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Okay, i need help here. I'm sure this question has been asked before and if it has, please refer me to it. But a story I have been working on for a while now revolves around the adventures of a colony on a alien earth-like planet many thousands of light years from earth. In the course of the story, which is set in the early 22nd century, the colony loses contact with earth (as the wormhole they have been using to transport new colonist and supplementary supplies to the colony collapses unexpectantly). The colony, at the time of the collapse, is decently developed and reasonably autonomous and self-sustaining, as in it has farms, water treatment facilities, government, etc., with a population around 8,000-10,000. My question is, how do I calculate how the population will grow in say a hundred years, assuming they have the means to live self-sufficiently? Please provide formula or some mathmatical process that can help me.

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

  • $\begingroup$ Offhand the population after 100 years should be between 10,000 and 2,000,000, depending on things that are not specified in the question. Are you looking for numbers, or just a process you can apply? $\endgroup$ – Cort Ammon Nov 20 '18 at 19:19
  • $\begingroup$ You have tagged this as hard science, so the answers are going to be expected to be quite mathematically sound $\endgroup$ – Cort Ammon Nov 20 '18 at 19:21
  • $\begingroup$ @Cort Ammon I am looking for a process i can apply for any population, whether its a hundred years or a thousand. $\endgroup$ – Noah Nov 20 '18 at 19:33
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    $\begingroup$ The math formula can be easily googled, like here. The biggest problem is estimation of population growth rate. $\endgroup$ – Alexander Nov 20 '18 at 19:53
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    $\begingroup$ @Noah you might want to switch the hard-science tag to science-based. The latter still expects realistic answers, but isn't as strict about proving correctness. Talking about extraterrestrial colonies doesn't exactly fit well with the hard-science tag, as humanity doesn't have any yet. Also, what are the conditions of the planet? A colony on Mars (think domed cities) will grow much more slowly than a colony on an Earthlike world. $\endgroup$ – Rob Watts Nov 20 '18 at 22:47
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Calculating a future population based on a given population growth rate is quite easy:

$populationFuture = populationCurrent \cdot e^{r \cdot t} $

where r is your annual population growth rate and t is the number of years in the future.

But while your colony may be self-sufficient, if it's a closed system it too will have to grow so the question becomes what is the limit your infrastructure places on your population growth rate?

And note that in this scenario life expectancy and death rates are being ignored.

You may find the following info useful: http://www.worldometers.info/world-population/#growthrate

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  • $\begingroup$ The colony was created to expand, so as time goes on, the society will grow and move out to form newer settlements. So really the sky is the limit. $\endgroup$ – Noah Nov 20 '18 at 22:20
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Ok, lets start with a population of 10,000 colonists.

I will assume that the population consists entirely of adults that have been selected to be of reproductive age, and that half the population is female.

Each woman will produce between 0 and 5 offspring. Assuming that they're voluntary colonists, it is likely that all available females are willing to produce children, so lets say that each female colonist produces at minimum 1 child. The average number of offspring in the first generation is 3.

So, generation 1 produces 15,000 children. I don't have a value for the rate of childhood mortality for this colony, but assuming 22nd century medical technology, it will likely be low.

Generation time for a 22nd century colony could reasonably be similar to that of late 20th/early 21st century developed nations, so: roughly 25 years.

Generation 2 (25 years): ~22,500 children.

Generation 3 (50 years): ~33,750 children.

Generation 4 (75 years): ~50,625 children.

Generation 5 (100 years): ~75,938 children.

So, the total number of births in 100 years is going to be around 197,813.

Death rates are a bit trickier, given that "22nd century technology" could have a massive effect on that. Have your colonists cracked the longevity problem? Is colonial life particularly dangerous? Are there alien pathogens around that kill off colonists?

Even without knowing these things, I think that ~200,000 colonists after 100 years is a good ballpark estimate.

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Consider the growth of the English colonies in New England and the Mid-Atlantic. These were societies that invested in their future, with strong families, churches, elementary schools, and communities. They even established small colleges to train preachers, linguists, and lawyers. They mostly used local resources, and built homes and terraformed their environment. They had reasonably stable governments, and the benefit of trade with another continent.

They had almost 300 years of steady population growth: About 2.5 percent per year natural population growth, and another 0.8 percent per year of net immigration. This is about the best possible population growth for a "K" society that invests in its society and infrastructure.

Now for the rule of 72: You can estimate the time it takes for an exponential phenomenon to double. Divide 72 percent by the growth rate. For example, 72 percent divided by (2.4 percent per year) is 30 years. If the English colonists had not had immigration, their population would double roughly every 30 years.

Another handy approximation is that 2 = 10 ^ (0.3). For example, a 10-fold increase in 100 years happens while repeatedly doubling every 30 years.

The formula for exponential growth is:

Population at time N = (Population at time 0) * (1 + growth rate)^N

For example:
Population after 100 years = (Starting population) * (1 + 0.024 per year)^(100 years)
100,000 = 10,000 * (1.024) ^ 100

There are societies today that have higher population growth rates, but they do not tend to invest in "slow-growth" terraforming. Consequently, they quickly come to rely on food imports.

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