# What is a reasonable amount of population growth for 900 years?

Starting with about 500 humans and allowing for 900 years to pass, about how many people would there be at the end? (Just asking for a rough estimate, maybe a range.) Assuming:

1. People are generally in good health, drink clean water and eat well.

2. People naturally live to around 60.

3. Contraceptives and abortions don't exist.

4. Incest isn't a problem.

5. No wars, minimal violent crime.

And what other factors might affect the end number? I know I've got to be missing some important ones.

• Is there any technological progress during these 900 years or is it stagnant ? It might also help to have the starting level of technology. From your description Considering you question I guess you have a level of tech form the 1950-1960. or it's alternate history ? Dec 9 '14 at 5:53
• – user
Dec 9 '14 at 9:05
• Could you explain why incest isn't a problem? With a population of 500 you're right up against the commonly estimated value for the lower bound of the Minimum Viable Population which assumes maximum genetic disparity in offspring. You'd quite literally need a breeding program to ensure that your population's gene pool remains diverse enough to guarantee long-term existence. While there's a lot of debate on the 500-number, Traill, Bradshaw and Brook find that "the MVP for most species will exceed a few thousand individuals". Dec 9 '14 at 16:06
• What sort of medical technology do your people have? Maternal death in childbirth was once a significant cause of death, and lots of children used to die young, but those aren't major factors anymore. If every woman can produce 10 healthy children, your population can increase by a factor of 5 every generation. Dec 9 '14 at 16:39
• @MikeNichols Correct, apparently I've had it explained to me incorrectly. The linked answer mentions a number as low as 80 as long as genetic diversity is optimised. As long as the population has the standard taboo on incest, 500 people is likely enough to establish a solid gene pool without too many deleterious genes working their way in. Dec 22 '14 at 21:02

Others have looked at this from various theoretical points of view. I want to look at it from the point of view of our own Earth's history, and what would be the impact if your population followed the same growth rate seen on Earth. Do note that for the purposes of this answer, I am ignoring the minimum viable population; in the real world, that's something you'd have to consider, and your 500 humans might not be large enough starting out particularly if they aren't selected based on genetic criteria for a maximally genetically diverse population and that diversity is carefully managed and monitored.

The Real Population Problem on the blog Do The Math comes in handy, as it gives population graphs and growth rates for the human population during different eras in a convenient form. The numbers are basically (all numbers are approximate, obviously):

• From 10000 BC to 3000 BC: about 0.03% population growth per year

Here is an alternative graph (Kremer and Vermillion) which shows the period 2500 BC to 2000 AD. Notice the negative population growth around 1300 AD, and that it dips to right around zero on several occasions. The dip around 1300 AD might be explainable by the beginnings of the Little Ice Age and the Bubonic plague, although strictly speaking that is speculation on my part.

The thesis posed in the Do The Math post to explain the jumps in growth rate is:

We can perhaps attribute the 1700 jump to the Renaissance and scientific progress. We learned to wash our hands after wrestling with our pigs, and that diseases were not caused by bad vapors conjured by impure thoughts. The jump around 1870 corresponds to the Industrial Revolution, in which coal transformed the production of steel (providing agricultural tools), rail transport of commodities, and began to mechanize agriculture in a limited way. 1950 marks the Green Revolution: full-scale petrolification of agriculture, accompanied by massive fertilizer campaigns using natural gas as the chemical feedstock.

This leads to a rather simple thesis: the surplus energy presented to us by fossil fuels allowed us to feed people more easily the world over. The bounty of fossil-fuel-turned-food encouraged an explosion in birth rates, as happens for virtually all organisms given similar circumstances. It’s so blindingly obvious that I am embarrassed to have belabored the point as long as I have.

We can also use these numbers to answer your question with a reasonable degree of accuracy. For example, for a Middle Ages level society, the human population will grow at about 0.12% every year on average. (If you want this to be realistic, don't forget to throw in the occasional boom year as well as the occasional plague or die-off due to a few years of crop failure!) Starting with 500 humans and letting 900 years pass, we end up with $$500 \times 1.0012^{900} \approx 1471$$ or let's round that off to 1,500 humans. Not a whole lot. Under those conditions, your colony is extremely vulnerable to die-off; a single serious disease can easily take out a large fraction of your entire population.

If instead we use the current era, during which we have gone to the Moon, (even human) spaceflight is so routine that it often isn't reported in the news, energy has been plentiful, intercontinental travel and trade is something the world could barely survive without, and so on, then the same calculation becomes $$500 \times 1.017^{900} \approx 1.94 \times 10^9$$ or just under two billion people.

