We consider spatially interacting Moran models and their diffusion limit which are interacting Fisher-Wright diffusions. For both models the historical process is constructed, which gives information about genealogies. 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 a look-down process. It will be discussed how this can be used to obtain new results on the long term behavior. In particular, we give representations for the equilibrium historical processes. Based on the latter the behavior 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.