Self-organization by bacterial cells often leads to the formation of a highly complex spatially-structured biofilm. In such a bacterial biofilm, cells adhere to each other and are embedded in a self-produced extracellular matrix (ECM). Bacillus substilis bacteria utilize localized cell-death patterns which focuses mechanical forces to form wrinkled sheet-like structures in three dimensions. A most intriguing feature underlying this biofilm formation is that vertical buckling and ridge location is biased to occur in region of high cell-death. Here we present a spatially extended model to investigate the role of the bacterial secreted ECM during the biofilm formation and the self-organization of cell-death. Using this reaction-diffusion model we show that the interaction between the cell's motion and the ECM concentration gives rise to a self-trapping instability, leading to variety of cell-death patterns. The resultant spot patterns generated by our model are shown to be in semi-quantitative agreement with recent experimental observation.