How to find the number of rabbits after an elapsed time if they increase each time they mate?

Question

The problem is as follows:

Jenny is given by her dad as a birthday gift two female rabbits and two male rabbits. Six months later each female rabbit gives birth three female rabbits and three male rabbits. If we assume that the same happens each six months. How many rabbits as maximum will she have when it has elapsed two years after received them as birthday present?

The alternatives in my book are as follows

$\begin{array}{ll} 1.&\textrm{1024}\\ 2.&\textrm{2048}\\ 3.&\textrm{512}\\ 4.&\textrm{2192}\\ \end{array}$

I often get confused in these sorts of problems. Can someone help me?. Is this related with the famous Fibonnaci sequence

I think the method which would help me the most is an approach which develops from step by step. The part where I'm stuck is how to account for the female rabbits already born? do they mate as well?. This is kind of confusing. Is there powers involved?.

From looking at the alternatives I can see that there are suspiciously powers of two. Why? what's the reason for this?. Because of this part, it would be very helpful for easing my understanding, explaining the basis of the solution.

I think let's suppose if I have two rabbits, and each one makes two, then in the next generation I will have four, and in the next eight, but I don't have an idea how to use this into the problem. All and all, can someone help me to better understand this?

Answer 1

Every six months, the number of rabbits of either gender gets quadrupled. Hence, the number of rabbits of either gender is $2\cdot4^{n}$ at the end of $n$ half years. Total rabbits$=2\cdot2\cdot4^n=4^{n+1}$. For your question, $n=4$.

Answer 2

I think the person who posed the problem meant for you to make the following assumptions:

• No rabbit ever dies during all this time.
• Each female rabbit gives birth every six months, repeatedly, during the entire time described in the problem.
• Just before the end of each six-month period, every female rabbit gives birth to three females and three males.
• When two years have elapsed, Jenny has all the rabbits she had after $18$ months, plus rabbits that were very recently born.

I don't blame you for questioning these assumptions. It is good that you question them; it shows you can think logically. In a better-written problem they would all have been stated explicitly.

A "brute force" way to do the calculation is to keep track of the following numbers:

• The number of female rabbits Jenny has after $n$ months have elapsed.
• The number of male rabbits Jenny has after $n$ months have elapsed.
• The number of female rabbits born in the $n$th month.
• The number of male rabbits born in the $n$th month.

Start with $n=0,$ then $n = 6,$ $n=12,$ $n = 18,$ and finally $n=24.$

For example, after $6$ months have elapsed, Jenny will have the female rabbits she had at the start ($n=0$) plus the female rabbits that were born in the $6$th month. Add those two numbers together, and you have the number of female rabbits Jenny had after $6$ months have elapsed.

As I said, this is a "brute force" method. It's a lot of numbers one after the other. Perhaps after working out the numbers for $n=6$ or $n=12$ you might see a pattern that you can explain and that will let you finish quicker.

Answer 3

For this solution the following is true:

• The population of male rabbits is always the same as the population of female rabbits. This is because every female rabbit gives birth to an equal amount of male and female rabbits and we start with an equal amount of males and females.
• Males have no direct influence on the growth of the population. This is because males don't give birth
• Rabbits never die and females give birth even after they've given it once before.

Since the population of female and male rabbits stays equal and males don't give birth we can just calculate the population of female rabbits and then double it at the end to get the complete population.

The key idea

In the problem it is stated that one female gives birth to 3 other females. This concludes that the the number of new borns is 3 times bigger then the population before 6 months.

Since the old population lives on and gives birth to new rabbits we have to add them to the new borns

Calculating the result

We can transform this into a recursive function $f(t)$ where every $t$ represents $6$ months.

($t=1 => 6months;$ $t=2 => 12months...$) :

$p(t+1) = p(t) + 3*p(t)$

The starting is two females so at $t=0$ the population is $2$:

$p(0) = 2$

This gives us the following results:

$p(1) = p(0) + 2*3 = 8$

$p(2) = p(1) + 8*3 = 32$

$p(3) = p(2) + 32*3 = 128$

$p(4) = p(3) + 128*3 = 512$

...

This however is only the population of female rabbits. As I've stated in the beginning we just have to double the population to also include the male rabbits. This gives us:

$2* p(4) = 2 * 512 = 1024$

So after $4 * 6$ months or $2$ years the population of the rabbits is at $1024$ rabbits.

To explain explain why the population always seems to be a power of 2 we can simply derive the explicit function of the recursive function wich gives us:

$f(t)=2*2*4^t$ or $ f(t)=2*2*(2*2)^t$

Connection to the fibonacii sequence:

As seen in this solution this problem can be solved via a recursive function. The common expression of the FibonaciiSequence is also a recursive function. So they are related in that way.