# Mathematical constraints in ecology and evolution, part 3: why selection is risk-averse

This is the third post in my series on the importance of mathematical constraints in ecology and evolution. See also parts 1 and 2.

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.

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I'm an ecologist at the University of Calgary. I study population and community dynamics, using mathematical models and experiments.

## 28 thoughts on “Mathematical constraints in ecology and evolution, part 3: why selection is risk-averse”

1. Life-history theory has a very long and distinguished history of looking at the evolution of phenotypes [ of many kinds] in the face of temporal and spatial variability, both predictable and unpredictable [except for the probability distribution]. And comparing these predicted results with the outcomes expected if one ignores variability, as I usually do. Jeremy is quite correct to claim the fitness functions alter in such cases; the next question is whether the expected outcomes are particularly sensitive to such alteration.
I suggest Derek Roff ,2002, LIFE HISTORY EVOLUTION, Sinauer, 527pp as an entre to the theoretical and empirical literature. Pg 358 gives a one pg summary of results, including the justification/implications of using ‘ geometric mean fitness’. What’s perhaps most interesting is the findings of just what phenotypes ( clutch size, reproductive effort,etc) are predicted to be real different in variable vs [assumed] constant environments. see roff for discussion.
Its a fun literature, with some surprizes. ESS sex ratio may not deviate from 1/2 in variable environments…. unless the beast can predict the time/ place of altered opportunities, respond accordingly, and the male /female fitness functions respond in different and somewhat special ways to the alteration. This underlies the theory of selection favoring GENOTYPIC vs ENVIRONMENTAL SEX DETERMINATION [ see Beukeboom & Perrin, 2014, THE EVOLUTION OF SEX DETERMINATION , Oxford, for recent discussion].
eric

2. Hmmm. There are several important concepts involved here but primarily I’d say this is a good example of the importance of precise language, and problematic math definitions. Mathematically, one should always avoid standardizing a subset of a larger set against the mean of that larger set. If you instead simply ratio genotype A’s fitness against (mean of all others genotypes – A), you will get a different result. The mathematics aren’t demonstrating really anything about nature–they’re only demonstrating the result of a definitional choice.

Orr states (p. 2999):
“So what can an allele, sayA1, do to increase its relative fitness?… the only thing that A1 can do to increase its relative fitness is to increase its absolute fitness, W1. But increasing W1 has less effect on w1 than one might guess. The reason is that there is a curve of diminishing returns between absolute fitness and relative fitness.”

Yes fine, but the reason there’s a curve of diminishing returns in the first place is due entirely to the way the mathematics were defined, i.e. normalizing relative to the overall mean. Nature doesn’t follow any rules of math. Rather math is a human invention, a language that attempts to describe nature quantitatively, and like any language, is only as good as the precision of the definitions used.

• Sorry Jim, no. If you relativize the fitness of every allele to the fittest allele (an alternative convention, also common in evolutionary biology), the results still hold. It’s still the case that increases in the absolute fitness of any given allele are subject to diminishing returns.

“Mathematically, one should always avoid standardizing a subset of a larger set against the mean of that larger set.”

No, that’s often precisely what one wants to do.

• Not standardization against the fittest, but rather against the complement, so that there’s no confounding of the two (or more) quantities.

“No, that’s often precisely what one wants to do.”
Not if you want the most accurate and non-confounded estimate of the actual distance between any two subsets of a population you don’t.

• An example:
parental: {p = q = 0.5}; offspring: {p = 0.75, q = 0.25}.
Then W1 = 1.5, W2 = 0.5 and W = 1, by Orr’s approach. If instead I normalize by the proportion complement of each, W1 = 1.5/0.5 = 3, W2 = 1/3 (and W is 5/3 but irrelevant). Then, I’m no longer sure that Orr’s statement (“In particular, it can be shown that natural selection prefers an allele that boosts mean absolute fitness while reducing the variance in absolute fitness”) is true. Maybe, but it would have to be demonstrated by simulation, and he doesn’t.

• Sorry, don’t follow. What do you mean by normalizing by the proportional complement? I don’t see how you think you can redefine the absolute fitnesses so that they equal 3 and 1/3.

See here for a brief overview of how absolute fitness is defined, and how to relate absolute and relative fitness: https://en.wikipedia.org/wiki/Fitness_(biology)#Relative_fitness

Edit: saw your recent comment above, afraid I still don’t follow. With respect, as best I can tell, in an attempt to avoid what you call “confounding”, you’ve defined quantities that aren’t fitnesses.

• No, they’re still relative fitnesses, just computed differently from either of his two methods. I didn’t explain it well. It’ll be a while before I can get back to this.

