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Math Help - population model, so confused

  1. #1
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    population model, so confused

    Suppose that a population model is the logistic model with harvesting term h, a constant.

    dN/dt=rN(1-(alpha)*(N))-h

    r,h,alpha +ve constants

    1.)assume r>4h(alpha).find and classify the steady states N_1 and N_2 where N_1<N_2.

    I can find steady states of normal differential equations and sub in x=whatever when f(x)=0 but i dont have a clue what this is asking of me. any useful sites would be much appreciated also.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    In general if dy/dx = f(y)

    Then the steady state solutions are the zeroes of f(y)

    Let a =alpha

    For dN/dt = rN(1-aN) - h

    Steady states are found when rN(1-aN) - h = 0

    Expanding -arN^2 + rN - h = 0

    Use the good ol' quadratic formula

    N = 1/2a + sqrt(r^2 - 4arh)/2ar

    this will clean up to 1/2a + sqrt(1-4ah/r)/2a

    or N = [1 + sqrt(1-4ah/r)]/2a

    since r > 4ah the solutions are real

    By considering a graph of dN/dt vs N you should be able to determine N1 unstable and N2 stable

    If you want see my web site : http://calculus7.com/id14.html
    Last edited by Calculus26; June 9th 2009 at 06:56 PM.
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