the problem with the world is that developing countries are the ones growing fastest. These countries will eventually hit their peak capacity and wont be able to grow any more.
If you look historically at population growth rates versus state of development, what usually happens is that people just voluntarily stop reproducing as much. Running out of resources and having mass-starvation is much less common.
When you have an undeveloped nation, the birth and death rates are both extremely high, especially for infants. Along comes industrialization and the death rate drops dramatically, while the birth rates stays the same. It takes many decades before medicine advances even further and women have access to contraceptives (and employment opportunities, which reduces the likelihood that they'll have children). At that point, the birth rate drops to match the lower death rate, and you end up with a country like Japan where population growth is essentially non-existent.
This is expected to happen for every country on Earth as long as they industrialize. We'll hit a maximum world population of 15-20 billion and then it'll stabilize.
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