Tags
children, dwarves, essay, extinction, fili, kili, lifespan, marriage, sex ratio, survival, thorin
With all the recent discussions on Dwarves following up The Hobbit: An Unexpected Journey there is one question that keeps popping up:
“How can Dwarves keep up their population with their low number of women?”
This essay will take a closer look at this question and other demographic issues concerning Tolkien’s Dwarves.
Let’s start with a quote from the Appendices in The Lord of the Rings:
“Dís was the daughter of Thráin II. She is the only dwarf-woman named in these histories. It was said by Gimli that there are few dwarf-women, probably no more than a third of the whole people. They seldom walk abroad except at great need. They are in voice and appearance, and in garb if they must go on a journey, so like to the dwarf-men that the eyes and ears of other peoples cannot tell them apart. This has given rise to the foolish opinion among Men that there are no dwarf-women, and that the Dwarves ‘grow out of stone’.
It is because of the fewness of women among them that the kind of the Dwarves increases slowly, and is in peril when they have no secure dwellings. For Dwarves take only one wife or husband each in their lives, and are jealous, as in all matters of their rights. The number of dwarf-men that marry is actually less than one-third. For not all women take husbands: some desire none; some desire one that they cannot get, and so will have no other. As for the men, very many also do not desire marriage, being engrossed in their crafts.”
So the much quoted “one third of the Dwarves are female” is not an exact number, but it is only estimated by ONE individual (Gimli) of the Dwarf race. For easier calculations though we will work with this exact number, assuming that one third, or 33.3% of all Dwarves are female.
A dwarf girl and a dwarf woman. http://www.wetanz.com/daniel-falconer/
What about sex ratio in the real world? Humans roughly conceive 50% male and 50% female babies (with a slight favour to males which we’ll neglect for easier assumptions). There are animals with much different sex ratios than this rather simple model. Just think of bees or ants where most individuals are female. Some animals need certain ecologic parameters to give birth to a certain sex. And the extreme example of course are parthenogenetic animals, such as certain crustaceans or stick insects of which males are completely unknown to science. Without getting too much into detail those species reproduce by “cloning” the mothering animal, making males completely unnecessary.
Whichever example one wants to take from the real world though, if the sex ratio leaves the classic 50/50 mark it always goes in the direction to favour female offspring. The reason is simple: females are more “valuable” to reproduction since they produce egg cells (which are a lot richer in resources than sperm cells) and also because they usually invest a lot more time and effort in raising their offspring (pregnancy, nursing, etc…) than males. Seahorse fathers are the laudable exception to this of course.
In real world species in theory it only takes one male individual to inseminate several females, which makes males a lot less valuable as well.
So the fact that Dwarven sex ratio distinctly favours males is odd. It does definitely help during warfare, but there is no biologic reason whatsoever to get to this awkward proportion. The only plausible explanations here could be that Y sperm cells are more common than X sperm cells (assuming Dwarves have the same XY-system as humans) that female Dwarves might be more sickly, female embryos die a lot more often during pregnancy and so on.
Of course all these assumptions would be a horrible initial situation for any species, and in the real world evolution would have definitely gone into a different direction.
So with this unfavourable point of departure, what would it take to keep a population at the same level? For humans (1:1 ratio) every woman statistically has to give birth to two children. There are women that can’t conceive, die early or simply choose not to have children, so in reality for women that DO give birth this number has to be slightly higher since the statistic also includes women that never have a child. Luckily there seems to be a lot of women getting three, four, five…. children because our number is constantly increasing.
For Dwarves however (1:2 ratio) every female statistically needs to give birth to three children to keep up the population. But in the case of Dwarves what are the conditions we have to consider in this statistic?
Lego dwarf family. http://www.flickr.com/photos/mooseteg/
Maybe we should first check how many Dwarves Tolkien actually ever gave a name throughout all of his works. Luckily there is a neat list of all his Dwarves on Wikipedia:
http://en.wikipedia.org/wiki/List_of_Middle-earth_Dwarves
The list gives 53 names, and we know both from the above quote and this list that Dís is the only female Dwarf ever mentioned. This gives us a horrible 1:52 ratio, which is a lot worse than the 1:2 we started with! However we do know that Tolkien didn’t mention names of a whole lot of female characters for all his peoples, so we can safely go back to keep our 1:2 ratio.
