I thought about that but was worried it would be to big a mechanical penalty
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Though I have said that, there are so many logic issues with D&D that you'd have to change half the game before coming to the "bite attack" issue.
So it's usually easier to come up with the fluff according to the result, if you're dead set on continuing with the system.
If you're still using six-second rounds it might be a good idea to watch some animal videos and time out six seconds during a fight, just to see how much activity that encompasses.
think your going to need to be more specific on the information you want noticed. Most animal fight just have them move into the equivalent of grapple as soon as they get a solid hit.