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IVP involving Eigenvalues and Population Dynamics

  1. Oct 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the interaction of two species of animals in a habitat. We are told that the change of the populations and can be modeled by the equations

    [itex]\frac{dx}{dt}[/itex]= 0.1x-0.8y
    [itex]\frac{dy}{dt}[/itex]=-0.2x+0.7y

    Find the solution to the above equations with initial values x(0)=9 and y(0)=7
    x(t)=
    y(t)=

    3. The attempt at a solution

    We are asked to compute the eigenvalues and I know then you have to find the corresponding eigenvectors. I did this but i am not sure I'm doing it correctly.
    This was my process: p.s. not sure how to use a template for a matrix so i'm just going to wing it...

    (A-λ)V = (R1) 0.1-λ -0.8
    (R2) -0.2 0.7-λ

    So I compute the characteristic polynomial and get λ2-0.8λ-0.9 which gives some pretty funky values for λ...

    I actually found the solutions to this exact question, except with different coefficients and it seems like there is a formula I am missing in order to get the particular solution for y(t).
    The general solution is supposed to look like (I'll do this using the vector Y=(x,y) for simplicity):
    Y=k1eλ1t*V1+k2eλ2t*V2
    where V is the eigenvector.

    The fishy part comes when I go to plug in the IV's and solve for the constants k1 and k2. It seems like for y(t) they are using an equation that looks like
    y(t)= k11-a)eλ1t+k22-a)eλ2t
    where a is the first coefficient in the dx/dt equation...or something along those lines...
    If anyone knows what i'm missing here, please just tell me straight up whats going on. I've been investigating this for two days and its due tomorrow at 2pm. I have a feeling it has to do with the fact that my coefficients are very small. Also, it's a population model so maybe it has something to do with the interactions of the two species..?
    Let me know if you want more information. any help is really appreciated.
    Thanks


    ****Click on the link to see the same question (different coefficients) including the answers. Scroll down to page 7, question#2 and you'll see the complete question. http://math.la.asu.edu/~cheng/MAT274F2010/T3P2.pdf
     
    Last edited: Oct 17, 2012
  2. jcsd
  3. Oct 17, 2012 #2

    Dick

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    For one thing your characteristic polynomial λ2-0.8λ-0.9 is just a little wrong. There is a decimal point in the wrong place. Fix it and you'll get nice eigenvalues.
     
  4. Oct 17, 2012 #3
    Thank you. Anything about the unique solution part? Thats where I really need help
    Thanks.
     
  5. Oct 17, 2012 #4

    Dick

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    First find the eigenvalues and eigenvectors.
     
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