As for this list, a totally useless number to know: one fourth of all male Dwarves EVER mentioned in Tolkien’s works are going on the Quest of Erebor!
But not only is just one third of the whole population capable of giving birth, no, not even everybody of this third chooses to do so! And what is even worse: if they marry (= choose a mate for reproduction, as they don’t bring forth children otherwise) and their husband dies early, they won’t choose another Dwarf man to wed and therefore let precious reproduction resources slip by.
However you may remember what I said earlier: to keep the population stable (I am not even talking about increasing yet!) every dwarf woman needs to give birth to three children. Now we know not all of them marry and some of them may lose the possibility to reproduce after the early death of their husband. We even have an almost exact number of the marriages in the above quote: Two thirds of all Dwarves are male, less than one third of those marry. Now some simple math: 2/3 * 1/3 = 2/9 = 22.2%. Each of those has a wife, so only (less than!!!) 2/9 = 22.2% of all Dwarves ever get the chance to even give birth!
This means our number of children per woman goes up enormously! To actually increase the Dwarven population in Middle-earth we have to assume every married Dwarf woman statistically has to give birth to at least five children (22.2% being slightly more than a fifth). Do they?
Dwarf family tree from the Lord of the Rings appendices
Unfortunately we don’t know too much about Dwarf families. We know of a couple of brother pairs (Oin – Gloin, Balin – Dwalin, etc.), but the only families we can safely say we know “as a whole” are Dís’ sons Fili and Kili, and Dís’ two siblings Frerin and Thorin. Of course, we do know Bofur and Bombur are brothers for example, but we don’t know whether there are more siblings. So the two families we know (almost) for sure as a whole bore two and three children. That is definitely not enough for our estimation!
The sad thing is that we really don’t know too much about family size. Brothers are never mentioned more than as a pair but then we never know about the rest of their families.
In fact the above quote even mentions that Dwarves’ numbers increase only slowly and even more so if they lost their homeland. Unfortunately though in Middle-earth history Dwarves have lost almost all their realms which would also explain why their numbers dwindle and why they so desperately try to get Erebor, Moria and even Gundabad back!
So is there anything left we could cling to? What about the longevity of Dwarves? Shouldn’t that help them with their reproduction somehow? Well, it would, if they decided to marry another partner should the first one die. It also would if they decided to marry a little bit earlier than they do.
Let’s look at this. Dwarves are considered battle-ready at the age of 30, but they rarely ever marry before they are 80! Just imagine how much could happen in those 50 years in between. I am going to use an unusual comparison here to make this clear: Galápagos tortoises are possibly the longest-lived animals in our recent world with ages above 150 years being not uncommon. Of course when having so much time at hand one’s whole development is really slow, so Galápagos tortoises don’t reach sexual maturity until roughly 30 years. Now while a 100 year old Galápagos tortoise is almost as indestructible as a 150 year old battle hardened Dwarf, both are very vulnerable when young! The tortoise might be eaten before it even had a chance to grow big enough and reach adulthood and the Dwarf, well… we all know how much they love battle. And a young yet untrained Dwarf is more likely to fall in battle, which is also before he had a chance to reproduce. Examples? Frerin died at the age of 48, Fili was aged 82, and Kili 77, all of them before they had a chance to marry. Dain was only 32 when he slew Azog (well, in the book at least…) and was lucky he survived the battle of Azanulbizar to be able to marry afterwards. But it shows that he also risked his life at an exceptionally young age.
To get back to those Galápagos tortoises: They actually live under the same dangerous conditions as Dwarves in Middle-earth when we think about their possibility of reproduction. And we all know that those tortoises are on the brink of extinction themselves.
So, the longevity WOULD help the Dwarves to bring forth more offspring but they simply don’t take that advantage.
