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I world-build for tabletop RPGs so I have Tolkien/DnD/Forgotten Realms like elves in mind. The assumptions are:

  • The ratio of humans to elves in the world has been more or less constant.
  • Humans live ~50 years and have 5-7 children per woman.
  • Elves reach sexual maturity at 80-100 years of age and live for 600-800 years.
  • Elves do not suffer infirmity until the very end of their lifespan.
  • Elven reproduction is human-like. Human-Elven hybrids are possible, even somewhat common (~5% of the population in some areas).

So, how poor does elven fertility need to be to keep the human/elf ratio near constant? Additionally, what is an internally consistent "mechanism" to describe this in-world? Are elven women fertile one day a year? How to explain human-elf couple having at least 2-3 children over their time together? Do half-elves necessarily almost always have a human mother to make this consistent?

EDIT: Assume child mortality rates slightly lower than late medieval period for humans (magical healing is easily accessible in large cities but not in the frontier or countryside). For elves, child mortality rate may be part of the answer. My gut reaction is that elves tend to have easier access to magic in most settings so their child mortality rate should not be higher than humans if at all possible.

EDIT-2: Here is what I learned from the many great answers below:

  • My assumption of 6 children in 50 years for humans is at best incomplete information, at worst inconsistent with the world I had in mind. I now assume 1% annual growth rate for humans.
  • Now, assuming elven reproductive period between ages 100-600, only 800 year old elves dying from old age that year, and 0.1% yearly mortality for all elves younger than 800; average elven woman needs to have 20 children over 500 years.
  • So, while elven fertility per woman per year is poor, fertility per woman is unintuitively high! The average elf has 10 maternal siblings and will have 10 more during her lifetime. Does this mean small elven communities are highly inbred?
  • The second part of the question about how to explain these numbers in-world while staying true to Tolkien/DnD elves is still open. Here are the ideas I like:
    • Magical birth control.
    • 1 elven child at a time (credit: Agent_L) per elven couple. Elven women do not ovulate unless they are in a committed relationship and not already bonded with a child. This will give at most 5 children per elven woman. Perhaps, I just need to increase human mortality rate to keep them in check :(
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    $\begingroup$ If you want to be able to tell that, you need to know how many children reach maturity, not how many are there born - and I assume that all of human children in your world don't do that, because with 6 children per woman that reach maturity you get EXPLOSIVE population growth. $\endgroup$ – Borsunho Mar 10 '17 at 12:25
  • $\begingroup$ I would assume Late Medieval age level child mortality. A settlement with 25,000 inhabitants is considered a "large city". In urban areas with easy access to magical healing, child mortality is lower. Elves generally have even easier access to magic, so their child mortality has to be less than humans'. $\endgroup$ – Solanacea Mar 10 '17 at 16:07
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    $\begingroup$ Humans living ~50 years may make sense as an average, which gets pulled down by child mortality, but even in the middle ages someone who made it to 20 could probably expect to make it to around 70. $\endgroup$ – KRyan Mar 10 '17 at 17:20
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    $\begingroup$ Also, not really compatible with your requirements, but an interesting approach that I’ve seen was an elven society in which people were not considered “adults” until they had reproduced and raised children of their own. I.e. the elves got married around 20 like humans, had children like humans, raised those children to around the time they got married in turn like humans, and then got to enjoy several centuries of life post-children, at which point they were considered mature enough to go do their own thing as adults. A thought, anyway. $\endgroup$ – KRyan Mar 10 '17 at 17:24
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    $\begingroup$ The question makes the assumption that elven fertility needs to be poor for this to happen, which is a fair assumption because it is the question posted. But it occurs to me that humans have a pretty compelling need to mate. The listed assumptions do not address how compulsory intercourse may or may not be in elven psychology or physiology. Perhaps their fertility is just fine and their behavior is what's different. Nerve cluster locations could be different, as could neurochemical response. $\endgroup$ – Sean Boddy Mar 11 '17 at 2:13

12 Answers 12

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Elven social norms allow for 1 kid at a time.

Elves reach sexual maturity at 80-100 years of age and live for 600-800 years.

