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Exhibiting cross-diffusion-induced patterns for reaction-diffusionsystems on evolving domains and surfaces
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The aim of this manuscript is to present for the first time the application of the finite element method for solvingreaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore wepresent pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains andsurfaces. A two-component reaction-diffusion system with linear cross-diffusion in bothuandvis presented. Thefinite element method is based on the approximation of the domain or surface by a triangulated domain or surfaceconsisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. Afinite element space of functions is then defined by taking the continuous functions which are linear affine on eachsimplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of patternformation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusionparameter 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; themethodology can deal with complicated evolution laws of the domain and surface, and these include uniformisotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing inthe domain or on the surface
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Madzvamuse, A., Barreira, R. (2014). Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces. Phys. Review E, 90. doi: 10.1103/PhysRevE.90.043307