14:15
We consider a system of interacting Fisher-Wright diffusions
which arise in population genetics as the diffusion limit of a spatial
particle model in which frequencies of genetic types are changing due to
migration and reproduction.
For both models the historical processes are constructed,
which record the family structure and the paths of descent through space.
For any fixed time, particle representations for the
historical process of a collection of Moran models with increasing particle
intensity and of the limiting interacting Fisher-Wright diffusions are
provided on one and the same probability space by means of Donnelly and
Kurtz's look-down construction.
It will be discussed how this can be used to obtain new
results on the long term behaviour. In particular, we give representations for
the equilibrium historical processes. Based on the latter the behaviour of
large finite systems in comparison with the infinite system is described on
the level of the historical processes.
The talk is based on joint work with Andreas Greven and Vlada
Limic.