Looking forward and backward in time - the concept of duality and applications to population genetics
- Martin Möhle (University of Mainz)
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.