What are examples of how the food chain can be broken between producer and consumer or predator and prey?

One way a food chain can become "broken" is when humans disrupt the food chain.

For example, in Yellowstone Park and nearby States, humans had killed off wolves in the area. Scientists noted how big game populations of deer and elk started increasing. After many decades, scientists noted that the increased big game ate most of the foliage, which changed the forests and soil. Trees started to die. Without trees and foliage, Yellowstone had more erosion especially near rivers. Rivers reacted to the higher soil quantities of runoff deposited into the water. Waters became more stagnant and flowed less vigorously.

Eventually, scientists decided to reintroduce wolves into the ecosystem. Wolves mated, had babies, thus increasing their population. Scientists added more wolves to prevent cycles of in-breeding genes. The wolves began to kill big game, lessening the ill and elderly populations of elk and deer. Foliage began to grow again. Wolves added nutrients to the soil from the wolves' wastes. New sapling trees started to grow again. But most interesting...the re-introduction of wolves changed the river! With more trees and foliage, there was less erosion. With less erosion, the rivers became less stagnant. The rivers began to flow faster and because the water erodes rocks, the river narrowed back into its original bed (rather than expanding when it was stagnant).

Really cool information to know!