How to nondimensionalize the equation $dN/dt=N[s - m(N - a)^2]$?

Question

I have a differential equation for population growth, which is

$$dN/dt=N[s - m(N - a)^2]$$

How to nondimensionalize the equation so it can depends on a single dimensionless parameter $k= a(m/s)^{1/2}$?

Answer 1

Define $N(t)=a x $ and $\tau=s t$. Then the differential equation becomes

$$\frac{\mbox{d}x}{\mbox{d}\tau} = x [1-k^2 (x-1)^2]$$

where $k^2=m a^2/s$ as desired.