Population Descent Math

[Malimar]

Okay, so here's the situation:

The locathah in my setting have a complex electoral system where eligibility to hold office and eligibility to vote are determined by descent from a pair of ancient locathah heroes.

Dagunas are those who are patrilineally descended from the hero Dagon Daguna; i.e., iff your father was a daguna, you are a daguna. Dagunas are eligible to hold office.

Atarathehs are those who are matrilineally descended from the heroine Atargatis Ataratheh; i.e., iff your mother was an ataratheh, you are an ataratheh. Atarathehs are eligible to vote.

(Those who are both daguna and ataratheh have great prestige in locathah society, can both vote and hold office, and are frequently elected to high office.)

Dagon Daguna and Atargatis Ataratheh may have had children with each other, and they may have had children with other locathah. Obviously, Dagon had at least one son, who had at least one son, etc, and Atargatis had at least one daughter, who had at least one daughter, etc.

Dagon and Atargatis lived roughly 1,000 years ago. Assuming a roughly human-like lifespan and breeding rate (unless any sources from any D&D edition offer different lifespans or breeding rates for locathah) and a stable population of let's say 50 million or so, my question is: how many dagunas and atarathehs would you expect there to be today? I'm really just looking for an order of magnitude or a rough percentage of the population.

[NichG]

There's an interesting phenomenon in population genetics where lineages tend to coalesce just from the statistics of neutral drift. The result is that given enough time, either everyone will be descended from them or no one will. Here's the wikipedia page on the subject: https://en.wikipedia.org/wiki/Coalescent_theory

So, would that have happened yet? The average number of generations to coalescence is equal to twice the population size, so probably not. That basically means you're going to have an incredibly wide distribution of these ancestors since (if these are neutral traits) you have a multiplicative random walk - like a stock market trajectory. So more or less, pick your favorite number I'd say.

[LokiRagnarok]

Okay, I am going to sleep right now, so my math is probably way off, but:

Any locathah that was alive 1000 years ago is either ancestor of all the population or none (assuming they are on the same continent). cf. http://gcbias.org/european-genealogy-faq/#q3

So every individual of your population alive today is descended from both of these individuals in a way. Now it's interesting whether they are in fact matrimonially/patrimonially descended.

You assume stable population for the last 1000 years (something I consider unrealistic, but okay), which means a reproduction rate of 1 child per individual. We can assume that 50% of them are female, 50% are male.

You have 1000 years. If a individual gets a child with 20, that's 50 generations. A given child has a 1/2 (50%) chance to be the "right" gender to keep the line going. So I think it would be 1 mio times (1/2)^50, which is 0.0000000888% for someone being e.g. a pure Daguan. Which is immensely low, so I am wodering if I made a mistake somewhere. You may want to simulate this for 3 generations or so, with each 2 children.

Note however that a society where one gender is really favored, people would try for a child again and again until they got the "right" gender.
Also, I assume people would be faking confirmation they are who they pretend to be all over the place.

[veti]

Also, I assume people would be faking confirmation they are who they pretend to be all over the place.

This right here is - worth thinking about. Claiming ancestry is one thing, proving it is another. Over the past 1000 years, there would have been a great deal of skulduggery gone into this - forgeries, lying about parentage, disputes about rights of natural vs legal descendants, inheritance... Almost certainly there would have been several major wars fought over the subject.

And many of the people who have the voting/ruling rights as of now, will have them basically by virtue of being descended from the winners of those wars. While other people who, genetically speaking, probably have just as good a claim - won't have the rights that go with them.

[Anxe]

Assuming a stable population? The most likely result is that there will be one of each. It's stable. Each couple has one son and one daughter.

To get a REAL mathematical answer to this question you need to either not have the population stable or apply a fitness factor. A fitness factor is a multiplier for how much better each of the special conditions is. Like dagunas and atarathehs are more desirable so they have twice as many kids. Then the fitness factor is 2. So what do you think the fitness factor is for them? Then we can answer the question.

[Malimar]

You assume stable population for the last 1000 years (something I consider unrealistic, but okay), which means a reproduction rate of 1 child per individual. We can assume that 50% of them are female, 50% are male.

Assuming a stable population? The most likely result is that there will be one of each. It's stable. Each couple has one son and one daughter.

OK, it transpires that you're completely correct to question my assumption of a stable population! I assumed it wouldn't change the math.

The population isn't stable. Not only because there are the regular issues of the expansion and contraction of civilization, but also because: 570 years ago, the world completely flooded, giving the sea-dwelling locathah massively more living space to expand into; then 20 years ago, the world partially unflooded, reducing their living space slightly.

