We simulate the two-dimensional patte formation in surfactant-mediated epitaxy using a kinetic model, in which the nucleation and growth of the stable islands are controlled by the exchange process between surfactant atoms and deposited atoms. Our model assumes the following. (1) The exchange barrier preventing an adatom from attaching to an existing island edge depends on the numbers of the adatoms nearest and next-nearest neighbours belonging to the island. (2) The exchange processes at the two different types of step are associated with different energy barriers. The simulations generate a series of surface morphologies ranging from fractal structures to compact triangular shapes, very similar to those observed in recent experiments.