Ok, really unscientific math example... #'s arnt real unit of measurements here just place holder examples of an amount vs a larger amount.
Brand new Uncycled Tank: the bioload the fish poop/ uneaten food/ decaying stuffs out put is 10, creating ammonia of 10...the amount the bacteria can process 0. equals really really high ammonia or in this case 10, as there is nothing to process it.
This works it way though to... now you have enough bacteria to process 10 ammonia, but not enough to process nitrates......to etc
A cycled tank is one that Has a stable health population of bacteria that can handle the tanks bio load.
The bio load output is 10, bacteria population can handle 10 ammonia, the bacteria different species population can handle 10 nitrates and hopefully the other species 10 nitrites as well(least a 9).
It's Balanced. Bio load in...handles bio load out. Ta Da Cycled tank
This is why we see new folks with OH it's cycled!....I added water..added a pinch of food waited a month. LFS said my water was perfect...whys my ammonia high and fish dead? or...
OH the reefer cycled, had two fish in there, they did ok...then got really exited purchased basically the tanks size max fish at once! Bioload is now around 25...for a tank with the population of bacteria that can handle 10.
Crap the reefer just took out that boring established LR and added a ton of uncured rock(but so much cooler) with all kinda of things that still need to decay! Or the reefer just stirred up a Whole bunch of stuff that was pocketed hiding deep in the sand..... bioload is now at 60! for a tank cycled to process at 10
Thats a Re-cycle, current bacteria can not keep up with the decay/waste going on in the tank..... tank will likely see another ammonia spike, then nitrite spike, then a nitrate..before settling out again. Often smaller spikes then the setting up a brand new tank, but if one really changes it all up then ya. Its going to be a problem.
Re-cycling is a Rebalancing of your tanks bacteria population vs you tanks bioload. In theory it should happen every time we add another bioload to the tank... but most of the time it so very small we done even see it with chem tests and the fish etc life is just fine.
And can go the Other way to...bacteria arnt books on a shelf. They dont get enough to eat they will died and reduce the populations to a more balanced level for them not your in you head tank wants. Which can cause issues for folks who have gone fallow...kicked a disease and want to add the fish back.