Thread: Finding parameters in a ODE system

1. 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

2. Originally Posted by laureanda
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?

3. Realize that you can integrate if you divide the two ODEs, i.e.

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

which separates and integrates to

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

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

4. 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