I have a pair of differential equations:
We can see that there are equilibrium points at which we can further determine are respectively unstable, stable, stable and a saddle point. Depending on the initial conditions, as , we expect the system to settle at one of the stable points (i.e. x or y goes 'extinct'). The separatrix is the curve which will determine the end behavior depending on which side of it the initial conditions lie. It should pass through the origin and the saddle point. I am wondering if it is possible to find the exact curve? I have tried a lot of things at this point and am beginning to think that it might not always be possible. I've attained a good approximation numerically, but would love to be able to find the exact curve so if anyone knows how to do so or if it's not possible I'd greatly appreciate the help!