We construct a two-zone model to describe hydrogen and helium burning on the surface of an accreting neutron star and use it to study the triggering of type I X-ray bursts. Although highly simplified, the model reproduces all of the bursting regimes seen in the more complete global linear stability analysis of Narayan & Heyl (2003), including the regime of delayed mixed bursts. The results are also consistent with observations of type I X-ray bursts. At accretion rates <~ 10% of the Eddingtion limit, thermonuclear helium burning via the well-known thin-shell thermal instability triggers bursts. As the accretion rate increases, however, the trigger mechanism evolves from the fast thermal instability to a slowly growing overstability involving both hydrogen and helium burning. The competition between nuclear heating via the beta-limited CNO cycle as well as the triple-alpha process and radiative cooling via photon diffusion and emission drives oscillations with a period approximately equal to the hydrogen-burning timescale. If these oscillations grow, the gradually rising temperature at the base of the helium layer eventually provokes a thin-shell thermal instability and hence a delayed mixed burst. This overstability closely resembles the delayed mixed bursts of Narayan & Heyl. For accretion rates >~ 25% of the Eddington limit, there is no instability or overstability, and there are no bursts.