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We investigate a generalized, non-dimensionalized Klausmeier model for vegetation dynamics that incorporates non-local plant dispersal on a finite, flat habitat. We show the model to be well-posed,  prove the existence of non-trivial stationary solutions, and derive criteria for their stability. In addition, we identify a critical threshold for the maximal biomass density that guarantees persistence and prevents extinction. In numerical experiments we calibrate variance of the dispersal kernels with the Laplacian operator and show that non-local models predict vegetation persistence in substantially smaller habitats than their classical local diffusion counterparts. This enhanced resilience is, thus, primarily caused by the higher-order moments of the dispersal kernel. In the regime of slow water diffusion, a clear morphological difference emerges: local models predict interior localization of biomass, whereas non-local models yield biomass concentrations near the habitat boundaries, a pattern that may be considered to be more realistic.

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Link to arxiv.org: (TO DO after submission)

This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the $n$-dimensional Euclidean space or a periodic environment modeled by $n$-dimensional torus. The analysis includes the study of spectrum of the linear, bounded operator in the considered equation, which is a scaled, non-local analogue of classical Laplacian with Neumann boundaries. In particular we show the explicit formulas for eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of eigenvalues. Within the context of the non-linear evolution problem, we establish the existence of an invariant region, give a criterion for convergence to the mean mass, and construct spatially heterogeneous steady states.

Link to DOI: https://www.sciencedirect.com/science/article/abs/pii/S0022039624007095

Link to arxiv.org: https://arxiv.org/abs/2403.11584