# Simple population growth problem

• Sep 27th 2011, 05:51 PM
process91
Simple population growth problem
I'm at my wits end, I don't understand why I'm having so much trouble, I think I know the answer and then it never matches the book.

At the beginning of the Gold Rush, the population of Coyote Gulch, Arizona was 365. From then on, the population would have grown by a factor of e each year, except for the high rate of "accidental" death, amounting to one victim per day among every 100 citizens. By solving an appropriate differential equation, determine, as function of time (a) the actual populations of Coyote Gulch t years from the day the Gold Rush began, and (b) the cumulative number of fatalities.

I got part (a) as follows:

$\frac{dP}{dt} = P - \frac{365}{100}P$

$P = 365e^{-2.65t}$

Where the 365 comes from the initial condition in the problem. This answer agrees with the book.

For part (b), I simply considered (a bit morbidly) the dead people as another population. Let F be the number of dead people at time t. Then

$\frac{dF}{dt} = \frac{365}{100}P$

$\frac{dF}{dt} = \frac{365^2}{100}e^{-2.65t}$

$F = \frac{-365^2}{265}e^{-2.65t}+\frac{265}{365^2}$

Where the fraction on the right comes from the initial condition that there are no fatalities at t=0.

The book, on the other hand, has that the answer is $365(1-e^{-2.65t})$ fatalities in t years, an answer they obviously got by setting $\frac{dF}{dt}=\left(\frac{365}{100}-1\right)P$. My question is, why? I'm having an interpretation issue here.

In my interpretation, the new population of interest is the dead people. They grow at a rate of 365/100 P per year. They don't "undie", or in any way get removed from the population, so why do you include the growth factor of the original population as your "death" factor here?

I can't seem to edit this post anymore, so I will have to post an addendum:

My answer of F above has a typo. It should be

$F=\frac{-365^2}{265} e^{-2.65t} + \frac{365^2}{265}$

This doesn't change my question or the fact that it doesn't match the book's answer.
• Oct 3rd 2011, 05:09 AM
Ackbeet
Re: Simple population growth problem
I think I would agree with your (final, including the fixed typo) answer, lock, stock, and barrel. If you had to include the growth factor of the population, you'd have already done that by plugging in the solution of the P differential equation into the F differential equation. The expression for P includes the growth and death factors.

So that's my opinion, though I could be missing something. I've pondered your problem over a few days! It's not a trivial interpretation problem.
• Oct 3rd 2011, 06:25 AM
tom@ballooncalculus
Re: Simple population growth problem
Don't they both have problems?

http://www.ballooncalculus.org/draw/graph/goldrush.png

I thought maybe the book isn't solving a DE at all, just subtracting the decline from the initial pop. But that doesn't allow any births. (I think?)

But your F never approaches the decline in P.
• Oct 5th 2011, 09:34 AM
process91
Re: Simple population growth problem
That graph seems pretty convincing, although shouldn't my F function approach 365^2/265 as t increases? Something seems off there...

Aside from that, focusing on the graph of the answer from the book (in green) it seems that they perhaps did subtract the decline from the initial population. I think you are correct, however, that doing so does not allow any births, since the book's answer approaches 365 as time increases, however this does not match the intuition that more people will have died over that span of time than just the initial population due to more people who were born.

Here's a graph which, I believe, correctly plots my function vs the population increase function. Intuitively, this one seems to make sense. More people die than the total population over time, and the increase/decrease seems to line up appropriately.

http://i55.tinypic.com/6gem9d.jpg
• Oct 5th 2011, 09:52 AM
process91
Re: Simple population growth problem
I just wanted to say thanks to Ackbeet as well, for all the time you have spent on it. I am working through this book on my own, so it is very helpful to have somewhere to turn with questions.
• Oct 5th 2011, 10:12 AM
tom@ballooncalculus
Re: Simple population growth problem
Quote:

Originally Posted by process91
shouldn't my F function approach 365^2/265 as t increases?

Yes of course it should, and if I'd had my wits about me I'd have checked my graph set up, which should have rendered it thus:

http://www.ballooncalculus.org/draw/graph/goldrusha.png

... just like yours. Sorry!