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Two forces at work here: (1) humans will get fewer children, leading to leveling off (2) climate change, the exhaustion of resources, breakdown of our ecosystem will lead to hunger, disease and political disruption and culling of our species.
I think (1) will have a much bigger impact on the human population than (2). We aren't going to exhaust Earth's resources in 200 years, and it will take much less time than that to switch over to renewables. The transition is already underway, and shows no signs of stopping.
I admire your optimism, but I definitely do not share it . You should read this; https://open.library.ubc.ca/soa/cIRcle/collections/facultyresearchandpublications/52383/items/1.0437324 There are many more scientific publications and newspaper articles and assessments about this topic - it looks very grim for humanity.