I have an Lotka Volterra type of system as seen below;
$$\begin{align} x' &= 10 x y -1/2 x - 1/10 x^2 \\ y' &= y - y^2 -x y \end{align}$$
I would like to plot the phase portrait of the system.
Critical points (points where $dx/dt$ and $dy/dt$ are simultaneously zero) are $(0,0) , (0,1), (95/101, 6/101)$. (there is $(-5,0)$ as well but as negative values are not meaningful for this system I ignore this Critical point.)
I see $(0,0)$ and $(0,1)$ are saddle points, and $(95/101, 6/101)$ is a stable spiral.
I think unstable manifold (Eigenvector) of $(0,1)$ is joining to spiral of $(95/101, 6/101)$. And no matter which initial points are chosen (except the ones on the eigenvectors) the system ends in $(95/101, 6/101)$. Is this true? I would appreciate if you could show me what the phase portrait of this system looks like.