Expert: Paul Klarreich - 11/30/2008

Question
Certain animal, current population 350, is growing 3% each year if left to itself, but hunters kill 15 of the animal each year.

1.   generate a difference equation which describes the population change year to year
2.   How many will there be in 15 years?
3.   Find the limit when the year tends to infinity
4.   Will the population eventually double? If so, how long this takes?
5.   Will the population die out? If so, how long this takes?  

Answer
Questioner:   Dag
Category:  Advanced Math
Private:  No
 
Subject:  Population growth or Decay
Question:  Certain animal, current population 350, is growing 3% each year if left to itself, but hunters kill 15 of the animal each year.

1. generate a difference equation which describes the population change year to year
2. How many will there be in 15 years?
3. Find the limit when the year tends to infinity
4. Will the population eventually double? If so, how long this takes?
5. Will the population die out? If so, how long this takes?
...............................................
Hi, Dag,

It will be hard for me to answer without knowing just what you are studying, but I'll try.  Did you read the instructions?

Always start with this word:

LET  k = the number of years from the start.
    P(k) = current population in year k, with P(0) = 350

1. generate a difference equation which describes the population change year to year

This means write  P(k+1) = something using P(k)

Now the growth means that you should have  P(k) *1.03.
But the hunting means you lose 15:

So P(k+1) = 1.03 P(k) - 15

I shall write  r = 1.03, to save typing.

Let's write out a few years' worth, and see if we deduce a pattern:

P(0) = 350

P(1) = 350r - 15

p(2) = (350r - 15)r - 15 = 350r^2 - 15r - 15

P(3) = ((350r - 15)r - 15)r - 15 = 350r^3 - 15r^2 - 15r - 15

p(k) = 350r^k - 15r^k-1 - 15r^k-2 - ... - 15r - 15
    = 350r^k - 15(r^k-1 +r^k-2 + ... + r + 1)

Now the sum inside the () is a geometric series with a known formula:
                            1 - r^k     r^k - 1
r^k-1 + r^k-2 +... + r + 1 = -------- or -------
                             1 - r       r - 1

So our final formula is:
                  r^k - 1
P(k) = 350r^k - 15 --------
                   r - 1

From this, you should be able to answer the other questions.