  We take interest in a reaction-diffusion system which has been recently
proposed [11] as a model for the effect of a road on propagation phenomena
arising in epidemiology and ecology. This system consists in coupling a
classical Fisher-KPP equation in a half-plane with a line with fast diffusion
accounting for a straight road. The effect of the line on spreading properties
of solutions (with compactly supported initial data) was investigated in a
series of works starting from [11]. We recover these earlier results in a more
general spatially periodic framework by exhibiting a threshold for road
diffusion above which the propagation is driven by the road and the global
speed is accelerated. We also discuss further applications of our approach,
which will rely on the construction of a suitable generalized principal
eigenvalue, and investigate in particular the spreading of solutions with
exponentially decaying initial data.
