In a very simple setup where we have the following expression:
$\frac{d B(t)}{d t} = a B(t)^2 + b B(t) + c$
evaluating the stability points is easy. I can evaluate the system where the derivative equals 0 and and solve the quadratic equation using the quadratic formula.
How should I go about evaluating the stability of the roots (one or two) i.e. which root is stable and which one is unstable? Assuming of course that at least 1 root exists.
Is there a simple trick to evaluate this?
Denote $f(x) = ax^2+bx+c$. Suppose you have two real roots, say $x_1<x_2$ when you solve the equation. Now assume $a>0$. Then, you can easily see that $f(x)$ is positive when $x<x_1$ and $x>x_2$ (meaning your function B(t) is increasing), but negative when $x_1<x<x_2$ (meaning B(t) is decreasing.) Therefore, solutions converge to $x_1$, but diverge from $x_2$. So $x_1$ is stable while $x_2$ is unstable.
You can do the same analysis when $a<0$.