That means when an Elven pair enters reproduction age at 80-100 and cares about their child until it's grown up, during 800 years they can raise at most 7-9 kids, comparable with your 5-7 for humans.

Upsides:

  • self balancing, loss of a child means the next one is borne soon.
  • can be explained with simple hormonal bonding trick (helps saving those eggs as well)
  • interesting social repercussions: a marriage has very strongly defined turning points. eg it can last for one child only and then gets automatically dissolved. Helps keeping the 800 years boredom at bay ; )

Downsides:

  • human-elf interbreeding gets VERY complicated. Male elf with human wife, can breed at human rate. Male human with elven wife is pretty much a reproductive failure (he won't live long enough to breed second time). Can be turned into advantage, a halfbreed with elven mother could be extremely rare one (and eg the only one fertile).

Caveat: to have more or less stable population, you need effective breeding rate at about 2 kids per pair/female of both species. From that 5-7 human children, only 2 will make it on average. If you really want elves to have low children mortality rate, you're looking at having 2-3 kids per entire 800 years of life. The only way I can defend my first proposition is for Elves to have some age-of-coming ritual which 3 out of 4 adolescent Elves don't survive (think "Trial of the Grasses"-like).

However, I don't like your math at all:

Humans live ~50 years and have 5-7 children per woman

But humans pretty much don't breed past 35-40, last 20-30% of life is effectively infertile. Let's look at humans: 15 sexual maturity, 40 menopause, 50 death (average). That means that childhood is at 30%, menopause onset at 80%. With 600-800 lifespan, an Elf should reach sexual maturity at 180-240 (15 in human years) and start losing fertility at 480-640. This means only 300-400 breeding years. Now, if Elves are more intelligent and magical creatures than humans, I would only expect them to have proportionally longer childhood, making a 300 years old still a teenager. And that can be easily solved with economy: make raising Elven kids expensive (that's at least 200 years of Elven schools of magic tuition, the horror!). When you're going to care about your kids for almost entirety of your adulthood, you won't be able to afford to support more than 2-3. And regular mortality will take care of the tiny excess, like it always does.

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  • $\begingroup$ This answer does not clearly deal with the fact that the long life span of the elves is relatively unimportant compared to their late maturity when considering population growth - or explain that given this elves must have far more offspring/year who go on to survive to adulthood than humans, therefore it is misleading. $\endgroup$ – user42528 Jan 25 at 0:00
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    $\begingroup$ Also to make that possible: elven magic allows for easy access to contraception, meaning that an elve will only have a kid if she really wants to, as opposed to humans $\endgroup$ – Hobbamok Mar 25 at 8:47
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The different causes of death are the biggest issue here.

  • Humans will lose a lot of children to disease, elves don't seem to suffer from this

  • Humans lose fertility early then die fairly young, again not a problem for elves

Most of the death rate for elves appears to be wars, in a time of war they'd need a birthrate approximating that of humans just to maintain a viable population around 100 years later.

However you're effectively looking at an advanced society in a primitive world. Humans have a lot of children to try to ensure that at least two survive to adulthood to reproduce and continue the species. Elves can be reasonably sure that all their children will survive to adulthood and hence, as in among wealthy westerners, their birth rate will inevitably fall. In spite of, or rather because of, their long lives, when their society was primitive and they were losing children to disease and injury they would have needed to have more children due to the greater age they need to survive to to reproduce. Humans need to survive a couple of decades, elves perhaps a century or more.

There's no need to impose any fertility restrictions, just let society take its natural course. The birth rate drops because there's no need to have more children.

