I’ve been thinking some more about this issue–the idea that selection should tend to favor those genotypes with the smallest temporal variations in fitness, for a given mean fitness value (above 1.00). It’s taken some time to work through this and get a grip on what’s going on and some additional points have emerged.

The first point is that although I surely don’t know the entire history, the idea appears to be strictly mathematically derived, from modeling: theoretical. At least, that’s how it appears from the several descriptions that I’ve read, including Orr’s, and this one. These all discuss mathematics–geometric and arithmetic means, absolute and relative fitness, etc., making no mention of any empirical origins.

The reason should be evident from Orr’s experimental description, in which he sets up ultra-simplified conditions in which the several other important factors that can alter genotype frequencies over generations, are made unvarying. The point is that in a real world experimental test you would also have to control for these things, either experimentally or statistically, and that would not be easy. It’s hard to see why anybody would go to such trouble if the theory weren’t there to suggest the possibility in the first place. There is *much* more to say on the issue of empirical evidence. Given that it’s an accepted idea, and that testing it as the generalization it claims to be is difficult, then the theoretical foundation had better be very solid. Well, I can readily conceive of two strictly theoretically-based reasons of why the idea might well be suspect. For time’s sake, I’ll focus on just one of those here.

The underlying basis of the argument is that, if a growth rate (interest rate, absolute fitness, whatever) is perfectly constant over time, the product of the series gives the total change at the final time point, but if it is made non-constant, by varying it around that rate, then the final value–and thus the geometric mean–will decline. The larger the variance around the point, the greater the decline. For example, suppose a 2% increase of quantity A(0) per unit time interval (g), that is, F = 1.020. Measuring time in generations here, after g = 35 generations, A(35) = F^g = 1.020^35 = 2.0; A is doubled in 35 generations. The geometric (and arithmetic) mean over the 35 years is 1.020, because all the yearly rates are identical. Now cause F to instead vary around 1.02 by setting it as the mean of a normal distribution with some arbitrarily chosen standard deviation, say 0.2. The geometric mean of the series will then drop (on average, asymptotically) to just below 1.0 (~ 0.9993). Since the geometric mean is what matters, genotype A will then not increase at all–it will instead stay about the same.

pstep = 0.00001; probs = seq(pstep, 1-pstep, pstep)
q = qnorm(p=probs, mean=1.02, sd=0.2)
gm = exp(mean((log(q)))); gm

This is a very informative result. Using and extending it, now imagine an idealized population with two genotypes, A and B, in a temporally unvarying selection environment, with equal starting frequencies, A = B = 0.50. Since the environment doesn’t vary, there is no selection on either, that is F.A = F.B = 1.0 and they will thus maintain equal relative frequencies over time. Now impose a varying selection environment where sometimes conditions favor survival of A, other times B. We would then repeat the above exercise, except that now the mean of the distribution we construct is 1.000, not 1.020. The resulting geometric mean fitness of each genotype is now 0.9788 (just replace 1.02 with 1.00 in the above code).

So what’s going to happen? Extinction, that’s what. After 35 generations, each will be down to 0.9788^35 = 0.473 of it’s starting value, on average, and on the way to zero. The generalization is that any population having genotypes of ~ equal *arithmetic *mean (absolute) fitness and normally distributed values around that mean, *will have all genotypes driven to extinction*, and at a rate proportional to the magnitude of the variance. If instead, one genotype has an arithmetic mean fitness above ~~1.00~~ a threshold value determined by it’s mean and variance, while all others are below it, then the former will be driven to fixation and the latter to extinction. These results *are not tenable*–this is decidedly *not* what we see in nature. We instead see lots of genetic variation, including vast amounts maintained over vast expanses of time. I grant that this is a fairly rough and crude test of the idea, but not an unreasonable one. Note that this also points up the potentially serious problem caused by using relative, instead of absolute, fitness, but I won’t get into that now.

Extinction of course happens in nature all the time, but what we *observe* in nature is the result of successful selection–populations and species that survived. We know, without question, that environments vary–wildly, any and all aspects thereof, at all scales, often. And we also know without question that selection certainly can and does filter out the most fit genotypes in those environments. Those processes are all operating but we don’t observe a world in which alleles are either eliminated or fixed. The above examples cannot be accurate mathematical descriptions of a surviving species’ variation in fitness over time–something’s wrong.

The “something wrong” is the designation of normally distributed variation, or more exactly, symmetrically distributed variation. To keep a geometric mean from departing from it’s no-variance value, one must skew the distribution around the mean value, such that values above it (x) are inverses ~~(1/x)~~ (mean/x) of those below it–that is the *only* way to create a stable geometric mean while varying the individual values. [*EDIT*: more accurately, the mean must equal the product of the values below the mean, multiplied by the mean divided by the product of the values above the mean, but the values will be skewed in any case.] Mathematically, the way to do so is to work with the logarithms of the original values–the log of the geometric mean is designated as the mean of normally distributed *logarithms *of the individual values, of whatever size variance one wants. Exponentiation of the sum of the logarithms will equal the product of the fitness series.

Hopefully, what I’m driving at is emerging. If the variance structure must obey this mathematical necessity to preserve a genotype’s mean fitness at 1.00, while still allowing the individual series values to vary…then why should we not expect the same to hold true when the mean geometric fitness is *not* equal to 1.00? I would argue that that’s exactly what we should expect, and that Gillespie’s original arguments–and Orr’s, and others’ summaries thereof–are not particularly defensible theoretical expectations of what is likely to be happening in nature. Specifically, the idea that the variance in fitness around an *arithmetic* mean should necessarily arise from symmetrically (normally) distributed values, is questionable.

As alluded to above, there is (at least) a second theoretical argument as well, but I don’t have time to get into it now (nor for this one for that matter). Suffice it to say that it involves simultaneous temporal changes in total population size and selective environments. All this without even broaching the entire hornet’s nest of empirically testing the idea, a topic reviewed five years ago by Simons. For starters, it’s not clear to me just how conservative “bet hedging” could ever be distinguished from the effects of phenotypic plasticity.

References

Simons, A.M. (2011) Modes of response to environmental change and the elusive empirical evidence for bet hedging. doi:10.1098/rspb.2011.0176

Other references are linked to in the previous post.