Lemme' tell you something about these. You can't. You just can't (or shouldn't) do these without drawing the phase portrait to see what's happening on the global level. Mathematica 7 has StreamPlot. Also easy to find the other equilibrium points:

Code:

In[37]:=
myEquilPts = {x, y} /.
Solve[{x - 3*x^2 - 3*x*y == 0,
y + y^2/2 - (x*y)/2 == 0}, {x, y}]
myEqSet = Graphics[{Red, PointSize[
0.015], Point[myEquilPts]}];
Show[{StreamPlot[{x - 3*x^2 - 3*x*y,
y + y^2/2 - (x*y)/2}, {x, -3, 3},
{y, -3, 3}], myEqSet}]
Out[37]= {{0, -2}, {0, 0}, {1/3, 0}, {7/6, -(5/6)}}

Yep, they're moving away from (1/3,0). I realize that's not helping with the analytic analysis but I think this "reinforces" such.