  How does the interplay between selection, mutation and horizontal gene
transfer modify the phenotypic distribution of a bacterial or cell population?
While horizontal gene transfer, which corresponds to the exchange of genetic
material between individuals, has a major role in the adaptation of many
organisms, its impact on the phenotypic density of populations is not yet fully
understood.
  We study an elliptic integro-differential equation describing the
evolutionary equilibrium of the phenotypic density of an asexual population. In
a regime of small mutational variance, we characterize the solution which
results from the balance between competition for a resource, mutation and
horizontal gene transfer. We show that in a certain range of parameters
polymorphic equilibria exist, which means that the phenotypic density may
concentrate around several dominant traits. Such polymorphic equilibria result
from an antagonist interplay between horizontal gene transfer and selection,
while similar models which neglect the transfer lead only to monomorphic
equilibria.
