Untangling the influence of human activities on food-web stability and persistence is complex given the large numbers of species and overwhelming number of interactions within ecosystems. Although biodiversity has been associated with stability, the actual structures and processes that confer stability to diverse food webs remain largely unknown. Here we show that real food webs are structured such that top predators act as couplers of distinct energy channels that differ in both productivity and turnover rate. Our theoretical analysis shows that coupled fast and slow channels convey both local and non-local stability to food webs. Alarmingly, the same human actions that have been implicated in the loss of biodiversity also directly erode the very structures and processes that we show to confer stability on food webs.