# Recurrence relations with rabbits pairs

1. Jul 9, 2014

### jonroberts74

A single pair (male and female) of rabbits is born at the beginning of the year. Assume the following:

1) Each pair is not fertile for their first month bet thereafter give birth to four new male/female pairs at the end of every month

2) no rabbits die

a) let $$r_{n}$$ be the number of pairs of rabbits alive at the end of each month n for each integer $$n \ge 1$$ find a recurrence relation for $$r_{0},r_{1},r_{2}......$$

b) how many rabbits will there be at the end of the year

Month | Babies (in pairs) | Adults (in pairs) | total Pairs (r_{n})
1 |1 |0 |1
_________________________________________________________________
2 |4 |1 |5
________________________________________________________________
3 |20 |5 |25
________________________________________________________________
4 |100 |25 |125
___________________________________________________________
5 |400 |125 |525
_____________________________________________________________
6 |2100 |525 |2625
_____________________________________________________________
7 |10500 |2625 |13125
______________________________________________________________
8 |52500 |13125 |65625
_______________________________________________________________
9 |262500 |65625 |328125
________________________________________________________________
10 |1312500 |328125 |1640625
_________________________________________________________________
11 |6562500 |1640625 |8203125
_________________________________________________________________
12 |328125000 |8203125 |41015625

(sorry for the ugly table)

the recurrence relation seems to be $$r_{n} = a+4a$$ where a = number of adults, for $$n \ge 1$$

is that correct?

and part b) would be 41,015,625 pairs so 82,031,250 rabbits

2. Jul 9, 2014

### micromass

Staff Emeritus
It is not correct. You have now counted the adults and the newborns, but not the rabbits which are now 1 month old.

It is also not in the form they are looking for. You have to express $r_n$ in terms of $r_k$ with $k<n$. So for example, a possible (but obviously incorrect) answer would be $r_n = 100r_{n-1} + 5\sqrt{r_{n-3}r_{n-2}}$.

You express it in terms of $a$. Can you perhaps express $a$ in terms of $r_k$?

3. Jul 10, 2014

### jonroberts74

aren't the rabbits that are 1 month old adults now because it says they are not fertile for the first month then give birth to 4 (male/female) pairs at the end of the month.

or am I incorrect in my computation

should it be

in the beginning theres

1 pair born, then the first month they have no births then the second month they make 4 pairs.

then the 3rd month: the original pair breeds 4 , the 4 pairs from the last month grow to maturity for a total of 9 pairs

then in month 4 theres 5 pairs that can breed, 4 pairs that mature, and 20 pairs born

Last edited: Jul 10, 2014
4. Jul 10, 2014

### jonroberts74

adult pairs = adults of previous month + maturing pairs of previous month

maturing pairs = baby pairs from previous month

baby pairs = 4(adult pairs from previous month + maturing pairs from previous month)

Last edited: Jul 10, 2014
5. Jul 10, 2014

### jonroberts74

| month | Adults | babies | maturing | total |
| ----- | ------ | ------ | --------- | ----- |
| 1 | 0 | 0 | 1 | 1 |
| 2 | 1 | 4 | 0 | 5 |
| 3 | 1 | 4 | 4 | 9 |
| 4 | 5 | 20 | 4 | 29 |
| 5 | 9 | 36 | 20 | 65 |
| 6 | 29 | 116 | 36 | 181 |
| 7 | 65 | 260 | 116 | 441 |
| 8 | 181 | 724 | 260 | 1165 |
| 9 | 441 | 1764 | 724 | 2929 |
| 10 | 1165 | 4660 | 1765 | 7589 |
| 11 | 2929 | 11716 | 4660 | 19305 |
| 12 | 7589 | 30356 | 11716 | 49661 |

Last edited: Jul 10, 2014
6. Jul 10, 2014

### Nathanael

Your table seems correct to me (I haven't checked it to the end)

Try to avoid using "a" (the number of adults), it's unnecessary.

How could you write it in terms of $r_n$? What do all the adult rabbits have in common?

7. Jul 10, 2014

### jonroberts74

I think the recurrence relation is

$$r_{n}=r_{n-2}+4r_{n-2} + r_{n-1} - r_{n-2}$$

which simplifies to

$$4r_{n-2} + r_{n-1}$$

8. Jul 10, 2014

### Nathanael

Looks good to me!

9. Jul 10, 2014

### micromass

Staff Emeritus
Seems ok!