Population Modeling using DE's

  • Thread starter prace
  • Start date
  • #1
102
0
Hi,

I was wondering if anyone out here on a Friday night could help me understand population modeling. Here is what I have as a problem (this is pretty simple because my goal here is to understand the thinking behind the madness :rolleyes: )

-
The population of a certain community is known to increase at a rate proportional to the number of people present at any time. If the population has doubled in 5 years, how long will it take to triple, to quadruple?
-

So I understand that the rate = dy/dt and proportional translates to "something" = "some constant" times "something", or (dy/dt)=Ky, or in this case, dP/dt = kP.

Solving this DE I get the equation P(t)= P(initial)e^(kt). So all I am told is that the population doubles in 5 years. So what can I do? I can't assume an arbitrary number as the initial population can I? So I have one equation with three unknowns, P(initial), P(t), and k.

Any insight would be great. Thanks!!
 

Answers and Replies

  • #2
1,235
1
you know that [tex] \frac{dP}{dt} = kP [/tex]. The solution is:

[tex] P(t) = P_{0}e^{kt} [/tex]

You also know that [tex] 2P_{0}= P_{0}e^{5k} [/tex]. What can you do from here? Solve for k:

[tex] e^{5k} = 2 [/tex]

[tex] k = \frac{\ln 2}{5} [/tex].



Then solve the equations for t:

[tex] 3P_{0} = P_{0}e^{(\frac{\ln 2}{5})t} [/tex]
[tex] 4P_{0} = P_{0}e^{(\frac{\ln 2}{5})t} [/tex]
 
Last edited:
  • #3
102
0
Oh man... So here the [tex] P_{0} [/tex]'s will cancel and all I have to do is take the natural logs of both sides to get t? man... I think I was making it a lot harder than it really was. Thank you for taking the time to show me this.
 

Related Threads on Population Modeling using DE's

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
2
Replies
34
Views
4K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
21
Views
5K
  • Last Post
Replies
2
Views
1K
Top