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    Radioactive logicpolice's Avatar
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    population dynamics for adam an eve

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    I,m trying to find a number for the greatest amount of humans that could have been produced by one man and one woman in eight-thousand years. Is there some kind of formula that can work that out? And would you have to come up with all kinds of info like, gestation time, age of reproductive maturity, infant mortality rate, average life spans and all that?

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    Adventuring Infidel Cyranothe2nd's Avatar
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    Are you wanting to include genetic deformities and low fertility? Because one pair would quickly produce inbred children that would eventually become unable to bear children themselves. In fact, population experts claim that it would take 150-160 humans to repopulate the world (there would need to be that much genetic diversity for the species to survive.)
    Or are you just looking for a general mathematical equation? Yes, you would have to figure out gestation, age of conception, how many children could be born by one woman in her lifetime, ect.
    "We will not be driven by fear into an age of unreason, if we dig deep in our history and our doctrine, and remember that we are not descended from fearful men ó not from men who feared to write, to associate, to speak and to defend causes that were, for the moment, unpopular."

    Edward R. Murrow

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    Cart-mod 2.0 Global Moderator Cartesiantheater's Avatar
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    I found two ways to represent it. Both of which are only approximate. But keep this in mind: birth rates are NOT static, and realistically, birth rates were probably much smaller in ancient times, since obviously more people died more often, or much larger, in the case that the species was bottlenecked and the survival of offspring and production of offspring was put at a premium.






    I did find the following formula that models growth that was derived from observing squirrel populations. This one does not take into consideration death rate (it only uses birth rate)

    Pn = Po (1 + f/2)^n

    Where

    Pn is the population at some year n


    Po is the population at year 0

    f is the birth rate

    n is the number of years of growth



    In your case, Po ≡ 2.

    The trick is to find your birth rate. Solving generally, your birthrate would be (in terms of the formula):

    f = 2( (Pn/Po)^(1/n) - 1)

    A HUGE THING TO CONSIDER!

    This model is VERY dependent upon what birth rate number you use.




    The current global birth rate is, according to wiki, about 20.3 births per year per 1000 total population, so numerically it is 0.0203.


    Using that value you get the following

    Code:
         n     |      Pn
    
    
        2              2.02
    
        4              2.08
    
       10              2.21
    
       20              2.45
    
       30              2.7
    
       50              3.3
    
       80              4.5
    
     500               311
    
     600               856
    
     700               2350
    
     800               6452
    
    1000               48, 629
    
    3000               28 700 000 000 000
    
    8000               2.44 x 10^35
    To show how dependent this is on the birth rate, consider what happens when you replace the GLOBAL birthrate (0.0203) with the US birthrate (0.014).

    Instead of 2.44 x 10^35 we get 2(1 + (.014/2))^8000 = 3.44 x 10^ 24

    or... EIGHT orders of magnitude smaller.



    Now, you must consider a couple of factors:

    This model does NOT take into consideration the fact that the environment will only support so many people. Nor does it account for random deaths, nor does it account for death rate.








    Another method:


    Just a basic model.


    N = number of people at a given instant

    t = some time interval

    r = rate of natural increase

    dN/dt = rN

    Solve this differential equation gives:

    dN/N = rdt

    ln|N| = rt + c

    N = e^(rt + c )

    N = e^c + e^rt

    * (note that e^c is just an arbitrary constant. We shall name it No for obvious reasons)

    ==>

    N = No e^rt

    No is the starting population

    N is the population after a certain time, t, has passed, and

    e is the natural number


    A HUGE THING TO CONSIDER!

    This model is VERY dependent upon what rate of natural increase you use.

    Letting r = 0.006, as per current US statistics.


    Code:
    
    t        |       N
    
    2             2.02
    
    4             2.05
    
    10            2.12
    
    50            2.70 
    
    100           3.64
    
    500           40.17
    
    1000          806.9 
    
    8000          1 400 000 000 000 000 000 000



    Anyway, the point is this:

    IF there are no negative factors and the population is granted OPTIMUM reproduction and survival, you are going to get HUGE number of people.

