TITLE: The number of birds after $n$ years
QUESTION [5 upvotes]: Problem
A pair of birds can produce between 3 to 4 birds each year. If we start with a pair of birds and if a bird usually lives for three years, then how many birds will we have after $n$ years?
What I have done
I assumed that the pair of birds that we started with were 1 year olds.
If we let $m\in\mathbb{N}$ and $3\leq m\leq 4$; $k_n$ be the number of birds after $n$ years, then
$$
\begin{align*}
k_1&=m+2\\
k_2&=\frac{1}{2}(m+2)^2-2\\
k_3&=\frac{1}{4}(m+2)^3-(m+2)\\
k_4&=\frac{1}{8}(m+2)^4-\frac{1}{2}(m+2)^2-m\\
k_5&=\frac{1}{16}(m+2)^5-\frac{1}{4}(m+2)^3-m(m+1)
\end{align*}
$$
and so on. The problem is that I can only see the pattern of the first term of $k_n$ and therefore can not find a general formula for $k_n$. Could you please help?

REPLY [2 votes]: If you assume that each pair of birds produces either $3$ or $4$ baby birds with equal probability $1/2$, then I'm pretty sure you cannot simply use a fixed value of $m$, like $m = 3.5$. To see this, suppose you start with $2$ birds. Then with probability $1/2$ each you will have either $5$ or $6$ birds after one generation. If $5$ birds, then you only have $2$ pairs. But if $6$ birds, then you have $3$ pairs. Thus there is a bias toward having more than $3.5$ times more birds per generation per pair of birds as time goes on. Yet the bias should eventually equalize so that the number of new birds is approximately $3.5/2$ times the number of birds in the previous generation that live, due to the central limit theorem. But you cannot ignore the effects in the very early generations. It seems like you want the EXPECTED number of birds per generation, and this seems like a very hard problem. You can get bounds by fixing $m = 3$ or $m = 4$, as noted in another answer.