If there was a constant inflow of flies into the room, after some time you would have an infinite number of flies. I never heard of this happening!
One could argue that there is a constant number of flies per volume outside the room and when you kill all the files in the room, there is a negative gradient of flies and the insects will "naturally" move to the room. But such a "fly pressure" would require the flies to bounce from each other, like atoms, which does not seem to happen.
Or maybe flies are territorial animals?