Is a 16 85 fast enough??

I think I may be rehashing a old question, but I'm doing more photography for our local zoo, when shooting zoo animals for their purpose, I use my D200, and 300 f4 non AFS, but I have been getting into more "function" shots and the lenses I shoot just don't focus and "lock on" fast enough, don't laugh but I use either one of three lenses, normal work its okay, but as for photojournalism  they don't cut the mustard, just to slow.

My three are Nikons old 24 50, and a old 35 135, and Sigma 17 70 2.8 4.0, I like the results all three give, that is under normal shooting, but in a hurry up mode with various zoo personnel moving and action, these fall way short of satisfactory, I've missed many shots because they focus to slow so that brings me to the big question?? what lens to get?? my first choice is the 17 55, but a little pricey, then the 3rd party 17 50s, but questionable quality, I had Tamron 17 50 2.8 no motor  sold it.

Another choice is Nikons 16 85, I was going to purchase this lens from the "tammy: but no one had in stock, so I bought the Sigma.

So can someone make suggestions as which may be the right ticket, I really would like to stay with my DX body, I have my D700, but I never intended it for that purpose.

Thanks for all and any help, I volunteer all my time[retired] and my photos, for them to use as they want and need, maybe this is my calling?

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