I have a question that concerns a story I am writing and I hope it is a valid one to ask. But in my story, several groups of colonists, each arriving on different ships at relatively the same time, land on a newly discovered habitable world with the goals of spreading and populating the new world while establishing new, yet individual, societies that fit their ideologies and views as quickly as possible. They start from complete scratch using the tech and resources they brought from earth. My question is: Initially, in detail and realistically, which would be more favorable; large or small settlements built with the in mind goal of quick expansion but also quick population growth?
Settlement size will depend much more on how habitable your "habitable" world is and how much they need in order to bootstrap their society, than on what would be better for quick expansion.
How fast a population can expand doesn't depend on how large your settlements are - it depends on how safe you can keep your people. A man and woman can have just as many children in a small settlement as they can in a large settlement as long as you have enough food for everyone and the ability to keep them safe.
How habitable is the new world? Does it already have plant life, a breathable atmosphere, seasons that don't get dangerously hot or cold, and no dangerous wildlife to worry about? If that's the case, then you could easily have small settlements similar to what happened early in the history of the U.S. This will allow for greater discovery and acquisition of resources.
What about if the world isn't quite so ideal? The more dangerous the planet is, the more that people will want to stay together for efficiency of resource use and security. For example, if the planet is mostly safe but doesn't yet have enough oxygen for us to breathe without assistance, you'll need to have larger settlements due to the difficulty in building controlled-atmosphere living spaces.
On the high-tech end of the spectrum, your settlers could have fabricators capable of mass to energy to mass conversion. The fabricators could consume any matter, use some of the energy produced to power themselves, and use the rest to produce whatever items the settlers need. The settlers won't need help from others, allowing them to spread out as much as they want.
If their tech level is fairly close to ours, they're going to want to start out with large settlements. Unless they're trying to "go back to nature", it takes a lot of infrastructure in order to have a modern society. For example, think about what it takes to produce electronics - you need to be able to find metal ores, process those ores, acquire sources of silicon, find sources of some rare earth elements, be able to produce the chemicals necessary in the manufacturing processes, build cleanroom facilities, and then manufacture the electronics (and I'm sure I've glossed over some key parts of the process). And that's just one aspect of a modern society. They'll need large settlements working hard in order to build up their infrastructure.
The easier it is to keep everyone safe, the more they can (and likely will) spread out. Finding a mate is likely not the biggest challenge the settlers will be facing.
Generally larger is better.
Bigger equals larger work force, more skills, bigger gene pool, more redundancy. The biggest downside is cost, more people also need more supplies, so you send the largest group you can afford to send.
However, you also usually have to consider what else you can send (like can I send more fabrication equipment if I send fewer people), but assuming everything else being equal then more people are better. You want to send as many people as you can without compromising something else to do it.
Other answers do not mention the importance of medicine.
Quick population growth requires high birth rates. If your colonists can produce babies only a natural way you need doctors and midwives. Childbirth is not that simple or safe for humans. Without an easily available medical care, you will face skyrocketing levels of child and maternal deaths. Especially, if you are pushing colonists to have as many babies as possible.
Every settlement should have at least one midwife. Ideally, it also should be within 30 minutes of a medical facility where urgent medical procedures can be performed.
It would also be wise to establish some kind of system of tracking genealogy to avoid inbreeding. This becomes especially important if you have a small number of original colonists.
Is this a planned economy or a market economy? Because in the later, the settlement size will be determined where people go. Let's look at that:
will be attractive to farmers and early settlers. If you are barely surviving, trust is vital and trust is easier to form in small communities where everyone knows everybody. If your initial economy is agriculture, it also means your field is close. The more people you add, the more outwards their fields will be (space is limited) and sooner or later it will be so far that additional farmers prefer the next, still smaller, settlement.
are more attractive to specialists and late settlers. A community needs a certain size to support people in professions that don't directly contribute to the basics of life. 10 families may simply not be able to feed a teacher, 100 might, 1000 will certainly, the advantage of education are bigger than 0,1% of your food. The larger a settlement grows, the more specialist jobs will appear out of necessity. Logistics and management will become issues that are better solved centrally.
The exact numbers depend much on the local economy. If the land is fertile, surpluses are bigger and non-food jobs appear earlier. If the economy is based not around agriculture and food is imported (say in a mining colony), that factor becomes the focus around which everything revolves.
