[TangoTrotFox]

This is the wrong way of looking at things. 2 is always the replacement rate except in the case of an immortal population. You're not considering that you reach an equilibrium between deaths and births after one generational cycle. Imagine for simplicity that everybody gives birth at 1 and dies at 4. Let's look at when people have two children, starting with a genesis population of 10.

----------------

Year 1:

age 1 = 10

births = 10

deaths = 0

----------------

Year 2:

age 1 = 10

age 2 = 10

births = 10

deaths = 0

----------------

Year 3:

age 1 = 10

age 2 = 10

age 3 = 10

births = 10

deaths = 0

----------------

Year 4:

age 1 = 10

age 2 = 10

age 3 = 10

age 4 = 10

births = 10

deaths = 10

----------------

Year 5:

age 1 = 10

age 2 = 10

age 3 = 10

age 4 = 10

births = 10

deaths = 10

----------------

Etc. The only way to see a longterm increase in population is with a birth rate > 2, and similarly the only way to see a longterm decline is a birth rate of < 2. 'Longterm' in this case meaning just one generation, at which point the equilibrium kicks in. Of course not everybody dies at the same time, but the effect is identical when you look at an average with some distribution - which is what we have.

However, this also makes the assumption that everybody makes it to the age of fertility and that everybody is fertile. Both assumptions are incorrect so, in reality, the equilibrium point for population stability is even greater than 2. In reality we would expect to see a decline in population if each couple had exactly two children.

[robocat]

Our lovely residents of Binary Town keep their average birth rate at 2 through enlightened and fair methods.

Then Professor Gemini discovers a cure for cancer, which doubles their lifespan.

Unfortunately very soon they are having an unexpected housing shortage.

How did that happen?

[TangoTrotFox]

Sure if you create an absurd example of suddenly doubling every single person's life expectancy you can start to see some impact from changes in such, but in reality this does not happen. Life expectancy changes extremely slowly, and it also goes in both directions on a worldwide scale further mitigating any effect on population levels.

It's like Big O in a way. Adding some stuff to O(1) doesn't really matter compared to something that's O(n^2) except when start adding some really really unreasonably huge things, though even there n^2 tends to catch up really fast. In either case it's not a practical issue in code or population.