




Renegade
____________/
Registered: May 2001
Location: Prague, Czech Republic


I posted this on another forum, but thought I might post it here for the edification of the maths wizards on this forum as well:
quote:  Okay, firstly these aren't homework questions: I've been reading "The Blind Watchmaker" recently and it's piqued my interest about the mechanics of evolution, so I'm really just asking these questions out of idle curiosity. Secondly, you shouldn't need to know much about evolution to answer these questions and the mathematics here shouldn't be too difficult, but I am absolutely useless with maths and wouldn't even know how to begin to set the equations up.
Question 1:
According to wikipedia, the odds of a person being spontaneously born with an extra digit are about 1 / 500. Presume that the gene for an extra digit is dominant (which is likely), so that a couple in which only one parent has an extra digit has about a 50% chance of producing a child with an extra digit and that the average couple has 2 children. Then presume that people born with an extra digit are 10% more likely to survive than those born without. In a population size of 1,000, how many generations will it take before the entire population has six digits? What if we presume a population size of 1,000,000 or 6 billion?
Question 2:
Presume that the mean length of the webbing on the hands of humans is 5mm. Presume the with each generation there is a natural 20% variance on the length of the webbing (i.e. the length of the webbing varies between 4mm and 6mm for the first generation) and that the data for each webbing length is evenly spread amongst the population. If we presume that each extra mm of webbing makes an individual 5% more likely to survive and reproduce, how many generations until the webbing of the entire population (again, let's say 1000, 1,000,000 and 6 billion) reaches all the way to the tops of the fingers (let's say, about 100mm)?
Question 3:
Similar to the last one, only assume it has to do with the height of animals. If the mean height of an animal is 100cm and there is a natural 5% variation either way (i.e. the range is 95cm  105cm in the first generation) and that each additional cm means that the animal is 1% more likely to survive then how many generations will it be until the animal is 200cm tall? How about 1,000 cm?
If there are any variables or assumptions I've missed in these scenarios, feel free to fill in the blanks yourselves or ask me about it. Thanks guys. 
Anyone have any ideas?
___________________
http://eschatonnow.blogspot.com/


