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General solution(s) to Logistic model with harvesting?

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi guys! I am trying to show the limiting population for this model:

    2. Relevant equations
    P'(t) = P(S - P) + H

    The harvesting, is actually immigration, so its a positive unknown. I have shown this before (without harvesting) by solving for P(t) using separation of variables. Then taking the limit t -> infinity. However i can't work this out, without an actual value for H. Problem is there isn't one!

    3. The attempt at a solution

    So far i attempted to change the formula to use separation and method of partial fractions:

    P'(t) = [ (S + sqrt{S^2 + 4H} - 2P)/2 ] [ (2P- S - sqrt{S^2 + 4H} )/2 ]

    then (2AP - AS - Asqrt{S^2 + 4H} + BS + Bsqrt{S^2 + 4H} - 2BP)/2

    is the numerator in my attempt at a partial fraction. Here's about as far is i get. Assuming S^2 + 4H is positive i can't seem to group to find a partial fraction. All i can obviously see is P(A - B) = 0

    Any help would be greatly appreciated.
  2. jcsd
  3. Feb 17, 2009 #2
    I managed to solve it, after a few more hours :)

    I just subbed in variables for the equilibria, rather than leave them in square root form. Taking the limit of t -> infinity showed population approaching the equilibrium value which made

    P''(equilibrium_value) < 0
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