Scale a population equation with Allee effect into dimensionless form. A population equation with Allee effect $$\dfrac{dN}{dt} = N[r-a(N-b)^2]$$ where $a$, $b$, $r$ are positive constants. Let $n=\dfrac{N}{c}$, $\tau =\dfrac{t}{d}$, substitute them into the equation and derive the dimensionless form $$\dfrac{dn}{d\tau} = n[1-\alpha(n-1)^2]$$ Express the $c$, $d$ $\alpha$ in terms of $a$, $b$, $r$. I tried to plug in $$\dfrac{d(nc)}{d(\tau d)} = nc[r-a(nc-b)^2]$$ but don't know what to do after this. Am I suppose to something like chain rules?

No need for a chain rule. Pure algebra. Since $c$ and $d$ are constants, you may safely pull them out of differentiation, that is,

$$\frac{c}{d}\dfrac{d(n)}{d(\tau )} = nc[r-a(nc-b)^2]$$

Cancel $c$ and multiply by $d$:

$$\dfrac{d(n)}{d(\tau )} = nd[r-a(nc-b)^2] = n[rd - adc^2(n - \frac{b}{c})^2]$$

Now all you need to do is to satisfy $rd = 1$ and $\frac{b}{c} = 1$. $adc^2$ becomes $\alpha$.

Final form is $$\dfrac{d(n)}{d(\tau )} = n[1 - \frac{ab^2}{r}(n - 1)^2]$$