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Recurrence relations with rabbits pairs

  1. Jul 9, 2014 #1
    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 [tex]r_{n}[/tex] be the number of pairs of rabbits alive at the end of each month n for each integer [tex]n \ge 1[/tex] find a recurrence relation for [tex]r_{0},r_{1},r_{2}......[/tex]

    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 [tex]r_{n} = a+4a[/tex] where a = number of adults, for [tex]n \ge 1[/tex]

    is that correct?

    and part b) would be 41,015,625 pairs so 82,031,250 rabbits
     
  2. jcsd
  3. Jul 9, 2014 #2

    micromass

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    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##?
     
  4. Jul 10, 2014 #3
    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
  5. Jul 10, 2014 #4
    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
  6. Jul 10, 2014 #5
    | 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
  7. Jul 10, 2014 #6

    Nathanael

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    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 [itex]r_n[/itex]? What do all the adult rabbits have in common?
     
  8. Jul 10, 2014 #7
    I think the recurrence relation is

    [tex]r_{n}=r_{n-2}+4r_{n-2} + r_{n-1} - r_{n-2} [/tex]

    which simplifies to

    [tex] 4r_{n-2} + r_{n-1}[/tex]
     
  9. Jul 10, 2014 #8

    Nathanael

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    Looks good to me!
     
  10. Jul 10, 2014 #9

    micromass

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    Seems ok!
     
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