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Math Help - Population Dynamics

  1. #1
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    Jul 2009
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    Population Dynamics

    I'm having some trouble with a Differential Equations Population problem.

    Here is the background for the question..

    The DE governing a fish pop. P(t) with harvesting proportional to the population is given by:
    P'(t)=(b-kP)P-hP
    where b>0 is birthrate, kP is deathrate, where k>0, and h is the harvesting rate. Model assumes that the death rate per individual is proportional to the pop. size. An equilibrium point for the DE is a value of P so that P'(t)=0.

    I worked out the integral to be the following:

    1 / [(b-h)-kP]P dp = dt

    equals:
    (lnP - ln(b-h-kP)) / (b-h) + C

    However, I'm having trouble answering this part of the question..
    Determine h so that Y is maximized, and find this Y. This is the maximum sustainable yield.

    How would I go about solving that part? Any help would be greatly appreciated.
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  2. #2
    Newbie
    Joined
    Jul 2009
    Posts
    2
    Quote Originally Posted by penney21 View Post
    I'm having some trouble with a Differential Equations Population problem.

    Here is the background for the question..

    The DE governing a fish pop. P(t) with harvesting proportional to the population is given by:
    P'(t)=(b-kP)P-hP
    where b>0 is birthrate, kP is deathrate, where k>0, and h is the harvesting rate. Model assumes that the death rate per individual is proportional to the pop. size. An equilibrium point for the DE is a value of P so that P'(t)=0.

    I worked out the integral to be the following:

    1 / [(b-h)-kP]P dp = dt

    equals:
    (lnP - ln(b-h-kP)) / (b-h) + C

    However, I'm having trouble answering this part of the question..
    Determine h so that Y is maximized, and find this Y. This is the maximum sustainable yield.

    How would I go about solving that part? Any help would be greatly appreciated.
    Moderator edit: This question has been posted and answered elsewhere. Thread closed.
    Last edited by mr fantastic; July 27th 2009 at 08:34 PM.
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