Those that favor reproduction eventually outbreed those that don't. Expansion will happen. If we can colonize one star we can colonize the galaxy.
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Or dissent groups that want to set up their own society. It will eventually drop to the point that it need not be a government effort.
I don't believe growth is essential. I believe growth will happen if it's practical.
All this assumes that starfaring species don't take active measures to limit their own expansion. But it seems to me it would be rational for them to do so.
I'm inclined to agree with Jokodo that the rational motive for transplanting your species to other stars is extinction avoidance -- the galaxy is a dangerous place and if you spread out to dozens of planets across hundreds of light years then the chance of your descendants being wiped out by a local disaster goes to near zero. So why spread further than that? Well, there are two answers. Either it's because you're concerned about your descendants being wiped out by some much larger, say, galactic-arm-scale disaster, or else it's because you want to for some irrational motive. So the questions become: what galactic-arm-scale threat is there to be rationally worried about, and what irrational motive could plausibly induce irrational expansion? When you look at it that way, you've already given the same answer to both questions: those that favor reproduction eventually outbreed those that don't. The disaster that could take out all the civilizations across thousands of light years is alien invasion -- invasion by some species that evolved a natural urge to just keep growing and growing despite the lack of any benefit to its members.
So let's say you're a species of starship builders and you occupy a stretch of space a couple hundred light years across, and your policymakers are debating whether to colonize any more worlds. The con side makes Jokodo's mathematical argument: you're already safe, your exponential growth rate has already gone to zero, and having more stars won't raise it above zero, so what's the point? The pro side holds up the specter of alien invasion -- if you don't grow, some other species eventually will, they'll outbreed your descendants, their realm will grow until it borders yours, and then they'll seize your planets for themselves and kill all your descendants. The only way to ensure survival of your descendants is to grow first and occupy the whole galaxy.
So what would the con side have to say against that? It seems to me they'd say the proposed preemptive defensive growth policy
will itself create the very threat it's meant to defend against. In a galaxy where starship-building species are evidently rare, who is more likely to eventually evolve into a species of dangerous irrational overbreeding starfarers? Some currently paleozoic mollusks in some alien ocean half the galaxy away? Or the descendants of their own colonists who they send out right now on a mission to occupy the galaxy? By far the most likely aliens to start an interstellar war over their living space are some future species of their own genus who value growth for the sake of growth and who decide they're entitled to go back to the ancestral worlds and replace the less evolved locals.
So they'd argue the best way to make sure their own descendants live on indefinitely is to prevent those who favor reproduction from outbreeding those who don't -- to maintain control of their species' genome. All their planets need to remain the same species indefinitely. They need to com-laser their genetic innovations to one another and introduce each planet's beneficial mutations into the gene pools of all the others. And it will be an awful lot easier to maintain genome-unity over two hundred light years than over twenty thousand.
So if private starships become affordable, and there are any dissent groups that want to go against government policy and start new colony planets beyond the current realm of their species, it seems to me it would be rational for the rulers of the existing planets to stop them by force. Seize any half-built starships and if necessary arrest the dissent groups' leaders. The best way for Vulcans to stay safe from Romulans is to stop the Romulans from separating from the Vulcans in the first place.