And now we get back to our initial question:
How can Dwarves keep up their population with their low number of women?
Well, with the given numbers they simply can’t! In fact considering all we know for sure, they are less likely to survive than the Giant Panda is in our world!
So is there no hope for Dwarves?
Dwarf families fleeing Erebor.
Yet there might be. Interesting enough Tolkien only gives us a sex ratio for Dwarf populations. Never at any point does he say what proportion of Dwarven population is of royal lineage. However all the numbers concerning family sizes we only know of the royal line of Durin’s Folk! So there really is only one way for Dwarves to survive their own extinction: The other six houses of Dwarves and/or the commoners must have much higher reproduction rates as those royal Longbeards. And this is actually a rather elegant solution, since Tolkien never really provides us with information about any other Dwarves, and this might well be true.
Royal Dwarves seem to be rare, but this was true of any royal families in our world as well. And just as it was in our case in historic monarchies it would be up to the common people among Dwarves to keep them from going extinct!
Please forgive me the impropriety, but is it possible (unless Tolkien expressly denied his dwarves would ever do such a thing) that dwarf women could have children without being married? It might not have been acceptable or encouraged, but as with our own cultures, no matter how much the powers that be may discourage it, it happens. Not to say that bastard children would have accounted for a huge percentage of the population of dwarves, but it might have added a percent or two.
I did indeed spend a thought on that myself but I doubt there is ever any hint on that topic whatsoever. Tolkien was very careful about the whole sex topic in his works (not so much in RL when you read his letters!) and didn’t really like to mention it. I mean, do we ever really read anything of that sort for men and hobbits? I suppose it does happen… for all peoples (yes, elves, hear me?), but there just won’t be any account of it.
On the other hand we do have to ask ourselves as well: Is intercourse a pleasure for dwarves? Cause in case it’s not, your percent or two might shrink to zero again. ;)
I will leave further research of that to you AC!
He specifically said Elves would never engage in premarital sex. No two Elves would ever think of it, and to an Elf maiden coerced by a Man, it would be tantamount to rape. And rape was so tragic to Elves that it would kill them, or at least destroy their lives so they were nearly dead (this is possibly what happened to Elrond’s wife when she was ravaged by Orcs and had to leave Middle-earth). Elves would absolutely never engage in sex outside of marriage.
Surely Men engaged in premarital sex, just as they do today, but it was probably more likely among the commoners than the high members of the Edain that we read about.
For Hobbits, I suppose it’s possible, but their society is so quaint and simple that any couple who falls in love would most likely just get married. That was their culture and what was expected of them, and it would give them yet another reason to throw a party. I doubt it happened very often among Hobbits, at least not in the Shire. For other Halfling societies, perhaps it was more common. We know nothing about them.
I really doubt Dwarves engaged in such practices. Not necessarily for moral reasons, but because they were so engaged in their crafts that most of them didn’t even bother marrying. It is unlikely that they would take the time out for romance. Since Tolkien modeled their culture after the ancient Hebrew culture, their marriages might have even been arranged, or at least more formally decided than those in Western cultures today.
I’ve heard around that both Dwalin and Bombur supposedly had large families later in their lives – like around 10 or 12 children each. I can’t see anything that says so online though.
Because Tolkien never wrote such a thing.
I distinctly remember reading that, too. At least about Bombur. But I can’t find it anywhere now, so it must not be true.
What an interesting topic! Thanks for this!
I’d always thought that when dwarves married, they tended to have very large families — I’d heard that Bombur had 13 children, and also read that in later life, Dwalin married and had 10 children, 7 of which were boys and 3 girls. However, when I went to check just now, I couldn’t find the references again in the wikis where I thought I’d seen them to begin with.
Would having large families when they did marry tend to make up for the low numbers of female dwarves?
Of course it would, but there is no record on any family this large.
I have no clue where those rumours about Bombur and Dwalin ever came from, it’s not Tolkien canon though.