So maybe call it a population of 10 million 1000 years ago, 50 million today?

To get a REAL mathematical answer to this question you need to either not have the population stable or apply a fitness factor. A fitness factor is a multiplier for how much better each of the special conditions is. Like dagunas and atarathehs are more desirable so they have twice as many kids. Then the fitness factor is 2. So what do you think the fitness factor is for them? Then we can answer the question.

Hm, yes. It is a good point, dagunas and atarathehs probably are more desirable as mates. Let's call it fitness factor 2, that sounds good, and we can tweak it from there if it leads us somewhere preposterous.

(If it matters, I don't know whether locathah society is monogamous or polygamous. Though their reproduction is certainly not like the indiscriminate spawning of fish, if they care about paternal and maternal lines of descent. Probably monogamous, or, if polygamous, at least careful about who fertilizes whose eggs.)

Also, I assume people would be faking confirmation they are who they pretend to be all over the place.

This right here is - worth thinking about. Claiming ancestry is one thing, proving it is another. Over the past 1000 years, there would have been a great deal of skulduggery gone into this - forgeries, lying about parentage, disputes about rights of natural vs legal descendants, inheritance... Almost certainly there would have been several major wars fought over the subject.

And many of the people who have the voting/ruling rights as of now, will have them basically by virtue of being descended from the winners of those wars. While other people who, genetically speaking, probably have just as good a claim - won't have the rights that go with them.

A good point.

Somehow I thought locathah tended to be Lawful, but nope, "Usually Neutral", so there wouldn't be any sort of "as a people, they're just less likely to cheat" business going on. They're probably just as likely to cheat on this issue as humans would be.

So yes, you're absolutely correct. There have undoubtedly been major wars fought over the legitimacy of descent claims, especially if it comes to close votes (e.g. is the person who cast the deciding vote even a legit ataratheh?) and e.g. whether the person the atarathehs have elected monarch is even a legit daguna.

[goes to add some locathah succession wars to the histories]

[OldTrees1]

For our purposes, until the most recent generation we can simplify it to the following case
A Daguna father can have Daguna sons or non Daguna daughters
A Ataratheh mother can have Ataratheh daughters or non Ataratheh sons
A non Daguna father can have non Daguna sons and daughters
A non Ataratheh mother can have non Ataratheh daughters and sons

As a result, to replace himself/herself a Daguna father/Ataratheh mother needs to on average have 1 son/daughter. At exactly 1 the pool of Daguna/Ataratheh will shrink relative to the entire population but would remain constant in number. Well, except that an average of 1 allows for fluctuations including 0. So right at the replacement rate will likely eventually cease to exist from this fluctuation (since fixation at 0% as a result of drift would have a higher chance than fixation at 100%).

Of course since Daguna/Ataratheh can produce daughters/sons, more non Daguna/Ataratheh are born than Daguna/Ataratheh (remember I am only counting those that can pass it on).

Assuming the reproduction rate is not increased/decreased by being a Daguna/Ataratheh, the population rate of increase per Daguna/Atartheh of the Daguna/Ataratheh would be half the rate of increase per Locathah of the Locathah. So take your P_L=P₀e^rt and solve for "r", halve that "r" and use it to calculate P_D&A.

Spoiler: Please check my math

Show

r=(1/t)Ln(P_L/P_0L)

P_D&A=P_0D&Ae^(1/2)rt

P_D&A=P_0D&Ae^{(1/2)t(1/t)Ln(P_L/P_0L)}

P_D&A=P_0D&Ae^{(1/2)Ln(P_L/P_0L)}

P_D&A=P_0D&Ae^{Ln((P_L/P_0L)1/2)}

P_D&A=P_0D&A(P_L/P_0L)^1/2
P_L=50 million as you said
P_0D&A=2 (Dagon and Atargatis)
What was the initial Locathah population size 1000 years ago? (P_0L)

Presumably they maintianed this number once the population became stable.

[Anxe]

Fitness factor of 2! Alright! That means that the number of dagunas and atarathehs should double with each generation relative to the general population. The 50 generations is pretty accurate. 2^50 is a grandiosely high number. Both traits should go to fixation within that time frame with all locathahs being dagunas or atarathehs. The population size doesn't matter in this case.

This would normally be complicated by people with the trait breeding with themselves and the chances of no one gaining the trait, but with the way you've set up the gendered traits, that wouldn't happen.

