  This paper is concerned with propagation phenomena for the solutions of the
Cauchy problem associated with a two-patch one-dimensional reaction-diffusion
model. It is assumed that each patch has a relatively well-defined structure
which is considered as homogeneous. A coupling interface condition between the
two patches is involved. We first study the spreading properties of solutions
in the case when the per capita growth rate in each patch is maximal at low
densities, a configuration which we call the KPP-KPP case, and which turns out
to have some analogies with the homogeneous KPP equation in the whole line.
Then, in the KPP-bistable case, we provide various conditions under which the
solutions show different dynamics in the bistable patch, that is, blocking,
virtual blocking (propagation with speed zero), or spreading with positive
speed. Moreover, when propagation occurs with positive speed, a global
stability result is proved. Finally, the analysis in the KPP-bistable frame is
extended to the bistable-bistable case.