• @Jim Bouldin:

Afraid I still don’t see how the numbers you calculated are fitnesses. Sorry, but I think if you’ve calculated something other than what’s on that Wikipedia page or in a standard evolutionary biology textbook, then what you’ve calculated are not fitnesses.

• It’s just standardization and you can standardize a set of numbers any number of ways–Orr himself discussed two different ways right in his article (Appendix).

The problem is that this discussion is traversing across several conceptual levels, from really broad philosophical stuff to very concrete mathematical operations. It’s hard to integrate all those levels to figure out just what it is you’re driving at, exactly.

If the point is that mathematics somehow “constrains nature”, which it seems to be–that conclusion I very much disagree with. Math doesn’t constrain anything, it follows a set of rules and describes things, just like any language does. If it describes things well it’s useful and if not, not.

• Jim, Orr shows the two standard, textbook ways of normalizing absolute fitnesses: by the mean absolute fitness of the population, or by the absolute fitness of the fittest type. And he shows how to interconvert between them, demonstrating their equivalence.

There are no deep philosophical issues here, sorry. If you can explain your own preferred normalization scheme and show that it’s equivalent to those shown by Orr, that’s fine. But in that case, you’ve merely recovered Orr’s results with a different normalization scheme.

Conversely, if your normalization scheme is not equivalent to Orr’s, it’s wrong, I’m afraid.

p.s. If you seriously think your normalization scheme is both non-equivalent to Orr’s, and somehow an improvement on it, then in all seriousness that’s the most revolutionary idea in evolutionary biology in over a century! The entire field would have to be reformulated from the ground up, substituting in your new definition of “fitness”.

I could try to be more specific, but I’m afraid I can’t follow your calculations. I don’t understand where the 1.5 comes from in “W1=1.5/0.5” (I presume 0.5 is p in the parental generation). And I don’t understand why W2 equals 1/3 in your calculation. Spell out your calculations and then I’ll be in a position to try to take this conversation in a more productive direction.

• “Jim, Orr shows the two standard, textbook ways of normalizing absolute fitnesses: by the mean absolute fitness of the population, or by the absolute fitness of the fittest type. And he shows how to interconvert between them, demonstrating their equivalence.”
I know, I read it fairly carefully, and as time allows I’m reading his ’09 review article also. I’m quite interested in the topic actually.

“There are no deep philosophical issues here, sorry.”
OK, this is a very puzzling assertion and I disagree with it about as much as is possible. Your series topic is, I thought, the insight(s) that mathematical logic provides about ecology/evolution. To me, that’s definitely deep.

“If you can explain your own preferred normalization scheme and show that it’s equivalent to those shown by Orr, that’s fine. But in that case, you’ve merely recovered Orr’s results with a different normalization scheme. Conversely, if your normalization scheme is not equivalent to Orr’s, it’s wrong, I’m afraid.”
The problem is I partially botched my explanation above, though I believe I was on the right track. I had to spend more time thinking it through more carefully.

“p.s. If you seriously think your normalization scheme is both non-equivalent to Orr’s, and somehow an improvement on it, then in all seriousness that’s the most revolutionary idea in evolutionary biology in over a century! The entire field would have to be reformulated from the ground up, substituting in your new definition of “fitness”.”
No, nothing like that, much more mundane. The matter is simply that just how one computes fitness determines whether one reaches Orr’s conclusion regarding the importance of minimizing variance in fitness, and hence “risk aversion” in nat selection. The problem is in the use of arithmetic means and variances: the mean of a fraction and it’s inverse does not have a mean of 1.0. That’s the root problem.

“I could try to be more specific, but I’m afraid I can’t follow your calculations. I don’t understand where the 1.5 comes from in “W1=1.5/0.5” (I presume 0.5 is p in the parental generation). And I don’t understand why W2 equals 1/3 in your calculation. Spell out your calculations and then I’ll be in a position to try to take this conversation in a more productive direction.”
Right and I don’t blame you, as I said, I botched it–and I did so because I was fuzzy in my thinking. May require a blog post to explain and show R code.

• And I really believe there are some very fundamentally important analytical issues represented here, both just within the scope of Orr’s argument, and also extending beyond it (e.g., the difficulty of detecting selection in natural populations generally and the tremendous difficulty of ever obtaining the empirical data that would be required to test the Gillespie/Orr hypothesis across taxa).

• Here’s the very short version in case I never get time to write it up, which is possible.

