Proof of slow linear growth instead of exponential in case of no children for people over 60 years old
Proof:
Let's denote y -- total population, m -- young population under 60 years, k1 is reproduction rate, k2 is transition rate into group "over 60 years old".
y' = m
m' = k1 *m - k2 *m = 0 when k1=k2 (two children per family)
m' = 0 => m = const => y (total population) growth is linear
Another proof:
If population under 60 years old is constant and equals N, then number of people transitioned into "over 60 years old" group cannot exceed N for any consequent 60 years. That's because if we suppose contrary then when the youngest of them was born, the population of people under 60 would have exceeded N.
Consequently, growth is linear and it's limited by N/60 * t.
Overall, the key is allowing to bear children only for constant number of people any time.
Linear population growth is stable: number of new people (who haven't their 2 children) is constant => growth remains to be linear.
Yet another proof:
UN states that Earth population would stabilize near 10 billion people.
Every 60 years, about 10 billion people would be born, and about 10 billion people would die.
Let's consider what happens if humans become immortal.
Again, every 60 years, about 10 billion people would be born, however about 10 billion people would not die. (people after 60 years aren't allowed to bear children).
In 30 years, Earth population would be 20 billion people, in 90 years it would be 30 billion. It's slow linear growth instead of exponential one.
The result, of course, is valid not only at 10 billion primary population of the Earth, but also at any other value.
It will be necessary to increase the yield in 4/3 times in the first 30 years, in the next 30 years - in 5/4, then in 6/5, 7/6 and so on.
Note.
Of course, if we change the restriction "people after 60 years aren't allowed to bear children"
to restriction "anyone can have no more than 2 children per whole life", population growth can only become even more slow and remain linear.