I have all of the wolfram aps, it does not do the graph I'm describing. Though I enjoy many of it's other functions... Here is a java app that does it:
XPP will do all this for you. Slope fields nullclines, equilibria, stability. Up to 1000 or more odes. It is free and pretty small. May have a bit of a learning curve. The ode syntax has been around for two decades on all the standard desktop platforms.