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# Scaling limits of a model for selection at two scales

 dc.contributor.author Luo, S dc.contributor.author Mattingly, Jonathan Christopher dc.date.accessioned 2015-07-28T19:31:51Z dc.date.accessioned 2015-07-28T19:32:28Z dc.identifier.uri http://hdl.handle.net/10161/10332 dc.description.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, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ 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. dc.format.extent 23 pages, 1 figure dc.relation.isversionof http://arxiv.org/abs/1507.00397v1 dc.relation.replaces http://hdl.handle.net/10161/10331 dc.relation.replaces 10161/10331 dc.relation.isreplacedby 10161/12939 dc.relation.isreplacedby http://hdl.handle.net/10161/12939 dc.subject math.PR dc.subject math.PR dc.subject math.DS dc.subject q-bio.PE dc.subject 37, 60 dc.title Scaling limits of a model for selection at two scales dc.type Journal article pubs.author-url http://arxiv.org/abs/1507.00397v1 pubs.organisational-group Duke pubs.organisational-group Mathematics pubs.organisational-group Trinity College of Arts & Sciences
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