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Thread: Finding parameters in a ODE system

  1. #1
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    Finding parameters in a ODE system

    Hello,
    I have a problem: I have the Lotka-Volterra system
    x'= x(A-By)
    y'= y(Cx -E)
    and sperimental data (I used some values of the parameters and I solved the system with procedure like "dsolve" of maple or similar).
    I would like to find again my parameters starting only to the datas.

    I find some book on this problem (Comincioli, Bard) but they say that I should find the explicit function:
    x(t)=.... (es. x(t)=Acos(5t)+Be^t -Csin(3t) + D)
    y(t)=....
    and then use the Least Mean Square to find parameters:

    Err= 1/2 (explicit x & y depending of A,B,C,E - datas)^2

    For this type of ODE, however, it is impossible.
    I tried to use Taylor's serie, but it doesn't depend to the parameters.
    Now I try to use the energy H (that is a constant in this particular case), but I am not sure that is a good idea: I should use a 5 parameter T:
    Err= 1/2 (H(A,B,C,E) - T)^2 and... I don't know T!!!

    Someone know this problem and can say me how find information (books, articles, web page...)?
    Thank you very much,

    Ilaria
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  2. #2
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    Quote Originally Posted by laureanda View Post
    Hello,
    I have a problem: I have the Lotka-Volterra system
    x'= x(A-By)
    y'= y(Cx -E)
    and sperimental data (I used some values of the parameters and I solved the system with procedure like "dsolve" of maple or similar).
    I would like to find again my parameters starting only to the datas.

    I find some book on this problem (Comincioli, Bard) but they say that I should find the explicit function:
    x(t)=.... (es. x(t)=Acos(5t)+Be^t -Csin(3t) + D)
    y(t)=....
    and then use the Least Mean Square to find parameters:

    Err= 1/2 (explicit x & y depending of A,B,C,E - datas)^2

    For this type of ODE, however, it is impossible.
    I tried to use Taylor's serie, but it doesn't depend to the parameters.
    Now I try to use the energy H (that is a constant in this particular case), but I am not sure that is a good idea: I should use a 5 parameter T:
    Err= 1/2 (H(A,B,C,E) - T)^2 and... I don't know T!!!

    Someone know this problem and can say me how find information (books, articles, web page...)?
    Thank you very much,

    Ilaria
    Are your experiment data points points only $\displaystyle (x,y)$ or $\displaystyle (x,y)$ for specific $\displaystyle t$ values?
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  3. #3
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    Realize that you can integrate if you divide the two ODEs, i.e.

    $\displaystyle
    \frac{dy}{dx} = \frac{y(cx - e)}{x(a-by)}
    $

    which separates and integrates to

    $\displaystyle
    c x - e \ln x + b y - a \ln y - k = 0
    $

    With only $\displaystyle (x,y)$ data points, you could preform a linear regression on this linear equation for the constants $\displaystyle a, b, c, e$ and $\displaystyle k$.
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  4. #4
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    My datas depend to time (how the 2 populations x and y growth in time t) and your formula is exactly the energy (my "T" is your "k").
    My problem is: ok, I know that there is k... but I don't know its value! So if I minimize the error... it minimize also k! But k strictly depends on parameters... I don't want the best value of k, but the best values of a,b,c,e. However, I don't know the true value of k (ok, in this case yes because I created the datas with some specific parameters, but in an hypotetic case the datas are "real" (and probably with errors) and I don't know at all parameters).
    This is the link for Wikipedia on the Lotka-Volterra equations
    Lotka?Volterra equation - Wikipedia, the free encyclopedia
    Thank you for your patience!
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  5. #5
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    Quote Originally Posted by laureanda View Post
    My datas depend to time (how the 2 populations x and y growth in time t) and your formula is exactly the energy (my "T" is your "k").
    My problem is: ok, I know that there is k... but I don't know its value! So if I minimize the error... it minimize also k! But k strictly depends on parameters... I don't want the best value of k, but the best values of a,b,c,e. However, I don't know the true value of k (ok, in this case yes because I created the datas with some specific parameters, but in an hypotetic case the datas are "real" (and probably with errors) and I don't know at all parameters).
    This is the link for Wikipedia on the Lotka-Volterra equations
    Lotka?Volterra equation - Wikipedia, the free encyclopedia
    Thank you for your patience!
    Maybe you could first approximate k. For this approximation, you might try picking 5 points and numerical solve for a - k. Repeat with a different set of data points then take the average. Then do a regression on the integrated equation for a, b, c, and e. Just an idea.
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