Bedrock rivers are characterized by conditions of insufficient sediment supply, wherein the amount of sediment supplied to the channel is less than the channel's sediment transport capacity. Consequently, bedrock channels generally exhibit discontinuous sediment cover, which current approaches to the morphodynamics of alluvial rivers are unable to handle. To fill this gap, we present a theoretical framework in which local sediment transport rates are proportional to the areal concentration of sediment available for transport on the bed and the sediment continuity equation is reformulated to account for temporal changes in areal concentration of sediment on nonalluviated surfaces. We then perform a linear stability analysis that shows that an initially uniform distribution of the areal concentration of sediment on the bed is unstable to small perturbations. This suggests that the sediment cover in bedrock channels tends to concentrate into regions which may eventually be alluviated, in general agreement with experimental observations.