Why doesn't predation cause extinction?

Answer 1

It has, in the past. The prey animals that survived are those who have a lot of babies perhaps every several months, who are very alert to the sound, sight, and smell of predators, and who are good at finding and staying somewhere that predators have difficulty getting to. Examples include burrows and thorn bushes.

Prey animals tend to live in large groups so that there are many animals alert to the sound and smell or sight of predators.

On the other hand, predators typically have only 1 or 2 babies only once a year or once every two years. Predators very often lie around a lot, so they need to eat fewer prey animals.