The above calculations assume no significant leaps in technology or society during the interim period as described by the blog post. For a reasonably developed society and over such a long period of time, this appears an unrealistic assumption. If we instead take the 900 year period from 1100 AD to 2000 AD and use the above figures, then the calculation becomes slightly more involved, but significantly more realistic.

• 600 years from 1100 AD to 1700 AD at 0.12%: $500 \times 1.0012^{600} \approx 1027$
• 170 years from 1700 AD to 1870 AD at 0.41%: $1027 \times 1.0041^{170} \approx 2059$
• 80 years from 1870 AD to 1950 AD at 0.82%: $2059 \times 1.0082^{80} \approx 3957$
• 50 years from 1950 AD to 2000 AD at 1.7%: $3957 \times 1.017^{50} \approx 9192$

for a final population of about 9,200 people. Frankly this sounds low, but that's part of the problem with the exponential function: it works slowly at first and with small inputs, then takes to the skies when the input grows. Note that the first doubling took about 600 years, whereas the last doubling happened in less than 50 years.

You can plug these equations into a spreadsheet and play with the numbers to see if you can get the effect you are after. For example, if instead of starting with 500 people at year 1100 AD we start with 10,000 people and calculate over the same period, the result becomes quite different:

• 600 years from 1100 AD to 1700 AD at 0.12%: $10000 \times 1.0012^{600} \approx 20535$
• 170 years from 1700 AD to 1870 AD at 0.41%: $20535 \times 1.0041^{170} \approx 41170$
• 80 years from 1870 AD to 1950 AD at 0.82%: $41170 \times 1.0082^{80} \approx 79125$
• 50 years from 1950 AD to 2000 AD at 1.7%: $79125 \times 1.017^{50} \approx 183807$

The population grows by the same factor in both cases (about 18x) but since the starting population size is larger, the resulting population also obviously is larger.

Looking at your median age at death of 60 years, we can also look at life expectancy variation over time and conclude that this is similar to a Medieval Britain (approximately 500 - 1500 AD) level of society (at age 21, life expectancy was to a total age of 64 years). Our handy-dandy table above doesn't include specific figures covering that period, but around 0.1% population growth per year appears to be a reasonable extrapolation based on the data we do have. That also appears to match reasonably well with the Kremer and Vermillion graph posted on the History Stack Exchange. In the present day, we see such life expectancies primarily in Africa south of the equator, and slightly longer life expectancies in Russia, including Asian Russia. The effect of diseases on life expectancy is particularly pronounced in Africa, however; according to World Health Organization data as quoted on Wikipedia, the life expectancy in Botswana and Zimbabwe would more than double were it not for HIV/AIDS. Current day countries which have a life expectancy of exactly 60 years at birth are Kenya, Rwanda and Afghanistan (again WHO data, dated 2012). Perhaps interestingly, neither of these show a large difference between genders; they are all listed as 59 years for men and 61 years for women.

It is also important to keep in mind that the population growth rate is going to be heavily impacted by culture as well. If the culture is one that encourages people to have lots of children, the overall population growth rate obviously goes up; if the culture actively or passively encourages people to have fewer children (as for example is the case with China's family planning policy), the number will be lower or could even lead to a population decrease over time. A population that reduces in size over the long haul is obviously not sustainable, but in extreme situations it could become necessary to take such measures in order to avoid even worse consequences of resource overshoot. The culture can also, of course, change over time; 900 years is a fairly long time for any society when looking at it in terms of a human lifespan.

You can decide on population growth rates for different periods and perform the same types of calculations yourself to get population sizes even down to per year if you are so inclined. If you do, I encourage you to make sure that there are die-offs as well; there are going to be years when quite a number of people die, especially in a society that doesn't have access to the advanced health care technology of today, and those are going to make a major dent in the population curve. For further added realism, consider demographics as well (maybe it is a disease that kills off the young more than the adults; also considering that you don't have many people who are actually old). The effects of such a die-off could last for decades before the population pyramid is somewhat back in shape.

