Hint
This bifurcation diagram is a little tricky, so let's think it through step by step.
First, what are the equilibria? You have a formula, above, for the equilibria as a function of $\alpha$ and $\mu$, but now you plug in $\alpha=0.5$ to get a formula for the equilibria as a function of $\mu$ alone, which is what you are going to plot. Since, you won't plot an equilibrium when it is not biologically, for some values of $\mu$, you will have two equilibria but for others you will have only one (biologically plausible) equilibrium.
One of the equilibria might change stability, being stable for some values of $\mu$ and unstable for other values of $\mu$. (That sounds like a bifurcation.) If this were to happen, as you plot the curve (or line) of equilibria, you would change from drawing a solid line to drawing a dashed line (or vice versa). Online, we don't have a way to change a curve from solid to dashed halfway through. Instead, you'll need to draw the two cases with two separate half-lines, so you can click one of the halves to turn it dotted while leaving the other half-line solid. In other words, you might need to use two separate segments for a single equilibrium in order to show that its stability changes.
Online, you specify the number of branches with $n_b$, which will give you that many half-lines, which you can drag to the proper spots. You may need to use more branches than the number of equilibria so that you can correctly draw all the parts where each equilibrium is stable and unstable.
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