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

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 equilibria 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 equilibrium, which we call “mixed state”, in which the equilibria are always unstable and take the form of a mix of two equilibria 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 behaviour of the Rietkerk model and the broader dynamics of vegetation patterns.