Abstract
Climate change has been shown to induce shifts in the timing of life‐history events. As a result, interactions between species can become disrupted, with potentially detrimental effects. Predicting these consequences has proven challenging. We apply structured population models to a well‐characterised great tit‐caterpillar model system and identify thresholds of temporal asynchrony, beyond which the predator population will rapidly go extinct. Our model suggests that phenotypic plasticity in predator breeding timing initially maintains temporal synchrony in the face of environmental change. However, under projections of climate change, predator plasticity was insufficient to keep pace with prey phenology. Directional evolution then accelerated, but could not prevent mismatch. Once predator phenology lagged behind prey by more than 24 days, rapid extinction was inevitable, despite previously stable population dynamics. Our projections suggest that current population stability could be masking a route to population collapse, if high greenhouse gas emissions continue.