May262006 09:29





PsyT
Melody Klein
Registered: Jan 2003
Location: Haifa


all the 999 members of this tribe are complete prudes, they will only reproduce with someone if that person has the same length of hand webbings as they do, they are completely monogamous, and they won't give birth to more than 2 offspring per couple, and to top that off, they're dying rapidly of a genetic disease to which the antibodies reside in the handwebbings, some of the former members of the tribe who didn't have handwebbings are all dead by now, they were estimated to have a 70% mortality rate, applicable before they had the chance to reproduce.. because that's when the disease attacks  before they can have an orgasm.
furthermore they discovered that each additional 1mm of handwebbing lowers their mortality rates by 5%.
two more quirky things about this tribe are that their offspring always come out half with an added 1mm to their handwebbings, and half with the same length as that of their parents, and the other thing, that comes somewhat incomprehensibly to my mind, is that currently their tribe is evenly divided to 3 groups  1 with 4mm webbedhands, 1 with 5mm webbedhands, and 1 with 6mm webbedhands.
the chief of the tribe sent me an email asking me for how many more generations will his tribe survive, if at all.
population 999
4mm 333
5mm 333
6mm 333
mortality
4mm 166
5mm 182
6mm 200
population 548
4mm produce 83 offspring with 5mm, and 83 with 4mm
5mm produce 91 offspring with 6mm, and 91 with 5mm
6mm produce 100 offspring with 7mm, and 100 with 6mm
4mm 83
5mm 174
6mm 191
7mm 100
population 548
mortality
4mm 41
5mm 95
6mm 125
7mm 65
population 326
4mm produce 20 with 5mm, 20 with 4mm
5mm produce 47 with 6mm, 47 with 5mm
6mm produce 62 with 7mm, 62 with 5mm
7mm produce 32 with 8mm, 32 with 7mm
4mm 20
5mm 67
6mm 109
7mm 94
8mm 32
population 322
mortality
4mm 10
5mm 37
6mm 66
7mm 62
8mm 22
population 197
4mm produce 5 with 5mm, 5 with 4mm
5mm produce 18 with 6mm, 18 with 5mm
6mm produce 33 with 7mm, 33 with 6mm
7mm produce 31 with 8mm, 31 with 7mm
8mm produce 11 with 9mm, 11 with 8mm
4mm 5
5mm 23
6mm 51
7mm 64
8mm 42
9mm 11
population 196
mortality
4mm 2
5mm 13
6mm 31
7mm 42
8mm 30
9mm 9
population 126
4mm produce 1 with 5mm, 1 with 4mm
5mm produce 6 with 6mm, 6 with 5mm
6mm produce 15 with 7mm, 15 with 6mm
7mm produce 21 with 8mm, 21 with 7mm
8mm produce 15 with 9mm, 15 with 8mm
9mm produce 4 with 10mm, 4 with 9mm
5mm 7
6mm 21
7mm 36
8mm 36
9mm 19
10mm 4
population 123
mortality
5mm 4
6mm 13
7mm 24
8mm 26
9mm 14
10mm 3
population 84
5mm produce 2 with 6mm, 2 with 5mm
6mm produce 6 with 7mm, 6 with 6mm
7mm produce 12 with 8mm, 12 with 7mm
8mm produce 13 with 9mm, 13 with 8mm
9mm produce 7 with 10mm, 7 with 9mm
10mm produce 1 with 11mm, 1 with 10mm
5mm 2
6mm 8
7mm 18
8mm 25
9mm 20
10mm 8
population 81
mortality
6mm 6
7mm 12
8mm 18
9mm 15
10mm 6
population 57
6mm produce 3 with 7mm, 3 with 6mm
7mm produce 6 with 8mm, 6 with 7mm
8mm produce 9 with 9mm, 9 with 8mm
9mm produce 7 with 10mm, 7 with 9mm
10mm produce 3 with 11mm, 3 with 10mm
6mm 3
7mm 9
8mm 15
9mm 16
10mm 10
11mm 3
population 56
mortality
6mm 2
7mm 6
8mm 11
9mm 12
10mm 8
11mm 2
population 42
6mm produce 1 with 7mm, 1 with 6mm
7mm produce 3 with 8mm, 3 with 7mm
8mm produce 5 with 9mm, 5 with 8mm
9mm produce 6 with 10mm, 6 with 9mm
10mm produce 4 with 11mm, 4 with 10mm
11mm produce 1 with 12mm, 1 with 11mm
7mm 4
8mm 8
9mm 11
10mm 10
11mm 5
population 38
mortality
7mm 3
8mm 6
9mm 7
10mm 8
11mm 4
population 28
7mm produce 1 with 8mm, 1 with 7mm
8mm produce 3 with 9mm, 3 with 8mm
9mm produce 3 with 10mm, 3 with 9mm
10mm produce 4 with 11mm, 4 with 10mm
11mm produce 2 with 12mm, 2 with 11mm
8mm 4
9mm 6
10mm 7
11mm 6
12mm 2
population 25
mortality
8mm 3
9mm 5
10mm 6
11mm 5
12mm 2
population 21
8mm produce 1 with 9mm, 1 with 8mm
9mm produce 2 with 10mm, 2 with 9mm
10mm produce 3 with 11mm, 3 with 10mm
11mm produce 2 with 12mm, 2 with 11mm
12mm produce 1 with 13mm, 1 with 12mm
9mm 3
10mm 5
11mm 5
12mm 3
population 16
mortality
9mm 2
10mm 4
11mm 4
12mm 3
population 13
9mm produce 1 with 10mm, 1 with 9mm
10mm produce 2 with 11mm, 2 with 10mm
11mm produce 2 with 12mm, 2 with 11mm
12mm produce 1 with 13mm, 1 with 12mm
10mm 3
11mm 4
12mm 3
population 10
mortality
10mm 2
11mm 3
12mm 3
population 8
10mm produce 1 with 11mm, 1 with 10mm
11mm produce 1 with 12mm, 1 with 11mm
12mm produce 1 with 13mm, 1 with 12mm
11mm 2
12mm 2
population 4
mortality
11mm 2
12mm 2
population 4
11mm produce 1 with 12mm, 1 with 11mm
12mm produce 1 with 13mm, 1 with 12mm
12mm 2
population 2
mortality
12mm 2
population 2
12mm produce 1 with 13mm, 1 with 12mm
as we can see here, despite the tribe's lowering mortality rates, the strictness of their rules led to their end within 14 generations, it's a sad tale.
___________________
People who own my ass: Citric Acid, Boomer187, Tribu, Sand Leaper,
Jackson, venomX, jamie, Renegade, Konjin, Akridrot, Miss Bliss.
PsyT  Down The Rabbit Hole (400minute long acid set)
Last edited by PsyT on May262006 at 19:20


