I have the ODE:
\begin{equation} \frac{dx}{dt} = ax - \frac{b}{b + x} x \end{equation}
where $a$ has units $[1/t]$ and $b$ has units $[x/t]$. $x$, in this example, is a resource, and the last term describes consumption rate, which cannot go above $x$.
I would like to non-dimensionalize the equation, but I'm running into problems with the consumption term. Here's my working:
Define the dimensionless variables:
\begin{equation} \hat{t} = \frac{t}{t^{*}}, \quad \hat{x} = \frac{x}{x^{*}} \end{equation}
Rewrite the ODE:
\begin{equation} \frac{d\hat{x}x^{*}}{d\hat{t}{t^{*}}} = a \hat{x}x^{*} - \frac{b}{b + \hat{x}x^{*}}\hat{x}x^{*} \end{equation}
\begin{equation} \frac{d\hat{x}}{d\hat{t}} = a t^{*} \hat{x} - t^{*} \frac{b}{b + \hat{x}x^{*}}\hat{x} \end{equation}
Define $t^{*} = a$ and $x^{*} = b/a$, then:
\begin{equation} \frac{d\hat{x}}{d\hat{t}} = a \hat{x} - \frac{1}{a} \frac{b}{b + \hat{x} \frac{b}{a}}\hat{x} \end{equation}
\begin{equation} \frac{d\hat{x}}{d\hat{t}} = a \hat{x} - \frac{1}{a} \frac{1}{1 + \hat{x} \frac{1}{a}}\hat{x} \end{equation}
\begin{equation} \frac{d\hat{x}}{d\hat{t}} = a \hat{x} - \frac{1}{a} \frac{1}{ \frac{a + \hat{x}}{a}}\hat{x} \end{equation}
\begin{equation} \frac{d\hat{x}}{d\hat{t}} = a \hat{x} - \frac{1}{a} \frac{a}{a + \hat{x}}\hat{x} \end{equation}
which gives us
\begin{equation} \frac{d\hat{x}}{d\hat{t}} = a \hat{x} - \frac{1}{a + \hat{x}}\hat{x} \end{equation}
However, the consumption term is not non-dimensionalized as it still has units of time.
I'd appreciate any advice.