• In your current era calculation, shouldn't the result be 2 billion people? Dec 9 '14 at 13:24
• Life expectancy at the high end has little direct influence on population growth. Assuming peak childbearing at 20 - 35, after about 55 adult death doesn't much affect things, since the youngest children will be grown by then. Jun 3 '15 at 4:23
• @WhatRoughBeast I'm not quite sure what you are referring to. This answer is based on historic overall population growth rates, which should already account for such factors as well as things like infant mortality. If you feel this answer is in need of improvement, then please state which part you are referring to so I can fix that.
– user
Jun 3 '15 at 7:59
• No, your analysis is fine. I'm just pointing out that specifying a "normal" lifespan" of 60 doesn't have a lot of effect. 50 would do just as well. As long as the parents live long enough for the kids to grow up, anything else is frosting on the cake. Jun 3 '15 at 12:13

From looking into my Crystal Ball - no, that's not Wikipedia on my laptop, now pay attention here - the answer to your question would be 'virtually any amount'. If you look at historical populations. (ok, fine, I admit it), particularly about half way down the page, you can see a list of estimated human populations through history. In 2000 BC, the estimated world population was 27 million. 1000 years later, 50 million. That's less than a 2 fold increase. And from year 1AD to 1000, it went 200 to 265 million - it didn't even double. But from 1000AD to 2000... it went from 265 million to 6 billion. A huge increase.

From that, you can see the range over 900 years might be from somewhere between virtually no increase in population, to 25 times or more. Obviously when talking human history you have wars and plagues to deal with. So you might raise the projections if your world doesn't have them. And technology - particularly medical technology - is an important factor. If people die after catching the sniffles, its going to have a serious impact on population growth.

My recommendation is to find a year range in that chart that best represents your world's level of development, and work from that as your base guess, adjusting as desired. You gave a life expectancy of 60, so you might base your estimates around real world population increases from when the real-world's life expectancy was 60.

Other factors might include available land/resources - is it a tough life or not? Also wealth, culture, luxury (modern 'western' nations have lower birth rates than other areas of the world).

Your question is quite general so I'll try to give you a range.

The absolute maximum would probably involve an industrial baby farm type system, where the limit is basically a woman's fertility (since even with only a few men there would be plenty of sperm). If you always have 50% women then you start with 250 of them. Let's say every woman can have 1 baby a year, every year, for 50 years of her life. This leads to an absurdly high population of at least 10^32, which would be impossible once you take into account environmental effects.

A more reasonable system where everyone has a large family: Suppose everyone pairs up when they're 20 and has 6 children, who also grow up til they're 20, pair up and have 6 children then die, turning 2 people into 6 people. Every 20 years your population multiplies by 3. Simply enough, after 900 = 45*20 years you have a population of 500*3^45 = 1.48*10^24, or 1.48 million billion billion. Still ridiculous.

A more likely scenario: Couples have on average 2.4 children. That's a population increase by a factor of 1.2 every generation. Again, assuming the generation is about 20 years, that's 500*1.2^45 = 1,828,630 or 1.83 million, a much more reasonable figure. That's probably your best estimate.

• There's something seriously wrong with your program if it ran into the billions after only a single decade. At that point, none of the new babies have reached reproductive age, so you'd only have 250 + 13 * 250 = 3500 people. To a rough approximation your 1st model can be represented as 16year generations (56 over 900 years) and each generation being 33x larger than the one before. This gives an estimated final population of 33^56 ~ 10^85. Dec 9 '14 at 14:09
• By the time @DanNeely mentions 10^85, I feel compelled to point out that the observable universe is estimated at 10^80 baryons. A baryon is composed of three quarks, with a quark of course being an elementary particle. In other words, we'd be incredibly likely to hit hard physical limits long before we even approach 10^85 people, even if we could figure out how to feed everyone.
– user
Nov 9 '15 at 19:34
• Just some data points - geneticists looking at human prehistory generally assume 29 years per generation. The age of the parents at the age of their average child not their first child, is what matters. Peak fertility per woman is about 7-8 per lifetime for a large population if not genetically twin-prone. Lifetime fertility is lower for hunter-gatherers due to longer nursing related infertility viz farmers. But, there's huge variation is mortality. About 2.1 kids per woman per lifetime is replacement rate today, but you need more historically with high rates of infant and childhood mortality. Jan 6 '17 at 1:12

Assuming no wars or serious plagues, anywhere from 0 to 18 quadrillion – or more.

If each couple has an average of four children, at about the age of twenty, then after 900 years the population will be 17,592,000,000,000,000. If each couple has an average of one child, at the age of twenty, humanity will end in about 250 years.

The problem with the 18 quadrillion estimate is the lack of land and resources. live science theorizes that the Earth can support a population of up to 10 billion, long-term. If people will completely ignore sustainability, I'd guess that the limit is about 100 billion.

