Disclaimer: This is for homework...

I'm attempting to understand how to find the first integral of the Volterra-Lotka Predator/ Prey model in order to plot the level curves in Matlab.

The model is defined as:

xdot = ax - bxy

ydot = bxy - cy

I've defined

1) x = x1 = dx1/dt

2) y = x2 = dx2/dt

After some work, I'm arriving at

3) x1*x2*(b*(1/2)*x1^2 - c - a + b*(1/2)*x2^2) = Constant

You can see my derivation at

http://imageshack.us/photo/my-images/337/img20120208171048.jpg/

Now, I've seen several write ups that explain that it has an "explicit integration" and they make the jump to here:

4) ((−c+b*x)/x )* dx/dt - ((a−b*y)/y)*dy/dt = 0

Which leads to:

5) d/dt (bx + by−clogx −alogy) = 0

What I'm not seeing is how they got to 4)...

I ask because implementing 5) is easier to do in Matlab than 3). Sort of.

Which leads to my next question... when writing the Matlab code, what do I set the variables a, b, and c to? Initial conditions? Or do I do a mesh/ for loop...?

Any help would be appreciated.

Thanks,

J