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Solving the logistic growth model

  1. Oct 4, 2007 #1
    The logistic growth model is the following:

    dx/dt=rx(1-x/K), with r and K and as constants, and x is a function of t.

    I'm really not sure where to begin. First I tried separation of variables, but that didn't work out (and I don't even know if I was doing it right). Should I even be looking for an integration factor in solving this? It looks simple....but I guess I'm rusty in this.

    The end result is supposed to be:

    x(t)=K/(1+ce^-rt) c=[K-x(0)]/x(0)

    Second, I tried deriving this equation and getting it to look like the previous equation, but I think I'm missing somethings.

    So just a tip or hint that can push me down the right track would be great. Thanks a lot!
     
  2. jcsd
  3. Oct 4, 2007 #2

    Dick

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    Separation of variables is the right way to go. It gives you an x integral you can do easily by partial fractions. The rest is algebra. Get started and if you get stuck let us know.
     
  4. Oct 4, 2007 #3
    So separation of variables in this case would be A(t)dt+B(x)dx=0. So I would have dx/rx=(1-x/K)dt. Then, I should integrate both sides? Is this the right track?
     
  5. Oct 4, 2007 #4

    Dick

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    Put ALL of the x's on one side with the dx.
     
  6. Oct 8, 2007 #5
    So I get dx/(rx(1-(x/K))=dt. Then I should use partial fractions to integrate?
     
  7. Oct 8, 2007 #6

    Dick

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    Exactly. Use partial fractions.
     
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