  We present a model for the dynamics of a population of bacteria with a
continuum of traits, who compete for resources and exchange horizontally
(transfer) an otherwise vertically inherited trait with possible mutations.
Competition influences individual demographics, affecting population size,
which feeds back on the dynamics of transfer. We consider a stochastic
individual-based pure jump process taking values in the space of point
measures, and whose jump events describe the individual reproduction, transfer
and death mechanisms. In a large population scale, the stochastic process is
proved to converge to the solution of a nonlinear integro-differential
equation. When there are only two different traits and no mutation, this
equation reduces to a non-standard two-dimensional dynamical system. We show
how crucial the forms of the transfer rates are for the long-term behavior of
its solutions. We describe the dynamics of invasion and fixation when one of
the two traits is initially rare, and compute the invasion probabilities. Then,
we study the process under the assumption of rare mutations. We prove that the
stochastic process at the mutation time scale converges to a jump process which
describes the successive invasions of successful mutants. We show that the
horizontal transfer can have a major impact on the distribution of the
successive mutational fixations, leading to dramatically different behaviors,
from expected evolution scenarios to evolutionary suicide. Simulations are
given to illustrate these phenomena.
