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I'm talking about life-bearing planets (so empty in terms of not having humans or other competing intelligent species on them). Kind of like the first human, maybe 40,000 years ago, walking into North America, could expand his tribe's population drastically.
Yes, but no evidence suggests that the original planets that mankind landed on were filled with predators that couldn't be efficiently dealt with any better than on earth.
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In the Koprulu Sector on the other hand, many of the colony planets were life-bearing and suitable for human habitations, but there were no other humans, or zerg or protoss there. At the same time these settling terrans would have had access to 22nd or later century knowledge (eg washing hands, bathing, boiling water, etc). Thus the perfect opportunity; free land, no competition, and a low death rate.
(Note that I said knowledge and not technology. I don't know how much "stuff" the settlers had.)
They might have had minimal technology. And your argument has no basis in a high death rate. If the death rate isn't high, then the growth rate is only higher. Terrans have no reason not to breed like rabbits in the early years.