    Now, if you want this to be a bit more accurate, I would suggest using a different birth rate as the years progress. It is not necessarily the case that the current birth rate is higher either. In fact, it would make sense that a population of TWO would try to have a very HIGH birth rate, to save the species, and a population of 6 billion would try to have a very LOW birth rate to save the planet.

    Why don't we try the first model with a birthrate of .05 for the first 500 years and a birth rate of 0.005 for the rest of the 8000 years?


    Pn = P500 + P7500

    P500 (with f = .05) = 460 217

    P7500 (with f = .005) = 271 574 936

    For a total population of about 272 million.


    In reality, the birth rate probably changes much more than twice within a span of 10000 years or so. The second birth rate given here is a little less than half of the current US birth rate.




    Hopefully this helps a little tiny bit. Tweak with different birth rates. Remember that if you break up the time interval into smaller time intervals with different birth rates, the total population will be the sum of each individual population calculation based upon each birth rate.

    Enjoy...
    "I was put on trial twice near Y2K for acting like Jesus and claiming to be the Messiah. Its not everyday that a man parks a Chariot of Fire in front of a tomb and stands against the US government with a bow and razor tipped arrows over his shoulder. I wore a suit of armor and was protected by an invisible bubble and my sharp tongue was more than the judicial system could handle."Jake
    "The toilet is more than a throne. It is a sacred chamber."-Anton LaVey, High Priest of Satanism

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    Cart-mod 2.0 Global Moderator Cartesiantheater's Avatar
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    Not workable? If you just calibrate the birth rate right this should work for you ...
    "I was put on trial twice near Y2K for acting like Jesus and claiming to be the Messiah. Its not everyday that a man parks a Chariot of Fire in front of a tomb and stands against the US government with a bow and razor tipped arrows over his shoulder. I wore a suit of armor and was protected by an invisible bubble and my sharp tongue was more than the judicial system could handle."Jake
    "The toilet is more than a throne. It is a sacred chamber."-Anton LaVey, High Priest of Satanism

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    Cart-mod 2.0 Global Moderator Cartesiantheater's Avatar
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    Here is a MUCH better way to do it

    EDIT- I can't help but be drawn back to this question... with this method not only can you get actual total population, you can know the number of kids, teens, adults, whatever you want. Additionally, it takes into account BOTH survival rates AND birth rates for each group, AND it takes into account that children do not have babies, and old people have much fewer babies than young people. Logicpolice, I do believe THIS is what you are looking for.



    Another thing you could do is use a Leslie matrix. It is much more accurate, but much more tedious for very long range population growth. Also, you have to have some basic linear algebra skills to use it, or you at least have to be familiar with the TI-83 calculator and it's matrix operations.




    Leslie devised a method for calculating the population growth of the females of a species that also includes age groups, survival rates, and (implicitly) reproduction rates. All the computation you need to do is (iterative) matrix multiplication. At the end of the computation, just multiply everything by two, since after all usually there are about as many females as males in a given population (although that is a function of time, but for the purposes of this I will neglect it)

    EDIT 2- I would suggest though that instead of multiplying the female population by two to get the correct results, multiply each age group by the correct amount of times the male group is larger or smaller than the female group. You can find these statistics pretty easily.





    As a first example, divide the population into the following ten year age groups:

    Children (1-10), teens (11-20), youths (21-30) and adults (31-40) - since the average lifespan wasn't much higher than 40 for a long time anyway.

    Then, let's make up some rates that each group will make it into the next age class:

    Children: 40% Teens: 50%; Youths: 60%; Adults: 0% (there is no class after adults. All adults are guaranteed to die before making it into the next class, which naturally is residency in the grave yard).

    Now we need a starting population.

    Let's start with 5 children, 2 teens, 2 young adults and 2 old adults.



    Now we need to know how many offspring each produce in on iteration (10 years).


    Children: 0; Teens: 2; Young Adults: 5; Old Adults; 2


    So, here are the two matrices we must multiply in general

    Code:
                             
    f1     f2     f3    ...
    
    s1     0      0     ...
       
    0     s2      0     ...
    times


    Code:
    n1
    
    n2
    
    n3
    
    ...

    where f1, f2, f3... are the number of females born each year per capita per group,
    s1, s2, s3... are fraction of individuals who survive into the next age class
    n1, n2, n3... are the number of individuals in each age class at the beginning of a particular iteration.


