Thus the proper model is to separately track the populations of sports and normals. So we begin with 10,000 members, 5000 females and 5000 males, with one of them being a sport. Say that in each generation 10,000 offspring are produced, and the sport has about 10% more success with females than do normals; thus produces an average of 2.2 offspring versus their 1.99996. Of course offspring are discrete, so this manifests itself in an extra sport within five generations. The mutation is at risk of dying out early on, but once there are five or more males with the mutation, numbers take over and the sport population will grow at the expense of the normals.

The result is not an average slight increase in beak length; the result is the mutation spreading to 100% of the population and thus a doubling of beak length. By analogy, our current human intelligence is not due to some ancient ancestor being 50 times smarter than us and passing down 2% of that genius. It is the result of some ancient ancestor being maybe 10% smarter than his peers, and using that slight advantage in understanding and strategy and planning to out-compete the dummies and out-mate them, and his children inherited ALL of his advantage and did the same to their peers, until the mutation spread throughout the population and became an ineffective advantage — At which point the next advantage become more important.

A) Can blended inheritance allow a beneficial mutation to have a lasting effect on the species? and
B) Can blended inheritance allow differences to persist among species members, without all species members eventually looking *the same*?

If a beneficial mutation spreads throughout the entire population (even in some weakened form as a result of the averaging of blended inheritance), then the answer to A can be yes while B is still no.

Jenkin’s model:
A) No and B) No

Davis’s model:
A) He thought yes, but your calculations show it’s No
B) He thought yes, but your calculations show it’s also No. (Assuming that the beneficial-mutation population grows without bound, dwarfing the original which stays constant, so that effectively “all members” of the species eventually look the same.)

Landsburg’s model:
A) You’re saying Yes
B) I think it’s not clear from your original post. You said: “Once again the total excess beak length goes to 2.38, but this time it’s shared among individuals of a fixed population size.” But, I had assumed that the excess beak length of 2.38 would eventually be evenly distributed among the whole population. Are you saying that it wouldn’t?

In other words, I think that your models can differ on the answer to question A, but I suspect that all 3 models say that the answer to question B is No — that blended inheritance causes us all to look the same eventually. (And so any such model using blended inheritance should have been self-evidently wrong, because we don’t all look the same.)

However, I think Dawkins’s essay makes a good point that is relevant here: Darwin and his contemporaries should have known that blended inheritance was almost certainly wrong anyway, because if that were the case, then we’d all look the same by now! (Well, maybe we wouldn’t look the same if you allow some Lamarckian development of traits during one’s lifetime and passing on of those traits to children. But we should still all look the same with regard to traits that don’t change much during our lives, like eye color, or with regard to traits that we don’t acquire as the result of actually doing anything, like hair color.)

Here, I think, are the necessary corrections and clarifications:

1) You are right that I’ve used the phrase “average sport value” inconsistently in the chart on page 3 of the link. In Jenkin’s case, the population eventually stabilizes at 2.38 while the sport value per individual continues to be halved each generation, so in the limit, the average sport value should be 0, not 2.38. I will correct this in the link, together with a footnote noting the original error so that your comments will still make sense.

2) Davis’s model, taken as literally as I have taken it, implies that the number of muggles eventually becomes negative (because the increasing population of sports continues to mate with muggles until eventually there are no muggles left to mate with each other). Obviously, one does not want to take this prediction too literally.

Thanks very much for making me get this right. (And do let me know if I’m still missing something.)

This is discussed in Richard Dawkins’s essay “Light Will Be Thrown” in his book “A Devil’s Chaplain”. Darwin’s evolutionary models were “pre-Mendelian” which means they assumed blended inheritance. But Darwin’s critics pointed out that blended inheritance was impossible, because it would mean that by now, all humans would look essentially alike, after all the averaging out in every generation! And it wasn’t until Mendel that people realized offspring simply either inherited a trait or they didn’t.

With traits either being inherited or not, I think it would be easy to show that a genetically dominant trait which adds 0.1 to your expected offspring count, will soon sweep the population.

For a recessive trait, I assume that what happens is that in the first individuals who possess it, it has no positive or negative effect on survival (because it’s recessive so it doesn’t manifest itself at all), so it just spreads through random genetic drift. But eventually you reach a tipping point where individuals who have the gene, have a realistic chance of mating with other individuals who have it, and so now for the first time they will produce offspring who actually *have* the trait and enjoying the advantages, and now it starts to spread faster and eventually sweeps through the population as well. (I don’t know anything about this though so I’m just assuming that’s what happens because it’s the only way to me that it seems logically possible. However, there might be another way that I just didn’t think of.)