Scaling and nondimensionalisation
Biological models use quantities with units: population may be measured in individuals, time in days, and rates per day. These quantities are essential for describing the real system. However, when analysing a model, we often want to see its relative behaviour rather than being distracted by a particular population size, time unit or numerical scale.
Why do we scale?
Suppose a population has carrying capacity \(K=1000\) individuals. A population of \(N=500\) can be described as 500 individuals, but we can also say that it is half of its carrying capacity:
\[ x=\frac{N}{K}=\frac{500}{1000}=0.5. \]The second description immediately tells us where the population lies relative to its natural maximum.
Thus we choose \(K\) as a population scale and define
\[ x=\frac{N}{K}. \]Now \(x=0.2\) means 20% of carrying capacity, \(x=0.5\) means 50%, and \(x=1\) means the population is at carrying capacity. Because \(N\) and \(K\) have the same units, those units cancel, so \(x\) is dimensionless.
Scaling time
If \(r\) is a growth rate with units time\(^{-1}\), then \(1/r\) has units of time and provides a natural time scale. We define
\[ \tau=rt=\frac{t}{1/r}. \]Thus \(\tau\) measures time relative to the natural growth time \(1/r\). For example, \(\tau=1\) means one growth time scale has passed.
How scaling produces nondimensionalisation
A dimensional quantity becomes dimensionless when it is compared with a suitable scale having the same units:
This is why scaling is needed. Scaling is the actual step that removes the units. Nondimensionalisation is the process of rewriting the whole model using those scaled variables.
Example: logistic growth
Consider
\[ \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right). \]Choose the natural population and time scales \(K\) and \(1/r\), and define
\[ x=\frac{N}{K},\qquad \tau=rt. \]Since \(N=Kx\) and \(d\tau/dt=r\),
\[ \frac{dN}{dt}=Kr\frac{dx}{d\tau}. \]Substitution gives
\[ Kr\frac{dx}{d\tau}=Kr\,x(1-x), \]so
\[ \boxed{\frac{dx}{d\tau}=x(1-x)}. \]What do the graphs show?
The two graphs below describe the same logistic trajectory. The first uses biological units; the second uses scaled variables. Only the axes have changed.
Original variables
Dimensionless variables
How to interpret the dimensionless equation
The equation
\[ \frac{dx}{d\tau}=x(1-x) \]is now written entirely in relative quantities. \(x\) is the fraction of carrying capacity, while \(\tau\) is time measured in natural growth-time units.
If \(x=0.5\), the population is at half its carrying capacity, and
\[ \frac{dx}{d\tau}=0.5(1-0.5)=0.25. \]This means the relative population is increasing at 0.25 per unit of scaled time.
As \(x\) approaches 1, the factor \(1-x\) approaches 0, so growth slows. At \(x=1\),
\[ \frac{dx}{d\tau}=0, \]meaning that the population has reached carrying capacity.
Why is this useful?
The original model contains the particular values \(r\) and \(K\). The dimensionless model focuses on the shape and relative behaviour of logistic growth. Two populations can have different carrying capacities and growth rates but still obey the same scaled equation.
Nondimensionalisation can therefore make models easier to interpret and compare, reduce the number of parameters appearing explicitly, and reveal which parameter combinations actually control the behaviour.
The parameters have not disappeared biologically. Their roles have been absorbed into the scales: \(K\) determines what counts as one unit of population and \(1/r\) determines what counts as one unit of time.
Returning to real biological quantities
After analysing the dimensionless model, we recover the biological quantities using
\[ N=Kx,\qquad t=\frac{\tau}{r}. \]For example, if \(K=1000\) and the dimensionless result gives \(x=0.8\), then
\[ N=1000(0.8)=800\text{ individuals}. \]