Traveling wave trains, pulled and pushed fronts in a reaction–diffusion model of seagrass meadows

Moreno‐Spiegelberg et al. [Proc. Natl. Acad. Sci., 122 (2025), no. 11] proposed a general model that incorporates positive feedback together with negative feedback mediated by an inhibitor, and successfully applied it to Posidonia oceanica meadows to explain the observed spatiotemporal phenomena. Their work via numerical simulations and theoretical analysis produced various spatiotemporal patterns, such as traveling pulses, wave trains, expanding rings, and spiral waves. Here, our work rigorously establishes the existence of traveling pulses, traveling fronts, wave trains, as well as pulled and pushed fronts for this model. Our approach, based on geometric singular perturbation theory, allows us to construct traveling waves far from equilibrium, and can reveal more organized underlying structures. This provides a rigorous mathematical foundation for the wave phenomena and contributes to a deeper understanding of the mechanisms generating complex spatiotemporal patterns in spatially extended ecological systems.