= Solution
The terms $u$ and $\alpha v$ represent intrinsic exponential population growth. The losses $-uv$ and $-\alpha uv$ describe competition between the species; $-\epsilon_1u^2$ and $-\alpha\epsilon_2v^2$ describe crowding within each species. The positive constant $\alpha$ sets the relative time scale of the second population.
The four <equilibrium points> in the nonnegative quadrant are
$$
(0,0),\quad (1/\epsilon_1,0),\quad(0,1/\epsilon_2),\quad
(u_*,v_*)=\left(\frac{1-\epsilon_2}{1-\epsilon_1\epsilon_2},
\frac{1-\epsilon_1}{1-\epsilon_1\epsilon_2}\right).
$$
The last follows by solving $\epsilon_1u+v=1$ and $u+\epsilon_2v=1$ and is strictly positive under the assumptions. The <Jacobian matrix> is
$$
J=\begin{pmatrix}1-v-2\epsilon_1u&-u\\-\alpha v&\alpha(1-u-2\epsilon_2v)\end{pmatrix}.
$$
At the origin its <eigenvalues> are $1,\alpha>0$, so it is unstable. At the first single-species equilibrium they are $-1$ and $\alpha(1-1/\epsilon_1)<0$; at the second they are $1-1/\epsilon_2<0$ and $-\alpha$. Both are asymptotically stable. At coexistence,
$$
J_* =\begin{pmatrix}-\epsilon_1u_*&-u_*\\-\alpha v_*&-\alpha\epsilon_2v_*\end{pmatrix},
\qquad \det J_* =\alpha u_*v_*(\epsilon_1\epsilon_2-1)<0.
$$
Thus \b[coexistence is a saddle; either species-only equilibrium is stable, and extinction is unstable]. The saddle's stable manifold separates the competing basins of attraction.
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