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    $\begingroup$ I came here to sneer at anyone who didn't use math, but it turns out this is the right answer. +1 $\endgroup$ – kingledion Mar 9 '17 at 20:49
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    $\begingroup$ Our birth rate dropped because we invented effective forms of birth control, it's not a natural progression of advancing societies. This answer does not address what I think is OP's question, assuming no birth control and similar sexual behavior as humans. $\endgroup$ – thegreatemu Mar 9 '17 at 21:54
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    $\begingroup$ Maybe the elves use magical birth control. It's still not 100% effective, though, for the sake of elvish PSAs about underage sex. $\endgroup$ – Steve-O Mar 9 '17 at 21:56
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    $\begingroup$ @bloer Are you insinuating that that beings as otherworldly and considered as Elves would stoop to entertaining themselves with mere sexual intercourse? My word! The very thought! * Drops monocle in tea* $\endgroup$ – Joe Bloggs Mar 10 '17 at 11:42
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    $\begingroup$ @EthanTheBrave Various things have been used to varying success for millennia. It's not a modern thing at all. $\endgroup$ – Izkata Mar 10 '17 at 19:08
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Human women are only fertile 1-2 days a month, so it isn't unreasonable to think that a very long lived elf would have a suitably delayed fertility cycle. Women are born with all the eggs they will ever have. The ovaries have finite space and an egg has a minimum size, but that still supports a million or so eggs at birth. But that number is culled significantly by puberty to a few hundred thousand, and during ovulation many eggs will involute instead of be released, so in the end a woman will only release a few hundred eggs to support their 12x40=480 ovulations in a lifetime. Some women don't ovulate every month, or have several ovulations, or undergo menopause earlier than others. If elven women go through a similar process, then it would be reasonable to limit their maximum number of ovulations to be similar to a human and just space them out proportionate to how long you want them to be fertile.

There are problems with prolonged fertility however. The longer you live, the more teratogens, mutagens, and carcinogens you are exposed to. Eggs hanging out for several hundred years could be substantially degraded, altered, or even infected. So a 750 year old elf, even if she is physically healthy, could be carrying eggs that are of poor quality after decades of exposure to teratogenic compounds, toxic metals, etc. Of course the same robust genetic repair mechanisms/magic that keep her healthy could keep the eggs healthy.

For humans, infertility is defined by not conceiving after a year of trying. Assuming sperm is present when the woman ovulates, fertilization occurs about 30% of the time, but this assumes perfect timing, which is difficult. A more reasonable estimate would be around 10% a month. A fertilized egg survives to become a live birth around 50-70% of the time. Of course these are often first trimester losses, so a woman can continue to ovulate and become pregnant for the rest of the year. The long and short of it is that healthy woman can expect to become pregnant every year if actively trying to become pregnant.

Most of this will apply to the elven female. So simply elongating their menstruation cycle proportionate to their fertility span ought to keep the birth rates comparable. But this ignores willful contraception, which will drastically decrease the birth rate. Studies show that the lower the infant/child mortality rate, the lower the birth rate, so that needs to be taken into account as well. If elven babies die at the same rate as human babies then one would expect elven women to be as aggressive in having children as human women.

But if elven infant mortality is low, similar to modern industrialized humans, then elven women, regardless of their natural fertility, could simply limit copulation during their fertile periods (there are many biological signals for ovulation) such that they have a replacement level birth rate, which is at least 2 for traditionally monogamous paired species (of course accidents, disease, and war will mean this has to be higher, so adjust it for whatever early death rate your society has). So their biological fertility is socially controlled (much like modern humans with access to birth control) and you don't need to have elaborate biological mechanisms for controlling the elven birth rate.

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    $\begingroup$ My thoughts were very similar. Just multiply every aspect of elven lifespan by 8 or 9, including pubescence and menstrual cycles. $\endgroup$ – cobaltduck Mar 9 '17 at 21:11
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    $\begingroup$ A 9 month long period? Sounds unpleasant. $\endgroup$ – SPavel Mar 10 '17 at 18:15
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    $\begingroup$ So you're saying this is likely: "Oh gosh, it's that time of the year again" - some elf male. $\endgroup$ – Jeutnarg Mar 10 '17 at 19:37
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    $\begingroup$ @JasonK Menstrual cramps some and go throughout a woman's period, and are not very fun (or so they say). $\endgroup$ – SPavel Mar 10 '17 at 20:28
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    $\begingroup$ @Solanacea they can act as elves in heat - throwing the entirety of their intellectual and physical resources towards process of securing a mate. Might be quite a sight. $\endgroup$ – alamar Mar 11 '17 at 8:06
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One thing not being taken into account is a difference in social norms. What age is considered 'their majority' or 'adulthood'? What age is common for Elves to marry? Are they more for monogamous relationships, or is there more emphasis on freedom considering their 'till death do we part' would mean living with the same flawed person for centuries?

Some examples to show what I mean.

In biblical times, you became 'a man' at thirteen. You were then expected to work, to marry, to start a family of your own. This in a time when life expectancy was... 30? 40?

In 'modern times' marrying at the age of 30 is 'normal', leaving roughly ten years to have one or two kids. With life expectancy at 80, more or less.

In both of the above-mentioned eras, infidelity in a monogamous relationship is (at best) frowned upon. Granted, there's more social pressure on the woman than the man to stay faithful, and equally granted that there are those (in increasing numbers) that do not see 'sleeping around' as infidelity. But the general rule still applies, if you are going to be unfaithful, it's usually done in a sneaky way.

What could this mean? Well, either Elves choose life-partners, and stay faithful to that partner... well, for life. (Done quite well in Dark Lycan) Or they could choose a partner for 'mating season', which might be more common for the more druidic types of Elven. They might even have an elder select a number of couples for mating season, to offset how many have died since the last such event.

This might even differ per Elven race, tribe, or region. After all, we humans aren't socially consistent throughout every country, race, or even religion. Why not offer your Elves the same respect?

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    $\begingroup$ When talking about historic life expectancy, you need to distinguish "life expectancy at birth" and "life expectancy at maturity". Until about the 1800s-1900s, there was a huge difference between the two: if you survived to adulthood, you would probably live to see 60, but you only had a 1 in 4 chance of living to adulthood. $\endgroup$ – Mark Mar 9 '17 at 22:00
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    $\begingroup$ @Mark Quite true, but I don't know those numbers, so I won't present otherwise in my answer. Kudos to you for bringing it up! $\endgroup$ – Fayth85 Mar 9 '17 at 22:06
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    $\begingroup$ Average age of death was about 35, but the modal age of death was 0. The overall average is not a valid value to use when you're looking at a bathtub curve. You need to look at the three curves that make up the bathtub separately, infant mortality, random/constant mortality and "wear out" mortality. $\endgroup$ – Separatrix Mar 10 '17 at 9:01
  • $\begingroup$ Not sure why paternity matters for this question, we are only interested in the number of offspring. Otherwise, you can assume elves becoming sexually active around age 100. For humans, you can assume roughly late medieval age norms. $\endgroup$ – Solanacea Mar 10 '17 at 16:30
  • $\begingroup$ @Solanacea The point was to offer lateral thinking. After all, it could be argued that they 'just are, because the DM says so'. Or it could be argued that the numbers are equal because Elves are pregnant for longer or take longer to raise each child. Or it could be argued that Elves and humans struck a treaty which stipulates that their numbers will remain equal to avoid potential wars. The importance of detail is in the eye of the beholder. And I happen to enjoy lateral thinking details. $\endgroup$ – Fayth85 Mar 10 '17 at 16:46
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Whatever you want it to be, it really doesn't matter.

Look up trends on number of babies per couple over time, and by country. In an educated society with good healthcare (magical), equal treatment of women, and a respect for the envionment, the limiting factors on the Elvish population will not be fertility or early death. It will simply be the number of children that each couple wants to have, which is likely to be somewhere between 2-3.

Elves may be able to have children at 80-100, but they will put that off as they are still studying - their legendary magic, swordfighting and art all arise from them having longer to practice than any others. With both parents having careers, and the children being expected to attend school instead of working in the fields, looking after any more than 2-3 children at a time would be difficult.

By the time those 2 kids have moved out (well after sexual maturity), the parents are likely 350. They don't then want to have another 2-3 kids, who would live with them at least until the parents are 500. If you're dying at 600-800, you certainly want to be retired by 500.

There's plenty of resources on the matter, here's one to get you started.

http://blogs.worldbank.org/health/female-education-and-childbearing-closer-look-data

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I made some mathematics and the results are unintuitive.

I am assuming here the following:

  • The living years are a mean, taking into account mortality due to facts beyond old age.
  • The 5-7 children per woman are a mean, and are how many children will a woman have at the end of her life.

Taking that into account, the Ratio of increase of Human population every year will be 1.08 (every year you multiply the population by 1.08). That goes as follows: Every 50 years, a couple will die leaving 6 children behind, that means that every year the number of Humans is increased by (6-2)/50 = 0.08

Now, if you want to keep the ratio between Humans and Elves equal, they population must increase equally fast. Doing the inverse calculations, we have that: (child-2) / years / 2 = 0.04; (child-2) / 700 = 0.04; child = 29.

Exactly: An elven woman will have 29 children in her lifetime. That is, if they achieve maturity at 90 years, one child every 20 years by mean (assuming she's having children while able).

Edit: Have to divide the ratio by 2, since there's need to people to have a son.

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  • $\begingroup$ The yearly increase of 0.08 is per pair of parents, right? I such case, the multiplier should be 1.04, not 1.08. $\endgroup$ – Angew Mar 10 '17 at 13:35
  • $\begingroup$ You are comparing two different time scales. For your comparison to be correct you need to take in consideration the number of children produced on the same quantity of time for both humans and elves. $\endgroup$ – Mur1lo Mar 10 '17 at 21:22
  • $\begingroup$ Perhaps my assumption of 6 children in 50 years is unrealistic for this setting. With an 8% annual increase, human population would double every 10 years. Very unintuitive indeed! Let's assume 500 elven couples, all 100 years of age. In the next 100 years, every couple will have 10 children each, increasing the elven population from 1000 to 6000. Mortality rates would have to be extremely high to keep the elven population in check, it seems. $\endgroup$ – Solanacea Mar 10 '17 at 21:23
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If we assume magic doesn't help, then most children do not make it to adulthood.

During the "high medieval clearances" of the wilderness, England over 250-odd years went from 1.5 million to 4.5 million or so. This rate of growth is 0.4% per year.

With an average generation length of 20 years, each girl child that survives until adulthood has to have 1.004^20 = 1.08 girl children on average to maintain this growth rate. This is roughly 2.16 children that "survive until adulthood" to in turn have children.

If they have an average of 6 children, that generates a rough childhood mortality rate of 64%.

For elves, I'll assume they don't die of old age and remain fertile.

They have a uniform 0.5%/year mortality rate (slightly higher than we have from accidents in the modern age), and have X children every 100 years relatively uniformly over their lives.

40% of elves die every century (60% live). To maintain replacement, each female elf has to produce .4 female elf children every century, or 0.8 elf children of either gender (assuming balance).

A 1.004 annual growth rate like humans corresponds to a 1.5 growth rate per century, upping the number of elf children to roughly 1.2 per century.

Adding an extended childhood period doesn't change these numbers all that much. Adding a "menopause" (cultural or not) that causes elves to stop having children could change these numbers significantly; you could roughly emulate it as an increase in mortality.

Warfare will be extremely hazardous for elves under such a model, unless they can produce a large bump in their birthrate, or avoid female casualties and accept significant serial or parallel polyandry.

A completely alternate approach would be to make elven biology extremely different than humans. For example, elves as a kind of plant spirit; the plant spawns an elf body, which explores the world and returns to its source tree when tired.

That permits "elves" that live for millenia (which are new spawns of the same ancient plant), the "elves" death not being as important as it might otherwise be for an immortal being (as it is just an expensive discardable body; you lose some experiences, not all), and explains why elves live in forests and care a whole bunch about them (the elves are the forest).

The question of growth rate now becomes limited by the territory the elves can plant new elves in. Elves become R-type reproducers.

