This is the extremely-belated fourth post in my series on mathematical constraints in ecology and evolution.
By “mathematical constraints” I mean the mathematical rules that all our numerical data and other scientific numbers have to obey. In previous posts, I discussed
- how the mathematical fact that species (or any other variables) can’t all covary negatively with one another affects the interpretation of covariation in the abundances of different species over time and/or space.
- how the mathematical fact that species richness in an area necessarily equals or exceeds species richness in a subset of that area affects the analysis of plots of “local” vs. “regional” species richness, in ways that ecologists had overlooked until recently
- how evolution by natural selection necessarily favors “risk aversion” (aka “bet hedging”), all else being equal, because of mathematical facts about the relationship between absolute and relative fitness.
In this post, I’ll explain the mathematical constraints on dimensions (units of measurement). Any theoretical model or data analysis that violates these constraints literally makes no sense. But on the plus side, you can take advantage of dimensional constraints to make analysis of your model easier, through the magic of “nondimensionalization”.
Note: this is going to be an informal introduction. It’s intended as an entry point into the relevant literature, not a substitute.
Most scientific numbers have measurement units (dimensions) attached, though the precise choice of units often is up to the investigator. For instance, length can be measured in cm, m, km, etc. Temperature can be measured in degrees K, degrees C, etc.
Dimensions have to obey certain mathematical rules.
- You can only compare the magnitudes of quantities of the same dimension. That is, the quantities on either side of an =, <, or > sign both have to have the same dimensions. 5 km = 5km, and 5 km > 3 km, but it makes no sense to compare the magnitudes of 5 km and 5 g. When people say that comparing “apples and oranges” doesn’t make sense, this is the dimensional rule they’re talking about. You often see this rule violated in newspaper business section articles that compare amounts of money (with units of dollars or euros or etc.) to flows of money (i.e. rates with units of dollars or euros or etc. per unit time). For instance, it makes no sense to compare Apple’s market capitalization (an amount) to the GDP of Chicago (an annual rate).
- Only quantities of the same dimensions can be added or subtracted. This is really just a restatement of the previous rule. 5 dollars + 5 dollars is 10 dollars. But 5 dollars + 5 m isn’t 10 of anything. Even 5 m + 5 cm isn’t 10 of anything–you first have to convert those to lengths to the same dimensions to add them.
- Dimensions multiply and divide like numbers. 5 km divided by 2 h is 2.5 km/h. And 5 km divided by 2 km is 2.5–a dimensionless ratio of two numbers with the same dimensions. (dimensionless ratios are going to be important later–remember them!)
- Exponents are dimensionless. If you have an area of 5 km², it has dimensions of km². The superscript “2” is part of the dimension, it doesn’t have dimensions of its own.
In ecology and evolution, we often take for granted that our numbers obey dimensional rules. But it’s worth being careful. I know of at least one high profile theoretical paper in evolutionary ecology that (according to a colleague who read the paper more closely than I did) disobeyed dimensional rules. Which if so would make that model meaningless, in the same way as “2 degrees C + 5 days = 7 somethings”.
We can take advantage of dimensional rules to simplify the analysis of our theoretical models using what’s known as nondimensionalization. Nondimensionalization begins with the observation that the choice of measurement units is arbitrary. Choosing different units doesn’t change the properties of the thing you’re measuring (unless it’s fractal, but leave that aside). 1 minute is same amount of time as 60 s, for instance. Which means that you can make some of the parameters in your model vanish via clever choice of measurement units. The trick is to measure the magnitudes of some parameters and variables in your model relative to other parameters and variables.
For instance, consider the logistic equation, dN/dt = rN(1-N/K). Each side has units of abundance/time. Which means that r has units of 1/time, and N and K have units of abundance. The number 1 has no units. We can of course choose whatever units for abundance and time we want, so long as we make the same choice throughout. So let’s measure time in whatever units needed to make r equal to 1. That is, let’s measure time in units of 1/r. For instance, some of the protists I study have r = 0.2/hour under the culture conditions I use. So if I were to measure r per 5 hours, r=1 (5×0.2). And let’s measure abundance in units of K, so that, when N=K, N=1 in the abundance units we’ve chosen. For instance, some of my protists have K of about 100,000 individuals in a 100 ml culture. So if we measure abundance in units of 100,000 individuals, N=1 when the population is at K, 0.5 when it’s at 50,000, etc.
Armed with these new measurement units, we can rewrite the logistic equation into a simpler, dimensionless form: dN’/dt’ = N'(1-N’), where we’ve used primes to remind ourselves that everything in this equation is now measured in new units. Notice that r and K have vanished! Ok, they haven’t literally vanished, they’re still there in a sense–but we don’t need to show them because they both equal 1, and anything multiplied or divided by 1 is equal to itself. And the lovely part is, we haven’t lost any information by doing this. For instance, we can still set the left hand size equal to 0 to solve for the equilibria, which reveals that N’*=0 and N’*=1. That is, the abundances at which population growth rate is 0 are 0, which equals 0 in the original measurement units, and 1, which equals whatever K was in the original measurement units. And the right-hand side is a humped quadratic function with a maximum at 1/2–i.e. population growth rate is maximal when the population size is 1/2 of K, just as in the original dimensional version of the logistic equation (see here for a more formal walkthrough of nondimensionalizing the logistic equation).
You shouldn’t see nondimensionalization as a magic trick. There’s a deep intuition behind it: that the magnitude of any quantity is only meaningful relative to the magnitudes of other quantities measured in the same units. For instance, in the logistic equation, abundance per se is not what matters for population growth rate. What matters is abundance relative to carrying capacity. Further, abundance relative to carrying capacity only determines population growth rate relative to the intrinsic rate of increase r. This is the intuition behind work like Eric Charnov’s on life history invariants. Different species have very different life histories. But if you scale away those differences via nondimensionalization, you might discover that all of that diversity is just variations on the same underlying theme. Just as different logistically-growing species might have very different r and K values–but if you scale away those differences in r and K, you’ll discover that they’re all described by the same nondimensional equation.
Now, there’s arguably not much point to reducing the number of parameters in the logistic equation via nondimensionalization, because the original equation is so simple. But for more complicated models, nondimensionalization and the associated reduction in the number of parameters can make your model much easier to analyze and interpret. For instance, Stephens and Dunbar (1993) use dimensional analysis to derive an analytical solution to the marginal value theorem for a class of cases for which the derivation would’ve been difficult otherwise.
You can even go a step further and use dimensional rules to make non-obvious predictions, without the need for any substantive biological assumptions. For instance, Stephens and Dunbar (1993) derive a hypothesis about the relationship between a foraging animal’s encounter rate with prey, its movement speed, the area it can scan simultaneously, and prey density, just from knowledge of the measurement units of those four variables. The same logic also reveals an error in previous theoretical work on optimal territory defense.*
As with the previous post in this series, dimensional analysis provides a nice example of how mathematical constraints lead to ecological insight. Nature has to obey the rules of math, and so if you know the rules of math you know some of the rules that nature plays by.
*Stephens and Dunbar (1993) is a great primer on dimensional analysis, this post is really just a plug for it. ht to a correspondent for bringing it to my attention.