Presented by:
Jonathan Mattingly Duke University
Date:
Friday 10th June 2016 - 11:00 to 12:00
Venue:
INI Seminar Room 2
Abstract:
The dynamics of a population undergoing selection is a
central topic in evolutionary biology. This question is particularly intriguing
in the case where selective forces act in opposing directions at two population
scales. For example, a fast-replicating virus strain outcompetes
slower-replicating strains at the within-host scale. However, if the
fast-replicating strain causes host morbidity and is less frequently
transmitted, it can be outcompeted by slower-replicating strains at the
between-host scale. Here we consider a stochastic ball-and-urn process which
models this type of phenomenon. We prove the weak convergence of this process
under two natural scalings. The first scaling leads to a deterministic
nonlinear integro-partial differential equation on the interval [0,1] with
dependence on a single parameter, λ. We show that the fixed points of this
differential equation are Beta distributions and that their stability depends
on λ and the behavior of the initial data around 1. The second scaling leads to
a measure-valued Fleming-Viot process, an infinite dimensional stochastic
process that is frequently associated with a population genetics.
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