The only reference that I know of about what Bombur got up to after The Hobbit is in Fellowship of the Ring, when Gimli is telling what happened:
“Of the ten companions who had survived the Battle of Five Armies seven were still with him [Dain]: Dwalin, Gloin, Dori, Nori, Bifur, Bofur, and Bombur. Bombur was now so fat that he could not move himself from his couch to his chair at table, and it took six young dwarves to lift him.”
There are new urban legends that have developed based on backstories the actors have given their characters (not to mention fanfiction). It can be a task to sort through it all to find the source since much of Tolkien’s “facts” are spread out over notes and texts scattered here and there.
Thanks, DJ — I could have sworn I read something in one of the Tolkien wikis, too, but apparently it has since been edited.
Basically, it seems to me like Tolkien’s information about dwarves was sketchy at best. But this article is a nice summary of what he did tell us. :)
Wow! That was really fascinating about dwarves’ reproduction! I think it is very curious that Tolkien chose not to have a lot of female dwarves in middle earth. That raises my question now that if male dwarves often don’t marry, then does that mean they have a low sex drive? That I’m sure would dash a lot of Thorin’s female fans fantasies, including mine.
Thanks so much for the essay. I learned a lot from it.
Low sex drive? There we are with the Giant Panda again. That is actually the most inconvenient idea evolution could bring forth, considering that strictly biologically speaking the true sense in life is reproduction. ;)
Are you saying the Giant Panda isn’t interested in other Giant Pandas?
I thought that was the main reason for them to be so rare. I keep hearing funny stories about how in zoos they try everything with them, even showing them panda porn (who would have thought…)
So mating and seeking company to each other is considered very rare and precious for them.
But yes, that only on a side note. And of course dwarves like the company of other dwarves. :D
Captive animals often have trouble reproducing, but that’s an effect of our meddling with them rather than evolution.
Great essay! You would so have kids flocking to a lecture on evolutionary biology with this take on it.
I’ve always wondered about the lack of familial women, too. Because what are these Dwarfs/Men fighting for if not for their loved ones, hearth and home, and their legacy to future generations. And they need women to have heirs.
P.S. Your essay also made me think of the Emperor penguins–in the documentary “March of the Penguins” narrated by Sigourney Weaver–trying to keep their egg warm while their mate went off to fetch food for several months. If the egg slipped outside of their body covering it,it instantly froze and the egg died. it was really sad to see the Penguins plight and loss. I wonder if Dwarven births were fragile as well as being rare.
You could also speculate that dwarves do not have menopause at the same relative point in their lifespans that humans do, allowing for a longer reproductive period. Tolkien points out that dwarves only show the disabilities of age in the last 10 years of their lives, so following the same logic, female dwarves may be able to safely bear children until late in life. Or, dwarves may not have menopause at all (random fact: although humans are primates, the majority of primates do not have menopause, or it occurs only at the end of the average lifespan, such as in chimpanzees).
Pretty sure Tolkien did not write about this topic either.
In fact menopause is rather uncommon in whole animal kingdom.
Dwarf women could definitely give birth for a longer time than they actually did. But you are right, Tolkien was a lousy biologist. :D
(I mean, where are the female orcs?)
Maybe non-orcs could not recognize female orcs, and nobody cared enough to ask them about it :)
I once read a wicked article on orcs being parthenogenetic, making Bolg actually Azog’s daughter. The idea was messed up, but really interesting. ;)
Well Tolkien in one letter confirmed that there exist orc-women:
“There must have been orc-women. But in stories that seldom if ever see the Orcs except as soldiers of armies in the service of the evil lords we naturally would not learn much about their lives. Not much was known.”
Then there is reference to “a little goblin-imp” in The Hobbit, ”nasssty young squeaker” that wandered into Gollum’s lair (and was eaten by him), which I assume must be an orc-child.
I’ve wondered about this too, and a couple of assumptions might help. First, multiple births might be common, for example Oin and Gloin could be twins and Dori, Nori and Ori triplets. Then the longer lifespan might allow more time for childbearing. Culturally the Uncle relationship could be very important, with those dwarves who have no children of their own helping raise a sister’s children or even adopting them as heirs, for example Thorin with Fili and Kili. None of this is canon, just my own musings.
