How Fibonacci sequence works in rabbits problem?

Question

I can't understand the explanation in my textbook.

In the following text $f_{n-1}$ is explained as the number on the island the previous month while $f_{n-2}$ is explained as the newborn pairs. But say $n=5,$

$$f_5=f_4+f_3,$$

in this case $f_3$ is not newborn pairs any more, it consists of both newborn rabbits and adult rabbits. Could anyone give me some advice?

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Edit:

Ok, my current idea is that, observe

$$f_n = f_{n-1}+f_{n-2},$$

• The difference between $f_{n-1}$ and $f_{n-2}$ should be the newborn rabbit pairs.

• The intersection between $f_{n-1}$ and $f_{n-2}$ is somewhat complicated, but pairs in this set should have been alive for at least 2 months, in the perspective of the current($n$-th) month.

Now the calculation of $f_{n-1}+f_{n-2}$ will double the intersection and include the newborn pairs.

So maybe this could be how it work?

Answer 1

You may be "overthinking" this. In the (very unrealistic) rabbit population model, the rabbits never die. So in month 5 all the pairs of rabbits alive in month 4 are still alive - this is $f_4$ pairs - plus there is a newborn pair born to each pair of rabbits that were alive in month 3 - so there are an additional $f_3$ pairs. Hence

$f_5 = f_4 + f_3$

and, in general

$f_n = f_{n-1}+f_{n-2}$

and the number of newborn pairs born in month $n$ is just $f_{n-2}$.