In this work, we report on the study of one-dimensional (1-D) and two-dimensional (2-D) cooperative arrangements of self-assembled Ge dots grown on patterned Si (001) substrates. Selective epitaxial growth (SEG) of Si mesas was first performed to form Si mesas before Ge deposition. Self-assembled growths of Ge dots with variable Ge amount have been carried out. It is found that the cooperative arrangements of Ge dots are dependent on Ge amount deposited. The cooperative arrangements is attributed to the self-regulation of the dot size and position, which is driven by the minimization of the total free energy including strain energy, surface energy and deformation repulsive energy. This cooperative arrangement promoted by the energetically preferential nucleation on Si mesas allows us to control the placement of the self-assembled dots at the specific positions.