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  • $\begingroup$ With a 0.5% constant annual mortality, an elf has only 5% chance of living to see 600. I need the average elf to live 600-800 years. Seems to me, each female needs to produce slightly more than 1 female child over her lifetime, regardless of other life history characteristics, unless pre-fertility mortality is high. I love the idea "Warfare will be extremely hazardous for elves under such a model, unless they can produce a large bump in their birthrate, or avoid female casualties and accept significant serial or parallel polyandry."; this gives me excellent cultural repercussions to use. $\endgroup$ – Solanacea Mar 10 '17 at 20:58
  • $\begingroup$ @Solanacea Do you want your elves to have old age, or not? A relatively constant accident rate can be tuned to an average lifespan and distribution of ages, but you'll always have a falloff. The average death age for a constant accident rate is 1/accident rate; the median is more complex. 0.5% is 200 year average age, but in a 1 million person population "on average" 1 person lives to 2500 years. For 600 average death age we require a 0.0017 death rate, and in a 1 million population 1 person lives to 7500. Old age changes this. $\endgroup$ – Yakk Mar 10 '17 at 21:14
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    $\begingroup$ Note that 1 million humans generate about 10,000 new soldiers per year (new adult males, average death age 50, population is steady). 1 million elves with an average death age of 600 generate only 800 new soldiers per year (also steady population). A human population can be 12 times more warlike without suffering worse consequences... even serious polyandry isn't going to mitigate this much. On the other hand, elves have the ability to replace war losses like humans can, without war they'd grow like crazy. This is hard to model. $\endgroup$ – Yakk Mar 10 '17 at 21:16
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If you want "The ratio of humans to elves in the world has been more or less constant", than the elven and human mean rate of population growth have to be more or less equal.

Let me be more formal. Take $e_t$ and $h_t$ as the even and human populations in a given year. The next year the populations are $$e_{t+1} = (1 + \alpha_e - \beta_e)e_t$$ $$h_{t+1} = (1 + \alpha_h - \beta_h)h_t$$ where the $\alpha$'s and $\beta$'s are associated to natality and mortality in each population.

If you want (more or less) constant ratios you need $$\frac{e_{t+1}}{h_{t+1}}\approx\frac{e_t}{h_t}$$ and as a consequence $\alpha_e - \beta_e\approx \alpha_h - \beta_h$.

The important thing to not here is that life expectancy plays no direct role in solving the problem. If enough time has passed and the initial population size is big, life expectancy does not affect neither the $\alpha$s nor the $\beta$'s.

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    $\begingroup$ Life expectancy does not affect mortality, but mortality certainly effects life expectancy. You have to assume much lower mortality for elves if they are to live till 600-800. $\endgroup$ – Solanacea Mar 10 '17 at 21:06
  • $\begingroup$ @Solanacea Ok, now I understood what you are talking about, and I think you are right. $\endgroup$ – Mur1lo Mar 10 '17 at 21:11
  • $\begingroup$ @Solanacea Now I disagree. Imagine that elves have the rule of not going to war if you are younger than 600 years old. In this case you can still have a large rate of mortality. The important thing is that that rate is happening only around older folks. $\endgroup$ – Mur1lo Mar 10 '17 at 21:18
  • $\begingroup$ Elves not going to war before 600 can be part of the answer. Could you edit your answer to show how the math would work? $\endgroup$ – Solanacea Mar 10 '17 at 21:30
  • $\begingroup$ The math is already there. My point is that for the population size ratio to remain constant you need th thing about the $\alpha$'s and $\beta$'s. The particularities of each civilization affect the ratio only in the proportion that they affect $\alpha$'s and $\beta$'s. And under my assumptions (large and long established populations) longevity plays no role on the maintenance of population size ratio. (but i can surely be wrong :-) ) $\endgroup$ – Mur1lo Mar 10 '17 at 22:01
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The answer to the question depends on a lot of factors. But one thing to keep in mind is that under most scenarios in which elvish lives are simply "stretched" human lives, elven fertility in terms of children-per-parent has to be significantly higher than that of humans, for the respective populations to maintain the same ratio. This is counterintuitive for most people.

To see how this is the case, assume that the human population doubles at each generation. This not too far from 13th century Europe, in the absence of plagues, serious wars etc. and it only requires a pair of humans to have $4$ children who grow up to have children of their own. Incidentally, this is the crucial aspect: it's exactly the same if a human couple has 4 children that all live to bear children, or it has 20, of which only 4 live to bear children.

