  Horizontal gene Transfer (HT) denotes the transmission of genetic material
between two living organisms, while the vertical transmission refers to a DNA
transfer from parents to their offspring. Consistent experimental evidence
report that this phenomenon plays an essential role in the evolution of certain
bacterias. In particular, HT is believed to be the main instrument of
developing the antibiotic resistance. In this work, we consider several models
which describe this phenomenon: a stochastic jump process (individual-based)
and the deterministic nonlinear integrod-ifferential equation obtained as a
limit for large populations. We also consider a Hamilton-Jacobi equation,
obtained as a limit of the deterministic model under the assumption of small
mutations. The goal of this paper is to compare these models with the help of
numerical simulations. More specifically, our goal is to understand to which
extent the Hamilton-Jacobi model reproduces the qualitative behavior of the
stochastic model and the phenomenon of evolutionary rescue in particular.
