= Solution
The <equilibrium points> are
$$
(0,0),\qquad(1,0),\qquad\left(\frac1\beta,1-\frac1\beta\right).
$$
The third is biologically interior only for $\beta>1$, and coincides with $(1,0)$ at $\beta=1$. The <Jacobian matrix> is
$$
J=\begin{pmatrix}r(1-2u-v)&-ru\\-\beta v&1-\beta u\end{pmatrix}.
$$
At $(0,0)$ the <eigenvalues> are $r,1$, so it is a repelling node. At $(1,0)$ they are $-r,1-\beta$, giving a stable node for $\beta>1$ and a saddle for $\beta<1$.
At the interior equilibrium,
$$
J_* =\begin{pmatrix}-r/\beta&-r/\beta\\-(\beta-1)&0\end{pmatrix},\qquad\det J_*=-\frac{r(\beta-1)}\beta<0,
$$
so coexistence is a saddle. The formal equilibrium with negative $v$ for $\beta<1$ is not a population state; there its trace is negative and <determinant> positive.
At $\beta=1$ the boundary equilibrium is nonhyperbolic. To determine its behavior into the positive quadrant, put $x=u-1$. A local <center manifold> has $x=-v+v^2/r+O(v^3)$, so $\dot v=v^2+O(v^3)>0$ for small positive $v$. Thus it is unstable to introduction of species $v$, though attracting along the $v=0$ axis. The boundary/interior branches undergo a <transcritical bifurcation> at this parameter.
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