Based on the density functional theory (DFT), we theoretically investigate the growth mechanism of GaN (0001) using the Metal-Organic Vapor-Phase Epitaxy (MOVPE). We first identify the structure of the growing Ga rich GaN (0001) surface, then study the adsorption of NHx (x= 0-3) on such surface. We find that NH2 and NH units spontaneously intervene in the Ga-Ga weak bonds on the Ga-rich GaN (0001) surface. A reaction pathway of decomposition of NH3 on Ga rich surface is revealed. During reaction, N is found to incorporate in the weak Ga-Ga bond and form - Ga - (NH) - Ga - structure. The activation barrier of NH3 is surprisingly small, just 0.63 eV. We also explore the decomposition of NH2 to N to form Ga-N network and find the plausible reaction pathway with the energy barrier of 2 eV. Taking into account the chemical potential of an H-2 molecule in the gas phase at the growth temperature, we find that this reaction can be overcome. This NH3 decomposition mechanism and the N incorporation on GaN is a new growth mechanism catalyzed by the growing surface.