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Math Help - Population recurrence relation problem.

  1. #1
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    Population recurrence relation problem.


    3. The populations of two somewhat interacting biological organisms Xand Y, generation
    after generation, follow the pattern

    X(n+1) = 6Xn - 4Yn, Y(n+1) = 2Xn,

    in which Xn
    and Yn are the populations of the nth generation of Xand Y, respectively.

    If X0=Y0=1
    , find out the populations Xnand Ynfor all nís, the positive integers.
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  2. #2
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    By substituting y_n=2x_{n-1} into the equation for x_{n+1}, one can express x_{n+1} through x_n and x_{n-1}. The result is similar to the equation for the Fibonacci numbers. Such equations are formally called second order linear homogeneous recurrence relations with constant coefficients. For a method of solving such equations, see this Wikipedia page.
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  3. #3
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    Umm

    I got eigenvalues of 4,2 and stuck again ...
    Can you help me ~~ ??
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  4. #4
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    As the Wikipedia page says, x_n=A\cdot2^n+B\cdot4^n for some constants A, B. Now, according to the problem statement, x_0=1 and y_0=1, so x_1=6x_0-4x_0=2. Substituting 0 and 1 for n in the first equation, we get 1 = A + B and 2 = 2A + 4B. From here, I get A = 1, B = 0.
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  5. #5
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    A quick observation - if xn = yn => x(n+1) = y(n+1) and hence by induction true for all 'n'
    so in our case x0 = y0 hence xn = yn = 2^n

    Is this correct?
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  6. #6
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    Yes, nice observation.
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