May262006 16:02





DrUg_Tit0
e^(i*pi)+1=0
Registered: Nov 2002
Location: Zagreb, Croatia


quote:  Originally posted by Renegade
I posted this on another forum, but thought I might post it here for the edification of the maths wizards on this forum as well:
Anyone have any ideas? 
Ok, for the first guys, if they are 10% more likely to survive, I suppose their ratio in the general population grows by 10% each generation. Now, I suppose there's a oneline formula, but we can also go step by step.
1st generation  2 6 fingered, 998 5 fingered
2nd  2.200000 997.800000
3rd  2.420000 997.580000
4th  2.662000 997.338000
5th  2.928200 997.071800
6th  3.221020 996.778980
7th  3.543122 996.456878
Hm..ok, this isn't really working..um..wait.
After n generations you have 2*(1.1)^n people with 6 fingers. So you wanna have a 1000 of them, then they'd exterminate the preceeding population. So, 2*(1.1)^n=1000 => 1.1^n=500 => n=log500/log1.1=65.8 generations. It's the same, regardless of the number of people (well, actually it isn't since with low populations you don't get whole numbers, but decimals  in that case every generation would have a 10% chance that one more 6 fingered kid will be born. So in that case the number of generations is partially based on luck)
The other two are pretty complicated to do. Like, for the webbings, you'd probably end up having all webbing dimensions, from 4100mm, and you'd have to count each separately. Additionally, if people with mixed webbing lengths breed, what's their length? Median? I suppose you'd end up with some sort of a gaussian curve that would move a bit towards 100mm every generation..
___________________
1+1=10


May262006 22:59





PsyT
Melody Klein
Registered: Jan 2003
Location: Haifa


quote:  Originally posted by DrUg_Tit0
Ok, for the first guys, if they are 10% more likely to survive, I suppose their ratio in the general population grows by 10% each generation. Now, I suppose there's a oneline formula, but we can also go step by step.
1st generation  2 6 fingered, 998 5 fingered
2nd  2.200000 997.800000
3rd  2.420000 997.580000
4th  2.662000 997.338000
5th  2.928200 997.071800
6th  3.221020 996.778980
7th  3.543122 996.456878
Hm..ok, this isn't really working..um..wait.
After n generations you have 2*(1.1)^n people with 6 fingers. So you wanna have a 1000 of them, then they'd exterminate the preceeding population. So, 2*(1.1)^n=1000 => 1.1^n=500 => n=log500/log1.1=65.8 generations. It's the same, regardless of the number of people (well, actually it isn't since with low populations you don't get whole numbers, but decimals  in that case every generation would have a 10% chance that one more 6 fingered kid will be born. So in that case the number of generations is partially based on luck) 
i'm not sure whether you worked with "a couple in which only one parent has an extra digit has about a 50% chance of producing a child with an extra digit and that the average couple has 2 children" as averages or imposed rules, because i don't understand the methods you used, but if it's the latter, i'm pretty confident the 6 fingered people can not rise signifinctly in number, but just in ratio to the 5 fingered people.
under the assumption that the statement quoted above is a rule, the only population rise for the 6fingered people can come from "the odds of a person being spontaneously born with an extra digit are about 1 / 500".
quote:  Originally posted by DrUg_Tit0
The other two are pretty complicated to do. Like, for the webbings, you'd probably end up having all webbing dimensions, from 4100mm, and you'd have to count each separately. Additionally, if people with mixed webbing lengths breed, what's their length? Median? I suppose you'd end up with some sort of a gaussian curve that would move a bit towards 100mm every generation.. 
what's wrong with the way i did that one other than the simplifications?
___________________
People who own my ass: Citric Acid, Boomer187, Tribu, Sand Leaper,
Jackson, venomX, jamie, Renegade, Konjin, Akridrot, Miss Bliss.
PsyT  Down The Rabbit Hole (400minute long acid set)


