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Math Help - Logistic Model with Harvesting

  1. #1
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    Logistic Model with Harvesting

    Logistic Model with Harvesting

    If the growth of a population follows the logistic model but is subject to "harvesting" (such as hunting or fishing), the model becomes



    where h(t) is the rate of harvesting.
    (a) Suppose that the harvest rate h is constant. Determine the maximum sustainable harvest.

    Hint: Set y'=0 and use the quadratic formula to find h_{max}

    I've tried using the hint and setting the equation to 0 then using quad. form., but haven't had any luck on this. If any one could point me in the right direction on what to do, I'd really appreciate it.

    What I have is:
    ry-ry^2/L-h=0
    ry-ry^2/L=h
    But then it says to use for quadratic formula, so I'm not sure if I should be solving for h or what..
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  2. #2
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    This is what I think it is:
    So if the population is increasing and I'm taking away h, how much h can I take away to just balance the amount being added to the population? That means, I'm taking away just enough so that the population is not changing at all. That means the rate \frac{dy}{dt}=0. Solving for that, I get the rate is unchanging when y=\frac{L}{2r}\left(r\pm\sqrt{r^2-4rh/L}\right). Now, what is the largest h can be? Surely, we don't wish imaginary numbers, so how large can h be before that expression in the square root becomes zero?
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