I'm not sure what the minimum population would need to be. To maintain the human race, I would think it would need to be a rather large sample, which would not be sustainable for quite a long while. After the colony is established, the population could be enlarged, but starting out it would need to be much smaller.
It would take years for any colony to become completely self-sustaining. The very first outpost is going to need large quantities of power and water. Mars has water, but most of it is currently frozen. Once a colony has been established, mining water from the polar caps will be an option, as well as vaporizing and condensing it from the soil. But at the beginning, water (or hydrogen to combine with oxygen generated by other processes) will be one of the most valuable resources. The power supply could be provided at first from methane/oxygen fueled generators, supplied by rather simple production facilities that have already been demonstrated, but this fuel might be better used for rockets and other vehicles. Mars soil is rich in silicon, which can later be used to produce solar panels, but that would come much later. Some areas of the planet are also young enough to be useful for geothermal power, but that also will be a resource that will take time to develop. I think the most useful power source for founding the colony would be a nuclear reactor. (However, just imagine the environmental lobby's reaction as soon as you mention you want to shoot a reactor into space... yeesh). Once you have sufficient water and power, you can begin to use native resources to expand the colony. With water and soil, you can create bricks to build structures. Dig a trench, line it with bricks, vault it over, and pile dirt on it, and you've got a structure that you can pressurize with breathable air, and will even be self sealing if leaks develop. Any air leaking out will contain moisture, which would freeze in the soil, sealing the leak. Food is another early concern. Large supplies of food will need to be imported early on. As materials are shipped, or become available from native sources, domes could be constructed, and pressurized for growing food crops. Importation of livestock, and methods of raising them, would be problematical at first. Meat could be shipped in from Earth, but I believe that the first colonists would need to get used to a very highly plant based diet.
Economic incentive? That's where the plan falls apart. It'll be quite a long time before the colony is able to show a profit. That's the main reason I doubt that any private company will get into the game anytime soon. However, if the colony could be established and maintained, new technologies for travel to and from will be developed, possibly making transportation prices drop significantly. Eventually, products that once needed to be imported will be replaced by local production. But it will be a long time.
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