The root problem is in the use of arithmetic averages of relative fitness over time. If you use those, then yes, Gillespie and Orr are correct. But there’s *no need* to use them, and if you don’t…then you don’t come to the same conclusion regarding selection acting “risk averse”. This is because the variance in fitness doesn’t then matter–only the mean (geometric) of the absolute fitness matters, just as one might naively first guess. Just simply use the *logarithms* of the *absolute* fitness values as your base data set–the variance therein does not then matter, only the mean matters. Arithmetic means and variances aren’t even involved.

• That’s what you’re getting at? Jim, I already pointed out in the original post that taking the arithmetic mean of relative fitness is equivalent to taking the geometric mean of absolute fitness! Quoting from the post:

“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.”

Orr is well aware of this too, so is everybody who works on this stuff. And since the geometric mean is approximately equal to the arithmetic mean minus 1/2 the variance, no, nothing in the post changes one whit if you prefer to express the results in terms of geometric means of absolute fitness. All else being equal, variance in fitness reduces your geometric mean absolute fitness. So, all else being equal, selection in a temporally varying environment is risk-averse, even if you think of it as favoring the type with the highest geometric mean absolute fitness.

Personally, I, like Orr and Alan Grafen, prefer to think in terms of the arithmetic mean of relative fitness, since that way of looking at the problem emphasizes that there’s nothing fundamentally different going on when absolute fitnesses fluctuate over time. All natural selection cares about is relative fitness. But to each his own; it’s fine if you prefer to look at geometric mean absolute fitness. Just so long as you recognize that everything in the post is still correct.

• Jeremy, we’re going round and round here. I do not recognize that everything in the post is correct and it seems pretty obvious to me that this is the case.

The post says exactly the following:
“…selection will favor whichever competing type (which allele, which species, whatever) has the lowest temporal variance in absolute fitness, all else being equal…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?…why does selection act as if it’s risk averse, preferring lower variance in absolute fitness, all else being equal?”

You’re very clearly talking about variance in absolute fitness there.

The temporal fluctuations (e.g. variance) in the values of a series of rates of length n are **unrelated** to the geometric mean of that series, since the geometric mean equals the nth root of the *product* of that series. Only the product matters; the individual values can assume an infinite number of possibilities to arrive there. They might all be identical or they might vary wildly year to year–it doesn’t matter as long as the cumulative product is the same.

Assume Orr’s conditions, 5 generations and that g1 = geometric mean of allele 1 and g2 same for allele 2. Then g2 = (1/g1^5)^(1/5) = 1/g1. Each pair of annual, absolute fitness values are also inverses of each other (what I was trying to express in my example above). So, you’ve still got relative fitnesses for the two alleles, but computed differently than either method that Orr explains, without using an arithmetic mean (or maximum) fitness.

The clear conclusion, if you define the issue this way, is that temporal variations in fitness are *irrelevant*; natural selection does *not* then act like a risk avoiding investor (assuming that is in fact how one acts), and the convex (and concave) curve of Orr’s Fig 1 disappears.

This is a simpler and cleaner way to address the topic. It obviates Gillespie and Orr’s arguments–there’s no need for them.

• “The temporal fluctuations (e.g. variance) in the values of a series of rates of length n are **unrelated** to the geometric mean of that series”

Um, so you deny that a geometric mean approximately equals the arithmetic mean minus one half the variance?

“The clear conclusion, if you define the issue this way, is that temporal variations in fitness are *irrelevant*”

Um, no. You’ve just claimed (among other things) that bet hedging is never selected for. Or that it’s selected for for reasons having nothing to do with the temporal variance in absolute fitness.

As I said, I think it’s fine if you prefer to look at geometric mean absolute fitness rather than arithmetic mean relative fitness–*so long as it doesn’t lead you into mistakes*. Bet hedging, for instance, is a real biological thing. A paper on it (Gremer & Venable EcoLetts) just won the Mercer Award. It’s correct to simply say that, when bet hedging is selected for, it’s because it maximizes geometric mean absolute fitness. But if I ask the follow-up question “WHY does it maximize geometric mean absolute fitness?”, your answer is “It just does, there’s nothing else to be said.” That’s incorrect, or at least maximally-unhelpful. Bet hedging is selected for because it reduces the temporal variance in absolute fitness, potentially even if this reduced variance comes at some cost to arithmetic mean fitness. For purposes of hypothesis development and testing, it is *extremely* useful to have a category for “traits that are favored because they reduced temporal variance in absolute fitness, even at some cost to arithmetic mean fitness”. If you think that category is totally artificial, that it’s just a meaningless artifact of an arbitrary mathematical choice…I’m sorry (and I do sincerely mean that), but I can’t for the life of me see why anyone would say that, and so I don’t know what to say in reply. Conversely, it is incorrect or at least maximally-unhelpful to decline to make any distinctions among traits that maximize fitness in terms of how they achieve this maximization. Would you also refuse to distinguish between, say, selection on fecundity vs. survival on the grounds that both are just components of average fitness?

