Hominin evolution is characterized by progressive regional differentiation, as well as migration waves, leading to anatomically modern humans that are assumed to have emerged in Africa and spread over the whole world. Why or whether Africa was the source region of modern humans and what caused their spread remains subject of ongoing debate. We present a spatially explicit, stochastic numerical model that includes ongoing mutations, demic diffusion, assortative mating and migration waves. Diffusion and assortative mating alone result in a structured population with relatively homogeneous regions bound by sharp clines. The addition of migration waves results in a power-law distribution of wave areas: for every large wave, many more small waves are expected to occur. This suggests that one or more out-of-Africa migrations would probably have been accompanied by numerous smaller migration waves across the world. The migration waves are considered "spontaneous", as the current model excludes environmental or other extrinsic factors. Large waves preferentially emanate from the central areas of large, compact inhabited areas. During the Pleistocene, Africa was the largest such area most of the time, making Africa the statistically most likely origin of anatomically modern humans, without a need to invoke additional environmental or ecological drivers.