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Math Help - Char. Eq.

  1. #1
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    Char. Eq.

    Given:

    M = ([0, f_2, f_3], [s_1, 0, 0], [0, s_2, 0])

    (Note: [0, f_2, f_3] is the 1st row)

    1.) Determine the characterist equation.

    2.) Given f_2 = 2, f_3 = 5, s_2 = 0.8, how large must s_1 be to get L >= 1 (L is lambda)

    3.) Given f_2 = 2, f_3 = 5, s_1 = 0.2, s_2 = 0.8, determine L with three-decimal-digit accuracy

    4.) What's the stable population ratios for the preceding parameter values
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  2. #2
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    I hope I didn't make an error in my calculations, but I got a charpoly of:

    s_{1}s_{2}f_{3}+{\lambda}f_{2}s_{1}-{\lambda}^{3}

    If you set the parameters accordingly, when s_{1}=\frac{1}{6}, then \lambda=1
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  3. #3
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    Hmm, yes 1/6 is correct I think, but I don't see how you get it
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by flash101 View Post
    Given:

    M = ([0, f_2, f_3], [s_1, 0, 0], [0, s_2, 0])

    (Note: [0, f_2, f_3] is the 1st row)

    1.) Determine the characterist equation.

    2.) Given f_2 = 2, f_3 = 5, s_2 = 0.8, how large must s_1 be to get L >= 1 (L is lambda)

    3.) Given f_2 = 2, f_3 = 5, s_1 = 0.2, s_2 = 0.8, determine L with three-decimal-digit accuracy

    4.) What's the stable population ratios for the preceding parameter values
    The characteristic polynomial is given by the equation:
    det(A - \lambda I) = 0

    So we have the matrix
    A = \left ( \begin{array}{ccc} 0 & f_2 & f_3 \\ s_1 & 0 & 0 \\ 0 & s_2 & 0 \end{array} \right )

    Thus the characteristic polynomial will be given by:
    \left | \begin{array}{ccc} -\lambda & f_2 & f_3 \\ s_1 & -\lambda & 0 \\ 0 & s_2 & -\lambda \end{array} \right | = 0

    -\lambda ^3  + f_2s_1 \lambda + f_3s_1s_2 = 0

    -Dan
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  5. #5
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    How do you get the s_1 = 1/6 though?
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by flash101 View Post
    How do you get the s_1 = 1/6 though?
    plug in the given values and solve for s_1. you were given the values of f_2 ...etc in the questions
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