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Math Help - Logistic Equation Help

  1. #1
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    Logistic Equation Help

    dy/dt = r(1-y/K)y - Ey

    r, K, and E are constants.

    This is a logistic model of fish population.

    a) Show that if E < r, then there are two equilibrium points y1 = 0
    and y2 = K(1 - E/r) > 0

    b) Show that y = y1 is unstable and y = y2 is asymptotically stable.

    c)A sustainable yield Y of the fisher is a rate at which fish can be caught
    indefinitely. It is the product of the effort E and the asymptotically stable
    population y2. Find Y as a function of the effort E.

    d) Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.


    Any hints of helps are appreciated. I'm having a real hard time with this .
    Thanks in advance!!
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  2. #2
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    Quote Originally Posted by aznstyles408 View Post
    dy/dt = r(1-y/K)y - Ey

    r, K, and E are constants.

    This is a logistic model of fish population.

    a) Show that if E < r, then there are two equilibrium points y1 = 0
    and y2 = K(1 - E/r) > 0

    Mr F says: By definition, solve 0 = r(1-y/K)y - Ey.

    b) Show that y = y1 is unstable and y = y2 is asymptotically stable.

    c)A sustainable yield Y of the fisher is a rate at which fish can be caught
    indefinitely. It is the product of the effort E and the asymptotically stable
    population y2. Find Y as a function of the effort E.

    d) Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.


    Any hints of helps are appreciated. I'm having a real hard time with this .
    Thanks in advance!!
    ..
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  3. #3
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    yea thanks for the help. About 2 mins after I made the post I realized that's all I had to do was make dy/dt = 0.

    I'm still having a hard time with D though.
    Where they saying determine E so that Y = EK(1-r/K) is maximized.
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  4. #4
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    Quote Originally Posted by aznstyles408 View Post
    yea thanks for the help. About 2 mins after I made the post I realized that's all I had to do was make dy/dt = 0.

    I'm still having a hard time with D though.
    Where they saying determine E so that Y = EK(1-r/K) is maximized. Mr F asks: Where has this expression for Y come from?
    "A sustainable yield Y ..... is the product of the effort E and the asymptotically stable population y2."

    So Y = E y2.

    "y2 = K(1 - E/r)"

    So clearly Y = KE(1 - E/r).

    This is a parabola with a maximum turning point. It's maximum occurs halfway between it's two E-intercepts ......
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  5. #5
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    The expression for Y was given in question c. Where it states that Y
    is equal to the product of E and y2. So Y = Ey2. And since y2 = K(1 - r/k), we can substitute in and get Y = EK(1 - r/K). And they ask to determine E so that it maximizes Y.
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  6. #6
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    thank you so much for all the help!!!
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  7. #7
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    ahhh!! it's actually Y = EK(1 - E/r). Sorry for the mistake
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