What is stability in tanks?

I think sometimes we overrate the importance of stability. The only thing thats really important to keep stable is the health of the system as many things even in the wild can rapidly change such as salinity and temp due to tides and floods. Do you want huge swings, prob not. Also depending on what you are keeping not all areas will be exposed to these kinds of swings in the wild.
As for why things will take off and then decline before taking off again, I am a firm believer that this is related to the many micro nutrients in our systems that we do not and sometimes can not measure. Different coral and algae types rely on different nutrients and chemicals for survival and they will boom when they are high until they use them up then start to decline and as they build up again in time take off again.