My questions get weirder and weirder. Here's one for you all.

Assume a colony world, populated initially in year 1 by 5000 people,
half male and half female, organizing into 2500 breeding couples. Each
couple has 6 children on average over the course of their
marriage/relationship.

16 years after the establishment of that colony, another 1000
newcomers enter the gene pool and population, also divided 500 men and
500 women, also entering into breeding, along with the 16 year old
children of the inital colonists. Assume that this 'generation' of
colonists stays at 6 children (it's a big planet and there's no cable,
so there's not much else to do). The child mortality rate is factored
into the above -- the assumption is 6  children surviving to menarche.

For the next 50 years, new colonists continue to emigrate to this
world, 1000 a year, and the children of earlier generations also enter
the population and breed. The average begins to go down, tending
towards 5 children by year 66.

As of year 66, colonization stops. Therefore, 55,000 colonists have
joined the world over the course of 66 years, with rough gender parity
and a continuing high birthrate.

Part 1 of the question is relatively simple. Assuming the birthrate
declines from 5 children per union on average in year 66 to 3 children
on average in the year 1200, and assuming no major plagues or famines,
and no lack of growing room, what would the population of the world be
in the year 1200.

Part 2 is genetic. Assume one of the new colonists who joins the
colony in year 16 is named Bob. Bob marries and has the average 6 kids
of his generation. Assuming normal propegation without mitigating
factors, what percentage of the population yielded in part 1 can trace
their lineage back to Bob? What percentage can trace their lineage
back to Bob by 2 different routes? By 4 routes? By 8?

Please "show the math" in any solution to this question. Thanks!

Hey, if nothing else, the questions aren't boring.

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Request for Question Clarification by voyager-ga on 25 Sep 2002 08:20 PDT

Hi scholarman,

I'm into writing SF stories, too, so I had to read your question
immediatly! I'm afraid there's still some info missing, which we will
need to answer your question:

- At what age do the children start to breed?
- How long between children?
- What about ...aeh... breeding patterns you would find on an isolated
colony world (inbreeding e.g., separation by distance, nation, relion,
whatever)? Should we just assume everybody has the same likelihood of
marrying any other person on the world?
- What about the decrease from an average of 5 to an average of 3
children per couple? Is this a sudden decreas... maybe linear... any
other option?

voyager-ga

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Clarification of Question by scholarman-ga on 25 Sep 2002 09:43 PDT

Good clarification requests, all. In order:
- At what age do the children start to breed? Let's assume 16 as an
average. Frontier societies tend to be a little more permissive than
ossified ones.
- How long between children? I would think 1.5 years or so in the
early years, to 3-5 years in later years. As the colony matures and
grows, the drive to have children lessens as larger families and
multiple children become more burdens than advantages.
- What about ...aeh... breeding patterns you would find on an isolated
colony world? The colony starts life with relative unity of viewpoint.
There is a single settlement radiating outward -- while over the
course of twelve hundred years it's reasonable to assume that the
entire planet ends up populated (look at what we've done in the last
250 years, population-wise), ease of movement from one segment of
society to another remains relatively high. As far as inbreeding goes
-- it's an obvious element in a population that starts from 55,000.
Let us assume that defective/non-breeding children are a part of the
model, and that the 5 children -> 3 children are specifically those
entering the gene pool actively, letting us handwave the problem a
little bit.
- What about the decrease from an average of 5 to an average of 3
children per couple? Is this a sudden decreas... maybe linear... any
other option? I am assuming this is linear, as a culture that values
large families (typical of frontier cultures) gives way to a culture
that slowly reduces the emphasis on larger families.
Hope this helps!

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Request for Question Clarification by nauster-ga on 25 Sep 2002 13:50 PDT

Do you need exact answers, or will a Monte Carlo (repeated
simulations, take the average) answer suffice?
It would be relatively easy to write a program that allowed various
inputs and then spit out average results over some number of
simulations.

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Clarification of Question by scholarman-ga on 25 Sep 2002 18:55 PDT

The Monte Carlo would be fine -- though I'd like to see some of the
results just to see the range of simulations. That could be private,
though.

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Request for Question Clarification by nauster-ga on 26 Sep 2002 13:32 PDT

I set up a program to let me simulate your world population. Oh my.
The situation very quickly gets totally out of hand. By year 193, you
are at 1 million and by year 334 you are at 1 billion. You've easily
filled the planet shoulder-to-shoulder by the end.

The numbers are extremely sensitive to the number of offspring per
breeding pair figure. You could fiddle with those numbers and produce
just about any final or intermediate results you could possibly want.
The central issue is that a 4-kids-per-couple rate doubles the
population every generation. 10 generations multiplies the population
by 1024.