    So, the starting n's are:

    Code:
    n1 = 5
    
    n2 = 2
    
    n3 = 2
    
    n4 = 2
    The fractional survival rates are (as we defined)

    Code:
    s1 = 0.40
    
    s2 = 0.50
    
    s3 = 0.60
    
    s4 = 0
    and the number born per capita will be:

    born by children: 0

    born by teens, young adult, and old adults: 2,5,2 respectfully.





    So, now plug these in and do some matrix multiplication. I'll use paint instead of code so it is easier to see. Here is the first matrix multiplication.




    (You can see on the left most matrix, the first row is the number of offspring each group contributes per iteration. The second, third and forth rows represent each group's chances of making it into the next age group (the zero's you see must be exactly where they are in order for this to work. Don't worry about them). The left most matrix WILL NOT EVER change, unless you want to adjust survival rates or birth rates (for example, if the survival rate of children is greater the tenth iteration than it was the first) The middle matrix (just to the left of the equal sign) is obviously the number of each group we started with (5 kids, 2 teens, 2 young adults, 2 old adults). The matrix on the right side of the equal sign is the number of each group AFTER ten years.)



    Now, the product of that multiplication (shown after the equal sign) will be the number of people in each age group after 10 years (one iteration). So, based upon our initial parameters, after ten years, you will have:


    18 Children

    2 Teenagers

    1 Young Adult

    ~1 Old Adult

    Keep doing this iteration, but each time replacing the matrix in the middle (or column vector, whatever you want to call it) with the matrix that was on the one on the far right (after the equal sign) the previous iteration.


    After 6 iterations, here is what the matrix will look like:





    Basically, if we assume that there are roughly the same amount of males as females in a human population, this means that after FIFTY years you will have:


    32 Children * 2 = 64



    11 Teenagers*2 = 22

    4 Young Adults*2 = 8

    ~1 Old Adult * 2 = 2

    For a total population of 96 people.

    and after SIXTY years you will have:



    46 Children *2 = 92

    13 Teenagers* 2 = 26

    6 Young Adults * 2 = 12

    ~ 2 Old Adults * 2 = 4

    For a total population of 134 people, most of whom will be young children.




    For Adam and Eve


    Now, if we wanted to start this off with only ONE young adult female (ala Adam and Eve), instead of two teenage females, two young adult females and two old adult females, we would have, after sixty years:





    Which amounts to

    11.4 children * 2 = 23

    4.6 teenagers * 2 = 9

    1.12 young adults * 2 = 2

    .715 Old adults * 2 = 1


    For a total population of 35 people after SIXTY years, with Grandfather Able recently deceased at the hands of Cain, and Cain and his wife dead from old age, Able's long suffering widow barely clinging to life. (of course, this is all based upon the realistic life spans we assigned rather than the ridiculous ones in the Bible. But all you have to do is adjust the initial parameters.)



    Now, if you want to calculate the amount after 8 thousand years, this WILL work, but it will take a lot of work on your part.

    To make it easier, find a program that will do iterative matrix multiplication, with the value multiplying changing every iteration.


    To make it more accurate, even after you find this program (I don't know if TI-83 will do this, btw), then you need to estimate how birthrates and survival rates changed over time. Basically, run with a matrix for as many years as you believe a given birthrate and survival rates are accurate. Then take your total value of the population and create a NEW leftmost matrix that represents each new value for those parameters and multiply it by the total population. Continue as before until the parameters change.

    As far as actual lifespans, and this is the huge part, that doesn't really matter! All you have to worry about is the birth rate WITHIN some given time frame. So, if instead of 40 years your lifespan is 80, all you have to do is adjust your groups.

    Instead of children, teens, young adults and old adults, you might have:

    Teens and children (1-20 years)

    Young adults (21-40 years)

    Middle age (41-60)

    Old adults (60-80)


    And then just adjust the birth rate for each group, now using a span of 20 years instead of 10, and PRESTO! Continue as you were. For example you might have the following birth and survival rates:

    Teens and children: Each contributes .7 births every 20 years: 65% chance of making it to the next level.

    Young Adults: Each contributes 6 births every 20 years: 50% chance of making it to the next level.

