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Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems [article]

Tony Wong, Michael J. Ward
2020 arXiv   pre-print
From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability  ...  Spatially localized 2-D spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio.  ...  solution for certain singularly perturbed RD systems.  ... 
arXiv:2002.01453v2 fatcat:o6djv76uwjdxncxxwagzgjtlp4

Dynamics of patchy vegetation patterns in the two-dimensional generalized Klausmeier model

Tony Wong, Michael J. Ward
2022 Discrete and Continuous Dynamical Systems. Series S  
<p style='text-indent:20px;'>We study the dynamical and steady-state behavior of self-organized spatially localized patches or "spots" of vegetation for the Klausmeier reaction-diffusion (RD) system of  ...  model in the singularly perturbed limit where the biomass diffusivity is much smaller than that of the water resource.  ...  We predict that the one-spot solution is unstable to a local peanut-shaped deformation.  ... 
doi:10.3934/dcdss.2022043 fatcat:vuwdqpd2orglxksap3estlshsy

The Stability and Slow Dynamics of Localized Spot Patterns for the 3-D Schnakenberg Reaction-Diffusion Model

J. C. Tzou, S. Xie, T. Kolokolnikov, M. J. Ward
2017 SIAM Journal on Applied Dynamical Systems  
By approximating each spot as a Coulomb singularity, a nonlinear system of equations is formulated for the strength of each spot.  ...  of the Schnakenberg activator-inhibitor model with bulk feed-rate A in the singularly perturbed limit of small diffusivity ε 2 of the activator component.  ...  singularly perturbed RD system.  ... 
doi:10.1137/16m108121x fatcat:rip3irvw3zbztfaehrz7x3mwwy

The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray–Scott Model

W. Chen, M. J. Ward
2011 SIAM Journal on Applied Dynamical Systems  
The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity ε of one  ...  A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates Sj and xj, for j = 1, . . . , k, which characterizes the slow dynamics of a spot pattern  ...  From this figure we observe that the 3 rd spot (the brightest one at t = 1) deforms into a peanut shape at t = 31, Table 3 .  ... 
doi:10.1137/09077357x fatcat:xcngbpd6njarbcmxrh7rwrcmlu

The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model [article]

Wan Chen, Michael J. Ward
2010 arXiv   pre-print
The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity ϵ of one  ...  A differential algebraic ODE system for the collective coordinates S_j and x_j for j=1,...,k is derived, which characterizes the slow dynamics of a spot pattern.  ...  With regards to the dynamics of spots, there are only a few analytical results characterizing spot dynamics for singularly perturbed RD systems in two-dimensional domains.  ... 
arXiv:1009.2805v1 fatcat:ymjekjtazndp7iqxrw2jwi3vdq

An asymptotic and numerical analysis of localized solutions to some linear and nonlinear pattern formation problems with heterogeneities

Ka Wah Wong
2021
To provide a theoretical basis for these observations, we derive an amplitude equation from a weakly nonlinear analysis which confirms that a peanut-shaped instability of a spot is subcritical.  ...  In many previous studies of spot instabilities, it has been observed from numerical PDE simulations that a linear shape-deforming instability of a localized spot is the trigger of a nonlinear spot-replication  ...  Acknowledgements First and foremost, I would like to thank my supervisors Colin Macdonald and Michael Ward for their continued support and encouragement throughout the completion of this thesis.  ... 
doi:10.14288/1.0401384 fatcat:kbk77yruwzej5lxsrgkehv3zr4

Localized patterns in the Gray-Scott model : an asymptotic and numerical study of dynamics and stability

Wan Chen
2009
Localized patterns have been observed in many reaction-diffusion systems.  ...  An asymptotic differential algebraic system of ODE's for the spot locations is derived to characterize the slow dynamics of a collection of spots.  ...  A General Class of Reaction-Diffusion Models An interesting extension of this thesis would be to extend the analysis to consider localized spot patterns arising in a class of singularly perturbed reaction-diffusion  ... 
doi:10.14288/1.0067324 fatcat:cm4jwpl2wrb4hjr64ii7vlv2bm