Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Madzvamuse, A and Barreira, R (2014) Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces. Physical Review E, 90. 043307-1-043307-14. ISSN 1539-3755

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Abstract

The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion systemwith cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science
Q Science > QA Mathematics
Q Science > QA Mathematics > QA0297 Numerical analysis
Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
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Depositing User: Anotida Madzvamuse
Date Deposited: 13 Nov 2014 09:12
Last Modified: 10 Mar 2017 18:13
URI: http://sro.sussex.ac.uk/id/eprint/51334

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Project NameSussex Project NumberFunderFunder Ref
Mathematical Modelling and Analysis of Spatial Patterning on Evolving SurfacesG0872EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCILEP/J016780/1
Unravelling new mathematics for 3D cell migrationG1438LEVERHULME TRUSTRPG-2014-149