Traveling wave formation enables strain coexistence in a spatial model of bacterial cross-feeding

Cross-feeding is a form of metabolic cooperation in microbial populations where one species or strain produces a byproduct that serves as a nutrient for another. This interaction can promote division of labor, resource efficiency, and species coexistence. The extent to which spatial heterogeneity and metabolite diffusion shape cross-feeding interactions is not yet well resolved. In this study, we examine the spatial dynamics of cross-feeding between two bacterial strains using a two-dimensional chemostat lattice model. Both strains are capable of growing on glucose and acetate, but differ in resource preference. One, which we term glucose specialist, primarily consumes glucose, while the other, which we term acetate specialist, primarily consumes acetate. Moreover, elevated acetate amount inhibits the growth of both strains. Metabolite dynamics are governed by reaction-diffusion equations, and bacterial motility is implemented through partially random local dispersal rules. Our simulations reveal that the emergence of traveling waves plays a critical role in enabling long-term coexistence of the two strains. Specifically, clusters of glucose specialists self-organize into wave-like structures that propagate toward regions of elevated acetate amount. These traveling waves suppress inhibitory acetate level ahead of the front, creating favorable conditions in their wake for the acetate specialists to grow. The resulting spatiotemporal patterns-characterized by merging wave fronts and sequential colonization-allow both strains to persist over the long term, despite competitive and inhibitory interactions. Mathematical analysis is used to support and interpret the simulation results. These findings demonstrate how spatial self-organization and reaction-diffusion dynamics can mediate coexistence in microbial cross-feeding systems, offering new insights into the ecological and evolutionary stability of microbial communities.