Interacting Fisher{wright Diiusions in a Catalytic Medium

We study the longtime behaviour of interacting systems in a randomly uctuating (space{ time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher{Wright diiusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium). Here we introduce a model of interacting Fisher{Wright diiusions where the local resam-pling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d{dimensional lattice. While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium.