The spatial organization of biomass resulting from plant-water feedbacks in arid ecosystems, "patterned vegetation,'' provides a macroscopic signal of nonlinear plant-water interactions and ecosystem health. Current models that reproduce such patterning assume diffusive biomass movement but do not account for realistic transport through seed dispersal. An adaptation of an existing three-equation model that accounts for the interactions between overland flow, subsurface flow, and biomass dynamics is used to investigate the impact of representing biomass spread with realistic dispersal kernels (a "kernel-based method''). Model results indicate that dispersion behavior changes the spatial organization of vegetation, destabilizing the regular patterns predicted by diffusion based models. The kernel-based approach provides a closer match to power spectra derived from a remotely sensed image of patterned vegetation when compared to their diffusion-based counterpart. Potential feedbacks between the presence of spatial patterns and selection of optimal seed dispersal length scales are also investigated.