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Math Help - differential equation problem

  1. #1
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    Question differential equation problem

    Hi, sorry this is quite a long question, but struggling to think where to even start!
    Thanks!

    To help determine a possible strategy for future whaling the population is modelled
    mathematically.
    (a) In the absence of any shing occurring the population of humpback whales, Y (t),
    can be approximated as obeying the logistic equation
    dY
    dt= rY (1 - Y/K) :
    If Y (0) = K=3 and Y (t). Hence fi nd the time t =  at which the population has
    doubled from its intial population.

    (b) The model can be extended to take "harvesting" of the whales into account. A
    simple model is to assume that the rate at which whales are caught, called the
    yield, is proportional to the population of whales. Speci cally the yield is taken
    to be EY (where E is a constant determined by the amount of resources devoted
    to catching the whales) and the model is then
    dY
    dt= rY (1 - Y/k) - EY�- Y=K) 􀀀 E
    (This is known as the Schaefer model.)

    Show that if E < r there are two equilibrium points Y = 0 and Y = K(1 - E/r)
    and that the first of these is unstable and the second stable.
    From this solution find the yield (ie: EY ) that will occur after a long time (this
    is called the sustainable yield).
    Find the value of E which gives the maximum sustainable yield (and hence the
    level of harvesting that will result in the maximum number of whales being caught
    on a sustainable basis).
    Comment on what might occur if we take E > r?
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  2. #2
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    Quote Originally Posted by fionais View Post
    Hi, sorry this is quite a long question, but struggling to think where to even start!
    Thanks!

    To help determine a possible strategy for future whaling the population is modelled
    mathematically.
    (a) In the absence of any shing occurring the population of humpback whales, Y (t),
    can be approximated as obeying the logistic equation
    dY
    dt= rY (1 - Y/K) :
    If Y (0) = K=3 and Y (t). Hence fi nd the time t =  at which the population has
    doubled from its intial population.

    (b) The model can be extended to take "harvesting" of the whales into account. A
    simple model is to assume that the rate at which whales are caught, called the
    yield, is proportional to the population of whales. Speci cally the yield is taken
    to be EY (where E is a constant determined by the amount of resources devoted
    to catching the whales) and the model is then
    dY
    dt= rY (1 - Y/k) - EY�- Y=K) �� E
    (This is known as the Schaefer model.)

    Show that if E < r there are two equilibrium points Y = 0 and Y = K(1 - E/r)
    and that the first of these is unstable and the second stable.
    From this solution find the yield (ie: EY ) that will occur after a long time (this
    is called the sustainable yield).
    Find the value of E which gives the maximum sustainable yield (and hence the
    level of harvesting that will result in the maximum number of whales being caught
    on a sustainable basis).
    Comment on what might occur if we take E > r?
    Teachers expect questions that form part of an assessment that contributes towards the final grade of a student to be the work of that student and not the work of others. For that reason, MHF policy is to not knowingly help with such questions.

    Your question presents in such a way as to suggest that it falls in this category. Thread closed.

    You can send me a pm to discuss the situation (but if I'm unconvinced, then the thread will remain closed).
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