Why? Given the random nature (in our experiment) of mutations, there is an equal chance that mutation will reach 1 or that mutation will reach 0 (where 1 and 0 are measures of 'closeness' to other members of the same species, with 1 being a new species and 0 being exactly the same as another member of the species). Say a mutation, with a 1x10^-8 chance of happening, occured which moved the organism from 0 to 0.1. There is an equal chance for the organism to move back to 0 given another random mutation, as there is that the organism will move from 0.1 to 0.2. ("Organism" in this sense I suppose being a transfered genetic code passed on to children).
Why? Given the random nature (in our experiment) of mutations, there is an equal chance that mutation will reach 1 or that mutation will reach 0 (where 1 and 0 are measures of 'closeness' to other members of the same species, with 1 being a new species and 0 being exactly the same as another member of the species). Say a mutation, with a 1x10^-8 chance of happening, occured which moved the organism from 0 to 0.1. There is an equal chance for the organism to move back to 0 given another random mutation, as there is that the organism will move from 0.1 to 0.2. ("Organism" in this sense I suppose being a transfered genetic code passed on to children).
Naturally. If this is what you said in your original post, then I apologise. We are on the same page after all.
ETA: Woah, hang on here. Boy did I read that post badly.
Why is it unlikely? Given a very long time, would there not be an equal chance of speciation as no speciation? Given that mutations are completely random in nature, and happen equally throughout a genome, and that there is no selective pressure.
The Drunkard's Walk theorem.
Given a random walk in a high dimensional space (such as gene-space), the probablity of the drunkard returning to his starting point is zero.
In particular, the expected distance from his starting point after N steps is increasing in N (I think the actual function is sqrt(N), but I don't remember that much from college topology. I had other things on my mind that semester.)
I'm not sure I understand why the probability would be zero. I'd appreciate an explanation, if you could give one.
There's some information available on Wikipedia:
http://en.wikipedia.org/wiki/Random_walk
I'm not sure I could reproduce Polya's proof. Let me rephrase that -- I don't remember Polya's proof. But there's a paper that seems to do some of the work available at
http://math.dartmouth.edu/~doyle/docs/thesis/thesis.pdf
This paper describes an electrical analogue to Polya's theorem.
Once you have Polya's theorem, though, the demonstration is pretty clear. In one or two dimensions, the drunkard will provably always return to his starting point given enough time. In three or more dimensions, there is a finite probability that the drunkard will not be able to return to his starting point. So even if he returns to his starting point a few times, eventually (since he has a positive escape probability) he will (with probablity one) "escape" and never return.
In genetic terms, we call this speciation.
I would accept that as an accurate summation of my arguments. Given enough time, the chance that genetic drift will result in a new species probably approaches one. But over moderate lengths of time, mutation alone wouldn't lead to speciation very rapidly. Even when it did, the old species would likely have been entirely "replaced" by the new form, so no additional species would have been produced - the definition of the earlier species would simply have changed.Ahh. I was a little confused as you were quoting my post.
I think, though, that Melendwyr is not considering selective pressure. I think his argument goes "by random mutation alone, speciation has a low probability of happening".
