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The question infers that mankind is the dominant species at the moment. Mankind is one of about 50 million species that have existed since the start of life and 90%+ of those are extinct. It is therefore highly likely that mankind will go extinct sooner or later, either through our actions or otherwise.
The idea of any species being 'dominant' is not the way that nature seems to work. Every species from a bacteria to a tree is 'dominant' in its environment, otherwise it wouldn't be there. It is a fallacy to think that evolution 'leads' to mankind; there is no progression, merely that in each generation, the ones that survive breed with minute differences caused by mutation and sexual recombination. You cannot say that a human is 'more evolved' than a bacteria; in fact a bacteria is considerably more evolved as its generation time is so much shorter (20 minutes vs. 20 years), and you should see what some of them can do! Fancy living at 120 degrees Centigrade?
Given long enough, i.e. maybe 50 million years even the species which we have destroyed will be replaced with species that occupy the same environmental niches.
The answer therefore is that the world will keep on turning, whether we are here or not.
I often imagine that the species that are left will heave a sigh of relief, if they could.
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- Thanks to everyone who took the time to answer my question. It has been interesting and amusing reading through all of your responses.