Today’s example of an important mathematical constraint is from evolutionary biology, though it has implications for community ecology as well. Community ecology, like evolutionary biology, is centrally concerned with the relative abundances and relative fitnesses of different types of organism (Vellend 2016). “Relative” is the key word here. As a matter of mathematical necessity, the relative abundances of all species you’re considering have to sum to 1. And the mean relative fitness (relative per-capita growth rate) of all species you’re considering has to equal 1. This ain’t Lake Wobegon; not everybody can be above average. These constraints on the values of relative abundances and relative fitnesses are purely mathematical, not biological. They’re as true of the relative abundances of rocks, and the relative “fitnesses” of linguistic variants, as they are of the relative abundances and fitnesses of species. But these mathematical constraints aren’t merely mathematical. They turn out to have important biological consequences, as Allen Orr (2007) showed in a wonderful little paper. From which I will now shamelessly steal (nothing below is original to me).
Keep reading even if you’re not an evolutionary biologist. This post is short, non-technical, and it’s about something really deep and cool.
The absolute fitnesses of organisms often fluctuate over time, for all sorts of reasons, most obviously environmental fluctuations. A well-known consequence is that selection will favor whichever competing type (which allele, which species, whatever) has the lowest temporal variance in absolute fitness, all else being equal. Which is really weird, when you first think about it. Why the heck does selection care about variance in fitness? When fitnesses fluctuate, why doesn’t selection only care about who’s fittest on average? Why does selection act differently when absolute fitnesses fluctuate over time?
Put another way, why does selection act as if it’s risk averse, preferring lower variance in absolute fitness, all else being equal? Why is selection like a cautious investor who prefers a more predictable (lower variance) in return on investment, all else being equal?
One answer is to note that natural selection doesn’t act any differently when absolute fitnesses fluctuate over time than it does when they’re constant. When absolute fitnesses fluctuate, the winning type will be the one with the highest relative fitness on average. That is, at every moment in time, selection’s just doing what it always does—favoring the fittest type relative to the others. Sometimes that might be one type, sometimes it might be another. But in the long run whichever type has the highest arithmetic mean relative fitness wins (Grafen 1999). Which, mathematically, turns out to be the same thing as favoring the type with the highest geometric mean absolute fitness, rather than arithmetic mean absolute fitness. And it turns out that that’s mathematically equivalent to favoring the type with the lowest variance in absolute fitness, all else being equal.
But that’s one of those answers that just changes the question. Why is the type with the highest average relative fitness the one that (all else being equal) has the lowest variance in absolute fitness?
The answer is: because relative abundances necessarily sum to 1, and the average relative fitness of all types necessarily equals 1. Selection is risk averse because of mathematical constraints.
Click through to Orr’s paper if you want to see the (very simple) math. I’ll try to give you the intuition behind it. Fluctuations in absolute fitness select for reduced variance in fitness because as a matter of mathematical necessity a fitter type increases in relative abundance (that’s what “fitter” means) and so necessarily reduces its own fitness advantage over the average member of the population. The higher your relative abundance, the less above-average your fitness can possibly be, all else being equal, because you and those like you comprise more of the average. So absolute fitness increases are subject to diminishing returns, which means that variance in absolute fitness is bad for you. A decrease in your absolute fitness reduces your relative fitness more than an equal increase in absolute fitness raises your relative fitness. So the more your absolute fitness fluctuates around a given arithmetic mean absolute fitness, the lower your arithmetic mean relative fitness will be. And the lower your arithmetic mean relative fitness is, the more you’re disfavored by selection when fitnesses fluctuate.
My two previous examples of mathematical constraints were both nuisances, or at least could plausibly be regarded as such. They arguably get in the way of biological insight. Today’s mathematical constraint is the opposite: a mathematical constraint that produces biological insight. Nature has to obey the rules of math. So if you know the rules of math, you know some of the rules that nature plays by.
p.s. to troll Brian and Jeff Houlahan🙂 : Orr (2007) is a fine example of conceptual insight or understanding independent of any prediction, never mind any correct prediction. The insight in Orr’s paper—the “a-ha!”—would still be there even if, in nature, fitnesses never fluctuated, or we couldn’t measure fitnesses so couldn’t test any predictions about them, or we had never realized that Orr’s insight implies empirically-testable predictions (e.g., regarding bet hedging). Further, the insight is still valid even though the assumption needed to reveal it–“all else is equal”–never actually holds. The effect Orr identifies still operates even if all else isn’t equal, it’s just harder to tell because other things are going on too. Just because you need clear water to see the fish in a river doesn’t mean the fish vanish when the water is muddy. Nor does Orr’s insight need any data to validate it. It’s not that it might be an insight, but we need data to check whether it really is. Finally, although the insight arises from math and is independent of data, that doesn’t mean it’s merely a mathematical insight, of interest only as mathematics but otherwise irrelevant. The question “What are the consequences of fluctuations in absolute fitness, and why?” is not a question of purely abstract mathematical interest. And part of the answer to that question is Orr’s answer: fluctuations in absolute fitness select for reduced variance in fitness because of the mathematical constraints to which relative fitness is subject.