Source-saddle pair in a quadratic planar system ($\dot{x}=x(2-x)-y,\dot{y}=y-kx,0<k<2$)

The nullclines meet at the origin and $P=(2-k,k(2-k))$. At the origin, the Jacobian matrix has trace three, determinant $2-k>0$ and discriminant $1+4k>0$, giving an unstable node. At $P$, its determinant is $k-2<0$, giving a saddle equilibrium. The horizontal component is positive below the parabolic nullcline and negative above it; the vertical component is positive above the straight nullcline and negative below it. These signs determine the directional regions of the phase portrait.

The quadrant is not positively invariant: nonzero points on either axis have an outward-pointing component. Restricting a drawing to that quadrant does not keep its trajectories there. For $k>2$, only the saddle at the origin remains in the quadrant, and an exponential Dulac multiplier for a quadratic planar system excludes nonconstant periodic orbits there.