in a world with still very few people to compete with

My anthropology knowledge is pretty paltry, but isn't that the opposite of how humanity spread? One slow expansion at a time, as a tribe grew too big and then split and settled in the next valley over seems to be more in line with how much of it worked. Some expansions may have been quite different, no doubt, like oceania or the east asia -> north america -> south america journey, but on average you would be relatively near other human tribes, right?