Life cycle events in plants and animals are typically adaptively tuned to anticipate predictable seasonal changes in environmental conditions or resources. Climate change is expected to affect the temporal component of species’ interactions, e.g. by creating a mismatch between a predator's breeding time (when ample food supply is critical) and the time when prey abundance is high. The demographic implications of such a mismatch remain unclear, however. Here we focussed on changes in the phenology of consumers relative to that of their food. We developed a model where reproductive output of the consumer up to offspring independence depended on mismatch and recruitment of the offspring to breeders depended on offspring density according to a Beverton–Holt function. Using a deterministic version of the model, we clarified how the effects of (constant) mismatch on equilibrium population size depended on the emergent strength of negative density dependence (DD). Using a stochastic, individual-based version, we showed that when the environment changed abruptly, the rate of population recovery was faster when heritability of seasonal timing was higher and DD was stronger. When the environment shifted continuously, the rate of decline in population size was inversely proportional to the rate of microevolution, but stronger DD slowed the rate of decline for a given heritability and thus effectively ‘bought time’ for evolutionary rescue. These results highlight the importance of negative DD, which interacts with the effects of trait heritability and stabilizing selection strength, in influencing the fate of populations experiencing environmental change. We emphasize, however, that outcomes in nature will depend crucially on the exact nature of DD, in particular whether population growth rate differences are greatest at low or high densities, highlighting the need for empirical comparisons of compensatory processes in different populations or species.