Scaling and nondimensionalisation
Biological models often contain quantities with very different units and numerical sizes. Scaling expresses them relative to characteristic values. Nondimensionalisation uses such scales to rewrite a model in terms of dimensionless quantities.
Scaling a variable
If \(N\) is a population and \(K\) is a characteristic population size, define
\[x=\frac{N}{K}.\]Both \(N\) and \(K\) have the same units, so \(x\) has no units. If \(N=K\), then \(x=1\).
Scaling time
If \(r\) is a rate with units time\(^{-1}\), then
\[\tau=rt\]is dimensionless because the units cancel.
Example: logistic growth
Consider
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).\]Using
\[x=\frac{N}{K},\qquad \tau=rt,\]the equation becomes
\[\frac{dx}{d\tau}=x(1-x).\]The original equation contains the parameters \(r\) and \(K\). In the dimensionless equation their roles have been absorbed into the definitions of population and time scales.
Why nondimensionalise?
Nondimensionalisation can reduce the number of independent parameters, reveal important combinations of parameters, make systems easier to compare, and identify natural scales controlling biological behaviour.
Key idea. Nondimensionalisation does not change the underlying model. It changes the units in which the system is described so that its mathematical structure is easier to see.