Abstract.
We study the longtime behaviour of interacting systems in a randomly fluctuating (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 diffusions. 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 diffusions where the local resampling 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.
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Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001
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Greven, A., Klenke, A. & Wakolbinger, A. Interacting Fisher–Wright diffusions in a catalytic medium. Probab Theory Relat Fields 120, 85–117 (2001). https://doi.org/10.1007/PL00008777
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DOI: https://doi.org/10.1007/PL00008777