system
Given the following post, respond as if you're commenting on Reddit:
human
How do we know the point where species truly divide to the point that they can no longer breed with each other? I was under the impression from biology class that to be a different species that means they can no longer breed with other species.

So looking at large cats and the equine family you can see that some members of the species can breed with each other. 

Some examples include mules which are sterile. We can also see ligers as well as other crossbreeds the feline family. So two questions: if two species within a genus can breed together why are they considered separate species? I thought that goes against the definition that defines each member within the taxonomy.

Askscience: what is the defining factor to make a species distinct if it isn't the inability to breed with another branch of the same family?
gpt
Defining a species can quickly become problematic so we actually have a few different 'species concepts' to try to accurately describe the relationships between closely related species. The concept that you're referencing is called the biological species concept. According to this idea, two organisms are conspecifics (the same species) if they can interbreed *and produce fertile offspring*. So the horse and the donkey are not conspecific because the mule is infertile. The same thing is usually true for the lion and tiger hybrids.    
   
But that concept doesn't cover a lot of things and there are some things that would be really hard to fit into really any concept. For instance, the biological species concept can't very easily define asexual species. And there's something called a 'ring species' which confounds all definitions. In that species complex there is a kind of gradient where any apparent conspecific can freely interbreed their neighbors over a large geographic range, but two individuals at the extremes of that range will not be able to mate.     
    
Other species concepts attempt to group less obvious species using their ecology, morphology, phylogeny, etc.   
As an example, the polar bear and the grizzly bear can produce fertile offspring and very rarely they do so in the wild. But they occupy very different habitats and the two populations are only very rarely in contact with one another; so even though they would fit the biological concept, they are different species because of their ecology.     
Mallard ducks can hybridize with around 50 other species of ducks and around 20% of the offspring are fertile. But again, because of ecology they are not likely to mate, or because of genital morphology they're just usually incapable of physically coupling.    
    
So to answer your question of what is the deciding factor, it really depends on why there is even a question. No one ever looked at two white tail deer and wondered if they were the same species, so it's never really been a problem to try to fit them into a 'species concept'. But asexual species can often only be distinguished by their ecological niches or with molecular genetics.    
    
   
So it really depends on the animals in question.