Spatial pattern formation in bursting systems can be modeled by systems of reaction diffusion equations with weak diffusive coupling. We study a simple two by two model of bursting systems in which stable limit cycle oscillations coexist with a stable steady state. This system combines features of oscillatory media and bistable media with two stable steady states. A natural question for such systems concerns the spatial propagation of "phase boundaries" between oscillatory, bursting states and stable steady states. Numerical computations and supporting analysis show a rather surprising behavior: a spatial wave extinguishes the oscillatory state and replaces it with an unstable steady state. The unstable steady state is connected to the stable steady state by a travelling wave.