Very interesting essay. Reading informations about Dwarves in Tolkien’s works, I thought also, that Dwarvish society had great respect for women. Imagine any human society with more men than women, where women have free will and may not marry anybody when they do not want! Especially in so called “pre-industrial” times, when marriages were usually arranged by families.
Nice! Thanks!! Dwarves didn’t talk much either…
a shameless plug I know.. .but as it is very much related to this, I believed it might be an idea to post it here as well.
For those that have an interest.. :)
http://dwarrowscholar.mymiddleearth.com/2013/07/30/counting-dwarves/
Greetings Kandral! Sorry I missed this comment until now. You have many interesting new essays which I would be happy to link to on my news page.
Hi there DJ,
Thank you for those kind words, as ever.
By all means, much appreciated.
cheers
By my beard … my favorite biologist.
So … you ‘are’ interested in Dwarven men. Well, why didn’t you just ask?
Foremost, we are honor bound to all of our commitments – so we tend to be conservatively traditional in regards to relationships.
True, our craftsmanship and warfare are prioritized over frivolous indulgences – except of course … good ale. And like a good ale … it’s not the quantity, but rather the quality of our interests in females that is so valuable.
Simply put … we are not playboys – they would be those clean shaven elven fellows. It takes time to find a good woman … who will carefully braid a Dwarf’s ever growing beard. And we won’t entrust our beards to just anyone who bats an eyelid, and makes us go all blush. Except perhaps … a favorite biologist.
Humans, by comparison, tend to overpopulate, and compromise their environment’s resources. So, fewer in number we may be – but the quality of our lives is richer. A good woman, like the Arkenstone … can make a Dwarf wealthy as a king.
For those of us who venture on quests to battle the forces of evil – there is too little time for flirtatious socializing. Gem cutters and gold hoarders tend to secure marriages long before an adventurer returns from slaying a drake in the north, or delving into the underworld of ancient tombs.
And so … we remain bachelors. All the more time for glorious battles.
Grim
Long, is my beard growing. To my belt … it will soon reach. A good Dwarven woman to braid it … I am questing to find.
Grim
Ahem… :)
Ahem … as in “My axe will cleave through your precious beard”
or
Ahem … “You’re still my favorite Greybeard”
Grim
By Durin’s blood … I would have thought the beard better appreciated.
I must have logged onto ‘Heirs of Thranduil’ by mistake.
Sorry.
Grim
Your beard is impressive Grim, and none could say otherwise, but judging by her icon, AC is something of a mustache aficionado, and…well, yours can’t compete with even a small monkey.
How demeaning … when Dwarves are compared to monkeys. I hate monkeys.
Grim
And I hate Thranduil, so we are even.
Touche’
Thranduil is a monkey anyway.
Depending on which Thranduil, if the movie one then yeah this one was basically made supporting villain, if the book then he is not so flat and one dimensional but more likeable :). And now look at that book which is essentially considered children’s book gives more nuanced character than movie HAHA! :)
Not sure of the original books, but the movies DID address this problem in the extra material. Bombur has a dozen children (maybe more), he is a “working class” dwarf aka. not royalty. And the brothers Dori, Nori, Ori are actually half-brothers of the same mother but three different fathers, explaining their drastically different appearance. So that probably means dwarves CAN marry more than once if widowed.
All of this is NOT book canon, therefore NOT Tolkien’s intention and and for that reason NOT included in this essay.
To be honest, a dwarf woman having three children from three different fathers is something that goes totally against Tolkien’s view on dwarves and their pride.