The settlement of the planet and surrounding space will actually depend on just how it is being settled.
Assuming you are using spacecraft, each individual spacecraft might be rather small and hold a small number of colonists plus tons of supplies and machinery. Smaller is generally easier to accelerate and decelerate, so this makes actually getting to the colony easier. A large number of small vessels might be sent out at once, in order to provide enough redundancy to ensure some will arrive and there will be enough viable colonists and machinery to make a successful settlement.
In this case, there might be the urge to make many small settlements using the spacecraft as the seed points, so no single disaster will engulf all of the colonists.
If the colonists arrived spaced out over time (i.e the launch device or fuelling station can only handle a small number of spacecraft at any give time), then the colony might resemble a seaport, with the first ship landing and building infrastructure, and subsequent ships arriving and landing there to take advantage of the already existing facilities.
The other consideration might well be points of origin. If the colonists are all coming from Japan, then the new colony ships being sent from Venus may well decide to avoid the Japanese zones, and create separate colonies elsewhere. Depending on the sort of technology they brought and the locations they choose. they might decide to settle in one large and powerful colony, or disperse into many small settlements which are hard to find and individually take lots of resources to track and overwhelm or absorb.
From a general survival POV, settling in a dispersed pattern of smaller settlements would seem to give you a better chance of avoiding being overwhelmed by issues like natural disasters or even the spread of disease or technological breakdowns (you can quarantine small locations which are separated). You also have a wider range of resources to choose from, and trade and economic development will also happen faster in these conditions.
Several small is better. Why go to another planet in the first place? Presumably because you want something (land or resources) there.
Look at how land is exploited on this planet: a country doesn't put its whole population in one mega-city, instead it's spread out over the whole (habitable portion of) the country.
Similarly on a new planet you might want small bits of population near whatever resource you went there to exploit.
The reasons for a large city (e.g. avoiding in-breeding, having specialist hospitals, or whatever) can be solved by tech that lets people travel (e.g. fly to a city centre occasionally when they need one).
Unless your target technology is to have 100% bio-plastic with 3D printable 3D printers chances are you need a large settlement around a nice place to make food and babies with most of the factories and satellite communities for resource extraction at the best sites within range of your transportation. Whether you count that as one settlement or many is semantics.
What local resources do you need? Are you going to mine copper, gold, iron, oil, silicon, tin, aluminum, lead, titanium, asbestos, coal, uranium or any of the dozens of other things we want from the earth? If you do chances are it behooves you to visit more than one site.
If you are building permanent infra structure like mines expected to produce for years it probably makes sense to have some people live close to it. So then it makes sense to put some support infrastructure near those people.
But if your capital machines are small and cheap it may be you can ship enough that making more than one such colony node makes sense. Generally it is assumed that more people give you a better chance of a colony not failing, but with a single settlement a hurricane might doom everyone. If you have geographically dispersion you can be pretty sure no local problem will get them all and it's easier to reclaim a damaged settlement from somewhere else on the planet than wait for more support from the home planet.
New is always better. No wait, large is always better.
Large settlements necessarily have a better genetic diversity, they are better at defending, they have more synergestic force to combine. It's easier to find an expert for a partiuclar thing, etc. etc..
Resources are presumably scarce on a previously uninhabitated, and vastly unexplored and only recently developped planet. More people need to eat more, and they need more water. This needs to come from somewhere.
They also need more space and breathable air (if the planet's atmosphere is not breathable or only to some extent breathable). If a habitat is being built, let's say a kind of "glass dome", it is increasingly more difficult to do so the larger it gets (square/cube law). You would need to build several smaller habitats and connect them with tunnels, but then, even though you pretend you don't, you effectively have many smaller settlements anyway.
Larger settlements also are more anonymous and have overproportionally more crime. In a society where everybody knows everybody else, crime rates are modest or virtually non-existent.
Also, on other planets, nature is generally not as indulgently forgiving as it is on our home planet. Where, except in some extreme regions, you can for the most part survive without technical gimmicks and sleep under the stars without having to worry about never waking up again.
On another planet, a single "average" thunderstorm or similar natural phenomenon could easily wipe out a complete settlement within hours, whether it be small or large. What's large by our standards is a grain of sand to planetary forces.
