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Math Help - 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 (x,y) or (x,y) for specific 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.

     <br />
\frac{dy}{dx} = \frac{y(cx - e)}{x(a-by)}<br />

    which separates and integrates to

     <br />
c x - e \ln x + b y - a \ln y - k = 0<br />

    With only (x,y) data points, you could preform a linear regression on this linear equation for the constants a, b, c, e and 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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