I am learning about non-dimensionalization, and how to apply it to ODEs. My question comes from the following predator-prey system:
\begin{align*} \frac{dx}{dt} &= r_1 x \Big(1-\frac{x}{K_1}\Big) - b_1 xy, \\[5pt] \frac{dy}{dt} &= r_2 y \Big(1-\frac{y}{K_2}\Big) - b_2 xy, \end{align*}
(Here, $r_1,r_2$ are per-capita growth rates, $K_1,K_2$ are carrying capacities, and $b_1$ and $b_2$ measure inter-species competition.) Let's say that $x(t) = $ number of foxes and $y(t) = $ number of rabbits at time $t$. My main question is: Are $x$ and $y$ dimensioned quantities? If so, would their dimensions be "number of foxes" and "number of rabbits", respectively?
In the examples I have come across in my textbook, the units of the variables have all been expressed in terms of the fundamental dimensions, time (T), length (L), and mass (M) (and sometimes temperature $\Theta$.) If rabbits and foxes could be considered the dimensions of $x$ and $y$, respectively, could we say that rabbits and foxes are also fundamental dimensions, since they cannot be expressed in terms of T,L,M?
In general, if we have a model which involves a number of "individuals"--be it atoms, cells, people, cars--are these always considered dimensioned quantities? Is there a general rule of thumb that we can use to say if a given quantity is dimensioned/dimensionless?
They're dimensionless. You can work out the parameters' dimensions now.