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We treat a model of population dynamics in a periodic environment presentinga fast diffusion line. This phenomenon is modelled via a "road-field" system,which is a system of coupled reaction-diffusion equations set in domains ofdifferent dimensions. Here, we consider for the first time the case of areaction term depending on a spatial variable in a periodic fashion, which isof great interest for both its mathematical difficulties and for itsapplications. We derive necessary and sufficient conditions for the survival ofthe species in terms of the sign of a suitable generalised principaleigenvalue. Moreover, we compare the long time behaviour of a population in thesame environment without the fast diffusion line, finding that this element hasno impact on the survival chances.

A Fisher-KPP model with a fast diffusion line in periodic media