The number of adults is equal to the number of adults in the prior month plus the number of children in the prior month (that mature to be adults).
a(n) = a(n-1) + c(n – 1)

Also, the number of children is just the number of adults in the prior month (because they all have offspring).
c(n) = a(n – 1)

Putting the two equations together you have:
a(n) = a(n – 1) + a(n – 2)

This is the classic Fibonacci sequence.

To answer the questions:
1) (21, 13) is followed by (34, 21)
2) (p, q) is followed by (p+q, p)
3) See below:

Let’s compare the ratios of adults to children in successive months.
p/q compared to (p+q)/p

Let’s say this does converge on a particular ratio.  I’ll call that ratio R.
R = p/q = (p+q)/p

Split the fraction at the end:
R = p/q = p/p + q/p
R = p/q = 1 + q/p

Substituting in 1/R for q/p:
R = 1 + 1/R

Multiply both sides by R:
R² = R + 1

Get everything on the left hand side:
R² – R – 1 = 0

This can be solved with the quadratic formula or completing the square.
R² – R  = 1
R² – R + 1/4 = 5/4
(R – 1/2)² = 5/4
(2R/2 – 1/2)² = 5/4
(2R – 1)²/4 = 5/4
(2R – 1)² = 5
2R – 1 = ± ?5
2R = 1 ± ?5
R = (1 ± ?5) / 2

In our example the ratio is always greater than 1 (more adults than children so we want:
R = (1 + ?5) / 2

That is the Golden Ratio:
R ? 1.618033988749894848204586834…[/spoiler]