The theory is that as ammonia goes down, nitrite goes up; then nitrite goes down and nitrate goes up. But in practise things often behave differently.
Can I just check with you:
You added a dose of ammonia on day 1 -
how much ammonia did you add?This dropped on day 13 so you added a second dose -
how much ammonia did you add the second time?Since then you have has constant 4ppm ammonia and 5ppm nitrite until today -
have you added any more ammonia apart from day 1 and day 13?Today (day 31) the ammonia reading was the same and the nitrite reading had dropped
Nitrate readings were higher on day 13 than on day 31.
The fact that the nitrite reading has suddenly dropped suggests that you have just grown a lot of nitrite eating bacteria. It is common to hear of cycles where the nitrite level stays constant then drops to almost zero overnight. If there was no nitrite being made it would stay constant until the bacteria grew, then suddenly drop.
The fact that the nitrite level stayed constant and that the ammonia level also stayed constant suggests that something has happened to your ammonia eaters. Once the ammonia level drops, with an accompanying rise in nitrite, further additions of ammonia should drop quickly and the nitrite level should go up further until the nitrite eaters grow.
So what has happened to your ammonia eaters? Have you done anything at all to the tank, added anything other than ammonia? Did you add more ammonia that you should have done on day 13? If you added the correct amount of ammonia and done nothing else to the tank, I'm at a loss

I think what I'd do in this situation would be a very large water change. Then test the ammonia level half an hour after you've filled up. If you have a reading, subtract that from 3ppm and add enough ammonia to make the reading up to 3ppm - do another test half an hour after adding the ammonia to confirm you have added the right amount. I know the API tester doesn't have a colour for 3ppm so aim for more green that 2ppm but less than 4ppm.
When have you been doing the tests? I know that the light under which you compare the colour of the tube to the chart can have an effect and give a false reading. I did an experiment with all the different bulbs in our house. Daylight is the best light source, though you need to be at home during the day for that. Old fashioned light bulbs and halogen light bulbs give the same reading as daylight. But fluorescent and compact fluorescent (energy saving) bulbs/tubes give a different reading from daylight. We didn't have any LED bulbs when I did the experiment so I don't know about those. If you read the tests under fluorescent light, it is possible you got a false reading for ammonia - and a too-high ammonia level causes the cycle to stall.