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.