= Source-saddle pair in a quadratic planar system
{title2=$\dot x=x(2-x)-y,\quad\dot y=y-kx,\quad0<k<2$}
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>.
\Image[/past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2017-phase-plane.png]
{title=A <source-saddle pair in a quadratic planar system> at $k=1$}
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.
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