I agree that we’re at an impasse and will have to agree to disagree. Thank you for taking the time to comment. I’m sorry the conversation wasn’t more productive.

3. Troll bait consumed.

Of course Orr made a prediction (the species with minimum variance in fitness will be most successful).

What he didn’t do is derive his prediction from an ecological mechanism. How he got his prediction is a whole different question. And I’m on record in journal articles and posts and comments in this blog that mechanisms are nice to have, but not must have. In particular I have a paper with Jeff Nekola arguing that all kinds of mathematical laws and constraints including the central limit theorem ARE mechanisms (or else we have mechanism free ecology – take your pick).

Prediction, however, is not optional.

• Thanks for taking the bait. 🙂 We’ll have to agree to disagree on this. Or perhaps it’s just a semantic disagreement. I don’t think of the statement “all else being equal, the type with the lowest variance in fitness will be favored” as a “prediction” (mechanistic or otherwise), at least not necessarily. You could of course treat it as such and go out and try to test it. But I don’t think you have to think of it that way or treat it that way for Orr’s result to count as an insight.

I am now on a self-imposed quest to identify a conceptual insight that you would agree does not count as a “prediction”. 🙂 How about the Price equation partitioning of evolutionary change in mean phenotype into additive components attributable to selection and transmission bias. Would you consider “there are two and only two components to directional evolutionary change in a closed population, selection and transmission bias” to be a “prediction”?

• I agree the partition is an insight but not a prediction.

A prediction is simply a statement about reality that can be measured/assessed/tested and have a yes/no (or goodness of fit) answer.

Are you saying that what I highlighted (species with least fitness variance will be most successful) is:
a) not a prediction
b) an uninteresting prediction
c) a prediction, but interesting for another reason
d) something else
?
I have to confess I can’t see how something could be tested if you want and at the same time not be a prediction.

• Depends what you mean by “tested”, I guess. Meaning that I don’t think it’s possible for any data to show that Orr is wrong. Selection does in fact favor lower variance in fitness, all else being equal–it *has* to. If your data suggest otherwise, that just shows that there’s something wrong with your data, that you’ve misunderstood the implications of your data, and/or that all else isn’t equal.

Put another way, Orr’s “prediction”, if that’s what you want to call it, follows from definitions of terms like “absolute fitness”, not from assumptions in the usual sense of “possibly-false statements about the world, which we assume are true for purposes of deriving the predictions that follow from those assumptions”.

The Price equation works the same way. The partitioning of directional evolutionary change into terms attributable to selection and transmission bias (and the interaction of the two, if you want to go to the finest-possible partitioning) doesn’t follow from any possibly-false assumptions. It follows from definitions of terms. It’s not that the equation might be true or false. It’s that it either applies or it doesn’t.

(Aside: just because a statement follows from definitions of terms doesn’t mean those definitions are arbitrary, in the sense that you can feel free to change them. You can change them, of course–but then you’re studying a different problem. The original definitions no longer apply.)

It seems to me that there’s an important difference between claims about the world that are necessarily true (because they follow from definitions of terms), and claims about the world that might be true or false because they follow from possibly-false assumptions. I guess you can lump together both sorts of claims and call them “predictions” if you want, but I don’t think that’s helpful. Because what we learn from testing predictions that follow from possibly-false assumptions is very different from what we learn by “testing” “predictions” that necessarily hold by definition.

• OK – I get where you’re coming from. There are really 2 assumptions to Orr’s paper – fitness working a certain way (which as you say is definitional) and “all other things being equal”. I guess it depends on how much you take the “all other things being equal” as a given (or a part of an experimental design) or as something worth testing for its truth/relevance in real world scenarios. In which case the prediction of smaller fitness variance performing well is a prediction deduced from the assumptions and falsifying that prediction would falsify one of the two assumptions (the logic connecting assumptions to predictions being mathematically 100% rigorous in this case).

4. As usual, I’m a little late to the party and I have to confess that while I have downloaded the Orr paper I haven’t read it yet so I’m just using the summary of it that you provided, Jeremy. That said, I don’t think I need a detailed understanding of the paper to make the point I want to make.

