  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.
