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So I have a religious group that settled a new continent and made a colony in 1630.

If that nation has thirty-six million people in the modern day (2020 at time of posting), how do I calculate the number of followers necessary for all of the modern day people to be related to/descendants of members of that original colony?

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    $\begingroup$ Has there been no immigration in those four centuries? A recent immigrant would definitely not be related to the initial colonists. Is the question assuming that the current population of 36 million is entirely descended from the initial colonists? Note that in this case the initial colonists must be unusually numerous... After 1800, population doubled every hundred years; in preindustrial times, a doubling every two centuries would be optimistic. Overall, to have 36 million today entirely descended from an initial population 4 centuries ago that population needs to be at least 4 or 5 million. $\endgroup$ – AlexP Jan 23 '20 at 16:13
  • $\begingroup$ According to the internet America only had a population of about 5.3 Million in 1800 and now has more than 300 Million. It seems like it's possible to have a smaller population than 4 or 5 Million in 1630. On the other hand America had a lot of immigration $\endgroup$ – YLong Jan 23 '20 at 16:59
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    $\begingroup$ Yes your numbers are correct. On the other hand, the vast majority of those 300 million people are descended from immigrants who arrived in America between 1850 and 1950, or from people who were incorporated in the U.S.A. by conquest or purchase -- in 1800 the USA was very much smaller than it is today. I understood that the question was about pure natural growth, with no immigration. $\endgroup$ – AlexP Jan 23 '20 at 17:01
  • $\begingroup$ Yes it was about pure natural growth sorry about that. I forgot about America's history of immigrants for a second there. $\endgroup$ – YLong Jan 23 '20 at 17:02
  • $\begingroup$ @AlexP it would be interesting to calculate the contribution of immigration to US population growth. Yes, vast majority of Americans have ancestors who arrived after 1850, but also the ancestors who arrived before 1850. $\endgroup$ – Alexander Jan 23 '20 at 17:19
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The simplest answer is to pick a population growth rate and apply the Rule of 72 in reverse. Modern (industrial) societies seem to be running about 3%, which means every 24 (72÷3) years you go back, you halve the population. If you have 36M people in 2020, that means 18M in 1996, 9M in 1972, 4.5M in 1948, and so on until you reach ~550 people way back in 1636 (if I did the math right).

However, as noted in a comment, that value is probably wildly optimistic for a pre-industrial society. If you reduce the growth rate to say, 1%, then you would only halve the population every 72 years, which means your original colony must be much larger—or you'd need a lot of immigration. OTOH, if your religion had childbearing as a major tenet, and the colony didn't send a significant fraction of its young people off to die in wars or experience regular famines or plagues (a good reason to colonize a new continent!), you might be able to do better.

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    $\begingroup$ Reality check: The population of the Qing empire, basically co-extensive with the modern People's Republic, was 400 million around 1900 and 300 million around 1800. If China managed only a 33% population growth in 100 years I strongly suspect that a natural increase of 3% per year is vastly exaggerated for pre-industrial societies... In Europe, the country with the fastest (naturally) growing population was France, which doubled its population in two hundred years, 1800-2000. $\endgroup$ – AlexP Jan 23 '20 at 16:58
  • $\begingroup$ @AlexP Does my edit address that sufficiently? $\endgroup$ – StephenS Jan 23 '20 at 17:13
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    $\begingroup$ Yes, it does. (About wars: in European history, only the Thirty Years' War, the Napoleonic wars, and the two World Wars are immediately visibile on a population graph, and even those only for a few countries.) $\endgroup$ – AlexP Jan 23 '20 at 17:17
  • $\begingroup$ So suppose childbirthing was a major tenet and famine and wars didn't kill a significant fraction of young people what do you think the growth rate should be. $\endgroup$ – YLong Jan 23 '20 at 17:43
  • $\begingroup$ @YLong 2-3% under ideal conditions. $\endgroup$ – StephenS Jan 23 '20 at 17:51
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Colony population growth is different from country, or the whole continent population growth. On one hand, colony may have a plenty of arable land and untapped natural resources (which a settled nation doesn't have), which can lead to population boom. On the other hand, it is particularly vulnerable to diseases and crop failures. The famous Mayflower pilgrims could have perished, one and all, and American history books would have to look for another group of forefathers.

Let's assume the following:

  • Conditions were very favorable for the colonists. There were no major crop failures or diseases;
  • The new land was suitable for agriculture and colonists had access to high-yield crops like corn and potatoes;
  • There was no significant immigration since the first group had arrived;
  • The colony had access to all technological inventions from the rest of the world;

In this case there will be a population boom, resulting in doubling the population every 20 years or so.

2020 - 1630 = 390 years

390 / 20 = 19.5 doubling cycles

If 2020 population is 36 millions, we had to start in 1630 with just 48.5 people.

So the math works, but sticking to my assumptions for all 390 years may be unrealistic.

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    $\begingroup$ Could you provide an example of a colony, anywhere and at any time, which increased its population by a factor of 32 in a hundred years though natural growth? I'll even settle for a factor of 10... $\endgroup$ – AlexP Jan 23 '20 at 17:46
  • $\begingroup$ No - because in historic times there probably wasn't any successful colony that was restricted to natural growth only for 100 years :) $\endgroup$ – Alexander Jan 23 '20 at 17:55
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The mathematics has been well explained in the comments etc but contradicts history. The maths does show double digits are potentially enough for a starting population, depending on growth rate assumptions.

Growth is rarely linear. The growth rate will change over time and won't be uniform, being typically higher on the frontier and less developed areas and when technology changes (antibiotics, mechanisation). https://ourworldindata.org/world-population-growth has various ways of looking at it

A more important reality check is avoiding inbreeding. Genetic research suggests there was a bottleneck event reducing our ancestors to about 10,000. Research on interstellar colonisation suggests 500-1000 drawn worldwide would provide suitable diversity.

This suggests your colony's starting population 'reality check' will dependent more on how widely it draws its initial population from, in the absence of immigration (or natives to assimilate), than any growth rate assumptions.

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