Get yourself at least one cannibal tribble and it'll help keep them under control, more or less. It will only munch on white tribbles so if a higher quality one appears then you'll have to manually pop them into the airlock to prevent the spread.
If you have a fancy tribble that you want to keep, such as the risa tribble for summer then stick it into a boffs gear slot until needed.
A caveat to this is that a Tribble in your inventory will consume the edibles, but, if the edibles would produce a unique Tribble, it will only produce one, but keep right on eating the stuff. This happened to a lot of our Jem'Hadar characters last year. They had loads of Ketracel White on them, and it just kept on disappearing silently once we had our Gamma Quadrant Tribbles.
This is an MMO, not a Star Trek episode simulator. That would make for a terrible game.
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If you have a fancy tribble that you want to keep, such as the risa tribble for summer then stick it into a boffs gear slot until needed.