  Horizontal gene transfer consists in exchanging genetic materials between
microorganisms during their lives. This is a major mechanism of bacterial
evolution and is believed to be of main importance in antibiotics resistance.
We consider a stochastic model for the evolution of a discrete population
structured by a trait taking finitely many values, with density-dependent
competition. Traits are vertically inherited unless a mutation occurs, and can
also be horizontally transferred by unilateral conjugation with frequency
dependent rate. Our goal is to analyze the trade-off between natural evolution
to higher birth rates on one side, and transfer which drives the population
towards lower birth rates on the other side. Simulations show that evolutionary
outcomes include evolutionary suicide or cyclic re-emergence of small
populations with well-adapted traits. We focus on a parameter scaling where
individual mutations are rare but the global mutation rate tends to infinity.
This implies that negligible sub-populations may have a strong contribution to
evolution. Our main result quantifies the asymptotic dynamics of subpopulation
sizes on a logarithmic scale. We characterize the possible evolutionary
outcomes with explicit criteria on the model parameters. An important
ingredient for the proofs lies in comparisons of the stochastic population
process with linear or logistic birth-death processes with immigration. For the
latter processes, we derive several results of independent interest.
