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.

Lilian Vanderveken et al.

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-1351', Robbin Bastiaansen, 20 Jul 2023
    • AC1: 'Reply on RC1', Lilian Vanderveken, 17 Aug 2023
  • RC2: 'Comment on egusphere-2023-1351', Anonymous Referee #2, 21 Jul 2023
    • AC2: 'Reply on RC2', Lilian Vanderveken, 17 Aug 2023
  • RC3: 'Comment on egusphere-2023-1351', Anonymous Referee #3, 27 Jul 2023
    • AC3: 'Reply on RC3', Lilian Vanderveken, 17 Aug 2023

Lilian Vanderveken et al.

Lilian Vanderveken et al.


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Short summary
In semi-arid regions, hydric stress affects plant growth. In these conditions, vegetation patterns develop and effectively allow for vegetation to persist under low water input. The formation of patterns, and the transtition between patterns can be studied with small models taking the form of dynamical sytems. Our study produces a full map of stable and unstable solutions in a canonical vegetation model, and show how they determine the transitions between different patterns.