In your example an elven generation is between $6$ times (if we look at age of reproduction) and $15$ times (if we look at age of death) that of a human. Let's go with $6$ (so most elves will have children between the age of $90$ and that of $300$, and enjoy the rest of their lives in peace :) Note that this is the scenario that requires each elven couple to have fewer children to achieve a given population growth, since if you have children early, they will reproduce earlier, and you'll have lots of grandchildren earlier etc. Now, in $6$ human generations (i.e. an elven generation) the human population will double $6$ times, i.e. grow by a factor $2\times2\times2\times2\times2\times2=64$. But then the elven population must grow by a factor $64$, too, in that same timespan. This means that every elven couple must have at least $64\times 2=128$ children who grow to bear children of their own!

You have a problem. And two ways out. The first is to assume that the human population remains relatively stable over an elven generation, growing by no more than a factor $2-3$ or so. If every human couple, and every elven couple, has exactly $2$ children who live to bear children of their own, the respective populations remain constant and in perfect proportion. If every human couple has $2.1-2.2$ children who live to bear children of their own, the human population will grow $20-50\%$ each century, and elven couples who bear $4-6$ children of their own will be able to match that. This requires some sort of pressure on the humans: a species who, in the absence of threats, is only able to bear at most $2.2$ children per couple is exceedingly fragile in the long term, and will not be able to recover quickly enough from plagues, natural disasters etc. But if you say that your humans can only marry if the prospective groom owns 10 acres of land, and the land is more or less all taken, voila, you have have your population check.

The second is for your elves to bear children much earlier, at an age possibly a little higher, but not significantly higher than that of humans. Ultimately, it does not really matter how long elves live, or in fact how many children they bear after the first few. What matters is how early they can get those few children, so that they may reproduce. That's the real handicap elves have against humans: humans reproduce early. Whether they also die off quickly after that, is immaterial. To see how this is the case, imagine a magical, immortal race where each couple has children (who will all bear children of their own) in batches of $4$, one batch exactly every $20$ years; being immortal every couple eventually bears millions of children over the aeons. Now imagine another, shorter-lived race, where each couple has children in batches of $6$, one batch after $20$ years... and that's it: the price of those $2$ extra children is death for the couple as soon the batch is born (though the orphans grow up with the relatives, all eventually having children of their own). You'll easily see that the two populations will maintain the same ratio; and this would be true regardless of whether the "time between batches" is $20$ years, $2$ years, or $2$ million years.

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As @Albert Masclans pointed out, keeping the human/elf - ratio constant means humans and elves need to have the same population growth. If we assume that elves are fertile from 100 to 600, Elven woman are fertile for $5/8$ of their lives. Humans my be fertile from 20 to 40, for $1/2$ of their life.

Next we need to know the population growth. If a human couple has 5 children over 20 years, starting at an age of 20, they grow by the factor of 2.5 every 40 years. That's a population growth of 2.3% every year.

If we take an Elvish population of 1000, half male/female, 2.3% growth means 23 + numberOfDeaths kids. Obviously, an elvish woman needs to have 2 children over her lifetime to keep the population constant when she and her mate die. How many additional kids would be needed to keep up with the human population growth? because we have already calculated the kids for keeping the population constant, we assume nobody dies, so we'll have 23 kids this year. $5/8$ of their woman are fertile, or 312.5, so the average of them gets $23/312.5 = 0.0736$ kids per year. They are fertile for 500 years, so they'll have $0.0736 * 500 + 2 = 38.8$ kids, or about one in 13 years.

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Besides the natural need to reproduce, which leads species (in general) to produce more children while the probability of them surviving decreases, a lot of the aspects of human sexuality are culture-driven. Think, for example, about the whole boys-not-girls-as-heirs-thing from the Middle Ages, the free love flower power everybody-happy age, and - even in some areas currently - father pastor knocking on the door of a newly-wed couple to check when if the Mrs is already pregnant and if you will please hurry a bit.

My point is: since you are writing the Elve's culture, you can have them handle sex any way you want and fine-tune the birth rate to exactly what you need in this way. Just to give a few examples that came to mind:

  • Do you need an Elven pair to have only one kid? That makes sense, because raising an Elf is a complicated and intensive process. Think of all the culture, all the magic, all the manners, the history, ... not to mention the fact that Elves are very generous and loving parents who do not wish to focus on raising more than one child in their lifetime. Basically, going through the process once, and delivering a well-respected member of society to carry on their bloodline and honour is the highest possible achievement and getting a second child ... well, just the thought!

