In a previous post, I suggested that mathematics is a powerful weapon against zombie ideas, ideas that should be dead, but aren’t. But mathematics is a double-edged sword. Unless it’s used carefully, math can create zombies as well as slay them.
Just as our choice of words can mislead us, so can our choice of math. Once you attach a number or mathematical symbol to something, it’s very easy to start thinking of that number or mathematical symbol as a real feature of the world. Sometimes that’s not a problem—for instance, the variable N in a population model, representing population size, is a real feature of the world. But sometimes it is a problem.
For instance, consider the familiar logistic equation:
This equation has two parameters, the intrinsic rate of increase r, which gives the per-capita growth rate of the population at low density, and the carrying capacity K, which gives the equilibrium population size. It’s standard to think of these parameters as real, measurable features of the population, at least if the population grows approximately logistically. Indeed, these parameters often are measured or estimated; I’ve done so myself. And a very important, textbook theory of life history evolution, the theory of r–K selection (MacArthur and Wilson 1967), treats these parameters as not just real but as evolvable. Species which are “r-selected” are thought to have evolved in low-density environments and so been selected for traits which confer high r, while “K-selected” species are thought to have evolved in high-density environments and so evolved traits which confer high K (Pianka 1970).
Now consider a second, less familiar model
This model also has two parameters, r, which still gives the intrinsic rate of increase, and α, which gives the per-capita strength of intraspecific competition, also known as density-dependence (i.e. the amount by which per-capita population growth rate declines when a single individual is added to the population). These two parameters are of course measurable or estimable, just like the parameters of the logistic equation.
And that’s because (2) is the logistic equation. It’s just a disguised version: a different, but mathematically-equivalent parameterization. If you set the right-hand sides of (1) and (2) equal and do a bit of algebra, you’ll discover that α in (2) equals r/K in (1). So if you set r in (2) equal to r in (1), and α in (2) equal to r/K in (1), equation (2) will exactly duplicate the dynamics of (1).
But so what? (I can hear some of you saying) Equation (1) is the real logistic equation, and (2) is just a sort of mathematical parasite, which only looks real because it’s designed to duplicate the real thing. Dressing someone up to look like Charles Darwin doesn’t make them Charles Darwin.
But there’s no reason why (1) should be accorded special status. The fact that it’s traditional and familiar doesn’t make it real. If (2) were traditional, would you argue that it was “real” and (1) was a just a mathematical fiction?*
Ok, fine. (I can hear some of you saying) But if they’re mathematically equivalent, why does it matter which one we use? I can always convert estimates of r and K into estimates of r and α, so what’s the problem with just sticking with the traditional form?
Glad you asked. There are a number of problems, but here’s a big one: the whole idea of r–K selection, which is arguably just an artifact of the mathematically-arbitrary choice to write the logistic equation in the form of (1) rather than (2) (or some other mathematically-equivalent form with the same number of parameters).
The ecological reality that the logistic equation describes, however you choose to write it mathematically, is a well-mixed population of identical organisms which grow and die continuously at a per-capita rate which declines linearly with increasing population size (One can derive this model from various underlying “microscopic” assumptions, but that doesn’t affect my point) Graphically, here’s what you’re assuming about reality:
To express that reality mathematically, you need to describe that negatively-sloped straight line with positive y-intercept. You need two parameters to do it. And which two parameters you choose is arbitrary. If you’re a traditionalist, you’ll use the y-intercept, and call it r, and the x-intercept, and call it K. If you prefer (2), you’ll still use the y-intercept and call it r, but you’ll also use the slope and call it α. You could even use the x-intercept and the slope.
