You have to linearize the system at each equilibrium point so it's not just dropping the non-linear terms. The linearized system is the Jacobian at each equilibrium point. You can calculate the Jacobian right? It's just the partials of the right side:

$\displaystyle \left(\begin{array}{cc}\displaystyle\frac{\partial f}{\partial x}(x_0,y_0) & \displaystyle\frac{\partial f}{\partial y}(x_0,y_0) \\ \displaystyle\frac{\partial g}{\partial x}(x_0,y_0) & \displaystyle\frac{\partial f}{\partial y}(x_0,y_0)\end{array}\right)$

At the point (0,0), the eigenvalues are $\displaystyle (-1\pm i)$. That's a spirial sink according to the standard naming convention.

Find "Differential Equations" by Blanchard, Devaney, and Hall. It's the best intro to the subject.

Now, use Mathematica to plot the phase portrait:

Code:

StreamPlot[{-x + y - y^2, -x - y}, {x, -2, 2}, {y, -5, 5}]

The origin looks like a spirial sink to me.