How many rats in one year? If you start with a pair, then each female reproduces 10 babies every 2 months?

Question

Let's assume 50% is female. Can someone work out the formula? We used to have rats, and used to think of this question, but never had it solved. From college math, I believe you use 10 to the power of something... arggh... my math has become so dull

Answer 1

After 2 months- 6 males, 6 females.
After 4 months- 36 males, 36 females.
After 6 months- 216 males, 216 females.
After 8 months- 1296 males, 1296 females.
After 10 months- 7776 males, 7776 females.
After 12 months- 46656 males, 46656 females.
Total after one year- 93312 rats, assuming that none of them die.
This should be right. I think the formula for the total is 2 * 6 to the (n/2) power with n being the number of months.

Answer 2

On the spur of the moment I'm not sure about the formula you are looking for, but I do have the answer - a total of 46, 656 pairs, that is 93,312 Rats @ 12 months.

Use:
(No. of Pair of Rats * 10) + (No. of Pair of Rats * No. of Parents) for each two Months.

@ 2 Months:
(1 x 10) + (1 x 2) = 10 + 2 = 12 = 6 pair.

@ 4 Months:
(6 x 10) + (6 x 2) = 60 + 12 = 72 = 36 pair.

@ 6 Months:
(36 x 10) + (36 x 2) = 360 + 72 = 432 = 216 pair.

@ 8 Months:
(216 x 10) + (216 x 2) = 2160 + 432 = 2592 = 1296 pair.

@ 10 Months:
(1296 x 10) + (1296 x 2) = 12960 + 2592 = 15552 = 7776 pair.

@ 12 Months:
(7776 x 10) + (7776 x 2) = 77760 + 15552 = 93312 = 46,656 pair.
That is, 93,312 Rats.

Answer 3

I seem to recall that answering a similar question (about rabbits rather than rats) 704 years ago led Leonardo of Pisa a.k.a. Fibonacci to discover the Fibonacci series. (0, 1, 1, 2, 3. 5. 8. 13, 21, 34, 55, 89, 144 etc.)

The generating formula for the series is F(n) = F(n−1) + F(n−2). i.e. each term of the series is the sum of the previous two.

I checked with Wikipedia on this and it told me:

"In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202) (the Book of the Abacus). He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:

(a) in the first month there is just one newly-born pair,
(b) new-born pairs become fertile from their second month on
(c) each month every fertile pair begets a new pair, and
(d) the rabbits never die

Let the population at month n be F(n). At this time, only rabbits who were alive at month n−2 are fertile and produce offspring, so F(n−2) pairs are added to the current population of F(n−1). Thus the total is F(n) = F(n−1) + F(n−2)."

COMMENT

It would seem reasonable to go along with Fibonacci's assumptions where they don't contradict your problem.

i.e. baby rats need to grow to maturity before they become fertile and a fixed time needs to be allowed for that eg the two months he suggests.

i.e. no rats ever die: the original breeding pair keep on reproducing throughout the duration of the study.

The new element you have introduced is 10 babies not 2 and we need to see what that means for a Fibonacci-type series.

Answer 4

I got 46656, not sure on the formula yet

After 2 months the 1 female has 10 babies, 5 of which are female. So there are 12 rats total (6 female). Those 6 females have 60 babies in the 4th month. The total is now 72 (36 female). Those 36 have 360 babies making the total 432. Those 216 have 2160 babies making the total 2592.....

Answer 5

I don't think rats mature sexually in less than a year, so you would have
N = 2 + 6*10
N = 62

If you assume instant maturity, you have
N0 = 2
N(2) = 12
N(4) = 72
N(6) = 432
N(8) = 2,592
N(10) = 15,552
N(12) = 93,312
N = 2(1 + 5 + 30 + 180 + 1080 + 6480 + 38880 + . . .)
N = 2(1 + 5(1 + 6 + 36 + 216 + 1296 + 7776 + . . .))
N = 2(1 + 5∑6^k), k = 0 to n
N = 2(1 + 5( 6^(n + 1) - 1)/5))
N = 2(1 + 6^(n + 1) - 1))
N = 2*6^(n + 1), n = {-1, 0 , 1, 2, 3, . . .)
Adjusting to months,
N = 2*6^(m/2), m = integers ≥ 0

Differing maturities will yield differing answers, of course.

Answer 6

To add to this discussion based on the biology of the typical American brown rat, it has a gestation cycle of 21 days and reaches maturity in 5 weeks. so assuming 1 litter a month and every litter is capable of breeding 6 weeks after birth, with 50% female rats born, in ideal condition, how many rats in 1 years time from the initial breeding pair?

Answer 7

2 + 10 + 6x10 + 36 + ...

Now we can see the pattern.

Let a0 =2 be the initial number, a1 be the number after the first reproduction, etc. We have

a1 = a0+(a0/2)(10) = 6 a0
a2 = a1+ (a1/2)(10) = 6^2 a0
...
a6 = 6^6 a0 = 93312 rats (There are 6 reproductions in a year.)

Also, we can find the formula for n times of reproductions,

an = 6^n a0


(The above has been edited.)

Answer 8

I don't know how to say it in real mathematical language but the answer i came up with is 30.
12 made into a half is 6 (every 2 months)
10 baby's and half are girls which is 5.
5x6=30

Answer 9

its going to be in the millions