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Once our population started growing following the agricultural revolution, we would have quickly outstripped fairies in terms of weight. In 2015, fairy biomass would be down to about 1.2% of human biomass, and by the mid-21st-century, they'd bottom out at less than 1%.
But as long as humans keep reproducing, the fairy population would keep growing. If we assume the human population will level off at around 9 billion partway through this century, then the share of mass occupied by fairies would continue rising steadily.
If our population stays at a stable 9 billion indefinitely, by 2100, they'll be back above 1% of human biomass, and they'll reach 2% by the year 3000. In the year 100,000, if our species is somehow still around, they'll outweigh us.
This makes things interesting. Let's assume fairies need to eat. If fairies weigh about as much as us, they'd presumably be consuming a similar share of food and water. So even if the Earth can support 400 million tons of human, it may not be able to support 400 million tons of human and another 400 million tons of fairy. This suggests that, sometime in the next hundred millennia, the growing fairy population would start to crowd out the human population.
But if fairies crowd out humans, that would in turn reduce the growth rate of the fairies, slowing down the replacement process. The end result (in this idealized model) would be that the fairy population growth would taper off as the human population declined. The situation would never quite reach a stable equilibrium, because every 1,728 human births would create one human's weight in fairies, reducing the Earth's total human carrying capacity by 1. But since the rate of fairy creation would slow down as the human population shrank, the process would stretch out for a very long time.
This odd situation would only exist because fairies—in Mira's scenario—are immortal. The scenario would change dramatically if we introduced a fairy death rate. Perhaps fairies don't age or experience natural deaths, but can still die from other causes.
What would kill the fairies? Who knows.[6] But ecosystems aren't static. If you mess with one part, the other parts change in response. Fairies could become the dominant species, but the system they learned to dominate wouldn't be there forever.
If fairies represent most of the biomass in the ecosystem, eventually, given enough time ...

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