A model of an animal cohort foraging on logistically growing food is analyzed. The problem is captured in three differential equations, one for food density and two for the state of the animal cohort, keeping track of body weight and number of individuals, respectively. When the animals efficiently exploit their food to low densities, the model produces cycles. The cycles differ markedly from those produced by traditional predator-prey models. Consumer decline is associated with starvation mortality when individuals lose too much weight. This condition causes a stepwise decline in individual number, each step corresponding to one cycle. Because the survivors of each starvation period grow and because larger animals have lower weight-specific metabolic rates, the nature of the cycles changes over time. They acquire a slow-fast character because of the increasing difference between food and consumer speed and show distinct catastrophic features, as the fast phases are caused by jumping between an over-and u...