The bell-shaped rate-pH curve coupled to production of base in the urea-urease reaction was utilized to give feedback-driven behavior: an acid-to-base pH clock (a kinetic switch), bistability and hysteresis between an acid/base state when the initial pH was adjusted by a strong acid, and aperiodic pH oscillations when the initial pH was adjusted by a weak acid in an open reactor. A simple model of the reaction reproduced most of the experimental results and provided insight into the role of self-buffering in the dynamics. This reaction suggests new possibilities in the development of biocompatible feedback to couple to pH-sensitive processes for bioinspired applications in medicine, engineering, or materials science.