Why are two breeds of dogs who can't mate without human assistance the same species, while two fish species, which can and do have fertile offspring, but which are intermediate in size and therefore not as good at obtaining resources as the parents, are different species?
The dog example is pre-zygotic isolation, and would seem prohibitive, if not for human assistance. The fish example is called "extrinsic post-zygotic isolation". So, we have that we consider populations who can't actually interbreed the same species, but those who really do mate are different species.
Personally, I can go either way (but do have a preference), but I really wish we could all agree to apply the Biological Species Concept a little more rigorously. My point is always that the BSC doesn't always work (as with (mostly) asexual species, such as bacteria), and other definitions should then be used. My view is that if two populations are different species by any of a set of good species definitions, then they should be called different species. This is an all-encompassing view of what species are.
Don't lose track of the fact that what we are trying to do when we designate something as species is to say something about biology. At the end of the day, species is a term that must say something about the clustering of genomes, and remember that it is possible to cluster a continuum.
Other good species definitions include the Ecological Species Concept, which classifies species as a set of organisms adapted to a particular set of resources, called a niche, in the environment. This definition* is more difficult to test for in natural populations, but that is neither here nor there when we talk about these basic theoretical questions.
And do note that here I am not even talking about the appropriateness of applying one definition when it doesn't match the actual process by which speciation occurred. Two populations may diverge and become different ecological species despite continuous interbreeding, and only after many generations become reproductively isolated (as in not able to have fertile offspring, for whatever reason, save physical isolation). Thus, saying that there are only different species many generations hence when some mutation occurs that changes the ability of sperm to enter the egg, say, makes no sense in the light of the adaptive processes that made the two groups different.
Let's call two populations different "reproductive" species or "ecological" species when the BSC or the ESC applies, and let's for Heaven's sake be rigorous when applying them!
P.S. If you want to have a crack at this, please don't think you can resolve this by putting meaning into the use of the words "breeds" and "species" in the examples above, 'kay?
* Oh why oh why must we call the definitions "concepts"? A "species" is a concept. Tsk, tsk, Mayr.
Rice, W., & Hostert, E. (1993). Laboratory Experiments on Speciation: What Have We Learned in 40 Years? Evolution, 47 (6) DOI: 10.2307/2410209
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