Testwiki:Reference desk/Archives/Mathematics/2011 October 26

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October 26

Lotka-Volterra equations

Hallo! I did ask this question already in the german reference desk but nobody could help me, so I try to reach out to more people with deep knowledge in analytical solutions of special ODEs. I want to know whether there are closed solution of the general lotka-volterra equations (closed in the meaning of I can write them down explicitly).
I can transform the general lotka-volterra equations to the form
$\left(\begin{array}{c}{x}^{\prime }=x(1-y)\\ y^{\prime }=-y\varrho (x-1)\end{array}\right)$
Also I know the first integral (invariant)
$log(x)-x=\varrho (-log(y)+y)+C$
with the integration constant $C$.
Also I know that the special case
$\left(\begin{array}{c}{x}^{\prime }=x(1-y)\\ y^{\prime }=-y(x-1)\end{array}\right)$
has a closed solution. These information I got from the following two articles: [1] and [2].
Can I somehow transform the general case into the special case? Can I somehow argue that it is enough to know the solution of the special case ($\varrho =1$) because I can change my units somehow? Can someone adjust the special solutions so that they fit the general problem? Any advises?
(I tried to plug one of the equations into the first integral but then I get a transcendental equation, I can't solve something like $\log x^{\prime }+1/x^{\prime }=f(x)$, perhaps somebody can?) Thanks for any advices!!!--Svebert (talk) 16:44, 26 October 2011 (UTC)