In many developing systems, the fate of a cell is determined by its position in a time-independent spatial distribution of a morphogen. However, during dorsal-ventral patterning in the Drosophila embryo, an initial low-level signal refines to a narrow, high-intensity band. This refinement suggests that cells respond to the local transient morphogen distribution that results from interactions between bone morphogenetic proteins (BMPs), their receptors, the BMP-binding proteins Sog and Tsg, the metalloprotease Tld, and a putative, positively regulated component that locally enhances surface binding of BMPs within the region of high signaling. We develop a computational model for dorsal surface patterning and show that, when positive feedback of a cell surface BMP-binding protein is incorporated, bistability in the kinetic interactions transduces the transient BMP distribution into a switch-like spatial distribution of the BMP-bound receptor. We also show that the inclusion of positive feedback leads to the observed contraction of signaling, because cells near the dorsal midline outcompete adjacent lateral cells for limited amounts of BMP. In the model, cells interpret the morphogen distribution by differentiating according to the history of their exposure rather than to a threshold concentration in a static spatial gradient of the morphogen.