Estimating numbers is always difficult in case of Tolkien, the genealogic trees show only specific bloodlines. From what Tolkien gives us it is not even clear how many dwarves were made in the beginning by Aule, since the Seven Fathers were the first made but they soon had some number of people to form the Seven Houses/clans. The original dwarves in one source were actually six pairs (six fathers had already their mates) and Durin, as being truly alone and so he must have married daughter of other royal family. So the progenitors were 13 dwarves both male and female and some unknown number of dwarves that formed their respective tribes, if to rationalize it. The beginning period of the dwarven race existance after their awakening (sometime after the elves but we don’t know WHEN) certainly it was many thousand years during the Years of Trees (accoridng to some estimation Feanor was about 3000 years old during his death and the First Age have not even begun by then. It is reasonable to assume that in the earliest periods the dwarves increased in numbers quickly and only later on they started to dwindle.
“The number of dwarf-men that marry is actually less than one-third.”
I always thought that to mean “Less than one-third of the whole population.”
If one-third of the dwarves are female, but not all of them marry, it stands to reason that of the two-thirds that are male, less than half of them actually marry… But it is very possible that I’m completely wrong, of course.
Wonderful essay, thank you for sharing this!
Nice essay :)
It would indeed be a problem for the dwarves if only ~20% or so marries and then has a small number of children. Does not mean it won’t happen though, species get extinct all the time.
I think the assumptions (1:2 female/male ratio; not all women get married; less than 1/3 of men gets married) do not apply to the shown family tree of Durin. Not to say either of those is ‘wrong'; I think this family is not representative for all dwarves, possibly because they are royal.
Most women aren’t shown in the family tree, but every dwarf must have had a mother. If you add one female dwarf for each male that has children, you already get a sex ratio of 3:2 instead of 2:1. This ratio gets even closer to 1:1 if some of the males at the end of the tree (Thorin III, Oin, Gimli, Balin, Dwalin) also got married (ratio would be 7:6 male/female if all 5 of them got married). Add to this some unmarried women (at least 1/3 of the women don’t get married), and then there are actually more females than males in the population (or the ratio is 1:1 if Thorin III, Gimli et cetera did not get married).
If sex determination works the same in dwarves as in humans, the 2:1 ratio would not be stable under the given conditions (monogamous lifestyle & most individuals reach maturity). If there are 2 times more males, a 2-fold higher percentage of females than males will procreate. As a result, a genetic propensity to produce more female offspring will be favoured: If your child is female, there is a 2 times higher chance she will have children than if she were male, so your genes are being spread 2 times more efficiently. (A dwarf that produces more sperm cells with an X chromosome compared to other dwarves, has an evolutionary advantage.) This results in more dwarves who also have this same genetic propensity, so the male/female ratio decreases and decreases… until more females than males are born, because then it is favourable to have a genetic propensity to produce more male offspring. On a population level this is how the 1:1 ratio is maintained (in humans and many other species): Simply because it is a stable equilibrium.
This is where the ‘in humans the ratio at birth is not exactly 1:1′ comes in: A trait can only be selected for (or against) if it influences procreation (more precisely, if it influences the chance of the gene that caused it to be passed on to the next generation). The stable equilibrium is thus not achieved when the ratio at birth is 1:1, but when the ratio at child-bearing age is 1:1. If half of the males die very young and all females reach fertile age, there is no advantage to producing 50% X-chromosome sperm instead of 33%, on the contrary. 33% would be optimal.
The XY sex determination system is theoretically possible for Tolkien’s dwarves, but then the male/female ratio at birth must be higher than 2:1. For example at birth it could be 3:1, at child-bearing age 1:1, and overall in the population it could then be 2:1. This means 2/3 of the males die prematurely. If this is the case, dwarves should have 4 children on average (so >>4 per marriage, as many don’t get married) to keep their population size stable. It is possible the family tree only shows the dwarves mentioned in Tolkien’s stories, leading to an underestimation of the average family size in dwarves. Also it could be that the royal dwarves have less children than the average dwarf. However, the family tree for sure does not look like 2/3 of the males die prematurely: Only 4 out of 25 die before the age of 120. This is probably not higher in the general dwarf population, since being of royal descent actually seems to predispose dwarves to dangerous adventures. It seems therefore unlikely that dwarves have the same sex determination system as humans.
I propose the following sex determination system for dwarves:
Facultative parthenogenesis.
