I just discovered the first word to this shout on my current play through and thought I'd give it a try but deer and rabbits run away for some reason, even though in the Wiki, it says that it can be used for hunting as the animal will not run or fight back, I know I'm in range of them, so I don't know what the problem is. I tried it on wolves that came for me and it worked fine?

Is Animal Allegiance the shout in which animals come and aid you? The shout you're looking for is probably Kyne's Peace.

I read UESP and another Wiki, both said the same for this specific shout, I also looked at what they said about Kyne's peace, as both shouts are similar.