Extensive experimental data and an accompanying theoretical model are presented for the self-limiting profiles and Ga segregation on patterned GaAs(111)B substrates during metalorganic vapor-phase epitaxy of A1(x)Ga1-As-x. Self-limiting widths and segregation of Ga produce quantum dots along the base of pyramidal recesses bounded by (111)A planes and quantum wires along the vertical axis of the template, respectively. Coupled reaction-diffusion equations for precursor and adatom kinetics reproduce the measured concentration and temperature dependence of the self-limiting width and segregation. Our model can be extended to other patterned systems, providing a new paradigm for predicting the morphology of surface nanostructures and inferring their quantum optical properties.