The dying off of the Common fox occurs more slowly than the reduction in the number of its prey – forest mouse-like rodents. Why?

The dying off of the Common fox occurs more slowly than the reduction in the number of its prey – forest mouse-like rodents. Why? How will the number of rodents change with the complete extermination or a sharp decrease in the number of this predator?

1) this is due to the high mobility and unspecialized nutrition of the predator;
2) first, the number of mouse-like rodents will increase, this will lead to depletion of food resources and the struggle for food, which will cause a subsequent sharp decrease in their number.