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Model of pattern formation in marsh ecosystems with nonlocal interactions. (English) Zbl 1435.92095
In the present work the authors propose a phenomenological model to describe the dynamics of the marsh edge in terms of two-way interactions between marsh grass Spartina alternifora and sedimentation.
The aim of this paper lies in understanding whether the well-known scale-dependent (nonlocal) feedback between marsh vegetation and sedimentation can lead to spatially variable shoreline configurations. They propose a system of reaction-diffusion equations with an additional integral term with a Mexican-hat kernel function that describes the nature of this scale-dependent feedback. The proposed system is highly cooperative; as cooperative systems in which lack the classic activator-inhibitor mechanism necessary for pattern formation, it becomes of interest how and under what conditions spatial patterns may develop. It is performed a biharmonic approximation of our system and carry out analysis on the simpler biharmonic system that expresses the kernel function as separate short-range and long-range diffusion terms. Using the more mathematically tractable biharmonic system, we are then able to derive general condition for the formation of spatial patterns in a cooperative system such as in the introduced model. Further, using numerical simulations, the authors establish that the biharmonic model, while an approximation, is consistent with the original model, and therefore we can apply the theoretical results from the biharmonic system to help gain insight into the formation of patterns in the original system. After parameterizing the kernel function using a set of reasonable parameters from literature and it is established that spatial patterns can develop, given that the scale-dependent interactions between marsh vegetation and sediment dynamics are strong enough.
92D40 Ecology
92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI
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