If the per capita birth rate of a population is given by $r[1 − a(N − b)^2]$ where $r$, $a$, and $b$ are positive parameters, write down a population model equation of the form $\frac{d N}{dt} = f(N)$. Nondimensionalise the equation so that the dynamics depend on a single dimensionless parameter $k = b(a/r)^{1/2}$. If $u$ is your nondimensional population, sketch $f(u)$ for $k > 1$ and $k < 1$ and discuss how the qualitative behaviour of the solution changes with $k$ and the initial condition.
I think we have
$$\frac{dN}{dt}=r[1 − a(N − b)^2].$$
I've tried some different ways when nondimensionalising this equation but I cannot find a way to have the model depending on single parameter $k=b(a/r)^{1/2}$. Could you please help me to find it out?