How do I draw a phase diagram for this system?

Question

The system of equations is as follows \begin{align} \frac{dx}{dt} &= 6x-3x^2-xy \\ \frac{dy}{dt} &= y-x+3xy \end{align} This is a question in my study guide and I cannot figure out how to draw the phase diagram as the first equation seems to be in the form of a logistic growth model but the second equation I have never encountered that form before and thus do not know how to draw the phase line.

Answer 1

You can start by drawing the nullclines of the ODE.

In each of the 9 regions you obtain the orbits move to the left or right and up or down.

Then you can either directly sketch the orbits, or first determine the stability of the fixed points of the ODE.

If you do determine the stability of the fixed points, you can sketch the stable and unstable orbits of the two saddles as extra guidelines. Since other orbits will start near (for various definitions of near) the stable orbit and end near the unstable orbit.