The Lapita culture (defined based on pottery) starts showing up on the islands of Melanesia around this time. The culture was almost certainly brought from outside, by mariners speaking an Austronesian language (Proto-Oceanic), who traced their roots back (immediately) to island Southeast Asia, and (earlier) to Taiwan. The nearer islands of Melanesia were already inhabited when the Lapitans arrived, by people similar to modern New Guineans, whose ancestors had been there for tens of thousands of years.
The Lapitans had advanced sailing skills, and also introduced some domesticated animals – pigs and chickens. But they don’t seem to have had a huge demographic edge over the earlier inhabitants, who had already developed agriculture on their own, and may have had more resistance to local diseases like malaria. The Lapitans mostly settled smaller, harder-to-reach islands, and established enclaves on larger islands. And one scholar calls the Austronesian expansion “an agricultural revolution that failed” because the pioneers abandoned the rice cultivation that their ancestors had been doing, although they quickly picked up local crops like taro and breadfruit. (In another part of the Austronesian world, island Southeast Asia, they gave up on agriculture altogether to become “fisher-foragers.”)
Curiously, Polynesians today get more than half of their patrilineally transmitted Y-chromosome DNA from Melanesia, while most of their matrilineally transmitted mitochondrial DNA, and even their bilaterally transmitted autosomal DNA, is from Taiwan/Southeast Asia.
There is good reason to think that ancestral Austronesians had a matrilineal social organization, with kin groups emphasizing descent through the female line. This might reflect a history in which men spent lots of time away from home, sailing, raiding, and trading, and chose to leave their households in charge of their sisters. In Micronesia, settled in a separate phase of the Austronesian expansion, matrilineal descent is the rule right up the present – a chief’s heir is his sister’s son, not his wife’s son. Matrilineal societies are often not very intense about policing female sexual behavior (compare the Middle East) and it seemed plausible that when the Lapita folk were passing through Melanesia, the women might have picked up some Melanesian Y chromosomes while their menfolk were off sailing.
But it now looks like the story is different. Ancient DNA from the very earliest Polynesian settlers in Tonga and Vanuatu shows no trace of Melanesian ancestry. Melanesian ancestry apparently came to Polynesia sometime after this initial settlement, perhaps as a result of conquest. And this fits with another aspect of Polynesian kinship: our best reconstructions suggest that Polynesians switched from matrilineal to patrilineal descent early in their history. The recent evidence suggests that a group of Melanesians, arriving later, perhaps as conquerors, may have been responsible for the shift, by setting themselves up as chiefs (in an already rank conscious society) and passing their privileged position on to their sons and their son’s sons – a story already familiar from other parts of the Old World.
A lot of what I know about kinship in Pacific island societies from my late colleague at the University of Utah, Per Hage.