= Solution
At an <equilibrium point of a dynamical system>[equilibrium], each product in
$$
\dot x=x(y-k-3x+x^2),
\qquad
\dot y=y(y-1-x)
$$
must vanish. This gives the two parameter-independent equilibria
$$
(0,0),\qquad(0,1),
$$
the equilibria on $y=0$
$$
\left(\frac{3\pm\sqrt{9+4k}}2,0\right)
\quad(k\geq-9/4),
$$
and the equilibria with both coordinates nonzero
$$
(1\pm\sqrt k,\,2\pm\sqrt k)
\quad(k\geq0),
$$
where matching signs are used.
The $y=0$ pair is born at $k=-9/4$, suggesting a <saddle-node bifurcation>. At $k=0$, one of these equilibria meets $(0,0)$, suggesting a <transcritical bifurcation>, while the two equilibria near $(1,2)$ are simultaneously born in a saddle-node. At $k=1$, the minus branch $(1-\sqrt k,2-\sqrt k)$ meets $(0,1)$, suggesting another transcritical bifurcation. At $k=4$, that branch meets the $y=0$ equilibrium $(-1,0)$, again suggesting a transcritical bifurcation. Thus the bifurcation parameter values are
$$
\boxed{k=-\frac94,\ 0,\ 1,\ 4}.
$$
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