\begin{align} \frac{\text{d} x}{\text{d} t} = f(x,y) &= - x+\gamma(b x+y)-\epsilon(b x+y)^3\\ \frac{\text{d} y}{\text{d} t} = g(x,y) &= - r y-\alpha b x \end{align} Based on real data, one can assume that $\alpha < r$. We will also introduce the notation $\mu = b \gamma$. Prove then that for $\mu_0 = \frac{1}{1-\frac{\alpha}{r}}$ the system undergoes a pitchfork bifurcation, i.e. the number of equilibria changes from 1 to 3.
I can't figure out if I'm supposse to substitute $\mu$ into $dx/dt$ or if because $dy/dt$ is linear integrate in order to find $y(t)$.
The question is, as stated, to investigate the equilibria of the system, in particular the number of different possible equilibria. An equilibrium is by definition constant in time, so if $(x_*,y_*)$ is an equilibrium of the system, that means that $f(x_*,y_*) = 0$ and $g(x_*,y_*)= 0$. I'm sure you can take things from here.