Say you have a race of human demigods that rule humanity. These demigods have a lifespan of about 5000 to 6000 years. They are similar in biology to humans except that they live longer. What is the best way to prevent their population from overtaking baseline humans, and at the same time keep their numbers up to prevent them from dying out?
If these demigods live 6000 years, their "childhood" could be 300 years long. The effort to maintain and educate such a "child" would put a strain on the parents, and while some would have more kids and some would have less, the "scale" of the lifespan would balance out.
Becoming a Demigod is dangerous.
Perhaps whatever method is used to achieve that long lifespan is inherently dangerous. You grow until 40 years old as a normal human would, but then the demigods go through a biological change which has a high failure rate. Maybe it's an external drug, maybe it's some sort of symbiosis with another form of life on the planet, or some chemistry in the environment, or maybe it's just the end of this species "puberty."
Surviving that long is dangerous.
Another option is that the ruling pyramid of the Demigods use younger demigods as tools, like western gods use angels. In order to make it to your first 1000 years, you have to survive being sent to the front lines of every holy war the demigods fight; surviving multiple trips back and forth to your realms, and your subjects learning that you're long lived but not immortal and trying to kill you.
Surviving your family is dangerous.
Maybe they DON'T do anything to reduce their child-rearing rate, but in order for a child to make it through to adulthood they have to live in a rough-and-tumble dog-eat-dog world, where only the strongest (and therefore healthiest) actually make it into the ranks of adulthood.
...and maybe this never stops, and even at 5500, you have to keep one eye open.
There are many reasons why a race of long lived demi gods / rulers might have a low reproduction rate.
- Any Children a "demi-god" might have would likely want power. So the more children one has, the more they have to share.
- The lesser races breed like vermin, one as pure as we are better than that. Ours are the purest, ours are the few, the great.
- This could manifest in a desire to not breed so heavily, or perhaps selectively breeding and weeding out of the weak. Ritualized killings, tribal initiations, breeding compacts, or just plain infanticide.
- Every child Zeus has weakens the position of Odin. Best to kill them before they reach maturity.
- They love selectively and love hard. Once paired, they can never pair with another. 5000 years is a long time, the chances of one dying is pretty high.
- The rite of bonding is a sacred thing. The One Who Made All would never countenance the sacrilege that is "free love".
- These "higher beings" require very specific environmental criteria to successfully pair. The moon must be in retrograde, bamboo in bloom, and a red headed orphan dancing the jig.
- Mating seasons. These demigods may have evolved from a lower creature and retained vestigial biological functions.
- High infant mortality rates.
- Perhaps they are not native to this world, and have issues adapting.
- Low fertility rate.
- Pregnancies last 500 years...
- It could be really really uncomfortable for one of the genders.
- Think angler fish for males (the males are absorbed by the females until only their testes remain, which the female then uses)
- For females, they may not have a designated orifice, and males create one with their barb. (think beetles have this)
Sexy sexy mortals
- Mortals are just so hot, and ever so ready to please their gods. Why go through a brutal 100 year courtship with Hera when I can turn into a swan and go to town.
- If the population is low enough and you live long enough you are going to hate literally everyone else. "No-one will not let go of that drunken rant I went on about the green wings 3000 years ago. I was only 200 years old, I've changed!"
- Zeus killed or sealed away anyone older than himself, then went and banged everything in sight. Now everyone is my sibling, and Zeus is probably banging them too. Some might be a little hesitant to dip into that gene pool. Also, Zeus ate half his kids.
- I got married at the age of 2000, and have another 3000 years of marriage to look forward to. Why would I possibly want to ruin that with kids. Maybe in a 1000 years or so.
As for keeping their population up, they live a very very long time. Even a low birth rate should keep their population numbers constant.
Cyclical fertility It's hard to convince intelligent creatures to reproduce for purely rational reasons. More power and autonomy will only exacerbate the problem. Continual reproductive impulse causes undesirable overpopulation. But cyclical fertility could solve the problem nicely.
When things are stable and normal there could be no reproduction for thousands of years. A fertility cycle could initiate a period of reproduction. Preventing extinction and stagnation.
This also also allows for feedback loops which allow circumstances to influence fertility rates. Catastrophic event kills off half the population? Increased duration and frequency of fertility cycles. Stable population? Reduced duration of fertility cycles and an increase in the pause between cycles.
You have quite a few options.
Little to no libido. If your beings are prudish (like many - especially abrahamic - gods) and are either exceedingly selective with who they want to mate with or only get horny every few thousand years then they obviously won't be having many offsprings.
Pratically infertile. If you want them to have the same sexual appitites as ordinary humans then you can make only a few out of every tens of thousand sexual interaction result in pregnancy.
They can't or don't want to procreate when they are rearing children. You didn't specify how their extended life-cycle works but if childhood lasts hundreds of years, you can simply make your creatures unwilling or unable to mate until their current cubs have been fully raised.
If you're worried that putting into place any of these solutions would make your species prone to extinction, say if the normal humans somehow started effectively rebeling and killing their overlords, simply make it so that under duress they becomes much more "passionate", fertile or much less dedicated to their current children (depending on what solution you choose). This is by no means fantastical, the reproductive behaviours of many species change when they are stressed out, including humans (human women produce more male children when they're stressed because it is an effective survival method when faced with famine or war, etc...).
In the demigod's society, harmony with nature and maintaining a sustainable population are cultural norms.
In modern fiction, we have seen several near-future-distopia works where having more than one child is a capital crime. In reality, we have several grass-roots organizations trying to convince us we should do this voluntarily. Most of these groups advocate two pregnancies per female, successful or not, as a means of slowly achieving a stable population. (The extremists among them want us to stop procreation and allow ourselves to go extinct.)
In Shard martin's world, no laws and no movements are needed. Through ages old tradition, it is just assumed that everyone will practice restraint. It is well acknowledge that every additional child is an additional consumer of limited resources. The responsible attitude to parenting considers all aspects of the potential new life on the world as a whole.