Show that the first quadrant of a dynamical system is invariant.

Question

I have the following dynamical system

$${dp \over dt} = p(1-p-q)$$ $${dq \over dt} = q(p-{1 \over 2}-q)$$

and I have to show that the first quadrant ( $p, q \ge 0$ ) is an invariant set. I know what this means but I'm having trouble coming up with a strategy to show it.

Would it be sufficient to show that each axis is an invariant set and by virtue of a trajectory not being able to cross an invariant set then each quadrant must also be invariant? Or is there a better way?

Answer 1

A diagram to show that the quadrant $(p>0,q>0)$ is invariant: