The self-organization of germanium islands on a silicon(001) substrate is studied using lattice-based kinetic Monte Carlo simulations. These islands form spontaneously via the Stranski-Krastanov mode during growth. The interplay of deposition flux and competing surface diffusion leads to a size and shape distribution of islands that varies with temperature and coverage. For the simulation parameters chosen, a kinetic regime of irreversible growth is observed at 500 K, and this changes to quasi-equilibrium growth at 600 K. At 550 K, we see that the surface roughness increases abruptly from a low value and crosses the roughness curve at 600 K. This behavior is explained on the basis of a change in the island formation mechanism. At 500 K, the island formation involves a nucleation barrier; whereas at 600 K this barrier is almost nonexistent. At an intermediate temperature, the stochastic effects due to the incoming flux initially slow down island growth, but the subsequent island nucleation rapidly increases the roughness. These results illustrate how island self-assembly is affected by mechanistic in addition to kinetic and energetic effects. Our results are discussed in the context of experiments on a Si-Ge system and show how the kMC models can be used to understand the processes in heteroepitaxial growth.