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Disclaimer: I am not a scientist.

I was wondering if there is a concern that the world will run out of breathable atmospheric oxygen as the number of dead zones in the ocean increases with global warming. This concern is based on the understanding that 60% (give or take) of our breathable oxygen is produced by phytoplankton. I have read a few peer reviewed articles to that effect, all by the same author, but haven't seen any articles that counter that concern.

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Yes, it is, though probably not immediately. Ocean and atmosphere deoxygenation is a thing, and many publications are out.

In a larger context, deoxygenation is indeed inevitable even if we ignore anthropogenic warming. Increasing solar flux alone (the sun get's hotter over its lifespan) could reduce the atmospheric oxygen to 1% of present day level within around a billion years, says this publication.

This publication estimates that almost 80% of ocean deoxygenation under current day CO2 levels have not been realized yet, but they say that oceanic surface deoxygenation will stop once CO2 emissions stop, while the deep ocean oxygen levels will continue to decrease.

A global warming by 6°C (which is at the upper end of current projections until the end of this century) could tip the oxygen production by phytoplankton, cancelling the ocean's role as a main oxygen producer.

Citing from the abstract:

[...] If, in the course of time, the oxygen production rate becomes too low or too high, the system’s dynamics changes abruptly, resulting in the oxygen depletion and plankton extinction. Our results indicate that the depletion of atmospheric oxygen on global scale (which, if happens, obviously can kill most of life on Earth) is another possible catastrophic consequence of the global warming [...]

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  • $\begingroup$ Thanks. To be clear, I was differentiating between deoxygenation in the oceans (many articles) and atmospheric deoxygenation on land. $\endgroup$
    – user21456
    Apr 26 '21 at 8:26
  • $\begingroup$ @user21456: I have cited the important part of what I think answers your question. The ocean is the main producer of atmospheric oxygen, one cannot isolate the one from other. And these things were potential drivers of past mass extinctions. $\endgroup$
    – user22279
    Apr 26 '21 at 9:36
  • $\begingroup$ @user21456: I see you've asked a related question and received a correct answer that was not accepted. Maybe there's missing information ? $\endgroup$
    – user22279
    Apr 26 '21 at 10:06

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