Simple population growth problem

[process91]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I got part (a) as follows:

[tex]\frac{dP}{dt} = P - \frac{365}{100}P[/tex]

[tex]P = 365e^{-2.65t}[/tex]

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

[tex]\frac{dF}{dt} = \frac{365}{100}P[/tex]

[tex]\frac{dF}{dt} = \frac{365^2}{100}e^{-2.65t}[/tex]

[tex]F = \frac{-365^2}{265}e^{-2.65t}+\frac{365^2}{265}[/tex]

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 [tex]365(1-e^{-2.65t})[/tex] fatalities in t years, an answer they obviously got by setting [tex]\frac{dF}{dt}=\left(\frac{365}{100}-1\right)P[/tex]. 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?

 

[HallsofIvy]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I got part (a) as follows:

[tex]\frac{dP}{dt} = P - \frac{365}{100}P[/tex]

[tex]P = 365e^{-2.65t}[/tex]

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

[tex]\frac{dF}{dt} = \frac{365}{100}P[/tex]

[tex]\frac{dF}{dt} = \frac{365^2}{100}e^{-2.65t}[/tex]

[tex]F = \frac{-365^2}{265}e^{-2.65t}+\frac{265}{365^2}[/tex]

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

? At t= 0, that becomes
[tex]F= -\frac{365^2}{265}+ \frac{265}{365^2}[/tex]
which is NOT 0.
Did you mean to write
[tex]F = \frac{-365^2}{265}e^{-2.65t}+\frac{365^2}{265}[/tex]

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

What I would have done is calculate the population if there were NO deaths. That would be, of course, the solution to dP/dt= P and so would be [itex]P(t)= 365e^{t}[/itex] (of course, the population was "growing by a factor of e each year"). Subtracting the actual population from the population if there had been no deaths gives the total dead.

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?

 

[process91]

? At t= 0, that becomes
[tex]F= -\frac{365^2}{265}+ \frac{265}{365^2}[/tex]
which is NOT 0.
Did you mean to write
[tex]F = \frac{-365^2}{265}e^{-2.65t}+\frac{365^2}{265}[/tex]

Yes I did - I fixed it above. Thanks for pointing it out.

What I would have done is calculate the population if there were NO deaths. That would be, of course, the solution to dP/dt= P and so would be [itex]P(t)= 365e^{t}[/itex] (of course, the population was "growing by a factor of e each year"). Subtracting the actual population from the population if there had been no deaths gives the total dead.

I'm not sure that works (please correct me if I am wrong). I did think about that, but came to the conclusion that the population calculated without deaths minus the population with deaths would be larger than the total number of fatalities. The reason is that the people who died only count as one total death, but their death affects the total population and therefore indirectly decreases the population as well.

Anecdotally, if someone dies they can't have children and therefore calculating the total population without the death and subtracting the total population with the deaths inherently includes the number of "forgone" children along with the total dead.

 

[process91]

Sorry to do this, but *bump*.

 

[process91]

I ended up determining that my solution was correct, and the book's was incorrect through inspection of the graphs of all three functions. Just wanted to post that here in case anyone else had been following.