One of the more peculiar species encountered, the defining characteristic of the Tribbles is their extreme reproductive rate. Over half of a Tribbles metabolism is devoted to reproduction, allowing them to bear a litter of young every twelve hours. Their young are born pregnant, and can produce young of their own in step with their parents. With an average litter of ten, a single Tribble can therefore create a population of 1,771,561 within three days, and an amazing 304,481,639,541,400,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 in thirty days!
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Fortunately, Tribbles can only breed if they have sufficient food. On their home planet their incredible reproduction is kept in check by a combination of a very limited food supply and some effective predators. If they are transported into an environment which lacks these checks, the effect can be devastating - there are examples of whole planets which have been practically overwhelmed by the Tribble. 
As a result the Tribble was illegal on most planets in the mid 22nd century. Despite this, Doctor Phlox kept Tribbles in the sickbay of the NX-01, using them as food for some of his other animals.  The species first came to the notice of the Federation in 2267 when Cyrano Jones, a trader in gemstones, Kivas and Trillium, transported several Tribbles to Space Station K-7. The small limb less balls of fur generate a purring sound when stroked which many species find pleasing, and quickly became a popular choice as pets. They quickly infested both the station and the USS Enterprise, which was there to safeguard the station's attempt to colonize Sherman's planet in competition with the Klingons. The Tribbles invaded the station's stock of Quadrotriticale, consuming it entirely. As luck would have it the grain had been poisoned by Klingon agents; the death of the Tribbles saved the Federation from a costly blunder which would have seen the planet ceded to its opponents. Captain Kirk was also able to use the Klingons violent allergy to Tribbles to reveal the identity of the agent who had poisoned the grain.
The Enterprise cleared out its own infestation of Tribbles by beaming them into the engine room of a Klingon battlecruiser, a move which caused the crew considerable distress on their trip home.  The Klingon empire responded by launching a major assault on the species, and the Tribbles were eventually wiped out completely.
In 2373 Arne Darvin, the Klingon Agent who had been uncovered by Kirk, travelled back in time in an attempt to assassinate the Captain. Captain Sisko and the crew of the Defiant was able to prevent the attempt, but on their return they accidentally brought some Tribbles with them. Deep Space Nine was quickly over-run,  and although the infestation was eventually cleared several ships had already carried Tribbles away from the station. Today the species has been firmly established on several dozen worlds.