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Preprints
https://doi.org/10.5194/egusphere-2023-1351
https://doi.org/10.5194/egusphere-2023-1351
22 Jun 2023
 | 22 Jun 2023

Existence and influence of mixed states in a model of vegetation patterns

Lilian Vanderveken, Marina Martínez Montero, and Michel Crucifix

Abstract. The Rietkerk vegetation model is a system of partial differential equations, which has been used to understand the formation and dynamics of spatial patterns in vegetation ecosystems, including desertification and biodiversity loss. Here, we provide an in-depth bifurcation analysis of the vegetation patterns produced by Rietkerk's model, based on a linear stability analysis of the homogeneous equilibrium of the system. Specifically, using a continuation method based on the Newton-Raphson algorithm, we obtain all the main heterogeneous solutions for a given size of the domain. We confirm that inhomogeneous vegetated states can exist and be stable, even for a value of rainfall for which no vegetation exists in the non-spatialized system. In addition, we evidence the existence of a new type of solution, which we called "mixed state", in which the equilibria are always unstable and take the form of a mix of two solutions from the main branches. Although these equilibria are unstable, they influence the dynamics of the transitions between distinct stable states, by slowing down the evolution of the system when it passes close to it. Our approach proves to be a helpful way to assess the existence of tipping points in spatially extended systems and disentangle the fate of the system in the Busse balloon. Overall, our findings represent a significant step forward in understanding the behavior of the Rietkerk model and the broader dynamics of vegetation patterns.

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Journal article(s) based on this preprint

11 Dec 2023
Existence and influence of mixed states in a model of vegetation patterns
Lilian Vanderveken, Marina Martínez Montero, and Michel Crucifix
Nonlin. Processes Geophys., 30, 585–599, https://doi.org/10.5194/npg-30-585-2023,https://doi.org/10.5194/npg-30-585-2023, 2023
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In semi-arid regions, hydric stress affects plant growth. In these conditions, vegetation...
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