= Solution
Let the conserved population be $S>0$. At equilibrium $Y'=-\beta YN=0$, so $Y$ and $N$ cannot both be positive. The other equations then force $U=0$. The only equilibria are
$$
E_Y=(S,0,0),\qquad E_N=(0,S,0).
$$
Near $E_Y$, using $(N,U)$ as coordinates gives the linearization
$$
\binom{N'}{U'}=
\begin{pmatrix}(\beta-\alpha)S&\zeta\\ \alpha S&-\zeta\end{pmatrix}
\binom NU.
$$
Its <determinant> is $-\beta S\zeta<0$, so $E_Y$ is a saddle and is unstable. Near $E_N$, coordinates $(Y,U)$ give the triangular linearization with <eigenvalues> $-\beta S$ and $-\zeta$. Hence $E_N$ is asymptotically stable.
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