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Informal Probability Workshop – Monday 26th February 2018

26 Feb 18

Speaker:              Aleksander Klimek, Mathematical Institute, University of Oxford

Date:                    Monday 26th February 2018, 12 noon in the Large Lecture Theatre, Department of Statistics

Title:                     Selection in fluctuating environment

Abstract:             We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment.

We consider a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. In dimensions greater than one, the ‘genetic drift’ disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic PDE driven by a noise that is white in time but coloured in space.

We discuss a possible extensions of this work, including description of the dynamics of a rare type in those models.

Joint with Niloy Biswas, Alison Etheridge and Jonathan Chetwynd-Diggle