Your nitrifying bacteria can die back without ammonia/ammonium, but it takes more than a few days and they also multiply pretty quickly.
BTW - do not think that your tank is cycled, as in done-done. You have the capacity to handle some fish waste, not to fully complete the nitrogen cycle nor to likely handle a dynamic event like a huge influx of food, fish or an accident or something. You still need to go slow as things develop and the tank takes steps to be truly, fully cycled which can take many months. Every day that you go along, the tank does get stronger, but you still need to be smart and go slow.