= Solution
The Jacobian is
$$
J=\begin{pmatrix}
\rho(1-2x-\alpha y)&-\rho\alpha x\\
-\beta y&1-\beta x
\end{pmatrix}.
$$
For $\beta>1$, the origin has eigenvalues $\rho,1$ and is an unstable node. At $(1,0)$ the eigenvalues are $-\rho$ and $1-\beta$, so it is a stable node. At the coexistence state,
$$
\det J=-\frac{\rho(\beta-1)}{\beta}<0,
$$
so it is a saddle.
The nullclines are $x=0$, $x+\alpha y=1$, $y=0$, and $x=1/\beta$. The stable manifold of the coexistence saddle separates trajectories tending to $(1,0)$ from trajectories on which $x\to0$ and $y$ grows without bound. The coordinate axes are invariant.
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