May272006 00:35





DrUg_Tit0
e^(i*pi)+1=0
Registered: Nov 2002
Location: Zagreb, Croatia


quote:  Originally posted by PsyT
i'm not sure whether you worked with "a couple in which only one parent has an extra digit has about a 50% chance of producing a child with an extra digit and that the average couple has 2 children" as averages or imposed rules, because i don't understand the methods you used, but if it's the latter, i'm pretty confident the 6 fingered people can not rise signifinctly in number, but just in ratio to the 5 fingered people. 
Hm, yeah, I really forgot to take notice of that part. Basically since the 5 fingered gene is recessive, it would result in a fact that there's going to be a much longer time until the 5 fingered gene disappears, since people can carry it and still have the 6fingered advantage. Infact it may not disappear at all, ever. But that can't be calculated exactly, it's a matter of chance.
quote:  under the assumption that the statement quoted above is a rule, the only population rise for the 6fingered people can come from "the odds of a person being spontaneously born with an extra digit are about 1 / 500". 
Yes, that and that they have a 10% advantage.
quote:  what's wrong with the way i did that one other than the simplifications? 
Well, nothing..except that humankind has not died out yet
___________________
1+1=10


May272006 03:41





PsyT
Melody Klein
Registered: Jan 2003
Location: Haifa


quote:  Originally posted by DrUg_Tit0
Well, nothing..except that humankind has not died out yet 
cheeky!
the tribe died mostly because i set it to very closed circumstances where they had to reproduce only twice per cycle, and because the sample size i chose was the smallest, i'm pretty sure they would have survived; in a different situation, i calculated the same with a few changes, most importantly 3 reproductions per couple aswell as much harsher mortality rates  in that one, already on the 4th generation the fast dwindling of the tribe turned to a population increase.. on the 9th (or maybe it was the 10th) generation, the illness precentage in the tribe was around 6% and falling; by the 11th the tribe had a larger population than it did in the begining.
___________________
People who own my ass: Citric Acid, Boomer187, Tribu, Sand Leaper,
Jackson, venomX, jamie, Renegade, Konjin, Akridrot, Miss Bliss.
PsyT  Down The Rabbit Hole (400minute long acid set)


May272006 04:45





trancaholic
Danish Prophet of Doom
Registered: Oct 2000
Location: Aalborg


quote:  Originally posted by Renegade
I posted this on another forum, but thought I might post it here for the edification of the maths wizards on this forum as well:
Anyone have any ideas? 
I haven't had time to think much about this, but it appears to me that your first question cannot be answered in the absolute, as it builds on a stochastic variable I(XY,Z) ("Is individual X born with an extra digit given that its parents Y and Z have extra digits"), which is identically distributed for any two individual given the same configuration of their parents. (As (0.5, 0.5) for configurations where one parent has the extra digit, (0.002, 0.998) for configurations where no parent has the extra digit, and (what?) for configurations where both parents have the extra digit.) Therefore, all you can hope to achieve is a distribution over the relative number R_N of extra digit carrying members of the population after N generations, and not any absolute values. If you're willing to set a threshold value, you could get something like "how many generations until the probability of R exceeding 0.99 itself exceeds 0.95", or something similar, which might be what you're looking for.
Of course, such calculations would be horrible, and I doubt that there's a closed form expression for the density of R_N given the number of generations N. So if I were you, I would run a lot of simulations and then calculate the empirical means and variances of R_N for each generation.
Last edited by trancaholic on May272006 at 10:11


May272006 08:14





 
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