| Abstract: |
| n this talk we report our results on a model, posed by Moreno-Spiegelberg et al. [Proc. Natl. Acad. Sci. 2025], 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. The original authors worked via numerical simulations and theoretical analysis and 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 contribute to a deeper understanding of the mechanisms generating complex spatiotemporal patterns in spatially extended ecological systems. This is a joint work with Qi Qiao and Baodong Zhang, which is based on our paper in J. London Math. Soc. (2) 2026; 113: e70428. |
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