    Middle age: Each contributes .7 births every 20 years 25% chance of making it to the next level.

    Old adults: Each contributes 0 births every 20 years: 0% chance of surviving death.


    I mean, this is so freakin' easy mathematically that it makes me tingle.


    Let's say after sixty years, Adam and Eve's family suddenly jumps to 80 year lifespans with the parameters listed. So what happens 80 years later?

    Start with the population of females we had after sixty years:

    11.4 children

    4.6 teenagers

    1.12 young adults

    .715 Old adults


    Now multiply the new left most matrix (made by the new parameters) by the population matrix after sixty years, and then iterate the new multiplication. So, after 1 NEW iteration:




    Which means after 80 more years we have:

    73 Children and teenagers, 15 young adults, 5 middle age adults, and 1 old folk barely clinging to life.

    Now, do this 3 more times, so...






    So, during that 320 year span, the population grew to:

    433 teens and children * 2 = 866

    128.7 young adults * 2 = 257

    23.55 middle age * 2 = 47

    2.98 old folks * 2 = 6



    Which means that the TOTAL population after 320 + 60 = 380 years is:

    866 + 257 + 47 + 6 = 1176 people.


    Just remember, the parameters you chose will have an affect on the final results. If you REALLY want to do this, do some careful research to determine birth rates and survival rates for each age group. I recommend studying modern tribal peoples as well as historical estimates on these parameters.


    However, if you put in the right numbers on this model and actually take the time to do the matrix iterations, I see no reason at all for you to not get a very reasonably accurate estimate.
    "I was put on trial twice near Y2K for acting like Jesus and claiming to be the Messiah. Its not everyday that a man parks a Chariot of Fire in front of a tomb and stands against the US government with a bow and razor tipped arrows over his shoulder. I wore a suit of armor and was protected by an invisible bubble and my sharp tongue was more than the judicial system could handle."Jake
    "The toilet is more than a throne. It is a sacred chamber."-Anton LaVey, High Priest of Satanism

  6. #6
    Woooooo! James Random's Avatar
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    Genesis 4 and 5 records the history of the Antediluvian world in a highly condensed account. From the genealogical list, which is probably complete without any gaps, the time between Adam and the flood of Noah---which occurred when the latter was 600 years old---can be calculated to be almost exactly 1656 years. If one takes Barry Setterfield's chronology as more accurate, the time period from Adam to the Flood was 2256 years.

    During this time period, man was much healthier than he is now; the gene pool, less corrupted by subsequent harmful mutations and other defects; and the environment on earth, was much more favorable to good health and long life, as can be seen by the recorded pre-flood longevities.

    While classical evolutionary theory presupposes earth's early atmosphere was a reducing atmosphere (devoid of Oxygen) newer scientific evidence confirms what Bible scholars had previously suspected: the earth's ancient atmosphere probably contained a larger fraction of oxygen than it does at present. It is even possible that photosynthesis in plant life was more efficient than it is now. A warmer average climate in ancient times would also mean a higher rate of oxygen generation by the more numerous plant life.

    At age 65, Enoch became the father of a son whom he named Methuselah, a name which means "when he dies it (the flood) shall come." Enoch went on to walk with God another 300 years and was taken up ("translated") into heaven by God without dying. Methuselah survived to age 969, the oldest man who ever lived. True to prediction, the flood came the year Methuselah died.

    Ray C. Stedman in his book Understanding Man has analyzed the meaning of the names of the descendants in the line of Seth to Noah and gives the following explanation:
    "a way of escape [for man] is indicated again in a most fascinating way in this chapter by the meaning of the names listed. There is some difference among authorities as to the meaning of these names, depending upon the root from which they are judged to be taken. But one authority gives a most interesting sequence of meanings. The list begins with Seth, which means 'Appointed.' Enosh, his son, means 'Mortal;' and his son, Kenan, means 'Sorrow.' His son Mahalalel, means 'The Blessed God.' He named his boy Jared which means 'Came Down,' and his boy, Enoch, means 'Teaching.' Methuselah, as we saw, means 'His death shall bring;' Lamech means 'Strength,' and Noah, 'Comfort.' Now put that all together:
    God has Appointed that Mortal man shall Sorrow;
    but The Blessed God, Came Down,
    Teaching, that His Death Shall Bring,
    Strength and Comfort.

