An analysis is made of a theoretical model of the formation of three-dimensional nanometer-size islands in molecular beam epitaxy. The kinetics of the self-organization processes are described using a lattice gas model of the adsorbate with self-consistent allowance for lateral interactions in the activation energies of the diffusion processes. It is shown that at below-critical temperatures in a certain range of thicknesses, decay of the spatially uniform state gives rise to arrays of three-dimensional nano-islands which do not participate in the coalescence process after growth has ceased. The average size of the islands, their geometric profile, and the spatial ordering depend strongly on the kinetic parameters of the model.