I would never dispute that work like Orr’s can add insight. What it does is identify the space in which empirical values COULD exist. This can be something trivial – absolute abundances can never be negative – or it can be something less intuitive (and therefore, in my opinion, less trivial) like ”the type with the lowest variance in relative fitness MUST be favored”. But these insights don’t tell us anything about how the world works – they identify the hard boundaries on the possible values that can be obtained. And maybe this comes down to how we would interpret the phrase “understand how the world works’. I would say that understanding is demonstrated by being able to make predictions about values in the state space that are better than random selections from that state space. The fact that logic or mathematics constrains the space to be something different than we had previously thought doesn’t mean we have increased understanding of how the world works. It means we have increased understanding about what logic or mathematics imply.
A trivial example clarifies this for me (if, possibly, for nobody else). If there was a time when we would have included negative values as possible values for absolute abundances but then we became aware that the state space was smaller than we had previously thought and didn’t include negative values it wouldn’t be an increase in our understanding of what drives absolute abundances. It would be an increase in our understanding of the logic/mathematics behind counting living organisms. If we used to think that the set of all possible values was X and now we know that due to logical/mathematical constraints the set of all possible values is Y I don’t consider that increased understanding of how the world works, I consider it increased understanding of how logic/mathematics works.
It’s a valuable insight and may ultimately lead to increased understanding of how the world works but it hasn’t increased our current understanding of how the world works.
What I will acknowledge is that I’ve defined ‘understanding of how the world works’ in a way that would prevent work like Orr’s from ever being considered to have increased our understanding of how the world works. So, if you would define ‘understanding the real world’ to include understanding the mathematical constraints then, of course, Orr has increased our understanding of how the world works. This then turns into a discussion of the two definitions.

Jeff

• Cheers for this Jeff. Wish I could say more than I said in my most recent reply to Brian above, but I don’t think I can, so I’ll just leave it at that.

I confess I’m unsure if we disagree substantively, or merely on semantics.

• Hi Jeff; My comment above seems to have been ignored.
Life history workers have known for decades that the measure of fitness that is theoretically maximized by selection in equilibrium [ or that gives the direction of selection] is different for constant versus variable environments . Indeed the question of ‘what’s FITNESS?’ under various conditions is one of the oldest LH questions [ think r/k selection, 2-sex cases,etc]. We combine this maximization with various LH trade-offs to predict oodles of LH stuff. We assume equilibrium.
All this is based on working out the consequences of population genetic[ read; evolutionary genetic] models , and thus whether those models adequately represent what goes on in nature. Read any modern LH book [roff,stearns ,charlesworth,charnov,82,93,etc] to decide if this works out and thus contributes to understanding how nature works. Obiviously I think it does.
Jeremy uses math constraints really to mean the consequences of conceptual assumptions[ sometimes sampling assumptions], carefully worked out; If the concepts don’t apply in nature, then the results may not be useful,( as you note). Some, like + population sizes are too trivial to argue about. others are more interesting; some are critical , some don’t matter much. that’s the art of theory making.
That begins the the fun: LH theorists have long asked if their predictions are at all sensitive to the assms of constant vs fluctuating pops. Some folks don’t like the assms of population genetics and insist on using quantitative genetics.
We all have favorites for critical vs not-critical assms [ and not just in pop fluctuations], and mine is a 1957 paper by Richard Shaw where he, before computers, showed that the equilibrium population sex ratio under natural selection was 1/2 for almost any autosomal genetic system controlling sex, and widely different from 1/2 for violation of the assm….. A life history trait invariant to the genetic assms. There are 3 wonderful experimental tests of this, and indeed it works quite well.
eric

5. Jeremy, Orr’s approach must strongly overlap with Chesson’s analyses of coexistence in variable environments. I personally can’t line up the concepts right now, but i’m sure you’ve thought about this, can you explain the connection?

• Actually, no, not really. I mean, they’re not in conflict or anything. But the storage effect and relative nonlinearity don’t necessarily have anything to do with bet-hedging. This is in part because Chesson’s models are about ecology, not evolution. Coexistence of species, not optimal life history strategies under unpredictable environmental variation.

Not that there aren’t any possible connections–check out Lawrence Venable’s work. He’s documented both bet hedging life histories and coexistence via the storage effect in desert annuals. In desert annuals, both bet hedging and the storage effect depend crucially on a seed bank and germination fractions <100%. But you could certainly imagine a desert annual system in which every species bet-hedges, but there's no coexistence via the storage effect (say, because every species grows and reproduces best in the same environments). Conversely, you can easily have a storage effect without bet-hedging, e.g.

There's a bit of work on evolution of the storage effect, which might be relevant here. From Snyder and Adler, also Peter Abrams, maybe a couple of other papers (an older one from Steve Ellner, I think). The short version is that you get different answers as to whether the storage effect can easily arise as a side effect of adaptive evolution in a fluctuating environment, depending on the details of model assumptions. Possibly, some of those papers have implications for the evolution of bet hedging as well.