Logarithmic first integral of the Lotka-Volterra equations
= Logarithmic first integral of the Lotka-Volterra equations
{c}
For the normalized equations $\dot x=x(1-y)$ and $\dot y=ry(x-1)$ on the <positive quadrant>,
$$
V(x,y)=r(x-\log x-1)+y-\log y-1
$$
is a <first integral>. It is nonnegative, <strictly convex>, and tends to infinity at the boundary and at infinity. Every positive regular level is therefore a compact simple closed curve traversed by one <periodic orbit>.