Variational Formulation of Models in Population Genetics

Variational Formulation of Models in Population Genetics, 

by Fabio A. C. C. Chalub, CMA/FCT/UNL.


Data: 14/5/2018

Hora: 15h30

Local: Sala de Seminários - Edifício VII, FCT-UNL




In this talk, we consider three different families of models frequently used in population genetics:

1. The Wright-Fisher process, a discrete-time finite population Markov chain. We also consider a continuous-time analogue.
2.The Kimura equation, a drift-diffusion partial differential equation of degenerated type.
3.The Replicator equation, a non-linear ordinary differential equation.

We start by providing an unified description of all these models, showing precise conditions such that 1 -> 2 -> 3. In particular, the Kimura equation is a good approximation of the Wright-Fisher process for large populations and weak selection for all time scales, while the Replicator equation approximates the Wright-Fisher process only for short times.
In the sequel, we show that all these models can be reformulated as gradient flows. We will achieve this goal using two different concepts that appear independently in the literature: "the fitness potential" and the "Shashahani distance".
The first provides useful informations in the associated dynamics of all these models, in a way that resembles the use of potentials in classical mechanics; the second provides the right metric for the reformulation of all models as gradient flows of the fitness potential.
We will finish by showing that all reformulations are compatible.

This is a joint work with Leonard Monsaingeon, Max O. Souza and Ana Margarida Ribeiro.