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Interacting Fisher–Wright diffusions in a catalytic medium
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  • Published: May 2001

Interacting Fisher–Wright diffusions in a catalytic medium

  • Andreas Greven1,
  • Achim Klenke1 &
  • Anton Wakolbinger2 

Probability Theory and Related Fields volume 120, pages 85–117 (2001)Cite this article

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  • 8 Citations

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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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Authors and Affiliations

  1. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraß 1½, D-91054 Erlangen, Germany. e-mail: math@aklenke.de, , , , , , DE

    Andreas Greven & Achim Klenke

  2. Fachbereich Mathematik, Universität Frankfurt, Robert-Mayer-Strae 6-10, D-60325 Frankfurt, Germany, , , , , , DE

    Anton Wakolbinger

Authors
  1. Andreas Greven
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  2. Achim Klenke
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  3. Anton Wakolbinger
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Additional information

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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  • Issue Date: May 2001

  • DOI: https://doi.org/10.1007/PL00008777

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  • Mathematics Subject Classification (2000): 60K35, 60J70; Secondary 60J60, 60H15
  • Key words or phrases: Interacting Fisher–Wright diffusions – Processes in a random environment – Catalytic medium – Longtime behaviour – Rescaling
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