What are the stylized facts of societies in the middle of the first demographic transition?

The answer is "A decline in mortality followed by a decline in fertility".

Demographic transition theory accept that fertility decrease is irreversible. This usually held presumption depends on perceptions of later and authentic decreases in ripeness that go with modernization and declining mortality. The irreversibility presumption, be that as it may, is profoundly suspect from a transformative perspective, since statistic attributes are at any rate in part impacted by hereditary qualities and are receptive to social and natural conditions. In any case, an unavoidable move from high mortality and richness to low mortality and ripeness is utilized as a directing structure for anticipating human populace sizes into what's to come.

most cajuns resided in acadiana, where their descendants are still predominant

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