# Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in $${\mathbb {R}}^2$$R2

@article{Iron2014LogarithmicEA,
title={Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in \$\$\{\mathbb \{R\}\}^2\$\$R2},
author={David Iron and John Rumsey and Michael J. Ward and Juncheng Wei},
journal={Journal of Nonlinear Science},
year={2014},
volume={24},
pages={857-912}
}
The linear stability of steady-state periodic patterns of localized spots in $${\mathbb {R}}^2$$R2 for the two-component Gierer–Meinhardt (GM) and Schnakenberg reaction–diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient $${\displaystyle \varepsilon }^2$$ε2 of the activator concentration. In the limit $${\displaystyle \varepsilon }\rightarrow 0$$ε→0, localized spots in the activator are centered at the lattice points… CONTINUE READING

#### Citations

##### Publications citing this paper.
SHOWING 1-8 OF 8 CITATIONS

## Refined stability thresholds for localized spot patterns for the Brusselator model in R2

VIEW 10 EXCERPTS
CITES METHODS & BACKGROUND

## Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

VIEW 7 EXCERPTS
CITES BACKGROUND & METHODS

## On accurately estimating stability thresholds for periodic spot patterns of reaction-diffusion systems in 2

VIEW 10 EXCERPTS
CITES BACKGROUND, METHODS & RESULTS

## Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in Two Dimensions

• SIAM J. Applied Dynamical Systems
• 2018
VIEW 2 EXCERPTS
CITES BACKGROUND

VIEW 1 EXCERPT
CITES METHODS

VIEW 2 EXCERPTS

## Numerical Approximation of Diffusive Capture Rates by Planar and Spherical Surfaces with Absorbing Pores

• SIAM Journal of Applied Mathematics
• 2017
VIEW 1 EXCERPT

## Effects of Open Systems on the Existence , Dynamics , and Stability of Spot Patterns in the 2 D Brusselator Model

VIEW 1 EXCERPT
CITES METHODS

#### References

##### Publications referenced by this paper.
SHOWING 1-10 OF 33 REFERENCES

## Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

• J. Nonlinear Science
• 2009
VIEW 4 EXCERPTS

## Fast algorithms for Helmholtz Green's functions

• Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• 2008
VIEW 6 EXCERPTS
HIGHLY INFLUENTIAL

## An Application of the Modular Function in Nonlocal Variational Problems

VIEW 6 EXCERPTS
HIGHLY INFLUENTIAL

## Existence and stability of multiple-spot solutions for the Gray–Scott model in R2

VIEW 7 EXCERPTS
HIGHLY INFLUENTIAL

VIEW 2 EXCERPTS

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

• SIAM J. Applied Dynamical Systems
• 2014

VIEW 1 EXCERPT

VIEW 1 EXCERPT

## An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

• Multiscale Modeling & Simulation
• 2010
VIEW 1 EXCERPT

VIEW 1 EXCERPT