This topic was inspired by a discussion which is not very politically correct. I will try to discuss it in a sanitary way. I saw a video where it was discussed how the shape of the male member was such that it could suck out the deposit of a previous donor. One of the commenters said that this meant that cheating was so common in the past that all men's members were shaped like this. I do not think this actually points towards a very high frequency of cheating, because even VERY small beneficial adaptations ought to become fully dispersed through the population with enough time. Another example is that most people have perfectly shaped eye-brows. I suppose this would point towards the fact that some of the people who failed to become our ancestors died because sweat got in their eyes.


Suppose there is a beneficial trait in the population which has proportion P (from values 0 to 1). "r" shows how much better the trait is at dispersing itself (it could be 0.01 for 1%).

P(n+1) = P(n) + P(n)*(1-P(n))*r


It looks like this because if the trait were neither beneficial nor harmful at all, then you'd expect its proportion in the next generation to be about the same as the previous one. That's the first term. The second term says that the added proportion in the next generation is proportional to the part of the population that already has it, but limited by the portion of the population that doesn't have it (you can't get P > 1).

If I were to write this in terms of calculus, you'd get

​
dP/dt = P*(1-P)*r

The solution is

P = P0 / (P0 + (1-P0) * exp(-r*t) )

"P0" means the original portion of the population.

This is an S curve. No matter how small "r" is, it will asymptotically approach 1, so long as r > 0


If you want to know how much time (in this case, how many generations) it takes for a trait to go from 1% expression to 99% expression, you get T = ln ( (Pf*(1 - P0) / (P0 - Pf*P0) )/r. "Pf" = final proportion in population. In the case of 0.01 - > 0.99, you get ~ln(.99^2/.01^2) = 9.2/r. If "r" is 0.01, (for 1% advantage over not having it), then it takes 920 generations. If a generation is 20 years, then that's 18400 years.


But presumably, beneficial genes arise from random mutation in one specific individual. So, let's calculate how long it takes to get from one individual to total saturation (there's only 1 individual left who doesn't have it). I get that the equation simplifies to 2*ln(S)/r, where "S" is the size of the population.

I googled that the historical population of homo erectus was ~20k. So, the time for a 1% beneficial mutation to completely disperse through the homo erectus population ought to be about 2000 generations, or 40k years. For a population of 10 billion (potential size of future human population), it ought to be about 4600 generations/ 90k years.

Now, this calculation assumes some things:

I think this calculation probably works for dominant genes. (a new recessive gene would basically never be expressed and so have no effect on its dispersal)
Gene dispersal is "smooth" (this would give unrealistic results for small populations, because you'd get things like 1.5 people having the given trait)
Gene dispersal is totally random within the population. In practice, gene dispersal is limited by geography, so the real result ought to take a bit more time than this theory.


Here are some qualitative conclusions I draw from this:

I have heard a recent theory that evolution is actually largely random. This would seem to be true. It is likely that the randomness of survival/reproduction is greater than the potential benefit of a small mutation, so, a lot of beneficial mutations were probably wiped out due to pure chance. There were probably some individuals who had beneficial mutations, but by bad luck, they died, or those genes were not passed onto their kids. This S curve is probably only accurate when the size of the population with the gene is already sufficient to withstand some randomness.

Even mutations with a very small benefit (if they can survive a period of chaotic randomness) will eventually become widely dispersed in a population, possibly in a period of tens of thousands of years. This seems to be consistent with the observation that one can easily discern the difference between different races of human, despite the fact that the migration out of Africa was ~ 60000 years ago.

The genes become so thoroughly mixed after a number of generations, that every individual from a distant time in the past can be the ancestor of every living individual. In long time scales, the species is like one genetic entity, and individuals don't matter that much.

And like I mentioned at the very start of this discussion, the ubiquity of a trait shows only that the trait is at least some-what positive. It is impossible to know how positive, because with enough time, all positive traits (not wiped out due to randomness) will become ubiquitous.