Cpr- "Cycling" is the process of building up nitrifying bacteria in your tank. As waste decomposes in the water (leftover food, poop, etc.) it creates ammonia. Ammonia is highly toxic to fish, shrimp, axolotls, ANYTHING that's living in the water. Ammonia toxicity also increases as the pH and temperature of a tank increase. Axies like cold water and a slightly higher than neutral pH, so people who say it's less toxic to them may be right, but it's still deadly if the levels get too high. Ammonia burns their skin and gills and eventually weakens their immune system, leaving them open to all kinds of scary infections. Columnaris, for one, loves to grow in cold, slow circulating water (ideal for axies!), and it's a quick, opportunistic killer that's hard to treat and eradicate afterwards so yeah... ammonia is dangerous!
The nitrifying bacteria, or the beneficial bacteria in a tank get rid of this toxic ammonia by converting it to nitrite. Nitrite is also deadly- it affects the oxygen carrying capabilities of the blood by reducing the blood's ability to absorb oxygen. This means your axolotl, who'd already be stressed from the ammonia peak, now has to contend with low oxygen levels in addition to a weakened immune system. Some airation in the tank would help a bit, but I personally wouldn't risk cycling a new tank with ANY animal in it. Better to wait till everything's settled!
The next and final step in the cycle is the conversion of nitrite into nitrate (watch these two similar words, people get them confused.) Nitrate is generally considered "safe" to aquatic life-forms, but you still have to monitor the nitrate levels and keep them under control. THIS is where your water changes come in.
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You'll read some places that you should put some zebra danios, or a few goldfish in a tank to start the cycle. This will work, because all fish produce waste for the bacteria to break down, but it's cruel and will probably kill the fish. Instead, throw a piece of shrimp into the tank, and leave it alone. Doing water changes during the cycling process lengthens the time the tank will take to cycle, and if there's nothing in the tank, you don't really have a reason to change water anyway, so DON'T. Your filter should be running at this point as well.
For a tank your size, it WILL take AT LEAST a month to cycle. Bigger tanks take longer than small ones, because they need to build a larger bacterial population. You'll know your cycle is complete when your ammonia levels read ZERO, your nitrite levels also read ZERO, and your nitrate reads LESS than 20. You should be testing often to monitor these levels throughout the cycle, so you'll know when it's done. You'll see an ammonia peak first, then the nitrites will skyrocket, and finally, no presence of either ammonia or nitrite.
20 or less nitrate is safe, but you'll need to do regular water changes to keep them under control- makes sense if you think about it. There's nothing converting the nitrate into a different form, so as more and more ammonia/nitrite gets converted, the nitrate just keeps building! Keep up with weekly 10-20% water changes, and you'll be fine. Water changes also help prevent buildups of any heavy metals or other toxins that you can't test for (these would build up if you were only topping up evaporated water, rather than removing old water)and will help keep algae under control.
You'll also need to vacuum the gravel lightly with each water change, to get the crud out. You should rinse your filter media (sponge, and carbon if you use it) in TANK water with water changes as well. I say to rinse these in tank water, because a lot of the biological filtration (bacteria) happens inside your filter. Bacteria grow in the sponge, and rinsing with TAP water would kill the bacteria. When adding new water to the tank, always make sure it's been dechlorinated, and try to match the temperature and pH of the new water with that of the tank. The less fluctuation in the system, the better, as a stable environment encourages a healthier, bigger bacterial population.
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It sounds like you're doing a great job researching your axolotls before getting them, which is great to see! Have you decided how many you'll get? I haven't had mine for long, but I'm already addicted and planning on getting another!
Good luck with yours, I'm sure you'll do a fantastic job!