And your arbitrary choice will, if you’re not very careful, shape your interpretation of the real world. Because presumably that’s why you built the model in the first place—to help you understand the real world. For instance, if you choose r and K as your parameters, you naturally start wondering what happens if one of those parameters changes, for instance due to natural selection. So let’s say selection increased r but left K unchanged. What’s happened in reality? Again, here it is graphically:
In reality, the y-intercept increased, which is what you wanted. And the x-intercept remained unchanged, which is what you wanted. And the slope of the line became more negative, indicating stronger intraspecific competition on a per-capita basis. Is that what you wanted? When you think of “r-selected” species, do you ordinarily think of species in which intraspecific competition is strong on a per-capita basis, so that adding a single individual to the population takes a big bite out of the population growth rate? Do you ordinarily think of environments which select for “high r” (e.g., disturbed environments, newly-colonized islands) as “selecting” for stronger intraspecific competitive ability? Do you ordinarily even think of r-selection as having the “side effect” of strengthening intraspecific competition? If you’re like most people, you probably don’t. For instance, r-selected species are classically thought of as small-bodied (Pianka 1970), and you’d think that adding a small individual to the population shouldn’t reduce population growth rate as much as adding a large individual would.
The point is not that (1) is misleading and (2) isn’t. Thinking in terms of r–α selection (or K–α selection!) isn’t any better (or worse) than thinking in terms of r–K selection. The point is that, if you’re even asking about the effect of selection on any parameter in either of these equations, you’re letting a mathematically-arbitrary choice dictate what questions you ask about reality.
It’s not just in the context of r–K selection that this issue crops up. For instance, what if you want to incorporate abiotic environmental fluctuations into the logistic equation? Conventionally, we think of abiotic environmental fluctuations as acting in a density-independent fashion. But if you incorporate such fluctuations into your model by allowing r in (1) to vary over time while holding K constant, you’re implicitly assuming that the strength of intraspecific competition varies over time. Which seems like pretty much the exact opposite of what you wanted to assume.
It’s absolutely legitimate and interesting to ask whether natural selection will favor different traits in low- and high-density populations. That’s the question MacArthur and Wilson (1967) asked. But that question needs to be asked in a way that makes sense independent of arbitrary mathematical choices. Your ideas should dictate your words and your math—not the other way around.**
*footnote: I actually do think there are situations where a mathematical model can be considered “real”, and another, mathematically-equivalent version merely “phenomenological”. For instance, the Holling type II functional response is mathematically equivalent to the Michaelis-Menten equation for enzyme kinetics. But the parameters of the Holling type II functional response (“attack rate” and “handling time”) both have straightforward mechanistic interpretations; they are natural parameters in the “microscopic” model of the behavior of individual predators from which the Holling type II functional response is derived. In contrast, the Michaelis-Menten equation contains a phenomenological parameter (the “half-saturation constant”), which is simply half of the maximum reaction rate. It’s perfectly estimable, but it would be somewhat awkward and artificial (at least to my mind) to use it as a parameter in a “microscopic” model of the behavior of individual molecules in an enzyme-catalyzed reaction. A more controversial case has to do with alternative, mathematically-equivalent formulations of group and kin selection models in evolutionary biology.
**footnote #2: In doing some background reading for this post, I stumbled on an old article by Eric Pianka (Pianka 1972) He explicitly recognizes that equations (1) and (2) are equivalent, although he seems to prefer (2) as a description of reality. He also argues against the alternative idea of “b and d selection” (Hairston et al. 1970), saying that this notion is “merely a matter of definition” springing from Hairston et al.’s choice of notation for the logistic equation, which restricts discussion of “biologic reality”. So my complaint about arbitrary notational choices shaping our thinking about real-world issues is hardly original to me, and in fact goes back to the earliest discussions of r and K selection. I still think it’s worth repeating. Indeed, Rueffler et al. (2008) make much the same complaint, arguing that the parameters in (1) and (2) are population-level parameters which have no straightforward mapping to the individual-level parameters on which selection actually acts, so that it’s misleading to think of selection as acting on r and K (or, presumably, on r and α). I wonder how many times this point will have to be repeated in order to kill off this zombie idea, so that discussions of selection in low- and high-density populations are no longer distorted by arbitrary notational choices.