Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

Stelt, S. van der; Doelman, Arjen; Hek, Geertje; Rademacher, Jens D. M.

doi:10.1007/s00332-012-9139-0

Cited by 116 publications

(143 citation statements)

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“…Supercriticality has been proven in an asymptotic scaling of (1) in one space dimension, 60 and numerically, it is found that this holds in a broad range of parameter space. Through Theorem 2, for advection c > 0, the Turing-Hopf bifurcation is a natural mechanism for the formation of striped or banded vegetation patterns.…”

Section: -7mentioning

confidence: 87%

“…In Figure 3, we plot the values of a T for 2c ¼ 0, 182.5, 365, 500, and 1000, together with a square root function since a T grows as ffiffi ffi c p for large c, in a certain scaling regime. 60 Now that we understand the influence of both parameters a and m, we can also fix c and infer the dependence a T ¼ a T (m) for free. The following approximation complementary to (25) can be made:…”

Section: -mentioning

confidence: 99%

“…26 The generalized form of (1) with the term a(1 À w) has been referred to as the generalized Klausmeier-Gray-Scott model; here also, the impact of nonlinear diffusion of the water component has been studied. 39,60 Both the Klausmeier model and the Gray-Scott model exhibit patterns. 31,40 We will study the influence of the advection parameter c on striped patterns.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

124

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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.

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Section: -7mentioning

confidence: 87%

Section: -mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

124

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“…In this paper, we presented a picture of pattern formation and collapse that occurs via two Turing bifurcations, but finite-wavelength Turing instability is not the only mechanism for vegetation collapse observed in models. In a study of the generalized Klausmeier model by van der Stelt et al [20], a patterned vegetated state collapses directly to desert via long-wavelength, sideband, and Hopf instabilities. Additionally, homoclinic snaking [21,22] has been proposed as a mechanism for the stabilization and motion of localized patterned states that emerge en route to desertification [23].…”

Section: Pattern Transitions In the Model By Von Hardenberg Et Almentioning

confidence: 99%

Transitions between patterned states in vegetation models for semiarid ecosystems

2014

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A feature common to many models of vegetation pattern formation in semi-arid ecosystems is a sequence of qualitatively different patterned states, "gaps → labyrinth → spots", that occurs as a parameter representing precipitation decreases. We explore the robustness of this "standard" sequence in the generic setting of a bifurcation problem on a hexagonal lattice, as well as in a particular reaction-diffusion model for vegetation pattern formation. Specifically, we consider a degeneracy of the bifurcation equations that creates a small bubble in parameter space in which stable small-amplitude patterned states may exist near two Turing bifurcations. Pattern transitions between these bifurcation points can then be analyzed in a weakly nonlinear framework. We find that a number of transition scenarios besides the standard sequence are generically possible, which calls into question the reliability of any particular pattern or sequence as a precursor to vegetation collapse. Additionally, we find that clues to the robustness of the standard sequence lie in the nonlinear details of a particular model.

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

Caviedes‐Voullième

Hinz

2020

Ecohydrology

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The multiscale nature of ecohydrological processes and feedbacks implies that vegetation patterns arising in water‐limited systems are directly linked to water redistribution processes occurring at much shorter timescales than vegetation growth. This in turn suggests that the initially available water in the system can play a role in determining the trajectory of the system, together with the well‐known role of the rainfall gradient. This work explores the role of initial hydrological conditions on vegetation dynamics and vegetation patterns. To do so, the HilleRisLambers–Rietkerk model was solved with different rainfall amounts and a large range of initial hydrological conditions spanning from near‐equilibrium to far‐from‐equilibrium conditions. The resulting vegetation patterns and ecohydrological signatures were quantitatively studied. The results show that not only do initial hydrological conditions play a role in the ecohydrological dynamics but also they can play a dominating one even resulting in divergent vegetation patterns that exhibit convergent mean‐field properties, including a new set of hybrid patterns. Our results highlight the relevance of assessing both global ecological and hydrological signatures and quantitatively assessing patterns to describe and understand system dynamics and in particular to determine if the systems are transient or steady. Furthermore, our analysis shows that the trajectories the system follows during its transient stages cannot be neglected to understand complex dependencies of the long‐term steady state to environmental factors and drivers.

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Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

Cited by 116 publications

References 81 publications

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

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

Transitions between patterned states in vegetation models for semiarid ecosystems

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

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