 Solution

For $x,y>0$, the signs needed for the phase portrait follow directly from the factorizations:The coordinate axes are invariant sets. On $y=0$, the logistic differential equation $\dot{x}=x(1-x)$ directs positive states toward the boundary saddle point $(1,0)$. On $x=0$, $\dot{y}=2y$ directs positive states upward. At the three equilibria, use the local directions found in part (ii); interior sample trajectories near $(2,1)$ spiral clockwise inward.

The nullclines prescribe where arrows become vertical or horizontal; they are generally not themselves trajectories. The image samples the vector field and several solutions, rather than assuming every nullcline is an invariant set.