Material degradation is currently one of the main technical issues in the development of polymer electrolyte membrane fuel cells (PEMFC). This degradation includes carbon corrosion of the catalyst support and catalyst migration that can occur at high electrochemical potentials. These high potentials can occur at start-up, shut-down or reactant starvation. This work reviews the known conditions that lead to high potentials in PEMFC and develops models that describe the phenomena concisely. Computations describing these cells in four steady starvation situations are presented: anode starvation; cathode starvation; and two types of local anode starvation. Reverse reactions, oxygen and hydrogen evolution, must be included to correctly identify the behaviour of fuel cells in these situations. Using logarithmic scaling of concentrations and adaptive grid refinement in the flow direction, computational implementations of the model can robustly handle the depletion of reactants in the flow and corresponding changes in potentials. The model is extended to time dependent computations using simpler descriptions of channel flow but including capacitance and stack-level electrical coupling effects. Simulations of fuel cell start-up are presented. In all cases, the results of the simulations are compared to literature and new experimental results. Strong qualitative agreement is obtained.