DeckardCanine asked, Why is this, anyway? Did solid trash cover all the water? Did air pollution get so thick it blocked precipitation?
One of the effects I've imagined to try to explain the film's unlikely situation is a loss of CO2. That, combined with toxic gases, would kill the plants, and indirectly most life. Especially land-based life, but most in the ocean as well.
The Earth suffers, and carbon dioxide (arguably the most crucial life-sustaining part of our atmosphere; we have hundreds of times as much oxygen) drops to too low a level to support remaining plants. Almost all plants, animals, and multi-cellular life forms are lost. Most bacteria too.
The changes to land cover affect humidity; grasslands and forests become deserts and dunes. The lack of CO2 combines with the lack of humidity (a much larger climate factor) to produce dry terrain almost worldwide. Rain becomes rare; in some parts of the world, it is not seen for centuries at a time.
Deep sea vents have collections of chemotropes, organisms who do not depend upon sunlight and photosynthesis at all. They live in water that reaches multiples of sea-level boiling points, and in highly corrosive chemical baths. In fact, they need these aspects for their ecology. It's unlikely that conditions elsewhere would affect them much. They survive.
Elsewhere, other tiny pockets of life are still viable; aerobic bacteria and other sorts locked in ice and reviving upon a thaw. It takes only a few.
Most die immediately. But, when the mixture of toxic gases is no longer fatal for them, some survive. They have little competition, and first the ocean is repopulated with them. CO2, gradually, begins to rise. Little is making use of it at this point. (Currently, CO2 on the Earth drops for five months of every year as the plant usage outpaces all production.)
Finally, some pockets of cyanobacteria (photosynthetic, CO2 users) appear. Eventually, bits of chance-preserved algae make it again, and these new forms suddenly dominate a sea largely stripped of competition and consumers.
The seas have no sharks; teleost (bony) fish are represented by a very few species that survived frozen as a few of them can. Their progeny will make the oceans a different place -- but they're edible. A few crustaceans and mollusks that had adapted to the chemotropic life begin to migrate upward from the deep sea vents; the sea is suddenly rich, from their perspective, and for a while they own most of the oceans.
Finally, the world's humidity begins to rise. There is little on land to take advantage of this; in the absence of humans, algae would need to evolve into something like plants once again. Perhaps they will, eventually.
In the meantime, after hundreds of years, the rains finally come. They don't make a paradise of life -- but they create a bacteria-rich ground fertile with possibilities. And waiting. It will be fatal to many humans that have spent generations away from exposure to such organisms; they've lost their defenses. But eventually, this live soil will be the foundation for a new beginning.
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