Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in search="http://marklogic

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• 作者：David Iron ; John Rumsey ; Michael J. Ward ; Juncheng Wei
• 关键词：Singular perturbations ; Localized spots ; Logarithmic expansions ; Bravais lattice ; Floquet–Bloch theory ; Green’s function ; Nonlocal eigenvalue problem
• 刊名：Journal of Nonlinear Science
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：24
• 期：5
• 页码：857-912
• 全文大小：912 KB
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• 作者单位：David Iron (1)
  John Rumsey (2)
  Michael J. Ward (3)
  Juncheng Wei (3) (4)
  
  1. Department of Mathematics, Dalhousie University, Halifax, NS?, B3H 3J5, Canada
  2. Faculty of Management, Dalhousie University, Halifax, NS?, B3H 3J5, Canada
  3. Department of Mathematics, University of British Columbia, Vancouver, BC?, V6T 1Z2, Canada
  4. Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
  
• ISSN：1432-1467

文摘

The linear stability of steady-state periodic patterns of localized spots in \({\mathbb {R}}^2\) 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 \({\displaystyle \varepsilon }^2\) of the activator concentration. In the limit \({\displaystyle \varepsilon }\rightarrow 0\) , localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area \(|\Omega |\) . To leading order in \(\nu ={-1/\log {\displaystyle \varepsilon }}\) , the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies \(D={D_0/\nu }\) for some \(D_0\) independent of the lattice and the Bloch wavevector \({\pmb k}\) . From a combination of the method of matched asymptotic expansions, Floquet–Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an \({\mathcal O}(\nu )\) neighborhood of the origin in the spectral plane is derived when \(D={D_0/\nu } + D_1\) , where \(D_1={\mathcal O}(1)\) is a detuning parameter. The periodic pattern is linearly stable when \(D_1\) is chosen small enough so that this continuous band is in the stable left half-plane \(\text{ Re }(\lambda ) for all \({\pmb k}\) . Moreover, for both the Schnakenberg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green’s function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of \(D\) . From a numerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green’s function that defines the objective function, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in \(D\) .