  We treat a model of population dynamics in a periodic environment presenting
a fast diffusion line. This phenomenon is modelled via a "road-field" system,
which is a system of coupled reaction-diffusion equations set in domains of
different dimensions. Here, we consider for the first time the case of a
reaction term depending on a spatial variable in a periodic fashion, which is
of great interest for both its mathematical difficulties and for its
applications. We derive necessary and sufficient conditions for the survival of
the species in terms of the sign of a suitable generalised principal
eigenvalue. Moreover, we compare the long time behaviour of a population in the
same environment without the fast diffusion line, finding that this element has
no impact on the survival chances.
