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.