    "Is this book (Genesis) from God? God has given you and me a life to watch just as Methuselah's generation watched his. It is your own life. God has written "Methuselah" on each one of us. "His death shall bring it," or "When he dies, it will come." How far is it till the end of the world for you? When you die. That is the end of the world. That is the end of man's day. Is it fifty years from now, ten, tomorrow? Who knows? But at any moment, when he dies, it will come."

    Growth of world population during various epochs of earth history can be calculated by a well-known formula:



    In this formula Pn is the population after n generations beginning with one man and one woman; n is the number of generations---found by dividing the total time period by the number of years per generation.

    The variable x can be thought of as the number of generations that are alive when P(n) is evaluated. Therefore, if x is 2, the generations that are alive are generations n and n-1. This means that only a generation and its parents are alive. It seems reasonable to choose x = 3 most of the time. Taking x = 3 means that when P(n) is evaluated generations n, n-1, and n-2 are all alive.

    C is half the number of children in the family. If each family has only two children, the population growth rate is zero, but a reasonable and conservative number of children per family is 2.1 to 2.5 as far as historical records are concerned. (The derivation of the above equation has been added as Note A at the end of this article).

    Allowing for famine, disease, war, and disaster, a few sample calculations will show that the earth's population could have easily reached several billions of people between the time of Adam and the time of the flood. It is even quite possible that the preflood population was much higher than it is now.

    Genesis 4:21-22 gives suggestions of the development of music and advanced technology during this period. Family reunions must have been spectacular affairs with average life-spans well over 900 years! Human culture and even technological achievements before the flood may well have been superior and dazzling in comparison to what we see now, even though evil in that society eventually increased to the point of that civilization's self-destruction. When the Flood destroyed the Antediluvian world only eight persons were rescued on the Ark of Noah.

    A home computer spread sheet or a hand calculator can be used to iterate world population growth rates for various realistic values of n, C, and x. This will soon convince the skeptic that the earth can be easily filled full of people in a few thousands of years.

    Henry Morris (Ref. 1) gives the following examples of possible population growth rates of the earth at various times in history:
    "...Assume that C = 2 and x = 2, which is equivalent to saying that the average family has 4 children who later have families of their own, and that each set of parents lives to see all their grandchildren. For these conditions which are not at all unreasonable, the population at the end of 5 generation would be 96, after 10 generations, 3,070; after 15 generations, 98,300; after 20 generations, 3,150,000; and after 30 generations, 3,220,000,000. In one more generation (31) the total would increase to 6.5 billion.

    "The next obvious question is: How long is a generation? Again, a reasonable assumption is that the average marriage occurs at age 25 and that the four children will have been born by age 35. Then the grandchildren will have been born the parents have lived their allotted span of 70 years. A generation is thus about 35 years. Many consider a generation to be only 30 years. This would mean that the entire present world population could have been produced in approximately 30 x 35, or 1,050 years.

    "The fact that it has actually taken considerably longer than this to bring the world population to its present size indicates that the average family is less than 4 children, or that the average life-span is less than 2 generations, or both. For comparison, let us assume then that the average family has only 3 children, and that the life-span is 1 generation (i.e., that C = 1.5 and x = 1). Then...in 10 generations the population would be 106 after 20 generations, 6,680; after 30 generations, 386,000; and after 52 generations, 4,340,000,000...At 35 years per generation, this would be only 1,820 years. Evidently even 3 children per family is too many for human history as a whole."

    With regard to the Old Testament and the time period between Adam and Noah, Morris says,
    "...the recorded average age of the nine antediluvian patriarchs was 912 years. Recorded ages at the births of their children ranged from 65 years (Mahalalel, Gen. 5:15; Enoch, Gen. 5:21) to 500 years (Noah, Gen. 5:32). Everyone of them is said to have had "sons and daughters" so that each family had at least 4 children, and probably more.

    "As an ultraconservative assumption, let C = 3, x = 5, and n = 16.56. These constants correspond to an average family of six children, an average generation of 100 years and an average lifespan of 500 years. On this basis the world population at the time of the Flood would have been 235 million people. This probably represents in a gross underestimate of the numbers who actually perished in the Flood.

