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Homework Help: Integrating for logistic growth model

  1. Feb 6, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi, can anyone help me solve a differential equation for the logistic growth model?

    2. Relevant equations



    It reads:

    M'(t) = M(S-M) + I, where M(t) represents the growth of a biomass. "I" represents immigration (in a coral reef) and there is no breeding.




    3. The attempt at a solution

    I've treated it as a separable differential equation, but get the term

    (integral of) 1/(M(S-M)+I) dM

    A solution is possible using wolfram mathematica, but it doesn't isolate M.

    I'd solve the equation using partial fractions, but the "I" term seems to make it impossible. Am I right in saying that?

    Also, going back a step, can anyone explain how to use partial fractions to derive the general logistic equation. Wikipedia has a section on it: http://en.wikipedia.org/wiki/Partial_fraction

    but there's a step I don't understand, namely: A=B, A=1/M, B=1/M. On what basis was this assumed?

    In trying to solve it myself, I let A=0 and got B= 1/P, then let B=0 and got A=1/(M-P).

    When these figures for A and B are substituted back into the original equation, I get 2/(P(M-P)) .

    Can anyone explain what I'm doing wrong?


    Thanks,

    Darkmisc
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 6, 2010 #2

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    Say you are trying to set up this partial fraction expansion and figure out A and B:

    [tex]\frac 1 {m(s-m)} = \frac A m+ \frac B {s-m}[/tex]

    Add the two fractions on the right to get:

    [tex]\frac 1 {m(s-m)} = \frac {A(s-m)+Bm}{(m)(s-m)}[/tex]

    For these to be equal, the numerators must be equal:

    [tex] 1 = A(s-m)+Bm[/tex]

    If you put m = 0 you get A = 1/s and if you put m =s you get B = 1/s. So your partial fraction expansion becomes:

    [tex]\frac 1 {m(s-m)} = \frac {\frac 1 s} m+ \frac {\frac 1 s} {s-m}[/tex]
     
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