The Fibonacci-sequence
f(n) = f(n-1) + f(n-2) for n > 2 with f(1) = f(2) = 1 ,
the first recursively defined sequence in human history (Leonardo of Pisa, 1170-1240), should be well known. A pair of rabbits that reproduces itself monthly as from the completed second month on will yield a stock of 144 pairs after the first 12 months
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 .

If we assume that each pair reproduces itself after two months for the first and last time and dies afterwards, we get a much more trivial sequence:

1, 1, 1, ... .

However the rabbits behind these numbers change. If we name them in the somewhat unimaginative but effective manner of the Old Romans, we get Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona, Decima, and so on.

A more interesting question is brought up, if the parent pair dies immediately after the birth of its second child pair. Then the births g(n) in month n can be traced back to pairs who have been born in months n-2 and n-3

g(n) = g(n-2) + g(n-3) .

The number f(n) of pairs in month n is given by those born in month n, i.e., g(n) and those already present in month n-1, i.e., f(n-1), minus those who died in month n (i.e., those who were born in month n-3):

f(n) = g(n) + f(n-1) - g(n-3) = g(n-2) + f(n-1)
g(n-2) = f(n) - f(n-1)
= g(n-4) + g(n-5)
= f(n-2) - f(n-3) + f(n-3) - f(n-4)
= f(n-2) - f(n-4) .

For n > 4 we have with f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 2 .

f(n) = f(n-1) + f(n-2) - f(n-4) .

The number of pairs during the first 12 months is

1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21 .

The sequence grows slower than the original one, but without enemies or other restrictions it will grow beyond every threshold. If we wait ℵo days (or use the trick that, facilitated by genetic evolution, the duration of pregnancy is halved in each step) we will get infinitely many pairs (although the set-theoretic limit of living pairs is empty because for every pair the date of death can be determined) – a nameless number, alas of nameless rabbits, because they cannot be distinguished. The set of all Old-Roman names has been exhausted already, and even all of Peano's New-Roman names S0, SS0, SSS0, ... have been passed over to pairs which already have passed away. (Since infinitely many have passed away during ℵo days, the set of names has been exhausted.) That is amazing, since none of the pairs of the original and much more abundant Fibonacci sequence has to miss a name.

So we obtain from set theory: The cultural assets of distinguishability of distinct objects by symbols, names, or thoughts do not belong to the properties of Cantor's paradise. Like in the book of genesis, before Adam began to name the animals, we have a nameless paradise – not mathematics though.

But this sequence with fatalities can also be obtained without fatalities (killings), namely if each pair has to pause for two months after each birth in order to breed again in the following month. Mathematically, there is no difference. (Pair P, that originally dies away after breeding its second child pair S, takes the position of S and pauses for two months like the fresh pair S would have done.) Set theory, however, yields a completely different limit in this case. The limit set of living rabbits is no longer empty, but it is infinite – and every rabbit has a name.

[W. Mückenheim: "Das Kalenderblatt 120412", de.sci.mathematik (11 Apr 2012)]

On 12/4/2021 3:25 AM, WM wrote:
> The Fibonacci-sequence
>
> f(n) = f(n-1) + f(n-2) for n > 2 with f(1) = f(2) = 1 ,
>
> the first recursively defined sequence in human history (Leonardo of Pisa, 1170-1240), should be well known. A pair of rabbits that reproduces itself monthly as from the completed second month on will yield a stock of 144 pairs after the first 12 months
>
> 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 .
>
> If we assume that each pair reproduces itself after two months for the first and last time and dies afterwards, we get a much more trivial sequence:
>
> 1, 1, 1, ... .
>
> However the rabbits behind these numbers change. If we name them in the somewhat unimaginative but effective manner of the Old Romans, we get Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona, Decima, and so on.
>
> A more interesting question is brought up, if the parent pair dies immediately after the birth of its second child pair. Then the births g(n) in month n can be traced back to pairs who have been born in months n-2 and n-3
>
> g(n) = g(n-2) + g(n-3) .
>
> The number f(n) of pairs in month n is given by those born in month n, i.e., g(n) and those already present in month n-1, i.e., f(n-1), minus those who died in month n (i.e., those who were born in month n-3):
>
> f(n) = g(n) + f(n-1) - g(n-3) = g(n-2) + f(n-1)
> g(n-2) = f(n) - f(n-1)
> = g(n-4) + g(n-5)
> = f(n-2) - f(n-3) + f(n-3) - f(n-4)
> = f(n-2) - f(n-4) .
>
> For n > 4 we have with f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 2 .
>
> f(n) = f(n-1) + f(n-2) - f(n-4) .
>
> The number of pairs during the first 12 months is
>
> 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21 .
>
> The sequence grows slower than the original one, but without enemies or other restrictions it will grow beyond every threshold. If we wait ℵo days (or use the trick that, facilitated by genetic evolution, the duration of pregnancy is halved in each step) we will get infinitely many pairs (although the set-theoretic limit of living pairs is empty because for every pair the date of death can be determined) –

your admitted failure to name the rabbits is noted, with the note sent to your university.

>a nameless number, alas of nameless rabbits, because they cannot be distinguished. The set of all Old-Roman names has been exhausted already, and even all of Peano's New-Roman names S0, SS0, SSS0, ... have been passed over to pairs which already have passed away. (Since infinitely many have passed away during ℵo days, the set of names has been exhausted.) That is amazing, since none of the pairs of the original and much more abundant Fibonacci sequence has to miss a name.
>
> So we obtain from set theory: <snip>
>
> But this sequence with fatalities can also be obtained without fatalities (killings), namely if each pair has to pause for two months after each birth in order to breed again in the following month. Mathematically,

you said mathematically, so supply equations!! supply the equation for the general size of the population given the 10 variables below

>there is no difference. (Pair P, that originally dies away after breeding its second child pair S, takes the position of S and pauses for two months like the fresh pair S would have done.) Set theory, however, yields a completely different limit in this case. The limit set of living rabbits is no longer empty, but it is infinite – and every rabbit has a name.
>
> [W. Mückenheim: "Das Kalenderblatt 120412", de.sci.mathematik (11 Apr 2012)]
>

your premise is wrong.
Rabbits have more than just 2 babies at a time, so rework the problem, that is how Australia got it wrong. Also include randomized litter size, and
gestation time, and randomized death time, and life time distributions, randomized food and/or water availability.

single point solutions are cute, but not useful at all.