However, if technology improves significantly, or many other planets are colonized, than 18 quadrillion is plausible. In fact, you could go even higher – a nontillion (1,000,000,000,000,000,000,000,000,000,000) people is doable with just six children per family.

Typical population growth assuming no limiting factors is roughly 1.7%. This is based on modern population growth I copied from Michael Kjörlings answer. The real maximum could be higher, maximum measured in the real world was 2.1%.

But here is the really important part: In practice the population growth has always been limited by some other factor. If you want to find out a plausible rate of population growth, you have to actually find out what is limiting the growth and derive the rate from that.

For most of history, until industrialization or so, population was limited by available food. When access was gained to new resources, more food, population could increase rapidly, but otherwise population would be relatively stable or even decline due to decreased acricultural productivity. Soil quality in acricultural areas would decline significantly due to erosion and, on irrigated areas, salinity. During antiquity and medieval times until the black death people actually considered world to be full, which had significant effects on society.

In urbanized societies where a significant parts of population lived in areas of high population density and trade was a significant factor disease would become significant factor. More in the form of occasional drops from epidemics than from adding a limit beyond the limits of available food. The drops would have an effect in the long term population growth, so for a period of 900 years you need to consider the level of medical technology and sanitation. Also a small starting population might cause an increase in the level of genetic disease, which could affect the growth for the first few centuries.

In modern societies social effects have become a major limiting factor on population growth. Even in societies where food is plentiful people postpone having children and choose not to have large families. This is essentially a life style choice. In other countries people are encouraged to have smaller families to prevent poverty and improve resource management.

From the parameters you give the colonists would seem to have a decent level of technology. So the limiting factor on the population growth would probably be the rate they can increase the available resources and expand their infrastructure. This could be close to that 2% mark, even above if the society is really focused on growing. Lack of contraceptives implies they would be, but that might not be sustainable over 900 years. Fast population growth without contraceptives or abortions would be pretty harsh on women. As such the limiting factor is probably going to be social. Note that the time women spend having children is time they can't contribute to increasing the available resources, so it would probably be counter-productive to reduce women into baby-making machines. Not to mention it would add an flavor you might not want into the society.

The rule given in GURPS Space: "On a wholly earth-like planet, with medical technology of at least TL5, a human population will increase by a factor of 10 every 100 years, up to the maximum population for the planet." This fits what I wrote above (giving independent confirmation) and gives a very simple formula, that is roughly equivalent of 2.3% growth a year. Which seems bit high, but given that I expected the number to be something slightly over 2% not implausible. The formula gives final population of 500,000,000,000. Essentially this means that provided that the colony becomes stable within the first century, it can grow to fill the maximum supported population of your planet.

• I wouldn't say at all that we currently have "no factors limiting population growth" (not an exact quote from your answer; paraphrasing).
– user
Dec 11 '14 at 10:58
• @MichaelKjörling I wouldn't either. Merely that current growth is a good guess for a value to use. (Which I forgot to give a reason for...) My answer specifically lists social factors that limit current growth. The paragraph start with "In modern societies..." Dec 11 '14 at 11:03

"Reasonable" depends of cultural expectations.

If you go for the max:

With in-vitro fertilization, you can select gender.

Female is fertile until 40 (increased chance of genetic defects). So let's assume 20 daughters and single son per generation, and 20 years per generation = 450 generations, and all females are fertile and willing to breed.

Each woman from your starting population can have 20**450 descendants. FYI that is approximately 2.9 followed by 585 zeroes. 2.9e+585 That's quite a lot mouths to feed. You will need huge robot army to work millions of solar systems all over galaxy. And they will be crowded.

Without IVF, if we assume 10 daughters per generation, you have 10**450 per original female.

Assuming a Doubling of the Population every 27.5 years (the medium Human Generation Time) with 32.72 Doublings from the original 500 will grow to 3,537,296,484,444 without me taking the time to calculate the exact population by only subtracting the deaths. But let's use the 1 Year Growth Rate the numbers are:

1. 1%: 3,874,417
2. 2%: 2,748,6825,799
3. 3%: 178,843,120,520,152

And then it just gets stupid.....

Doubling the Population calculated by Generation Times result in:

1. 10 years: 618,970,019,642,690,137,449,562,112,000
2. 25 years: 34,359,738,368,000
3. 27.5 years (as above): 3,537,296,484,444
4. 30 yeas: 524,288,000
5. 50 years: 131,072,000

Again, I did not calculate death-rates.