# Determine a conserved quantity in a dynamical system Lotka-Volterra

I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:

$\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right)$

$\frac{dZ}{dt}=c\cdot PZ-d\cdot Z=Z\left(cP-d\right)$

Multiplying both sides by $\frac{1}{PZ}$ :

$\frac{dP}{dt}\frac{1}{PZ}=a\frac{1}{Z}-b$

$\frac{dZ}{dt}\frac{1}{PZ}=c-d\frac{1}{P}$

Multiplying with the original system equation gives:

$\frac{1}{PZ}\frac{dP}{dt}\frac{dZ}{dt}=\frac{dZ}{dt}\left(a\frac{1}{Z}-b\right)$

$\frac{1}{PZ}\frac{dP}{dt}\frac{dZ}{dt}=\frac{dP}{dt}\left(c-d\frac{1}{P}\right)$

Taking the difference of the last two equations gives: $0=\frac{dZ}{dt}\left(a\frac{1}{Z}-b\right)+\frac{dP}{dt}\left(d\frac{1}{P}-c\right)$

By the chain rule, the total time derivative of my conserved quantity is: $\frac{d}{dt}E\left(P,Z\right)=\frac{dZ}{dt}\frac{\partial}{\partial Z}E\left(P,Z\right)+\frac{dP}{dt}\frac{\partial}{\partial P}E\left(P,Z\right)$

so then the following should hold:

$\frac{\partial}{\partial Z}E\left(P,Z\right)=\left(a\frac{1}{Z}-b\right)$ and $\frac{\partial}{\partial P}E\left(P,Z\right)=\left(d\frac{1}{P}-c\right)$

so

$E\left(P,Z\right)=\int\left(a\frac{1}{Z}-b\right)\partial Z=\int\left(d\frac{1}{P}-c\right)\partial P$ $=a\ln{Z}-bZ+C_{Z}\left(P\right)=d\ln{P}-cP+C_{P}\left(Z\right)$

Finally: $E\left(P,Z\right)=a\ln{Z}+dln{P}-bZ-cP$

Edit and this quantity is conserved! I had a bad calculus mistake.

-
A nice context introduction would help. –  Weltschmerz Oct 1 '10 at 22:42
Thanks for the suggestion. –  Gus Oct 1 '10 at 23:01
You can go ahead and delete the question, I believe. –  Aryabhata Oct 2 '10 at 0:12