The gut microbiome is a highly complex microbial community that strongly impacts human health and disease. The two dominant phyla in healthy humans are Bacteroidetes and Firmicutes, with minor phyla such as Proteobacteria having elevated abundances in various disease states. While the gut microbiome has been widely studied, relatively little is known about the role of interspecies interactions in promoting microbiome stability and function. We developed a biofilm metabolic model of a very simple gut microbiome community consisting of a representative bacteroidete (Bacteroides thetaiotaomicron), firmicute (Faecalibacterium prausnitzii) and proteobacterium (Escherichia coli) to investigate the putative role of metabolic byproduct cross feeding between species on community stability, robustness and flexibility. The model predicted coexistence of the three species only if four essential cross-feeding relationships were present. We found that cross feeding allowed coexistence to be robustly maintained for large variations in biofilm thickness and nutrient levels. However, the model predicted that community composition and short chain fatty acid levels could be strongly affected only over small ranges of byproduct uptake rates, indicating a possible lack of flexibility in our cross-feeding mechanism. Our model predictions provide new insights into the impact of byproduct cross feeding and yield experimentally testable hypotheses about gut microbiome community stability.