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Math Help - Lotka-Volterra Equations

  1. #1
    Member Maccaman's Avatar
    Joined
    Sep 2008
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    85

    Unhappy Lotka-Volterra Equations

    Hi everyone,
    I have 2 lotka-volterra equations

     x' = ax(1 - x) - bxy
     y' = cy(1 - y) - dxy

    where x and y is the population of the species normalised by the carrying capacity of the ecosystem. a,b,c,d are positive constants.

    How do I determine the equilibrium points of the system, and whether or not these points are stable?
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    To find the equilibrium points solve:

    ax(1-x)-bxy=0
    cy(1-y)-dxy=0

    simultaneously. Next need to linearize them and then solve for the eigenvalues of the resulting matrix. The form of the eigenvalues determines the type of equilibrium point. May I suggest "Differential Equations" by Blanchard, Devaney, and Hall". I think it does a nice job of explaining these in great detail at a confortable level. I'll try and work on it today and see what I get.
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