Model of pattern formation in marsh ecosystems with nonlocal interactions
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Smooth cordgrass Spartina alterniflora is a grass species commonly found in tidal marshes. It is an ecosystem engineer, capable of modifying the structure of its surrounding environment through various feedbacks. The scale-dependent feedback between marsh grass and sediment volume is particularly of interest. Locally, the marsh vegetation attenuates hydrodynamic energy, enhancing sediment accretion and promoting further vegetation growth. In turn, the diverted water flow promotes the formation of erosion troughs over longer distances. This scale-dependent feedback may explain the characteristic spatially varying marsh shoreline, commonly observed in nature. We propose a mathematical framework to model grass–sediment dynamics as a system of reaction–diffusion equations with an additional nonlocal term quantifying the short-range positive and long-range negative grass–sediment interactions. We use a Mexican-hat kernel function to model this scale-dependent feedback. We perform a steady state biharmonic approximation of our system and derive conditions for the emergence of spatial patterns, corresponding to a spatially varying marsh shoreline. We find that the emergence of such patterns depends on the spatial scale and strength of the scale-dependent feedback, specified by the width and amplitude of the Mexican-hat kernel function.
KeywordsPattern formation Nonlocal interactions Marsh ecosystem Reaction–diffusion Steady state Cooperation
Mathematics Subject ClassificationPrimary 92D40 92D25 35K57 Secondary 35B36
This work is partially supported by NSF Grant DMS-1715651 and DMS-1313093. We also thank Romuald N. Lipcius and Matthew L. Kirwan for helpful discussion.
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