Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Abstract

For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns. V C 2015 AIP Publishing LLC. This paper has been motivated by studies in one space dimension of a scaled phenomenological model for vegetation on possibly sloped planes in arid ecosystems. 53,60 One-dimensional patterns ideally represent striped patterns in two space dimensions by trivially extending them into a transversal direction. Such patterns are referred to as banded vegetation and have received considerable attention after reports of widespread observations. 8,58 Understanding the appearance and disappearance of vegetation bands may ultimately help prevent land degradation. The restriction to one space dimension may overestimate stability: patterns that are stable against 1D perturbations are not necessarily stable against all 2D perturbations. Natural questions to pose are:• Which of the 1D stable patterns extend to 2D stable striped patterns? • In case of destabilization by 2D perturbations, which mechanisms are responsible?In this paper, we answer these questions for the arid ecosystem model and determine the impact of slope induced advection of water. The influence of advection on striped pattern formation is studied in a more general setting. This approach provides a clear argumentation that is unobscured by model-specific details. Equally important, the results will be applicable to a wide range of models. Applicability to the arid ecosystem model is carefully checked though, assuring that the abstract requirements can in fact be met in practice.

The effect of climate change on the resilience of ecosystems with adaptive spatial pattern formation

“…For simplicity, in this article we restrict the model to one spatial dimension and focus on constantly sloped terrains, that is, . Note that general behaviour observed in the 1D model typically corresponds well to the behaviour of 2D patterns (Siero et al, ).…”

From nonequilibrium initial conditions to steady dryland vegetation patterns: How trajectories matter

“…Nonetheless, using idealized set‐ups with steady rainfall, many authors have analysed the transient dynamics resulting from these inherently transient partial differential equation (PDE) models and drawn insights on different issues, such as the rates of transitions (Yizhaq et al, ), transitions of patterns on the basis of rainfall (Roitberg & Shoshany, ; Zelnik & Meron, ), response to gradual environmental change (Dagbovie & Sherratt, ; Siteur et al, ), succession dynamics (Pueyo et al, ), the pseudosteady migration of vegetation bands (Rietkerk et al, ; Sherratt & Lord, ), colonization of a slope (Sherratt, ), or the relevance of the transient history of patterns (Sherratt, ). This has further been studied using the concepts of pattern selection and wavelength selection (Dagbovie & Sherratt, ; Siero et al, ; Zelnik & Tzuk, ) used to describe the process under which a given pattern (with a particular characteristic wavelength) is established. Pattern (wavelength) selection has been shown to be history dependent (Dagbovie & Sherratt, ; Zelnik & Tzuk, ).…”

“…History dependence (alternatively, trajectory dependence) and the transient dynamics of these systems are the result of time‐dependent rainfall (Baudena, Boni, Ferraris, von Hardenberg, & Provenzale, ; Baudena et al, ; D'Odorico, Laio, & Ridolfi, ; Guttal & Jayaprakash, ; Kletter et al, ; Zhao & Wang, ) and many complex responses to spatial heterogeneities (McGrath, Paik, & Hinz, ; Siero et al, ; Siteur et al, ; Sun, Li, Yu, & Jin, ; Tietjen, ; van der Stelt et al, ; Yizhaq et al, ; Zhao & Wang, ). It has been shown that many of these factors can coerce patterns (Bastiaansen et al, ) and can strengthen or weaken the ecosystem (Kletter et al, ).…”

Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation

“…We note that the consideration of the one-dimensional model might overestimate the stability [43]. As we consider small perturbations, we can ignore terms that are nonlinear in terms of n and w .…”