How do predator and prey influence each other

Depending on the concept and context, but if prey pop. is low, the the food source for predators is low and thus both populations will be low, a sudden increase in the population of prey wont necessarily mean a boom in the population of predators right away, initially there will be a steady growth and then exponential as the time spent hunting will be greatly diminshed and they have more time to reproduce, this would lead to a fluctuating 2 way graph between prey population ad predator population with a time lag between the two values. Hope i helped!