Female dwarves can reproduce asexually (via parthenogenesis) and sexually (in which case they have only (or mostly) male progeny). (From a biologist’s perspective I think this is a lame system, since a lack of males will result in… only new females being produced. Usually it works the other way around (e.g. in bees). But it’s not impossible.)
~2/3 of the population is male, and ~1/3 of the males get married –> ~2/3 of the females get married.
~1/3 of the females does not get married. Many of them do have children and these are all daugthers (not necessarily clones; only if the women herself was a result of parthenogenesis will her daughter be a clone). Twice as many females do get married, and have only/mostly male progeny (either there is something in the sperm that suppresses parthenogenesis, or married dwarves do sometimes have daughters).
If married and unmarried women on average have the same number of children, this will lead to a 2:1 male to female ratio at birth. If the Durin family is respresentative for the dwarf population with respect to lifespan expectancy, the sex ratio can be close to 2:1 at all ages.
Fertilization with sperm always leads to a male embryo for one of the following reasons:
– Males have XY (females XX), but only produce sperm cell with a Y chromosome (e.g. the Y chromosome contains a gene encoding a product that interferes with meiosis I and destroys the sister cell with the X)
– Males are diploid, females are haploid (in this case parthenogenesis is always clonal expansion). Unfertilized egg cell leads to haploid (hence female) offspring, fertilization leads to diploid (male) offspring.
– Sex determination is epigenetic. There are no sex chromosomes, gene expression is different depending on the origin (from father or from mother) of the genes. Two chromosome sets from the mother give rise to a female child. If one set originated from the mother and one from the father, the gene expression is different and the embryo develops as a male.
This system could theoretically reach an equilibrium at almost any sex ratio, but it seems reasonable to assume that when there are fewer males, females become more interested in them (law of scarcity). In that case relatively little women choose parthenogenesis over marriage, and the relative number of male births increases. With the resulting increase in the male/female ratio, the interest of females in marriage decreases. It is the strength of the psychological ‘law of scarcity’ effect that determines at what sex ratio there is a stable equilibrium, and this appears to be at ~2:1 male/female. (More than 67% males –> females become less interested in marriage –> more parthenogenesis and less sexual reproduction –> more females and less males are born –> less than 67 males –> females become more interested in marriage –> less parthenogenesis and more sexual reproduction –> more males and less females are born –> more than 67% males.)
On second thought… There would be no gene transfer from male to female (as females never have a father). This cannot be good for the genetic diversity amongst females.
New solution:
Dwarves have an XY system like humans. Female dwarves can also reproduce asexually, giving rise to haploid offspring. These have a male appearance but are sterile (and not interested in sex or marriage, only in working/mining/fighting). The haploids make up ~1/3 of the population, so if there is a 1:1 ratio of XY-males to females (stable equilibrium, not influenced by the number of sterile males), the male/female ratio is 2:1. (It may seem simpler to assume there is no asexual reproduction and 50% of the (XY) males are sterile, but natural selection would decrease that percentage.) There is an evolutionary benefit to a dwarf woman, if not interested in marriage (possibly because the man she desires is not interested, or he dies before they get married), of giving birth to sterile sons, instead of having no children at all. The sons will not procreate (so not directly pass on her genes) but they will fight to protect their family, increasing the chance her siblings will successfully pass on their genes (50% similar to her genes).
Dwarf women then need to have 3 children on average to keep the population stable. (Average married female –> 1.5 female and 1.5 fertile male, of which 1 female replaces her mother in the population, 1 male replaces his father, and 0.5 female and 0.5 male do not get married. Average un-married female –> 3 haploid offspring.) Because 2/3 of the females marry, the average dwarf woman gives rise to 1 haploid offspring, 1 female offspring and 1 fertile male offspring. Only 1 out of 3 males will marry, as stated in the essay.
In the Durin family tree, dwarves of whom we know they had children, have 1.44 sons on average, which is close to the expected 1.5. They could have had equal numbers of daughters, but these are not shown (because they are female).