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Discrete logistic equation, restricted growth model

  1. Aug 26, 2014 #1
    1. The problem statement, all variables and given/known data
    I need to write
    \begin{align*} N_{k+1} = \frac{\lambda N_{k} }{1+aN_{k} } \end{align*}
    in the form
    \begin{align*} N_{k+1} = N_{k} + R(N_{k})N_{k} \end{align*}

    2. Relevant equations
    As above

    3. The attempt at a solution
    I know that
    \begin{align*} N_{k+1} = N_{k} + R(N_{k})N_{k} \end{align*}
    and that
    \begin{align*} R(N_{k}) = -\frac{r}{K}N_{k}+r \end{align*}
    where r is the growth rate, and K is the limiting factor.
    Taking lambdaNk out the front, I get
    \begin{align*} N_{k+1} = \lambda N_{k}\frac{1 }{1+aN_{k} } \end{align*}
    Which looks a lot closer and simple as anything, but what am I trying to get to?


    It all seems simple enough, and I can do the rest of the work required, but I'm just having trouble grasping what the equation should look like and how to get there...
    This is some homework for my IT class on computer modelling, I've done 4/6 questions but having trouble on the first two, this one being the first. There are other questions to this part, but I think I can handle them if I can grasp this bit.
    Any ideas? Help will be greatly appreciated :)
     
    Last edited: Aug 26, 2014
  2. jcsd
  3. Aug 27, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    There is no "r" or "K" in your original formula. You say they are the "growth rate" and "limiting factor" so presumably you have some knowledge of how they affect Nk that you haven't told us.
     
  4. Aug 27, 2014 #3
    Like the title says, this is about the logistic equation. It's a well known model of population growth. How "r" and "K"
    affect it is given in the most basic understanding of the model.
    I've given you just as much information as I've received. "r" and "K" are obviously constants, just as lambda and "a" are also. And like I said, I'm just having trouble rewriting my first equation in the required form to actually get the information I need from it.
     
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