v(x,L,t)\!-\!\m u(x,t) & \!\!\text{for } x\in{{\mathbb{R}}}, & \!\! t>0 \\
  \!\!\!v_t(x,y,t)-d\D v(x,y,t)=f(v) & \!\!\text{for } \!(x,y)\!\in\!{{\mathbb{R}}}\!\times\!(-\infty,L), & \!\! t>0  \\ 
  \!\!\!dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \!\!\text{for } x\in{{\mathbb{R}}}, & \!\! t>0.
  \end{array} \right.$$ was introduced to model the evolution of the species $v(x,y,t)$ in a *field* ${{\mathbb{R}}}\times(-\infty,L)$ which is bounded at the level $y=L$ by a *road* where part of the same species, $u(x,t)$, diffuses with coefficient $D>0$, which in principle may be different from the diffusion coefficient in the field $d>0$. A reaction of Fisher-KPP type takes place in the field, i.e. $f\in{\mathcal}{C}^1([0,+\infty))$ satisfies $$\label{12}
f(0)=0=f(1), \qquad 0<f(s)<f'(0)s \;\; \text{ in } (0,1), \qquad f<0 \text{ in } (1,+\infty).$$ On the contrary, no reaction occurs on the road, where the density of the species varies only because a fraction $\mu>0$ of the population jumps from the road to the field while a fraction $\nu>0$ of the population jumps from the field to the road.

This model was motivated by empirical observations of wolves moving along seismic lines in Canada (see [@McK]) or insects like the *Aedes albopictus* (tiger mosquito) spreading in the United States along highways (see [@MM]). Another example of this phenomenon is the diffusion of diseases along commercial 