From a recent paper by Sebastian Galiani and Raul A. Sosa:

Fertility rates have fallen below replacement in most countries, fueling predictions of demographic collapse. We show these forecasts overlook a crucial fact: societies are not homogeneous. Using the Bisin–Verdier model of cultural transmission with endogenous fertility and direct socialization, calibrated to U.S. and global data, we find that high-fertility, high-retention groups persist, gain share, and lead the total population to grow. Even if fertility remains below replacement in every country, extinction is unlikely. Simulations imply continued growth with pronounced compositional change, driven especially by religious communities with high fertility. In our ten-generation world calibration, Muslims become the largest tradition.

I am pleased to hear that extinction is unlikely.