Water produced in a polymer electrolyte membrane (PEM) fuel cell enhances membrane proton conductivity; this positive feedback loop can lead to current ignition. Gas-phase convection and membrane diffusion of water coupled with water production in a simplified two-dimensional PEM fuel cell leads to localized ignition and current density front propagation in the cell. Co-current gas flow in the anode and cathode channels causes ignition at the cell outlet, and membrane diffusion causes the front to slowly propagate toward the inlet; counter-current flow in the anode and cathode channels causes ignition in the interior of the cell, with the current density fronts subsequently, spreading toward both inlets. The basic chemistry and physics of the spatiotemporal nonlinear dynamics of the two-dimensional fuel cell current can be captured by extending a simple one-dimensional stirred tank reactor model to a "tanks-in-series" model.