1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Differential Equations - Population Dynamics

  1. Jul 26, 2009 #1
    1. The problem statement
    The DE governing a fish pop. P(t) with harvesting proportional to the population is given by:
    P'(t)=(b-kP)P-hP
    where b>0 is birthrate, kP is deathrate, where k>0, and h is the harvesting rate. Model assumes that the death rate per individual is proportional to the pop. size. An equilibrium point for the DE is a value of P so that P'(t)=0.

    Find general solution of the DE, when..
    a) h>b
    b) h=b
    c) h<b

    3. The attempt at a solution
    I'm having problems figuring out how to set up parts a) and c). I'm horrible at DE, so if any one could help point me in the right direction, it would be greatly appreciated.
     
  2. jcsd
  3. Jul 26, 2009 #2

    rock.freak667

    User Avatar
    Homework Helper

    P'(t)=(b-kP)P-hP
    P'(t)=bP-kP2-hP
    P'(t)= (b-h)P-kP2

    P'(t)= ((b-h)-kP)P

    P'(t)= dP/dt

    so put it in the form

    f(P) dP= f(t) dt

    then integrate both sides.
     
  4. Jul 26, 2009 #3
    okay, i worked the integral of dp/dt = ((b-h)-kP)P out as...



    1/(b-h) * ln(p/b-h-kP) + C = t

    is that correct?
     
  5. Jul 26, 2009 #4

    djeitnstine

    User Avatar
    Gold Member

    Umm you have a [tex]P^2[/tex] in there. You should try partial fractions.
     
  6. Jul 26, 2009 #5
    I did use partial fractions.

    1/((b-h)-kP)P dp Let a = b-h

    integral of 1/(a-kP)P = integral of A/a-kP + B/P

    Solved for A & B, A = k/a, B = 1/a

    So integral (k/a)/(a-kP) + (1/a)/p

    end up with -(1/a)ln(a-kP) + (1/a)lnP
    ==> 1/a ln(P/(a-kP)) + C

    sub back a = b-h

    1/(b-h) * ln(p/(b-h-kP)) + C

    Did I do something wrong?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Similar Discussions: Differential Equations - Population Dynamics
  1. Differential Equation (Replies: 12)

  2. Differential Equations (Replies: 1)

Loading...