Consider a dynamical system of the following form:
$$ \dot{x} = x(1-x)-xy $$ $$ \dot{y} = y\left(1-\frac{y}{x}\right) $$
Would the point $(0,0)$ be a fixed point of the system? Clearly, the vector field is discontinuous at this point, which leads me to believe that the point is not a fixed point. However, using PPlane to display the phase portrait of the system, I noted that some trajectories asymptotically converge to $(0,0)$: Phase portrait of the above system
More generally, what should one do when faced with a problem where the vector field is discontinuous?