Suppose a new species is created from a random mutation that happened during an instance of reproduction in an existing species. How can that new species survive and flourish if there only exists one of it's kind and therefore it's not able to reproduce? By definition a species can only reproduce with others of the same species, no? It seems to me the only way it could is if a mate is created due to a separate occurance of a similar mutation, and the chances of both mutations occurring at around the same point in timespace must be very close to zero. And even if it were to happen, it seems to me the offspring wouldn't be able to reproduce due to problems with inbreeding.
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3Your question implies some misunderstandings about what mutation and speciation mean. I recommend https://evolution.berkeley.edu/ - it contains a good short summary of the basics of evolution. – Armatus Mar 28 '18 at 08:24
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1By definition a species can only reproduce with others of the same species, no? No, and this shows a profound misunderstanding of science in general and evolution in particular. In any case, even if that was true, it applies to "species", not "incipient species". You should read the link that Armatus pointed to to begin to correct your misunderstandings. – iayork Mar 28 '18 at 15:15
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Virtually all cats can interbreed. (Lions, Tigers...) So can almost all dogs. Interbreeding is an ancestral condition, lost on a case-by-case condition. – Karl Kjer Mar 28 '18 at 16:15
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One single mutation won't create a species. Suppose instead you have an initial population that is separated into groups by geographic barriers - say they're on the Galapagos Islands, for instance. Then each group will experience random changes, different beneficial ones will be selected for given the different conditions on the islands, and eventually the populations will diverge enough to be considered new species. – jamesqf Mar 28 '18 at 19:08
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1"this shows a profound misunderstanding of science in general" thanks @iayork for the hasty generalization / just plain insult. – Jonah Mar 28 '18 at 22:53
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Suppose a new species is created from a random mutation that happened during an instance of reproduction in an existing species.
No honest informed person would define species in this manner.
This is a better analogy

swbarnes2
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1So the key here is that a mutation that contributes to the evolution of a new species doesn't make the recipient of the mutation reproductively incompatible with it's immediate ancestors, but it may make the recipient reproductively incompatible with its ancestors farther up the line due to the cumulative effect of this and previous mutations of ancestors in-between? – Jonah Mar 28 '18 at 23:17
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1Yes. But also, speciation is not about individuals mutating. It is about populations changing. The population, due to mutations, changes so much from the ancestral population that we label it a new species. – swbarnes2 Mar 28 '18 at 23:31
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When I saw your answer, with this ugly text posted as an image, I was getting ready to complain! Of course, then I read the answer. The analogy is good and simple enough. +1 – Remi.b Mar 29 '18 at 00:59
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1@Jonah you are still confused about the difference between a definition and reality. There are many different definitions of "species"; you've chosen one of the dozens and are upset because it doesn't cover everything in biology. You're debating semantics that have little to do with science. – iayork Mar 29 '18 at 01:34
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@Jonah For more information about the difficulties of the concept of species, you can have a look at this post. – Remi.b Mar 30 '18 at 05:26
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@iayork I'm not upset about anything, I was just asking an honest question. I understand now that a species is not soley defined by reproductive compatibility. Really I already new that it's possible for two different species to have fertile offspring (e.g. Homo sapiens), I just gave an oversimplified definition as a postulate I guess. The confusion I had is more about what makes certain individuals reproductively compatible. My assumption that determining which two animals can reproduce with each other can be done with some hard-and-fast rules like chromosome count is wrong ... – Jonah Mar 30 '18 at 16:17
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I thought a single mutation would have to "make the jump" between reproductively incompatible groups of animals. Really my question should of asked how any two animals can be rep. comp. if mutating into a new species would always ensure 0 compatible mates. I understand now that reproductive boundaries are not so clear – Jonah Mar 30 '18 at 16:17