I've studied all the critical points of the following system using Hartman's Theorem, but for the (1,0) doesn`t apply because it's not hyperbolic. So I would like to find a strict Lyapunov function to determine the asimptotical stability.
$$\begin{cases}\frac{d}{dt}x=x(1-x+y)\\ \frac{d}{dt}y=y(1-x-y)\end{cases}$$
$$ Let\,\, U_{(1,0)}\subset R^{2}\,\,be\,\, a\,\, neighbourhood\,\, of\,\, the\,\, point\,\, (1,0) \\$$ I've tried $$V(x,y)=x^{2}+y^{2}-1$$ Satisfies $$V(1,0)=0$$ $$V(x,y)>0,\,\,\forall(x,y)\in R^{2}\setminus U(1,0)$$ but I have problems to determine if $$V'(x,y)<0,\,\,\forall(x,y)\in U$$
Note: The point (1,0) can't be a centre because it lies on the invariant line y=0.