The question is as follows. Consider the Lotka-Volterra equations:
$\dot{x}=\alpha x-\beta xy$
$\dot{y}=-\gamma y+\delta xy$
Use $V(x,y)=\delta x-\gamma ln(x)+\beta y-\alpha ln(y)$ to show that the populations undergo a limit-cycle if $x(0),y(0)>0$ is not an equilibrium.
It is my understanding that in order to prove a limit cycle exists I need to show that $\dot{V}\leq0$ for some set $M_1$ and $\dot{V}\geq0$ for some set $M_2$ and then I should get an Invariant set $M=M_{1}\cap M_{2}$ for which there is a limit cycle(Poincare - Bendixson). Only problem is that for the given $V(x,y)$, $\dot{V}=0$. So I'm not sure what my approach should be.