The linear stability of a vertically-distributed, Nutrient-Phytoplankton-Zooplankton (NPZ) ocean ecosystem model is analyzed to understand how vertical mixing influences biological dynamics. In the absence of vertical diffusion, the model generally exhibits both stable fixed point and limit cycle behavior, depending on the depth and choice of parameters. Diffusion couples the dynamics of nearby levels and can induce stable profiles as well as oscillatory dynamical trajectories that become vertically phase-locked for large mixing levels. Calculations of the Lyapunov exponent reveal that vertical diffusion can drive this model into a chaotic state, though this occurs only for levels of diffusion well below those found in nature. The dynamics of the model, assuming macrozooplankton are the dominant grazers in the ecosystem, are compared to those in which microzooplankton dominate, with a faster grazing rate and poor assimilation efficiency. While the coupled physical-macrozooplanton system has a stable profile, the coupled microzooplankton profile remains unstable, even at large mixing levels. Fluctuations occur on time scales varying between a few days and a few months, depending on the parameters and magnitude of diffusion.