Polynesia had ongoing population problems, they were solved by emigration, endemic warfare, blood feuds and in times of drought cannibalism which in some places was formalised. Warfare became a part of everyday life, all men were trained warriors and sudden death was a constant threat.
Genocide was not unknown although not normal, but whole districts were known to have been totally depopulated. Whole tribes became extinct in the larger Islands.
Some small Polynesian islands lost their whole population due to lack of resources when interisland trade collapsed due to warfare on other islands.
Micronesia there are tiny islands which could not survive population pressures without trade, and some practiced infanticide and/or killing mothers to prevent them having more kids (according to legends).
So population can be managed, but it's not pretty.
Polynesia still has population problems which are now solved by poor health care and bad food being dumped there, rampant obesity and emigration. But if for some reason they were once again isolated, it would in short order become a deadly struggle over resources again I would posit.
One other issue no one has touched is there is no continental land, this means that some resources are actually pretty scarce in most places in terms of minerals, metals and things like that, you can have lots of trees with none really suited to building on a particular Island, this is how it works on earth anyway. Most of Micronesia, Melanesia (not so much since some of that is ex-continental) and Polynesia have this problem which was partially solved by trade and by replacing the native flora with imports, most of their staple foods in terms of plants were imported. Very few are native to the Islands. Take away just a few like taro, yams and breadfruit, and humans can't survive in any great number. Coconuts can survive a sea journey on their own for about three months, but none of the others can. So you need an equivalent of those for a start.