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Math Help - mathematical models

  1. #1
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    mathematical models

    Show that the solution of the logistic equation dN/dt = N(a-bN)
    has an inflection point at a population equal to 1/2 the saturation level. (population level a/b called the saturation level)

    Explain why the sketches in fig 38.2 do not have inflection points
    ( it is a simple logistic growth model but has no inflection points, only a line with only growth and another with only decay)

    Thanks for any help!
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  2. #2
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    Quote Originally Posted by dixie View Post
    Show that the solution of the logistic equation dN/dt = N(a-bN)
    has an inflection point at a population equal to 1/2 the saturation level. (population level a/b called the saturation level)

    Explain why the sketches in fig 38.2 do not have inflection points
    ( it is a simple logistic growth model but has no inflection points, only a line with only growth and another with only decay)

    Thanks for any help!
    Option 1:

    To find points of inflection you solve \frac{d^2 N}{dt^2} = 0 and test the nature of the solution.

     \frac{dN}{dt} = N(a - bN) \Rightarrow \frac{d^2 N}{dt^2} = \frac{d}{dt} \left[N(a - bN)\right] =  \frac{d}{dN} \left[N(a - bN)\right] \cdot \frac{dN}{dt} = (a - 2bN) \cdot \frac{dN}{dt}.

    Therefore 0 = a - 2bN \Rightarrow N = \frac{a}{2b}.

    To justify the existence of a point of inflection at N = \frac{a}{2b} you need to show that \frac{dN}{dt} has a turning at N = \frac{a}{2b}.

    Note that \frac{dN}{dt} \neq 0 (why?)


    Option 2:

    If \frac{dN}{dt} has a turning point then N = N(t) has a point of inflection. The point of inflection occurs at the value of N corresponding to the N-coordinate of the turning point of \frac{dN}{dt}.

    Since \frac{dN}{dt} is a quadratic function it's not difficult to:

    1. establish that \frac{dN}{dt} has a turning point.
    2. find the N-coordinate of the turning point.
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