I didn't proceed to part 2 yet because I'd like to give you an
opportunity to comment, guide, or redirect my efforts in light of the
above figures.

nauster-ga

Hi - 

I think that I've come up with an answer to your question that should
allow you to modify the parameters for the answer (and I think that
you'll want to modify them a bit...).  The answer that I calculate,
assuming 5000 initial settlers (age 16), 6 children each
(instantaneously), settlers of 1000 per year for 49 years between year
17 and 66 (each at age 16 with 6 children resulting) and birth rate
declining linearly from year 66 to year 1200, with death at 80 years
is 7.69 ^ 30 or 7,690,000,000,000,000,000,000,000,000,000 people - a
number for which I don't even know the name.

Here's how you can set this up yourself (as I cannot attach a file to
my answer).  I used Excel, but, any spreadsheet should probably work.

Label columns 
Year, population, immigration, birth, death, and now columns 0-80 for
age range

For rows - put a 1 in the first column under year and increment your
way up to 1200 (the easiest way is to put a '1' in the first entry (a2
in Excel) and a formula like '+a2+1' into the next column (a3) - now
copy that cell and copy it across the range.

For population, insert whatever numbers you want into cell
corresponding to the ages that you want in the first row.

Now, for the fun bits.

Go to row 3 (year 2):
  :input into b3 : +sum(g3:cl3) - copy this cell into all rows b to
1201
  :input into d3 : +g3 - this assumes that all children are born at 0
  :input into e3 : +ci2 - this gets the folks that would've been 81
  :input into g3 : +6*(w2/3) - this says, take all the 16 year olds,
divide by
                               two (couples), and multiply by 6
children -
                               change the column (w) to another to
vary age at
                               time children are born, or vary with
                               +(w2+y2+aa2+ac2+ae2+ag2)/2 to have
children born
                               at two year increments
  :input into h3 : +g2 - copy this cell into all cells from h3 to ci3
  :input into w3 : +v3+c3 - this makes all immigrants come at age 16 -
to vary
                            replace any cell corresponding to age with
a
                            +v?+c?  where ? is the row number of your
current
                            row - make sure to copy this through the
extent of
                            the immigration range

now copy g3 to ci3 and paste this into g4 to ci1201.

Enter immigration numbers into the column for immigration for the
years that you want immigration.

Now, for years 66 and above, since the birthrate is changing, we need
to change the rate of birth.  This is determined by column g (the
number of individuals popping up at age 0).  The following formula
will give you a constant decline from 5 to 3: input
'+W67*ROUNDUP((5-((A67/1200)*2)),0)/2' into g67.  Copy g67 and paste
into g68 to g1201.  To vary the linear birthrate decline, the '5' in
the formula corresponds to the starting birthrate and the '2' being
multiplied by '(A67/1200)' gives you the decrease in birthrate over
the period (in this case 2 years between 5 and 3).

The final population can be read off of the end of the spreadsheet.

I will gladly answer part two of your question, but think that you
might want to revise part 1 first.

Let me know if you have difficulty with the spreadsheet or would like
me to run several simulations for you.  I will answer part two at any
point that you want (when you're satisfied with the population
estimate from part 1)

I think that Nonatillions of people is a bit high for my purposes. :)
Really, this answered my question in the specific way I needed --
namely, it told me I can set whatever population and whatever
percentage of population I wish for the purposes of my story and know
it's within a reasonable realm of possibility. Thanks for all your
help!

I probably won't be able to answer the question, but I think some
information is missing from your question.

You mentioned they have 6 children over the course of the
relationship.  Is that having children every 2 years?  Having them all
at once (probably easier to calculate)?

What is the closest relationship that the parents can have?  Siblings
(any)?  First cousins?
Does age matter in deciding a relationship?
Arbitrary between all unattached (probably simpler).

How long do they live?

Is it a linear decline in the birthrate over the range?

Also, did you mean 64 (16*4) instead of 66?

>You mentioned they have 6 children over the course of the
>relationship.  Is that having children every 2 years?  Having them
all
>at once (probably easier to calculate)?

That's an average for family size. If we assume breeding begins at 16
and ends at 36 (arbitrary guidelines), that's twenty years for (on
later average) 5 children, or 1 every four years. However, in the
early years, children would be coming every "9 months and ten minutes
apart," as the frontiersmen say. I think the easiest thing to do is
assume they have them all at once. There's only so much granularity
the story needs for plausibility.
 
>What is the closest relationship that the parents can have?  Siblings
>(any)?  First cousins?

Intentionally left vague. I suspect that second cousins will be the
closest relationships for the most part, but mores can change a lot
over 1,200 years.

>Does age matter in deciding a relationship? 

No.
 
>How long do they live? 
 
Let's call it 80 years on average.

>Is it a linear decline in the birthrate over the range? 
 
Yes.