If there are several smaller settlements, the positive side of having a settlement anihilated is that there still remain some.
It is a trade-off between benefits.
Large settlements allow more efficient collective action, trade, etc. It is easier to dig one water well than 10 water wells.
If I want all settlements to be fenced, say for protection against local wildlife, then it is 1/3 the effort to fence one large settlement versus fencing 10 smaller settlements. (Because area enclosed goes up with the square of the fence length).
A single large population allows market fluidity: In ten settlements of some distance apart (to allow for growth), you need more doctors, so simultaneous illness or emergency in multiple settlements can all get the attention they deserve. You just need fewer of each kind of specialist in a single large settlement, and this in turn allows for more types of specialist: If you have three doctors in one settlement instead of ten doctors in ten settlements, then the other seven people that could have been doctors can be researchers, teachers, or of some other profession that utilizes their intellectual capabilities, bettering the community as a whole.
But there are drawbacks to a single large community. Diseases can infect everybody. A fire can devastate or kill more people. A flood or drought can wipe everybody out.
Basically there are many problems with a large concentration of people in one place, including getting supplies (food and water) into the place, and waste out of it. Sewage and garbage disposal presented a large problem when small villages became larger cities, sanitation and running water became more difficult than when everybody had their own well and their own piping. Power generation and distribution can become a problem, too.
In many smaller villages (historically on Earth) the solution to pollution was dilution. There was enough air, water and fields that waste and garbage wasn't a problem. But as they grew larger, the distance to disposal increased and crap piled up (literally).
It is easier to grow a small village (or start another small village) than it is to grow a large settlement, because growth of a large settlement serves to make the center of it increasingly barricaded and distant from the edges, where we need to go to dispose of waste (if we aren't going to dump it on our neighbors).
You have a trade-off between the complexity of a large settlement and its advantages, and the simplicity of small settlements and forgoing those advantages.
I would say smaller settlements are better, but keep them relatively close to each other. Why? You solve two problems with this.
You avoid the "all your eggs in one basket" problem". A Plague, drought, and so on might be easier to contain and less impact on the world as a whole. You can more easily quarantine a small town than a neighborhood in a large city. If one community has a water system failure they can disperse. If a large city has a water system failure, well...
The reason you keep them close together is so that each community can aid and support those nearest to it. In the example of a water system failure, 2 or 3 nearby communities should be able to handle the excess population without much problems. If you have one big city, where do those people go?
There are many other benefits to a slightly dispersed populace. Space, for one thing. They are easier to manage.
If you initially set up small communities, each within a day or so travelling distance from each other, one is going to have some advantages over some of the others. That one is going to start growing, probably faster than the others. This is a good thing. It will likely, over the course of a few generations, turn into a trade hub. If conditions are favorable, it will eventually grow to the point of enveloping it's nearest neighbors. That's how your large city should come about, and then you get the advantages, You will have had time to actually plan for things like waste removal and so on. The quicker a city grows, a lot of the stuff we take for granted can blossom into epic problems.
In short, spread out at first and then let your large city grow organically, over time.
If you take a cue from how bacteria or fungus spread, smaller distributed colonies will always spread faster than larger colonies. It comes down to simply a matter of perimeter of the expanding front of the colonies. Many small colonies will have a combined much greater external perimeter area than one large colony. (Since circles maximize interior space.) This actually is exactly why many fungus spread spores instead of just spreading across the ground.
This is all granted that people aren't actually a bacterial infection... of course if you are the planet you might not be so happy that you've developed a case of the humans.
Many small is probably better than one big, considering a possibility that something happens (probably biological, but could be as simple as a fire) and the settlement gets wiped out. Survivors can be rescued by the people from the other settlements nearby, but they must exist.
Also, keep in mind that the ships that bring the colonists need to be built, equipped, and get safely to the new world. Several large settlements would require either ships that are huge in size, or a giant fleet of smaller ones. It will become a millennium-long project to construct enough ships for multiple large settlements. Such long term projects have a way of right-sizing themselves, unless they totally die, of course. So, the big+many idea probably wouldn't work due to simple budgetary constraints.
Final conclusion: several smaller settlements would see more pragmatic and provide the necessary redundancy.