When Cyrano Jones took some tribbles out of their predator-filled environment, they ended up overrunning an entire space station (not to mention the Enterprise).
New York City has squirrels in every park – and no natural predators for squirrels. So why is there no population explosion of squirrels there?
Remembering something I’d seen about peregrine falcons in Melbourne, VIC led me to this.
Fearless and Well-Fed, New York City’s Red-Tailed Hawks Are Flourishing.
Not a birder, but a city like this strikes me as being a raptor’s paradise - lots of high places to swoop down from, sheltered ledges for nests, updrafts just about everywhere.
For the more down-to-earth predators, there’s plenty of dogs and (especially) cats - as I understand it, feral cats are all over the place. And, of course, lots of cars to run down the slow, old, and/or stupid ones.
Perhaps we could get a NYC resident or two to comment.
There are plenty of predators in New York parks. Red-tailed Hawks, Sharp-shinned and Cooper’s Hawks, Kestrels, dogs, feral cats, owls, snakes. Coyotes too, now. And cars take a toll.
But predators aren’t the only thing that control population. Food can be limiting in places the squirrels aren’t being fed, as can be shelter in winter during cold spells. They also can be affected by disease.
Also, remember that the portrayal of animal growth in most movie and TV science fiction is really, really, really, really stupid. Including that of tribbles.
Also, tribbles reproduce asexually all year 'round, so long as they’ve got food. I suppose that they’re more like bacteria than anything else. Being born pregnant is quite the timesaver, too. 
I think this is true - in parts of New York where these predators are excluded, the population of small furry critters just booms.
NYC squirrels also have lots of competition for their limited food supply. There are rats, mice, robust roaches, and even raccoons and opossums.