> And your chosen simulation starting point highly influences the outcome of the simulation.
Yes, it does. The effect I am describing is real, and I simply haven't thought it through in detail until now.
1. The starting point for my simulation has both the cicadas and the predators hatching out simultaneously, in year zero. This obviously happens in nature as well as in my simulation (but only rarely).
2. Given that starting condition, what is the likelihood that the predator, with a range of reproduction cycles, will succeed in synchronizing with the cicadas again, or (the best of all worlds) over and over in perpetuity?
3. The answer is that, if the cicadas adopt a prime-numbered reproduction cycle, the predators are handicapped by the fact that a typical prime number has fewer divisors than a composite number, and divisors are required for synchronization:
4. For cases in which the predators and the cicadas are out of synchronization, the outcome for a random difference in years is neutral -- not advantageous, not disadvantageous. I've shown this by adding a loop to my program that tests against a range of differences in years between cicadas and predators. When I do this, the difference between a prime and a composite nearly vanishes ("nearly" because the advantage of a prime numbered cycle is still present, but is almost completely buried by the much greater number of years where there's no advantage).
5. It is only when cicadas and predators hatch out in synchrony (for any one of a number of reasons) does this prime-number effect have any effect on the outcome.
6. In terms of natural selection, this leaves the prime-number effect, plus an initial synchronization between cicada and predator, as the only selection advantage, for which the prime-number effect works against the predators.
7. Nature, and natural selection, does something my simulation doesn't -- over time, by rewarding reproductive fitness, it amplifies the effect of small advantages, in some cases turning a chance, small fitness advantage into the entire population of a species in the future.
8. I could obviously write a more complex simulation that imitates natural selection, one in which a small advantage is eventually turned into a new species, but because of the complexity of such a simulation and the amount of tuning required, I don't think this would represent anything but confirmation bias.
My simulation greatly exaggerates the advantage conferred by a prime number, and it assumes something that's almost never true in nature -- that cicadas and predators hatch out in synchrony at the outset. But that set of circumstances does represent a small fitness advantage, and it probably explains the two primary reproduction cycles of the cicada species my paper describes (i.e. 13 and 17 years).
Yes, I certainly did not disagree that the effect is real. But at the same time the effect shown by the simulation is due to an artifact of the simulation.
Consider the two possible groups of predators with cycle 2:
year 0123456789 ...
predator1 X X X X X X ...
predator2 X X X X X X ...
In your simulation, you are only taking into account predator1, i.e. the predator that hatches at even years. You are not taking into account predator2, the one that hatches at odd years. This artificially punishes cicada that have cycles that are multiples of 2. Your argument is essentially that if you had a cicada with an even cycle length, then it would get eaten by predator 1. But one could equally validly say that an even cycle length is an advantage because predator 2 will never be able to eat it. For example the cicada cycle 4:
year 0123456789 ...
predator1 X X X X X X ...
predator2 X X X X X X ...
cicada X X X X
This cicada gets eaten by predator1 but not by predator2. So by assuming that all predators with cycle length 2 are of type 1 you are artificially punishing cicada with cycle 4. If you do the simulation with equal amounts of predator1 and predator2 you would not see any difference between prime and non prime cicada.
Now, it could still be that prime numbers have an evolutionary advantage; this kind of simulation is just insufficient to show it. For example I could think of a reason that has to do with natural selection of the predators. Because predators that are in exact dis-synchrony with the cicada have less food, they are less successful. So over time predators of type 1 will increase, and predators of type 2 will decrease. So predators naturally start to synchronize with cicada. That's why prime years are better for cicada, because there are fewer predator types that can synchronize with that. But the synchronization should be an emergent property of the simulation, you can't a priori assume that predators that share a prime factor are always synchronized with the cicada as you have done here.
That's the theory, but in practice I can think of any number of reasons why in practice even this is a very flawed model. The primary reason why cicada all hatch together is because then the predators cannot catch them all because there are simply too many cicada. So if at one point in history the cicada had a cycle of 6 years, then they will not adapt to a cycle of 7 years through evolution. Because the lone cicada that evolves to hatch every 7 years will easily be eaten. This strategy only works when a bunch of them do it together. So it is very possible that 7 years is just a historical accident.
> In your simulation, you are only taking into account predator1, i.e. the predator that hatches at even years.
1. No, not even years necessarily, but any scenario in which cicadas and predators hatch out in synchrony, regardless of the numbering of the years (as long as both cicadas and predators have the same year number applied).
2. Did you miss that I tried adding one more loop that tested all the relationships between cicadas and predators WRT year differential, and that shows only a tiny advantage for prime-numbered years? The size of the effect being discussed only works if predators and cicadas hatch out in synchrony, so my having set both to zero ends up emphasizing this specific case to a perhaps absurd degree. If it weren't for the fact that natural selection tends to amplify small fitness advantages, this relatively small effect might not be be worth discussing.
> The primary reason why cicada all hatch together is because then the predators cannot catch them all because there are simply too many cicada.
Yes, true, but this doesn't address the timing of the synchronous hatch-out. Why 13 and 17 years, and no other cycles in nature? My paper only addresses the timing of the hatch-out, not the fact of it.
> you can't a priori assume that predators that share a prime factor are always synchronized with the cicada as you have done here.
