So this problem is from my homework and I've been having some trouble with it. We have the following model of population growth where a and b are positive constants and r is a positive growth rate:
$u' = ru - \frac{au}{1+bu}$
The goal is to get a non dimensionalized version that has a single dimensionless parameter h. I honestly have no idea what to use for h. I imagine it has to be something from the second term but it seems difficult since I'm not even sure about the units there. It just seems like a/b must have units of members/time but I'm not sure about that either. If anyone could help me figure this out or at least find out what to set h to I'd appreciate it.
Being a number, the dependent variable $u$ is dimensionless. This implies that $u'$ has units of inverse time. Hence $r$ has units of inverse time as well, which makes sense since it is a rate. In the denominator of the fraction the term $bu$ is added to $1$, so $bu$ must be dimensionless, implying that $b$ is dimensionless too. This leaves $a$, which must have units of inverse time as well.
Hence we know that $\tfrac{1}{r}$ is a time. This suggests to introduce a new dimensionless time $\tau := rt$ and a new dependent variable $v(\tau) := u(t(\tau))$. Now you can work out the differential equation for $v$, using the differential equation that you have for $u$.
Since this is homework, I leave it up to you to show from there that $w(\tau) := bv(\tau)$ and $h := \tfrac{a}{r}$ are suitable choices for the final dimensionless dependent variable and single parameter, respectively.