  The system under study is a reaction-diffusion equation in a horizontal
strip, coupled to a diffusion equation on its upper boundary via an exchange
condition of the Robin type. This class of models was introduced by H.
Berestycki, L. Rossi and the second author in order to model biological
invasions directed by lines of fast diffusion. They proved, in particular, that
the speed of invasion was enhanced by a fast diffusion on the line, the
spreading velocity being asymptotically proportional to the square root of the
fast diffusion coefficient. These results could be reduced, in the logistic
case, to explicit algebraic computations. The goal of this paper is to prove
that the same phenomenon holds, with a different type of nonlinearity, which
precludes explicit computations. We discover a new transition phenomenon, that
we explain in detail.
