Stability of first order nonlinear ODE Consider the following ODE:
$$\frac{dy}{dt}=(y-1)(3-y)-ry,\quad r,y\in\mathbb{R^+}$$
$(a)$ Find the positive fixed points $y^*\geq0$ and determine their stability for all $r\geq 0$.
$(b)$ Sketch the values of $y^*$ against $r$ to produce the bifurcation diagram and find the value $r_c$ at which any bifurcation happens classifying the bifurcation found.
So I found the fixed points as $$y^*=\frac{4-r\pm \sqrt{r^2-8r+4}}{2}$$
But I'm really confused as to how to determine their stability. Also for the second part I know how to classify the bifurcation once its found - but obviously finding it is the hard part.
Could someone please offer some hints/what steps I need to do to get the finished answer (though I wouldn't mind if someone gave a full solution if they wanted to)
 A: Here's to hopin' no one writes a full answer thus robbing you of the distinct pleasure of working through the details.
Some remarks to get you on track:
To determine stability, it helps to have some physical intuition first.  Why is a point called a fixed point in the first place?  Well because if you are at that point at some time, then because $dy/dt = 0$, you'll just stay there.  
But what happens if you move every so slightly away from that fixed point?  Say you move to the right of it slightly.  Will the time-evolution of the equation move you back to toward the original point, or will it cause you to move away?  Well, if you move to the right slightly, then you'll move back to toward the fixed point if $dy/dt$ is negative there, and you'll move away if $dy/dt$ is positive there.  Can you see how it works if you move slightly to the left?
If the time-evolution of the equation tries to move you back to the fixed point when you move away slightly, it's called a stable fixed point, but if it tries to move you further away, then it's called unstable.  There could also be cases when there is stability in the rightward direction but not to the left, or vice versa.
A bifurcation happens when the fixed points change qualitatively.  For example, if fixed point(s) are created or destroyed at certain values of $r$, or if fixed points' stabilities change at certain value(s) of $r$.  You know what a fixed point is, and you know what its stability means now, so hopefully you can try to fill in the details from here.
Advice: Graph the right hand side of the ODE you wrote down.  Since the value of the right hand side equals the value of $dy/dt$, it can help answer questions like: when is that negative or positive? which is the type question you need to answer for determining stability.
