I don't know where you got that idea. Some species critically depend on wildly unstable birth rates (grasshoppers and cicadas, but probably also deer and many other prey populations).
Stable populations are completely irrelevant at the microscopic levels; InBev would fold within a week if yeast populations were stable.
And both grasshoppers and cicadas are famous for suddenly disappearing across enormous areas. Which is a situation only very rarely referred to as "stable".
I know there's a joke in here about this being literally in the bible, with God using such an insect birth rate swing as a punishment for an entire state. That's how "stable" it is.
Stable populations are completely irrelevant at the microscopic levels; InBev would fold within a week if yeast populations were stable.