Rat Population

Date: 04/27/99 at 14:09:33
From: Kimberly Cook
Subject: Rat population 

This is a word problem.  Help!

Two rats, one male and one female, scampered on board a ship anchored 
at a local dock.

The ship set sail across the ocean. When it reached a deserted island 
in late December, the two rats abandoned the ship to make their home 
on the island.

Under these ideal conditions, estimate the number of offsping produced 
from this pair in one year. Make the following assumptions:

1. The number of young produced in every litter is six, and three of 
   those six are females.

2. The original female gives birth to six young on January 1, and 
   produces another litter of six 40 days later and every 40 days 
   thereafter as long as she lives.

3. Each female born on the island will produce her first litter 120 
   days after her birth, and will then produce a new litter every 40 
   days thereafter.

4. The rats have no natural enemies and plenty of food; therefore, in 
   this first year, there are only births, no deaths.

What will be the total number of rats by next January 1, including the 
original pair?

My write-up 

I know the original female rat will only produce 54 offspring (or nine 
litters in a year), because if you divide 365 days by 40 day cycles 
you will see she can only have 9 cycles a year.  9 cycles x 6 rats = 
54 rats. I know half of the 54 will be females, 27 female rats.

27 female rats with 245 days to produce. If you take 365 days in a 
year - 120 days before first litter = 245 days left in the year to 
produce. 245 days divided by 40 day cycles = 6 cycles. 6 cycles x 6 in 
a litter x 3 rats from the Jan. 1 litter =  108 rats. Half are female; 
54 female rats.

Then I am lost.

Date: 04/27/99 at 17:45:20
From: Doctor Tom
Subject: Re: Rat population 

Hi Kimberly,

Here's how I'd go about it. If we look at the population every 40 
days, there is the first birth (on Jan 1) and then 9 more births.  
There are 4 kinds of rats - newborns, rats 40 days old, rats 80 days 
old, and adults (breeders).

We can make a list like this to show the populations. After the births 
on January first:

(6,0,0,2) -- 6 newborns, 0 40-day-olds, 0 80-day-olds, and 2 breeders.

If the list looks like this:

(w,x,y,z) -- w newborns, x 40-day-olds, etc., then at the next period, 
there will be:

(3z, w, x, y+z) rats, right? Each z makes 3 babies (2 parents make 6,
so each makes three), everybody advances to the next age group, except
the 80-day olds become breeders, and the breeders remain breeders.

Let's call Jan 1 the zero-th breeding, and there are 9 more:

     0:  (6,0,0,2)
     1:  (6,6,0,2)
     2:  (6,6,6,2)
     3:  (24,6,6,8)
     4:  (42,24,6,14)
     5:  (60,42,24,20)
     6:  (132,60,42,44)
     7:  (258,132,60,86)
     8:  (438,258,132,146)
     9:  (834,438,258,278)

The grand total is 834 + 438 + 258 + 278 = 1808

Maybe. I did all the calculations above in my head so you should check 
them. I think most are right, but you're the one who's getting the 
grade. :^)

There's a fast way to do this based on matrix multiplication. Look up 
"Leslie matrix" in any book on population biology.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/