What would happen if the bobcats would go extinct?

If bobcats were to go extinct rodents, rabbits and other small pests would have one less predator to manage their populations, this in-turn would cause other imbalances within the ecosystem to occur. It's always bad for an ecosystem's health when a top predator species is removed. Bobcats are already considered an endangered species in some states, Man is the bobcats worst enemy.