Are you adding ammonia to the tank? I see no reason for the ammonia levels to go up ever unless there is something making ammonia.
Basically, your numbers are moving up and down without any explanation for why this might be happening. To have nitrite levels rise you must have ammonia being added in some way.
Also, live plant want ammonium (NH4) while the bacteria prefer ammonia (NH3). The amount of each of these two forms in water depends upon the pH and temperature of the water. The higher these are, the more of the ammonia that will be in the NH3 form.
While the plants can use ammonium much faster than the bacteria can use NH3, the more plants one has, the less NH3 that will be available for the bacteria. Bear in mind that if either NH3 or NH4 is removed, the balance dictated by the pH and temperature will cause things to rebalance quickly.
To understand this all one needs to realize that in a tank without live plants, only two things can take up ammonia/ammonium one is obviously the nitrifying bacteria but the other can be algae. However, ignoring algae for a minute we come to see this. One can have a tank with no live plants and the result is that the bacteria will develop and multiply and will handle the ammonia. However, no matter how many live plants we might have in a tank, there will still be some amount of bacteria present and working as well.
In water where all of the ammonia is in the form of NH4 (as one approaches pH 6.0 and under this will be the case), the bacteria can still use this, but they will do soo much less efficiently than they can process NH3.
However, for either plants (or algae) and bacteria to thrive, they need Nitrogen and that usually comes from ammonia in a tank.
So back to the tank in this thread, my question becomes are you dosing ammonia? If not is there something continuously dying and rotting in the tank? If you are not adding ammonia, then once it begins to drop it shold go to zero since no more is being added or created. And this means that no increase in nitrite or nitrate should happen.
The numbers you are reporting are not possible in the absence of regular ammonia additions. These could come from your adding the ammonia or ammonium chloride, the presence of live animals- from snails to fish and then from decomposing organic matter. But you are not recounting any of these things being done or ongoing. So I see no reason for the numbers you report.
Here is the best part of cycling, it is a process and it has to follow a series of steps. And when it comes to the typical test kit methods, we are reading on the total Ion scale. What this means is the as ammonia turns to nitrite. There will nbe more ppms of nitrite than there were for ammonia, And when that nitrite turns to nitrate, the ppms again get higher.
The cycle is complete when, for the amount of ammoina being added there is no reading created for nitrite and the nitrate will be the only number rising. While ammonia might evaporate, nitrate will not. So unless we have denitrification involved (nit coomion in most tanks) then the nitrate must be handled. This can be done bu live plants and/or water changes.
You plants are processing some of the nitrogen compounds but clearly nowhere near all of them. However, from where is the ammonia coming? Not only are you not adding any, both the plants and evaporation would be using the ammonia as well as any bacteria. And any ammonia used by plants or reduced from evaporation will not create any nitrite and thus no nitrate either.
Next, as time passes whith ammonia constantly being supplied, the bavcteria that handle nitrite will also multiply and then you see nitrite dropt to and stay at 0. The one wild card in your tank is how much nitrate might be created and then how fast the plants or water changes might use it. Evaporation will reduce ammonia some, but not nitrate. Only plants (or lagae )using it and/or water changes will lower nitrate. Most tanks will never have nough denitrification at work to make a serious dent in Nitrate levels.