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Math Help - Predict without iterating?

  1. #1
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    Predict without iterating?

    When P0 is greater than .5Q0 Q will reach "0" first and become extinct. If P0 is less than .5Q0 P will reach "0" first and become extinct. However, if P0 = .5Q0, the population will eventually reach an equilibrium with 200 Ps and 400 Qs. How could this be proven without using iteration?

    Population model:

    Pn+1 = Pn -.25Qn + 100

    Qn+1 = -Pn + Q n+ 200
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  2. #2
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    Quote Originally Posted by Mitchell View Post
    When P0 is greater than .5Q0 “Q” will reach "0" first and become extinct. If P0 is less than .5Q0 “P” will reach "0" first and become extinct. However, if P0 = .5Q0, the population will eventually reach an equilibrium with 200 “P’s” and 400 “Q’s.” How could this be proven without using iteration?

    Population model:

    Pn+1 = Pn -.25Qn + 100

    Qn+1 = -Pn + Q n+ 200
    That can be written as the matrix equation \begin{bmatrix} P_{n+1} \\ Q_{n+ 1}\end{bmatrix}= \begin{bmatrix}1 & -.25 \\ -1 & 1\end{bmatrix}\begin{bmatrix} P_n \\ Q_n\end{bmatrix}+ \begin{bmatrix}100 \\ 200\end{bmatrix}

    I would start by finding the eigenvalues and eigenvectors of that matrix.
    Last edited by HallsofIvy; April 28th 2009 at 03:34 AM.
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