kingman 1982), appear if and only if

If this condition is relaxed then the corresponding asymptotic coalescent process Y allows for simultaneous and multiple mergers of ancestral lines. This reflects the presence of individuals with large offspring sizes. The talk will finish with an outlook on recent research for diploid models and models with varying environment.