Dimensional analysis
Dimensional analysis checks whether mathematical expressions are consistent with the kinds of quantities being measured. It is one of the simplest ways to detect errors in a biological model.
Dimensions and units
A dimension describes the type of quantity, such as time or population. A unit specifies how it is measured, such as days or hours.
| Quantity | Possible unit |
|---|---|
| Time \(t\) | day |
| Population \(N\) | individuals |
| Per-capita rate \(r\) | day\(^{-1}\) |
Dimensional consistency
Consider
\[\frac{dN}{dt}=rN.\]The left-hand side has units
\[\frac{\text{individuals}}{\text{day}}.\]If \(r\) has units day\(^{-1}\), then the right-hand side has units
\[\frac{1}{\text{day}}\times\text{individuals}=\frac{\text{individuals}}{\text{day}}.\]The two sides are therefore dimensionally consistent.
Basic rule. Quantities that are added or subtracted must have compatible dimensions, and the two sides of an equation must have the same overall dimensions.
Why this matters
An equation can look algebraically reasonable but still be biologically impossible because its units do not match. Dimensional analysis cannot prove that a model is biologically correct, but dimensional inconsistency usually reveals that something is wrong.
Key idea. Always attach meaning and units to model quantities. Checking units should be a routine part of constructing and interpreting mathematical models.