For a comparison, a fitness factor of 1.1 would get you 117.4 dagunas and 117.4 atarathehs proportionally in the final generation. The population has increased by 5x, giving you 587 of each. Plus, the previous generations that are still living adds on 534 of each.

That might be a more reasonable fitness factor for what you want.

[Malimar]

I didn't want to say "this is the number of atarathehs and dagunas I want there to be, show me how to finesse the math until we get that number", but... I guess kind of that's what I'm saying anyway.

I mean, a medieval-technology voting system (where all the voters get together in one place for a moot for every vote) gets impractical if there's fewer than 100 or more than 1000 voters (or candidates, but not all dagunas will be standing for every election, so a high number of dagunas is less problematic than a high number of atarathehs), so I guess as long as the math is finessable (by adjusting the fitness factor), I guess we might as well finesse the math to get a number in that range.

[OldTrees1]

I didn't want to say "this is the number of atarathehs and dagunas I want there to be, show me how to finesse the math until we get that number", but... I guess kind of that's what I'm saying anyway.

I mean, a medieval-technology voting system (where all the voters get together in one place for a moot for every vote) gets impractical if there's fewer than 100 or more than 1000 voters (or candidates, but not all dagunas will be standing for every election, so a high number of dagunas is less problematic than a high number of atarathehs), so I guess as long as the math is finessable (by adjusting the fitness factor), I guess we might as well finesse the math to get a number in that range.

When fitness factor = 1:
P_D&A=P_0D&A(P_L/P_0L)^1/2
P_L=50 million as you said
P_0D&A=2 (Dagon and Atargatis)
What was the initial Locathah population size 1000 years ago? (P_0L)

Presumably they maintianed this number once the population became stable.

100/2 = (50000000/X)^(1/2)
50^2 = 50000000/X
X=50000000/2500
X=20,000

So if your initial Locathah population 1,000 years ago was 20,000 and there is no fitness difference, then the stable population of 50 million Locathah will have roughly 100 Daguna fathers and Atarathehs mothers. So there will be probably 200 total when including the female Daguna and male Ataratheh.

1000/2 = (50000000/X)^(1/2)
500^2 = 50000000/X
X=50000000/250000
X=200

So if your initial Locathah population 1,000 years ago was 200 and there is no fitness difference, then the stable population of 50 million Locathah will have roughly 1000 Daguna fathers and Atarathehs mothers. So there will be probably 2000 total when including the female Daguna and male Ataratheh.

So pick an initial population of Locathah between 200 and 20,000 to get your 100-1000 Daguna and 100-1000 Ataratheh.

[Segev]

There would be tremendous sociological pressure to have Daguna males marry Arteleth females. And incentive, as well, to have those who are neither seek the opposite sex's heir-type as mates, so they can achieve children who have at least one type. Whether officially condoned or encouraged or not, harems (at least de facto ones) would be relatively common, with at least one "prime spouse" who is an heir of his or her sex's type married to a "prime spouse" of their type. Their children will be the highest-prestige sort, and all will be heavily encouraged to marry others of the appropriate sort.

The harems would then consist of those who were not the right kind of heir to pass on their heritage, seeking to have at least some kids who would have the right heritage so that they can marry their kids into better families and have peripheral prestige.

This would tend to cause it to spread considerably; it is probable that a majority of people are at least one sort of heir, and that it is hard for those who are not the right kind of heir to pass on their heritage to their offspring have a harder time finding mates.

[MesiDoomstalker]

There would be tremendous sociological pressure to have Daguna males marry Arteleth females. And incentive, as well, to have those who are neither seek the opposite sex's heir-type as mates, so they can achieve children who have at least one type. Whether officially condoned or encouraged or not, harems (at least de facto ones) would be relatively common, with at least one "prime spouse" who is an heir of his or her sex's type married to a "prime spouse" of their type. Their children will be the highest-prestige sort, and all will be heavily encouraged to marry others of the appropriate sort.

The harems would then consist of those who were not the right kind of heir to pass on their heritage, seeking to have at least some kids who would have the right heritage so that they can marry their kids into better families and have peripheral prestige.

This would tend to cause it to spread considerably; it is probable that a majority of people are at least one sort of heir, and that it is hard for those who are not the right kind of heir to pass on their heritage to their offspring have a harder time finding mates.

Adding on to this, we quickly fall into blue blood territory. We'll get all the people in power being related to one another. Who spawn tykes, who are related to each other and will eventually take the positions of power when they come of age. Even if we assume that this in-breeding trend waited till the population of Daguna and Arteleth's was sufficiently diverse, it won't take too many generations to get cousins marrying cousins.