I am writing a series of books/text adventure games I am tentatively calling the Sublight series. In the stories, even though mankind has cracked the powers of Magic, they still run up against the speed of causality. (Which we know as the speed of light). They also use the colors spectrum to describe how their magic system works.
Series 1 takes place in the Solar System. Series 2 takes place on a ship travelling to a distant star. Series 3 takes place on a settlement a few generations after it is established, and for which Earth is just a memory passed on by their ancestors. (It starts with the death of the last person alive who launched on the seed ship.)
The vessel for series 2 (and which is preserved as a shrine in series 3) is the Arthur Charles Clarke. It is a block-3 vessel of Project Iliad, launched from the Solar system in that universe's 22nd century. It was sent to settle the 18 Scorpii system, 47 light years from Sol.
In the late 21st century, the ISTO is engaging in a pattern of encirclement. The Krasnovian Empire won the early space race, and has now settled Alpha Centauri and has expressed that it will oppose any settlement within 10 light years of Sol with military force if need be.
Limitations in the settlement approach has prevented Krasnovia from actually building a sustainable population in any of their remote star systems. They insist on shipping packaged food instead of allowing their colonies to engage in their own agriculture. They are occupying these star systems mainly to keep ISTO from occupying them.
The ISTO has adapted the Iliad Block-1 from the venerable Einstein class. And the Einstein as basically a Van Gogh with a G-drive propulsion system. The Block-1 Iliads extended the 5 year endurance of the Einstein to 10 years. Redundant systems, structural reinforcements, extreme quality control, and extra corrosion measures did make the vessels extremely expensive compared to their Einstein step-sisters.
The Block-2 ships were the first to get beyond the 10 light year range imposed by Krasnovia. They were designed for a round trip. While both missions ultimately succeeded, by the time the vessels returned to the Solar System decades had passed, and it was only through a combination of sheer luck and perseverance that the crews made it back alive. Crews also had trouble dealing with how changed the world had become since their vessels launched and returned.
The concept for block-3 vessels was to just make the journey a one-way trip. The vessel would be equipped with everything that was needed to bootstrap civilization in another star system. (Material source locally in the remote system allowing.) While block 1 and block 2 were missions with only two vessels, block three was far more ambitious. ISTO would be sending ships to every star within 50 light years that seemed to harbor commercially exploitable resources. (i.e. A sun-like star with a metal content that indicates the likelihood of an asteroid belt.)
Given the number of ships they seek to build, the ISTO was forced to economize on crew. At the start of the voyage, the ship has only 600 crew to operate the ship. The hope was to use population growth to bolster those numbers into a viable population in the remote star system.
I built a computer model to work out the working population vs. total population for a generation ship:
The blue line is the total population. The orange line is the workforce population. For the purposes of this analysis, a woman having a child withdraws from the workforce for 1 year. Thus the dip in the workforce immediately after the ship launches, even though the overall population is climbing.
This graph was generated from the this spreadsheet. See the Population and Population Model tabs.
The model is a statistical model. It divides the population into 7 year buckets, and further divides those buckets by gender. Every year several members of the bucket die. 1/7th of those that do not are removed from each bucket and promoted to the next bucket.
The number of people who enter the lowest age bucket are the children born. The birth rates are calculated based on the the age of the women on board. Women are most likely to become mothers in their late 20s, with some starting earlier, and some starting later.
The numbers for mortality and birth rates is derived from historical population data in the United States during the 20th century. So your mileage may vary. But the overall behavior should be a plausible estimate.
Just to repeat: this is a statistical model. Int only works on a population level. It does not attempt to predict individual behavior. It is numbers in an actuary table. Note that all of those columns have decimal points!
As stated, the ship launched with 600 crew.
By year 35, 5 years before the ship arrives at the remote star system, the workforce nearly doubles to 1100. The overall population is higher, at 2500. The overall population includes 964 children below the age of 21, and 370 retirees over the age of 55.
The workforce peaks at year 54, at ~1427. By this point the ship and the local economy have been focused on developing the 18 Scorpii system for 10 years. They are most likely mining that system for resources, and building new ships and settlements. The overall population is 2917. There are 1079 children under the age of 21, and 671 retirees.
The overall population peaks at year 80, at ~3130. The population is probably spread across several settlements by now. The workforce has shrunk to 1273, and it will continue to shrink as people retire. The population is settling into a steady state, with 996 children and about 1072 retirees. There are just about as many people entering the workforce as leaving.
This population model required no draconian birth mandates. I had provisions for some sort of agency to ration out birth slots. But it turns out it if you have the right mix of ages in the crew at the mission start, you can tune your expected growth without any heavy handed tactics. Basically: if you want a shipload of babies, pack the vessel with 18 year olds. If you want a sane growth pattern, have 50% of the population between 21 and 27, 40% between 28 and 34, and 10% between 35 and 41.
If you start with a balanced distribution of ages, the system maintains a balanced population. The concept with this particular choice of crew was to minimize the drain an outgoing ship would place on the working population of the civilization it was leaving.