You guys are really, really underestimating human population growth. The only reason we sat at a billion people for so long was infant mortality, we kept dying off before we could grow up for thousands of years. The moment we hit the twentieth century, the population started climbing, and in little over a hundred years it's hit 7.9 Billion. You might be tempted to point out that we're living in a time of relative peace, but that's actually inconsequential to my point since the greatest areas of population growth are third-world countries that live in conditions much worse than The Persean Sector. Mogadishu is so bad every two weeks I ask someone "Hey have you heard about that terrorist attack in Mogadishu a few days ago?" Without looking at the news and I've yet to be wrong.
Even with epidemics, being ravaged by war and genocide, and a phenomenally low life-expectancy, the population of Africa went from 177 million to 1.2 Billion in Fifty years The biggest reason for that was infant mortality rate dropping so suddenly. And that's still not hitting the resource cap. It's expected to hit 4.7 billion in 2100. That's more than half our current population on one continent.
You all seem to forget that when the going gets tough, humans screw like rabbits. The worse conditions are, the more children people have. Even though we might not have as many children at a time as other species, female humans are capable of pumping out a kid every year starting at like sixteen until their fifties (depending on the individual). Go watch the Duggars if you want to see what that looks like with any kind of access to modern medicine. Of course, that's not taking safety into consideration, but as life gets more dangerous and harder, people start to care less about the 'safety risks' of things like pregnancy. Even with access to birth control, regions with less wealth and less safety still have higher birth rates, as exemplified by every ghetto in the US.
For the Sector to be sitting around a billion people after two hundred years, either conditions are bordering on 'Antartica in the middle of a blizzard with medieval level medicine' bad, or every major population center is enjoying the kind of success of a first-world country while also having hit their resource caps. As long as people aren't getting nuked, it doesn't really matter how many fleets get blown up, it's inconsequential to the population.
To put it in perspective, if we average it together, our population is currently growing at an average of a bit over 1% per year, compounding annually (and decreasing as we approach our resource limit). Even if the Persean Sector's population grew at the same rate, that's at least 1.7 million up to 17.7 million people a year. That's 590 to 5,900 3000 person fleets being destroyed per year with a hundred percent mortality rate. Despite what you might think, humanity just doesn't have the economic prowess to wage a war that can outpace our birthrate unless you specifically target population centers with intent to wipe them out, as evidenced by, once again, Africa, which I will remind you has basically stayed in a perpetual state of war since the turn of the century.
To put it simply, the numbers are in, and Alex got his wrong.
That's part of it, the other part is increased lifespans. Not only are children surviving to adulthood far more often, they're also remaining alive through the next 3 to 5 generations after them far more frequently.
That said, the Persian sector isn't just Mogadishu. It's mostly environments that are completely uninhabitable by humans without technological assistance, and that assistance is breaking down. A new person in Africa (or anywhere else on Earth in the 20th century) needs water, food, some degree of shelter, etc. These might be hard to come by, during some periods a lot of people will suffer and die from their lack, but the capacity to produce far more of these basic resources than were used existed. Unfortunately that almost certainly won't be the case in the 21st century.
A new human on almost any Persian world needs the same resources, along with other resources like oxygen that we don't really think about on Earth. In the Persian sector though, not only is the maximum capacity of those resources very clearly limited on most worlds, it's often shrinking. It's not just a matter of people producing more of the basic goods - people *can't* produce more of those goods due to technological limitations. A new human in the Persian sector means everyone else gets by with a bit less, or someone else has to die.
This hit especially hard after the collapse, when worlds lost access to imports necessary to meet basic needs. Vast numbers of people died because the worlds in the sector not only weren't producing the resources needed, they in fact didn't have the capacity to produce those resources at all without additional terraforming, and that terraforming ground to a halt soon after the collapse for the same resource reasons.
I'd argue that most of the sector lives in conditions *FAR WORSE* than Mogadishu (and briefly to your suggestion that Mogadishu represents the parts of Africa with significant population growth, take a look - most of the population growth in Africa is in relatively modern, stable megacities
https://africa.businessinsider.com/local/lifestyle/fastest-growing-cities-in-africa-2021/b97e271?op=1). Not only are they worse off than Mogadishu, the maximum productive capacity of most Sector worlds has already been met and is declining, whereas in Africa and elsewhere on Earth, maximum short term production potential is far higher than actual production. In addition, Africa benefited from significant foreign aid and investment in the 20th century, which also isn't available in the Sector. Without efforts to restart terraforming, or to colonize the few *habitable* worlds that aren't already colonized, there's very little room for population growth.
Terraforming and colonization are extremely high risk investments when the Sector is unstable and in conflict. It's much easier to grab some guns and take what someone else already has - and that is also a net negative for population growth.
Edit: Another way to look at it - On Earth, an average human creates more value than they consume during their life. A new human is going to provide more resources in net than they need to survive, and they're able to do this generally through extraction of resources from finite reserves and making them available for circulation in the economy (usually through a specialized economy where only some people and tech do that extraction). The 19th and 20th centuries also saw enormous resource availability gains through innovation, but *almost all* of that innovation came from increasing our ability to access limited resources, not in finding ways to do more with less resources. This enormous growth in resource availability was a significant driver of population growth.
In the sector, this is almost certainly not the case. A new human doesn't have the means to extract more than they consume from the environment. Technology is in decline, and most people don't have the means to access resource reserves on barren or inhospitable worlds. The technology that does allow extraction in those harsh environments is failing, which means the net output of humanity and it's technology is declining, and a new person adds to the consumption, rather than the production, of resources.