  • Do you need less than one kid per couple? The society could be compartmentalized, and only some groups are expected to have children. For example, practitioners of magic have a very high status and if you have the skills to become a Great Magician you definitely want to go for that. Unfortunately, that requires so much devotion and dedication that having a family and children is absolutely irreconcilable with this career path. Similarly, the Traveller Elves that go around the world and collect information on all other races, document their historical events and interact with the world on behalf of the Elvish race, are in no position to have families. You could even turn it around: raising an Elfling is such a complicated and delicate task that only a select subgroup of Elves is even trusted to do that, so unless you are clearly born a Mother having kids is not even an option.

Finally note that we humans mostly have sex for pleasure. As mentioned elsewhere, this is probably the reason that we spent the last 4000 years or so trying to come up with reliable birth control. A race that does not have sex just because it's cold outside and they have nothing better to do after sunset, does not get nearly as much children. In fact I can imagine that Elves - being the sophisticated race they are - don't have a dirty process involving all kinds of sticky fluids flying around the place, but creating a child is a beautiful public ceremony where a couple joins minds with flashes of light and bright stars swooshing through the night sky in the middle of a circle of their beloved and most trusted community members. If you need a really really low birth rate, just make this a special event that only happens every Blue Moon (which as we all know, is once every year... or 10 years ... or even every century) at which one (or was it 5? Or 10?) select couples are given the honour of partaking in the mating ceremony.

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  • $\begingroup$ Excellent answer; however, I need my elves to be more or less similar to Tolkien / Dungeons and Dragons / Forgotten Realms elves. Their reproductive biology needs to be compatible with humans to be able to produce half-elves, for example. $\endgroup$ – Solanacea Mar 12 '17 at 19:37
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I was thinking very much in the line of Albert: you have to account not just the children-per-couple, but the children-per-couple-per-generation, since elves and humans have very different lifespans. A human couple has 5 to 7 children, then die, all in about 60-70 years. An elven couple has to have as many children PER HUMAN GENERATION, so is, every 60-70 years, to match the number of human children who survive to adulthood. Even if it's only 2 or 3, like Separatrix has noted, they are going to have 20 to 30 children each.

I'll expand the math as a Fibonacci-like function: let's start with a couple, Adam and Eve, at a certain time T(0)

T(0) = 2 (Adam & Eve)

Adam and Eve had 3 children who get to adulthood and marry (we ignore their spouses):

T(1) = 5 (2 parents, 3 children)

Some time after that, their children have children. Assuming an average of three survivors to adulthood:

T(2) = 14 (2 grandparents, 3 children, 9 grandchildren)

Adam and Eve die about the time their grandchildren are having children (again, ignoring spouses):

T(3) = 39 (3 grandparents, 9 parents, 27 great-grandchildren)

You can follow the progression:

T(4) = 117(9 grandparents, 27 parents, 81 grandchildren)

If you want to know how many offspring Adam and Eve have after 10 generations, you have to calculate T(10), which is 3*T(9) + T(9) + T(8) = 4*T(9) + T(8). The result is 28431 descendants of Adam and Eve after ten generations.

Now the thing is, if Adam and Eve are a human couple, they died between T(2) and T(3), when they were 60 to 80 years old, and T(10) happens about 240 years after T(0). If Adam and Eve are an elven couple, 240 years is only T(1). Or to put in a different perspective: after 250 years, the descendents of human Adam and Eve surpass the descendants of elven Adam and Eve by 28431... to 3.

This is, of course, a replacement ratio of 1.5, but is the very same thing with a replacement ratio of 1.08; at any case, the ratio will be 10 times slower for elves if they have the same number of children than humans, since they live 10 times more years. They necessarily must have ten times more children than humans (counting only those who make it into adulthood) or in six or seven generations - which are less than an elven life - they will be outnumbered.

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    $\begingroup$ This does not seem to add up. If each elf has 20 children that reach adulthood, elven population will explode. This does not seem to be dependent on what other species live on the same planet. Humans in this setting are reproducing barely above replacement levels, their population is very slowly increasing. $\endgroup$ – Solanacea Mar 10 '17 at 21:02

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