ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability Analysis and Parameter Classification of a Reaction-Diffusion Model on an Annulus

Journal of Applied Nonlinear Dynamics 9(4) (2020) 589--617 | DOI:10.5890/JAND.2020.12.006

Institute of Cancer and Genomic Sciences, Centre for Computational Biology, University of Birmingham, Edgbaston, B15 2TT, UK

Abstract

This work explores the influence of domain-size on the evolution of pattern formation modelled by an \textit{activator-depleted} reaction-diffusion system on a flat-ring (annulus). A closed form expression is derived for the spectrum of the Laplace operator on the domain. Spectral method is used to depict the close form solution on the domain. The bifurcation analysis of \textit{activator-depleted} reaction-diffusion system is conducted on the admissible parameter space under the influence of domain-size. The admissible parameter space is partitioned under a set of proposed conditions relating the reaction-diffusion constants with the domain-size. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived domain-dependent conditions.

Acknowledgments

WS acknowledges support of the School of Mathematical and Physical Sciences Doctoral Training studentship. AM acknowledges support from the Leverhulme Trust Research Project Grant (RPG-2014-149) and the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642866. AM's work was partially supported by the Engineering and Physical Sciences Research Council, UK grant (EP/J016780/1). The authors (WS, AM) thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme (Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation; EPSRC EP/K032208/1). AM was partially supported by a fellowship from the Simons Foundation. AM is a Royal Society Wolfson Research Merit Award Holder, generously supported by the Wolfson Trust.

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