But I don't do that. My simulation addresses the case in which cicadas and predators hatch out in synchrony (advantage goes to the cicada), versus times when they don't (no advantage or disadvantage). It's not an assumption, it addresses the fact that cicadas and predators sometimes do hatch out in synchrony. There is no "always" assumption at work, only that, when there is a synchronous hatch-out, this works to the advantage of prime-numbered reproduction cycles. Other year relationships between cicadas and predators show no advantage or disadvantage.
It's true that my simulation greatly exaggerates the effect, since such a synchronous hatch-out is rare in nature. But I already discussed why designing as full-on natural selection simulation would probably produce nothing more than an exhibition of confirmation bias.
The effect being discussed is real, it is much smaller than my simulation would lead one to believe, but natural selection often amplifies small differences in fitness.
> Did you miss that I tried adding one more loop that tested all the relationships between cicadas and predators WRT year differential, and that shows only a tiny advantage for prime-numbered years?
You asserted this, but I do not believe this is true:
1. Your code so far has bugs that I pointed out that you did not address.
2. My corrected version shows no advantage as you'd intuitively expect.
> It's not an assumption, it addresses the fact that cicadas and predators sometimes do hatch out in synchrony.
You can't assume that because something sometimes happens it always happens. You need to take all possibilities into account evenly.
Anyway, I do not think that further discussion is useful. I have explained my point enough times that repeating it one more time is not going to cause you to see what I mean...so I'm out of this discussion for now.
> Your code so far has bugs that I pointed out that you did not address.
I updated the code. This was not something I remembered to mention, since it didn't change the results for the zero synchrony case -- but it certainly proved your point that a difference in starting year wiped out the advantage, something I then discussed at length.
The current code (the old links should still work for this same version):
> My corrected version shows no advantage as you'd intuitively expect.
Not so for the case my simulation models, and other assumptions produce no advantage or disadvantage (they're neutral).
> You can't assume that because something sometimes happens it always happens.
We already covered this. My simulation doesn't make that assumption, it shows the advantage that is present when that happens, however rare that might be.
> You need to take all possibilities into account evenly.
I did. That was the point of my last post. The synchronous hatch-out produces an advantage, no other circumstance does. Therefore the overall advantage is equal to the synchronous advantage multiplied by its probability. This doesn't take into account the fact that natural selection can amplify small fitness advantages, it only describes its genesis -- no feedback effects are considered.
> I have explained my point enough times that repeating it one more time is not going to cause you to see what I mean
I do see what you mean, as I explained at length in my last post. The effect being described is a factor that may explain the two cicada reproduction cycles seen in nature, as my article points out.
Dude, the code you posted again contains the exact same problem that started off this discussion! Even though I've already posted the solution! But of course that solution doesn't suit you because it shows that your previous results are wrong.
I'm not sure if you're saying here that now that you've been made aware of the problem you're just going to say that it's intentional, and that you're introducing this arbitrary assumption because that's what your simulation is supposed to investigate. If that's the case then fine, but the results of your simulation are still meaningless. The reason the prime numbers come out is still a purely synthetic artifact.
I can understand that you're bummed that your article is misleading and don't want to admit it. Just let the article be. I'm probably the first and the last person to ever notice that it's wrong. Has been very effective for psychologists ;-)
Yes, it does. The effect I am describing is real, and I simply haven't thought it through in detail until now.
1. The starting point for my simulation has both the cicadas and the predators hatching out simultaneously, in year zero. This obviously happens in nature as well as in my simulation (but only rarely).
2. Given that starting condition, what is the likelihood that the predator, with a range of reproduction cycles, will succeed in synchronizing with the cicadas again, or (the best of all worlds) over and over in perpetuity?
3. The answer is that, if the cicadas adopt a prime-numbered reproduction cycle, the predators are handicapped by the fact that a typical prime number has fewer divisors than a composite number, and divisors are required for synchronization:
http://en.wikipedia.org/wiki/Table_of_divisors
4. For cases in which the predators and the cicadas are out of synchronization, the outcome for a random difference in years is neutral -- not advantageous, not disadvantageous. I've shown this by adding a loop to my program that tests against a range of differences in years between cicadas and predators. When I do this, the difference between a prime and a composite nearly vanishes ("nearly" because the advantage of a prime numbered cycle is still present, but is almost completely buried by the much greater number of years where there's no advantage).
5. It is only when cicadas and predators hatch out in synchrony (for any one of a number of reasons) does this prime-number effect have any effect on the outcome.
6. In terms of natural selection, this leaves the prime-number effect, plus an initial synchronization between cicada and predator, as the only selection advantage, for which the prime-number effect works against the predators.
7. Nature, and natural selection, does something my simulation doesn't -- over time, by rewarding reproductive fitness, it amplifies the effect of small advantages, in some cases turning a chance, small fitness advantage into the entire population of a species in the future.
8. I could obviously write a more complex simulation that imitates natural selection, one in which a small advantage is eventually turned into a new species, but because of the complexity of such a simulation and the amount of tuning required, I don't think this would represent anything but confirmation bias.
My simulation greatly exaggerates the advantage conferred by a prime number, and it assumes something that's almost never true in nature -- that cicadas and predators hatch out in synchrony at the outset. But that set of circumstances does represent a small fitness advantage, and it probably explains the two primary reproduction cycles of the cicada species my paper describes (i.e. 13 and 17 years).