In many consumer-resource systems the consumer has synchronized reproduction at regular intervals (e.g., years), but consumes the resource and dies continuously, while the resource grows continuously or has overlapping generations that are short relative to the time between consumer reproductive events. Such systems require models that are a mix of discrete and continuous components. Some ‘semi-discrete models’ have appeared, but their behavior is not well understood. We analyze a model with logistic resource growth and a linear functional response in the discretely breeding consumer. This model sets up a framework for understanding dynamical behavior of more complex, semi-discrete models. The model can display stable equilibria, consumer-resource cycles, and single-species-like overcompensation cycles. Cycles are induced by high maximum fecundity in the consumer. We show that the model can be reduced to a single-species discrete-time model of the consumer, which can exhibit overcompensation cycles, if the resource grows rapidly and the consumer has high maximum fecundity. By contrast, such cycles in discrete consumer-resource models typically occur only in the resource once the consumer is extinct. Also unlike a common class of discrete models that do not display consumer-resource cycles with periods below 4 years, semi-discrete models can exhibit consumer-resource cycles with periods as short as 2.