Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach

Doelman, Arjen; Gardner, Rod; Kaper, Tasso J.

doi:10.1016/s0167-2789(98)00180-8

Cited by 163 publications

(358 citation statements)

References 21 publications

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“…Note that this agrees completely with the critical saddle-node/self-replication value of e as can be deduced from Ref. 13, that was obtained by careful numerical experiments on the critical magnitude of e for which the methods developed there, and used here, are valid.…”

Section: Appendix C: No Advection: Existence Of Long Wavelength Patternssupporting

confidence: 88%

“…Note that this expands and confirms the arguments in Ref. 13 about the instability of spatially periodic patterns by setting' ¼ 0 (in the more special scaling there).…”

supporting

confidence: 89%

“…13 In these cases, k pole ðl;'Þ typically merges with another eigenvalue as' > 0 decreases and forms a pair of complex conjugate eigenvalues that cross through the imaginary axis as' decreases further, see Ref. 35 for a much more detailed analysis of the spectral curves kð1; 'Þ associated to the stability of a homoclinic stripe, i.e., the limit l # 1.…”

Section: B No Advection: Transverse Instability Of Long Wavelength Smentioning

confidence: 99%

“…13,17 This does, however, not influence the proof of the result that can be directly copied from Ref. 17.…”

Section: Appendix C: No Advection: Existence Of Long Wavelength Patternsmentioning

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: Appendix C: No Advection: Existence Of Long Wavelength Patternssupporting

confidence: 88%

“…Note that this expands and confirms the arguments in Ref. 13 about the instability of spatially periodic patterns by setting' ¼ 0 (in the more special scaling there).…”

supporting

confidence: 89%

Section: B No Advection: Transverse Instability Of Long Wavelength Smentioning

confidence: 99%

“…13,17 This does, however, not influence the proof of the result that can be directly copied from Ref. 17.…”

Section: Appendix C: No Advection: Existence Of Long Wavelength Patternsmentioning

confidence: 99%

See 2 more Smart Citations

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

View full text Add to dashboard Cite

show abstract

“…Selfreplicating patterns have been also observed numerically in a reaction diffusion system ( [1], [5], [8], [9]). Several interesting analytical works have also appeared recently: For instance, construction of single-spot solution to the Gray-Scott model (see below) and its stability has been done by [6] with the aid of formal matched asymptotic analysis, which is closely related to the splitting phenomenon; A rigorous analysis concerning the existence and stability of steady single pulse as well as nonexistence of traveling pulses has been done quite recently by [10] and [11]. Although we have found a nice collection of exciting dynamics, we don't know yet how and why the transitions occur from one dynamics to another, especially what kind of mathematical mechanism causes STC.…”

mentioning

confidence: 99%

Global bifurcational approach to the onset of spatio-temporal chaos in reaction diffusion systems

Nishiura¹

2001

Methods and Applications of Analysis

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Abstract. A new criterion for the onset of spatio-temporal chaos arising in the Gray-Scott model is presented. This is based on the interrelationship of global bifurcating branches of stationary or periodic solutions with respect to the removal rate contained in the model, especially their locations of saddle-node points and the Hopf bifurcation point of a constant state play a key role. At the onset point there exists a generalized heteroclinic cycle on the whole line and spatio-temporal chaos emerges by unfolding this cycle.

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Introduction

Wei

Winter

2014

Mathematical Aspects of Pattern Formation in Biological Systems

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Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach

Cited by 163 publications

References 21 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

Global bifurcational approach to the onset of spatio-temporal chaos in reaction diffusion systems

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