The Stability of Localized Spot Patterns for the Brusselator on the Sphere

Rozada, Ignacio; Ruuth, Steven J.; Ward, Michael J.

doi:10.1137/130934696

Cited by 40 publications

(147 citation statements)

References 48 publications

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“…This analysis is an extension to the analysis performed in [4], where it was shown that a 1-D localized stripe pattern of active-ROP in a 2-D domain will, generically, exhibit breakup into spatially localized spots. To analyze the subsequent dynamics and stability of these localized spots in the presence of the auxin gradient, we will extend the hybrid asymptotic-numerical methodology developed in [25,35] for prototypical RD systems with spatially homogeneous coefficients. This analysis will lead to a novel finite-dimensional dynamical system characterizing slow spot evolution.…”

Section: Parameter Set a Parameter Set Bmentioning

confidence: 99%

“…However, depending on the parameter regime, these localized patterns can also exhibit fast O(1) time-scale instabilities, leading either to spot creation or destruction. For prototypical RD systems such as the Gierer-Meinhardt, Gray-Scott, Schnakenberg, and Brusselator models, the slow dynamics of localized solutions and the possibility of self-replication and competition instabilities, leading either to spot creation or destruction, respectively, have been studied using a hybrid asymptotic-numerical approach in [26,35]. In addition, in the large inhibitor diffusion limit, a leading-orderin-−1/ log ε analysis, shows that the linear stability problem for localized spot patterns characterizing competition instabilities can be reduced to the study of classes of nonlocal eigenvalue problems (NLEPs) (cf.…”

Section: Parameter Set a Parameter Set Bmentioning

confidence: 99%

“…Our previous studies in [6] of 1-D spike patterns and in [4] of 1-D stripe patterns have extended these previous NLEP analyses of prototypical RD systems to the more complicated, and biologically realistic, plant root-hair system (1.2). In the analysis below we extend the 2-D hybrid asymptotic-numerical methodology developed in [26,35] for prototypical RD systems to (1.2). The primary new feature in (1.2), not considered in these previous works, is to analyze the effect of a spatial gradient in the right-hand sides of (1.2), which represents variations in the auxin distribution.…”

Section: Parameter Set a Parameter Set Bmentioning

confidence: 99%

“…Although it is beyond the scope of this paper to give a detailed analysis of a competition instability for our plant RD model (1.2), based on analogies with other RD models (cf. [9,25,26,35]), this instability typically occurs when the source parameter of a particular spot is below some threshold or, equivalently, when there are too many spots that can be supported in the domain for the given substrate diffusivity. In essence, a competition instability is an overcrowding instability.…”

Section: Numerical Illustrations Of O(1)mentioning

confidence: 99%

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Spot Dynamics in a Reaction-Diffusion Model of Plant Root Hair Initiation

Avitabile¹,

Breña‐Medina²,

Ward³

2018

SIAM J. Appl. Math.

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Abstract. We study pattern formation in a 2-D reaction-diffusion (RD) sub-cellular model characterizing the effect of a spatial gradient of a plant hormone distribution on a family of G-proteins associated with root-hair (RH) initiation in the plant cell Arabidopsis thaliana. The activation of these G-proteins, known as the Rho of Plants (ROPs), by the plant hormone auxin, is known to promote certain protuberances on root hair cells, which are crucial for both anchorage and the uptake of nutrients from the soil. Our mathematical model for the activation of ROPs by the auxin gradient is an extension of the model of Payne and Grierson [PLoS ONE, 12(4), (2009)], and consists of a twocomponent Schnakenberg-type RD system with spatially heterogeneous coefficients on a 2-D domain. The nonlinear kinetics in this RD system model the nonlinear interactions between the active and inactive forms of ROPs. By using a singular perturbation analysis to study 2-D localized spatial patterns of active ROPs, it is shown that the spatial variations in the nonlinear reaction kinetics, due to the auxin gradient, lead to a slow spatial alignment of the localized regions of active ROPs along the longitudinal midline of the plant cell. Numerical bifurcation analysis, together with time-dependent numerical simulations of the RD system are used to illustrate both 2-D localized patterns in the model, and the spatial alignment of localized structures.1. Introduction. We examine the effect of a spatially-dependent plant hormone distribution on a family of proteins associated with root hair (RH) initiation in a specific plant cell. This process is modeled by a generalized Schnakenberg reaction-diffusion (RD) system on a 2-D domain with both source and loss terms, and with a spatial gradient modeling the spatially inhomogeneous distribution of the plant hormone auxin. This system is an extension of a model proposed by Payne and Grierson in [33], and analyzed in a 1-D context in the companion articles [4,6]. The new goal of this paper, in comparison with [4,6], is to analyze 2-D localized spot patterns in the RD system (1.2), and how these 2-D patterns are influenced by the spatially inhomogeneous auxin distribution.We now give a brief description of the biology underlying the RD model. In this model, an on-and-off switching process of a small G-protein subfamily, called the Rho of Plants (ROPs), is assumed to occur in a RH cell of the plant Arabidopsis thaliana. ROPs are known to be involved in RH cell morphogenesis at several distinct stages (see [13,23] for details). Such a biochemical process is believed to be catalyzed by a plant hormone called auxin (cf. [33]). Typically, auxin-transport models are formulated to study polarization events between cells (cf. [14]). However, little is known about specific details of auxin flow within a cell. In [4,6,33] a simple transport process is assumed to govern the auxin flux through a RH cell, which postulates that auxin diffuses much faster than ROPs in the cell, owing partially to the in-and out-pum...

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Section: Parameter Set a Parameter Set Bmentioning

confidence: 99%

Section: Parameter Set a Parameter Set Bmentioning

confidence: 99%

Section: Parameter Set a Parameter Set Bmentioning

confidence: 99%

Section: Numerical Illustrations Of O(1)mentioning

confidence: 99%

See 2 more Smart Citations

Spot Dynamics in a Reaction-Diffusion Model of Plant Root Hair Initiation

Avitabile¹,

Breña‐Medina²,

Ward³

2018

SIAM J. Appl. Math.

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“…Li and Bi [31] proposed the enveloping slow-fast analysis that could be used to explain the bursting phenomenon in the system with three time scales. Work about the Brusselator with different time scales, has been found in [8,32]. However, there is little analytical study on the slow-fast phenomenon or its generation mechanism of a Brusselator due to the coupling of the fast and slow subsystems.…”

Section: Introductionmentioning

confidence: 99%

Slow-fast effect and generation mechanism of brusselator based on coordinate transformation

Hou

Shen

2016

Open Physics

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Abstract:The Brusselator with different time scales, which behaves in the classical slow-fast effect, is investigated, and is characterized by the coupling of the quiescent and spiking states. In order to reveal the generation mechanism by using the slow-fast analysis method, the coordinate transformation is introduced into the classical Brusselator, so that the transformed system can be divided into the fast and slow subsystems. Furthermore, the stability condition and bifurcation phenomenon of the fast subsystem are analyzed, and the attraction domains of different equilibria are presented by theoretical analysis and numerical simulation respectively. Based on the transformed system, it could be found that the generation mechanism between the quiescent and spiking states is Fold bifurcation and change of the attraction domain of the fast subsystem. The results may also be helpful to the similar system with multiple time scales.

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