Rat Population

Date: 03/14/97 at 19:47:23
From: Adam Hilton
Subject: Estimating rat population

Two rats, one male and one female, live on an island and mate on 
January 1. The number of young produced in every litter is 6, and 3 
of those 6 are females. The original female gives birth to 6 young on 
Jan. 1, and produces another litter of six 40 days later and every 40 
days thereafter as long as she lives.

Each female born on the island will produce her first litter 120 days 
after her birth, and then produce a new litter every 40 days 
thereafter. The rats are on an island with no natural enemies and 
plenty of food.  Therefore, in this first year, there are only births 
so no rats die.

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

My mom and I tried to plot out the generations, using a calendar, 
labeling the generations alphabetically.  We came up with 3,590.  I 
have to do a report showing how I did this, start to finish, with 
graphs or anything else I used.  The teacher doesn't really care if I 
get the answer right, she wants to see the math I have used.

Thanks,
Adam Hilton

Date: 03/14/97 at 23:13:30
From: Doctor Steven
Subject: Re: Estimating rat population

This is a problem in regression, which is a complicated way of saying 
that the number of rats at a point in time depends on the number of 
rats at a previous point in time.

We start with 8 rats, so we'll call the number of rats at our first 
point in time N1.  N1 = 8.  (We have 8 because the first pair had a 
litter on Jan 1st.)  

Forty days later, we'll have another 6 rats.  So N2 = 14. Forty days 
after this we'll have another 6 rats, so N3 = 20. Another forty days 
later, our first litter of rats will now be mature so they will be 
contributing to the rat population. Now for each PAIR of mature rats 
we add six, so for EACH rat we contribute only 3. This means that 
N4 = N3 + 6*N(4-3) (the number of rats in the previous time period 
plus 3 times the number of rats three time periods ago).

In general, for a number greater than 4 (denoted by R), we have:

NR = N(R-1) + 3*N(R-3). 

Now the problem is to find how many time periods we go through 
before the next Jan 1st.  Well, there are 365 days in a year so we 
divide this by 40 to get the number of time periods we go through.  
So 365/40 = 9.125.  We only want whole time periods so the last 
increase in rats before Jan 1st next year is the ninth increase, or 
the 10th point in time. So the problem is asking us to find N10:

N1 = 8
N2 = 14
N3 = 20
N4 = N3 + 3*N1 = 20 + 3*8 = 20 + 24 = 44
N5 = N4 + 3*N2 = 44 + 3*14 = 44 + 42 = 86
. 
.
.
N10 = N9 + 3*N7 = ....


The result for N10 is what your teacher wants.  The first three N's 
(N1, N2, N3) are called the initial conditions, because they are not 
found using the formula given for N's greater than 3 (N4, N5, ...).

Hope this helps.

-Doctor Steven,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/