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Math Help - matrix question

  1. #1
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    matrix question

    The population of rats in my farm moves constantly back and forth between the house and the field. Every week 40% of the rats in the field move to my house while 30% of those in my house move to the field. Suppose that, initially, the distribution of this population is: 70% in the field and 30% in my house.

    (a) Find the migration matrix and set up a difference equation for this situation.
    (b) What is the ratsí distribution after two week in my farm?

    a) I get this migration matrix:

    0.70 0.40
    0.30 0.60

    Is this correct? If so, how do I set up a difference equation?

    b)

    0.70 0.40
    0.30 0.60

    multiply this matrix by:

    0.70
    0.30

    this will give the rats distribution after 1 week, and multiply by that again to get the distribution after two weeks. Is this correct?

    Please can someone give me some guidance on this problem.
    Last edited by dumbledore; December 4th 2012 at 04:29 AM.
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  2. #2
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    Re: matrix question

    Just write what you are doing in matrix form

    \left[ \begin{array}{cc}H\\F \end{array} \right]_{n+1}=\left[ \begin{array}{cc}0.7 & 0.4\\0.3 & 0.6 \end{array} \right] \left[ \begin{array}{cc}H\\F \end{array} \right]_{n}.
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  3. #3
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    Re: matrix question

    Yes I did write it in matrix form, which is why I said "I get this migration matrix" and then wrote it in matrix form.

    Can someone help me with this problem please?
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  4. #4
    MHF Contributor

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    Re: matrix question

    The "difference equations" are just H_{n+1}= .7H_n+ .4Fn and F_{n+1}= .3H_n+ .6F_n or, if you prefer the " \Delta H= H_{n+1}- H_n" and " \Delta F= F_{n+1}- F_n" form then \Delta H= (.7- 1)H+ .4F= -.3H+ .4F and \Delta F= .3H+ (.6- 1)F= .3H- .4F, for each n.
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