A loser in both environments can survive by switching between them

How can a species persist in an environment where it is always outcompeted? Using a minimal predator-prey model with environment-dependent parameters, we show that a predator driven to extinction in each of two static environments can survive indefinitely once the environment alternates between them fast enough. We derive the critical switching rate above which persistence occurs, and show that random (Poisson) switching needs to be faster than periodic switching in order to offset prolonged spells in the unfavorable environment. We then generalize the mechanism to any two-species system, and can predict persistence solely based on the sign of a single ``switching rescue function" assembled from the two boundary vector fields. This general result has broad reaching consequences: for instance, when applied to a standard model of viral dynamics, it predicts that two drugs which each clear a pathogen on their own can fail when alternated, giving a non-resistance-based explanation for the failure of drug-cycling strategies. Our results demonstrate that the tempo of environment change, as opposed to the environments themselves, can lead to species survival.