    "Multiplication was probably more rapid than assumed in this calculation, especially in the earliest centuries of the antediluvian epoch. For example, if the average family size were 8, instead of 6, and the length of a generation 93 years, instead of 100, the population at the time of Adam's death, 930 years after his creation, would already have been 2,800,000. At these rates, the population at the time of the Deluge would have been 137 billion! Even if we use rates appropriate for the present world (x = 1 and C = 1.5), over 3 billion people could easily have been on the earth at the time of Noah."

    With regard to the effects of plagues, wars, and disasters on population growth rates, Ian T. Taylor (Ref. 2.) notes,

    "The use of formulas gives the maximum figure possible from the variables that have been selected, and it is cogently argued that natural disasters have always played a hand in keeping human population in check; the long-term picture is thus seen to be one of population stability. History shows, for example, that the Justinian plague, A.D. 540-90, took 100 million lives; the Black Death, A.D. 1348-80, swept away 150 million from Europe alone; and even as late as 1918-19, the influenza epidemic took 25 million lives (Wallace 1969; Webster 1799)... the awful figures for natural disasters are very quickly made up for by the subsequent rates of increase among the survivors (Langer 1964)." Taylor gives the following typical recovery curve after a plague for which data is available:


    Consider the descendants of Jacob (Israel) who numbered 70 persons (Ex. 1:5, Dt. 10:2) when they went down to stay there while Joseph was Prime Minister. They remained 400 years (Gen. 15:13, Acts 7:6; Ex. 12:41 says "430 years"), and numbered 600,000 able-bodied men, plus women and children when they left under Moses (Ex. 12:37, Nu. 11:21). If a generation was 40 years, then 10 generations is the total. A total population of 2 million would be generated, starting with only couple, if the average number of children per family was 8, which is an entirely reasonable number, since Genesis 47:27 says the Jews "multiplied exceedingly" during their sojourn in Egypt. If a generation were 30 years, then the number of children per family would have averaged 5.6. The lifespans of the average person were evidently longer than today, Moses lived 120 years (Ex. 33:39) and his brother Aaron 123. Their father Amram lived to be 137 (Ex. 6:20).

    The above formula readily shows the absurdity of evolutionary time scales for mankind. In one million years, if n = 23,256 generations, C = 1.25, and x = 3, the present population of the world would be

    P = 3.7 x 102091 persons.

    In contrast the total number of electrons in the universe is only 1090!

    Assuming that man has been on the earth for a million years or so, as the evolutionist adamantly insists, we calculate that the entire universe would now be filled full of dead bodies! A population of 1090 in one million years requires that the number of children per family be less than 2.0176.





    The total surface area of the earth is about 5 x 1014 square meters. If we allowed every man, woman and child a square meter and filled all the land masses with people the earth would hold no more than 1014 persons. (That is, one hundred thousand billion persons). In one million years this number would be reached only if the average number of children per family were less than 2.0026. The average number of children per family over the past 2000 years has been of the order of 2.1.



    The following chart assumes the human race began with two persons, Adam and Eve, relatively recently. Population growth was very rapid for 1656 years until the Flood of Noah reduced the population to eight persons (4 couples). I have arbitrarily chosen the population at the time of the Flood as 9 billion, though as shown above this may be too conservative. Very little data on world population is available until recent times, so a few intermediate points have been selected. I have guessed the world population at the time of Abraham at 5 million. For example there seems to be broad agreement that the world population at the time of Christ was between 200 and 300 million. The latest demographic data used to plot this chart is available on the Internet and is referenced below.

    In order to show the narrow range of values of C which will generate very large populations in a short time, my associate Gordon A. Hunt of Stanford University (gordo@sun-valley.Stanford.edu) has plotted sets of curves from the standard population for x = 2 and x = 3 and for several values of C. His plots are shown below in Note C.

    Note D has been added as a comment on the uncertainty of world population at the time of Christ.
    We talk about civilization as though itís a static state. There are no civilized people yet, itís a process thatís constantly going on. As long as you have war, police, prisons, crime, you are in the early stages of civilization.

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