I have this dynamical system in $\mathbb{R}^2$:
\begin{aligned} \dot{x}&=x \left( 1-x-\frac{1}{3} y\right)\\ \dot{y}&=y \left( \frac{3}{4}-y-\frac{1}{2}x \right) \end{aligned}
I found where $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are $0$ and the nature of these points, after that I studied the global dynamical of the system now; I have to show that every points s.t. $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are $0$ in that point is asymptotically stable in the future using an appropriate Lyapunov function, and then discuss the evolution (in the future) of a generic ($x_0$,$y_0$) with both $x_0$ and $y_0$ > $0$.
I really have a problem in finding the Lyapunov, anybody can help me?
Thanks