What is the brink of extinction?

A brink is the very edge of a cliff or a similar structure. It could be the edge of the roof of a flat-roofed building. But either way, standing on the brink of the cliff or roof means that one step forward will mean a long and probably fatal fall.

The expression "on the brink" is often used figuratively, as a metaphor. Thus, "We are on the brink of nuclear war" means that we are one step away from falling into the Horror of war.