I did google persistence and gifted and did not see a "nature" link, it appeared to be nutured. In fact there were several things that came up and mentioned teaching persistence to gifted children.
Ren
When very young children show persistence as a trait, it's probably at least part of their nature. Again, the fact that you can find a reference to something being taught does not mean that there is no link to giftedness; there are scads of references, some of which come up in those starter searches, showing that there are indeed links between persistence and giftedness.
A sample that literally took me less than five seconds to pull up, from one of the starting links I gave you:
http://gcq.sagepub.com/content/31/2/70.short"Gifted students showed more persistence in regard to the difficult task than EMR students and also chose more often to repeat the difficult task."
