Find the fixed points of the following systems. Linearize the system about each fixed point and determine the nature and stability in the neighborhood of each fixed point, when possible. Verify your findings by plotting phase portraits using a computer.

a.

\[\begin{aligned} x^{\prime} & =x(100-x-2 y) \\ y^{\prime} & =y(150-x-6 y) \end{aligned}\]

b.

\[\begin{aligned} x^{\prime} & =x+x^{3} \\ y^{\prime} & =y+y^{3} \end{aligned}\]

c.

\[\begin{aligned} x^{\prime} & =x-x^{2}+x y \\ y^{\prime} & =2 y-x y-6 y^{2} \end{aligned}\]

d.

\[\begin{aligned} x^{\prime} & =-2 x y \\ y^{\prime} & =-x+y+x y-y^{3} \end{aligned}\]