Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

chambek

  • 3 years ago

Let's say there is a stable population size Q that p(t) approaches as time passes. Thus the speed at which the population is growing will approach zero as the population size approaches Q. One way to model this is via the differential equation p' = kp(Q-p) p(0) = A. The solution of this initial value problem is p(t) =

  • This Question is Closed
  1. hewsmike
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If \[\frac{dp}{dt}=kp(Q-p)\]then\[\frac{dt}{dp}=\frac{1}{k}\frac{1}{p(Q-p)}=\frac{1}{k}[\frac{1}{Q(Q-p)}+\frac{1}{Qp}]\]thus integrate both sides with respect to p\[t=\frac{1}{kQ}[ln(p)-ln(Q-P)]=\frac{1}{kQ}ln(\frac{p}{Q-p})\]so\[kQt = ln(\frac{p}{Q-p})\]\[e^{kQt}=\frac{p}{Q-p}\]or\[p=\frac{Qe^{kQt}}{1+e^{kQt}}\]at t = 0\[p(0) = A = \frac{Q 1}{1+1}=Q/2\]or Q = 2A, so\[p(t)= \frac{2A e^{kQt}}{1+e^{kQt}}\]

  2. chambek
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ohhh i see where i went wrong, thanks for the help!

  3. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy