Pulse Solutions for an Extended Klausmeier Model with Spatially Varying Coefficients

Bastiaansen, Robbin; Chirilus‐Bruckner, Martina; Doelman, Arjen

doi:10.1137/19m1255665

Cited by 19 publications

(18 citation statements)

References 47 publications

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“…Thus, the phase portrait M N has one (attracting) equilibrium: the configuration in which the N vegetation patches are regularly distributed. Moreover for more complex topographiesthat fall outside the scope of this articlethe pulse location differential equation can explain many of the (from a simple model's perspective counter-intuitive) observations like downhill migration of vegetation patterns (Bastiaansen et al, 2018a;Bastiaansen & Doelman, 2019).…”

Section: Migration Of Vegetation Patchesmentioning

confidence: 99%

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

et al. 2020

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In a rapidly changing world, quantifying ecosystem resilience is an important challenge. Historically, resilience has been defined via models that do not take spatial effects into account. These systems can only adapt via uniform adjustments. In reality, however, the response is not necessarily uniform, and can lead to the formation of (self-organised) spatial patternstypically localised vegetation patches. Classical measures of resilience cannot capture the emerging dynamics in spatially self-organised systems, including transitions between patterned states that have limited impact on ecosystem structure and productivity. We present a framework of interlinked phase portraits that appropriately quantifies the resilience of patterned states, which depends on the number of patches, the distances between them and environmental conditions. We show how classical resilience concepts fail to distinguish between small and large pattern transitions, and find that the variance in interpatch distances provides a suitable indicator for the type of imminent transition. Subsequently, we describe the dependency of ecosystem degradation based on the rate of climatic change: slow change leads to sporadic, large transitions, whereas fast change causes a rapid sequence of smaller transitions. Finally, we discuss how pre-emptive removal of patches can minimise productivity losses during pattern transitions, constituting a viable conservation strategy.

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Section: Migration Of Vegetation Patchesmentioning

confidence: 99%

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

et al. 2020

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“…There are several numerical and observational results in this direction [22], but little is known analytically. A first analytical step towards this can be found in [1], in which the impact of non-trivial topographies on 1D stripe patterns is studied.…”

Section: Discussionmentioning

confidence: 99%

“…The traveling wave ODE (1.7) is a two-fast-one-slow system. We obtain the fast subsystem or layer problem by setting ε = 0 in (1.7), which results in the system 1) or, equivalently, the collection of planar ODEs…”

Section: Critical Manifoldsmentioning

confidence: 99%

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

2019

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In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. We consider a twocomponent reaction-diffusion-advection model of Klausmeier type describing the interplay of vegetation and water resources and the resulting dynamics of these patterns. We focus on the large advection limit on constantly sloped terrain, in which the diffusion of water is neglected in favor of advection of water downslope. Planar vegetation pattern solutions are shown to satisfy an associated singularly perturbed traveling wave equation, and we construct a variety of traveling stripe and front solutions using methods of geometric singular perturbation theory. In contrast to prior studies of similar models, we show that the resulting patterns are spectrally stable to perturbations in two spatial dimensions using exponential dichotomies and Lin's method. We also discuss implications for the appearance of curved stripe patterns on slopes in the absence of terrain curvature.

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“…Especially in models of drylands, many patterned states have been analyzed, including coexistence states [37,59], although spatial heterogeneities are not often taken into account, with few exceptions [60,61]. However, the models considered are typically more complicated than (4) and more advanced mathematical techniques are used to analyze them that are beyond the scope of this article.…”

Section: Ecosystemsmentioning

confidence: 99%

Partial tipping in a spatially heterogeneous world

Bastiaansen¹,

Dijkstra²,

Heydt³

2021

Preprint

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Many climate subsystems are thought to be susceptible to tipping -and some might be close to a tipping point. The general belief and intuition, based on simple conceptual models of tipping elements, is that tipping leads to reorganization of the full (sub)system. Here, we explore tipping in conceptual, but spatially extended and spatially heterogenous models. These are extensions of conceptual models taken from all sorts of climate system components on multiple spatial scales. By analysis of the bifurcation structure of such systems, special stable equilibrium states are revealed: coexistence states with part of the spatial domain in one state, and part in another, with a spatial interface between these regions. These coexistence states critically depend on the size and the spatial heterogeneity of the (sub)system. In particular, in these systems a tipping point might lead to a partial tipping of the full (sub)system, in which only part of the spatial domain undergoes reorganization, limiting the impact of these events on the system's functioning.

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Pulse Solutions for an Extended Klausmeier Model with Spatially Varying Coefficients

Cited by 19 publications

References 47 publications

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

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

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

Partial tipping in a spatially heterogeneous world

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