$$\dot{x} = x(1-\mu x-y), \quad \quad \dot{y} = y(-1+x-y)$$
where $\mu > 0$ is a constant. The number and type of equilibria depend on the value of μ; in fact there are three essentially different cases. Find the three corresponding ranges of μ and the number, type and stability of the equilibria in each case. (Ignore a borderline case with a zero eigenvalue, but look carefully at any other borderline cases.)
I've already worked out the following:
$$(x_0, y_0) = (0,0), \quad (0,-1), \quad (\frac{1}{\mu},0), \quad (\frac{2}{\mu +1},\frac{1-\mu}{\mu+1})$$
$\therefore$ If $\mu = -1$ or $\mu = 0$ then there exist only 3 equilibrium points. However, if $[\mu \in \mathbb{R} : \mu \ne 0, -1]$ then 4 equilibria exists.
Thus from here, assume $\mu \ne 0,-1$
Jacobian:
$$J = \begin{bmatrix} 1-2\mu x -y & -x\\y & -1+x-2y\end{bmatrix}$$
I've also worked out the Jacobian for $(\frac{1}{\mu},0)$ and found the polynomial for the EVal's to be $\frac{-b \pm \sqrt{b^2 - 4b}}{2}$ where $b = \frac{\mu -1}{\mu}$.
Now where do I go from here to complete the question? Also please do keep in mind that I'm studying Maths not Physics.
Many thanks.