The initial stages of multilayer Co thin film grown on Cu(111) surface were simulated by means of kinetic Monte Carlo (KMC) method, where the realistic growth model and physical parameters were presented. The effects of edge diffusion along the islands and mass transport between interlayers were included in the simulation model. Emphasis was placed on revealing the transition of growth morphology in heteroepitaxial Co/Cu(111) system with the changing of surface temperature. The simulation results show that the dendritic islands form at low temperature (T=210 K), while compact islands grow at room temperature (RT). The Volmer-Webber (three-dimensional, 3D) growth mode is presented due to the relative higher Ehrlich-Schwoebel (ES) barrier. Our simulation results are in good agreement with the real scanning tunneling microscopy (STM) experiments. The initial stages of multilayer Co thin film grown on Cu (111) surfaces were simulated by means of kinetic Monte Carlo (KMC) method, where the realistic growth model and physical parameters were presented. The effects of edge diffusion along the islands and mass transport Between interlayers were included in the simulation model. Emphasis was placed on revealing the transition of growth morphology in heteroepitaxial Co / Cu (111) system with the changing of surface temperature. The simulation results show that the dendritic islands form at low temperature (T = 210 K) while compact islands grow at room temperature (RT). The Volmer-Webber (three-dimensional, 3D) growth mode is presented due to the relative higher Ehrlich-Schwoebel (ES) barrier. Our simulation results are in good agreement with the real scanning tunneling microscopy (STM) experiments.