The objective of the current paper is to investigate the dynamics of a new predator–prey model, where the prey species obeys the law of logistic growth and is subjected to a non-smooth switched harvest: when the density of the prey is below a switched value, the harvest has a linear rate. Otherwise, the harvesting rate is constant. The equilibria of the proposed system are described, and the boundedness of its solutions is examined. We discuss the existence of periodic solutions; we show the appearance of two limit cycles, an unstable inner limit cycle and a stable outer one. As the values of the model parameters vary, several kinds of bifurcation for the model are detected, such as transcritical, saddle–node, and Hopf bifurcations. Finally, some numerical examples of the model are performed to confirm the theoretical results obtained.