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Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems
[article]
2020
arXiv
pre-print
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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
2022
Discrete and Continuous Dynamical Systems. Series S
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<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
2017
SIAM Journal on Applied Dynamical Systems
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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
2011
SIAM Journal on Applied Dynamical Systems
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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]
2010
arXiv
pre-print
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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
2021
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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
2009
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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