Birth rates are dropping globally and we’re all living longer. The local answer for counties like South Korea is immigration from areas that have higher birth rates. With climate change likely to affect the habitability of many of our current settlements, as a species we’ll need to support massive migration shifts over the next few hundred years. So the problem kind of solves itself if we manage migration and resourc…
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Apart from that, simply replacing the population also presents quite a few challenges that are likely to result in a very infected and polarized debate regarding the future direction of the societies that such replacement takes place in.
That being said, I believe you're right about the fact that there's going to be a lot of migrations going on around the Globe for the next few hundred years. It's just not automatically going to solve the birth rate issue.